C 1 > β q C_{3}>C_{1}>\\beta^{q} . This hierarchy aligns with the underlying physics: the dominant sensitivity of the global fit comes from the charged-current R D ( ∗ ) R_{D^{(*)}} observables (controlled by C 3 C_{3} ), followed by the neutral-current B → K ( ∗ ) ν ν ¯ B\\to K^{(*)}\\nu\\bar{\\nu} modes (controlled by C 1 C_{1} ), while the quark-mixing angle β q \\beta^{q} modulates both sectors without dominating. The agreement between the data-driven SHAP ranking and the expected operator structure confirms that the surrogate captures the correct EFT dynamics rather than overfitting numerical noise."},{"type":"para","text":"Using a sample of 1.5 × 10 4 1.5\\times 10^{4} points generated with the emulator, we run the SMEFT-to-WET renormalization-group evolution at each point and compute the Pearson correlation coefficients between the resulting low-energy observables (Figure 4 ). Because no NP enters the lepton sector in Scenario III ( α ℓ = β ℓ = 0 \\alpha^{\\ell}=\\beta^{\\ell}=0 , so that C 10 μ = 0 C_{10}^{\\mu}=0 and R K ( ∗ ) = 1 R_{K^{(*)}}=1 exactly), the b → s ℓ + ℓ − b\\to s\\ell^{+}\\ell^{-} ratios and the B ( s ) → μ + μ − B_{(s)}\\to\\mu^{+}\\mu^{-} branching ratios remain at their SM values and show no variation across the sample; the matrix therefore retains only the observables that genuinely respond to ( C 1 , C 3 , β q ) (C_{1},C_{3},\\beta^{q}) . The b → c τ ν b\\to c\\tau\\nu ratios R D R_{D} and R J / ψ R_{J/\\psi} are perfectly correlated, reflecting their common origin in the single coefficient C V L ∝ C 3 C_{V_{L}}\\propto C_{3} , while the neutral-current modes BR ( B + → K + ν ν ¯ ) \\mathrm{BR}(B^{+}\\!\\to K^{+}\\nu\\bar{\\nu}) and BR ( K + → π + ν ν ¯ ) \\mathrm{BR}(K^{+}\\!\\to\\pi^{+}\\nu\\bar{\\nu}) respond to the orthogonal combination C 1 − C 3 C_{1}-C_{3} . The central result concerns the interplay between R D R_{D} and BR ( B + → K + ν ν ¯ ) \\mathrm{BR}(B^{+}\\!\\to K^{+}\\nu\\bar{\\nu}) : the almost-perfect correlation between these two observables found in Scenario II [ Alda:2021rgt ] disappears (it drops to ≈ 0.06 \\approx 0.06 ) once C 1 C_{1} and C 3 C_{3} are allowed to vary independently, since BR ( B + → K + ν ν ¯ ) ∝ ( C 1 − C 3 ) \\mathrm{BR}(B^{+}\\!\\to K^{+}\\nu\\bar{\\nu})\\propto(C_{1}-C_{3}) while R D ∝ C 3 R_{D}\\propto C_{3} . The lack of a fixed correlation between the enhanced B + → K + ν ν ¯ B^{+}\\!\\to K^{+}\\nu\\bar{\\nu} rate and the b → c τ ν b\\to c\\tau\\nu ratios is a distinctive prediction of Scenario III and a target for future experimental scrutiny."},{"type":"heading","level":2,"text":"3.2 Light New Physics"},{"type":"para","text":"If the NP candidate is light and feebly interacting, there is a variety of new challenges, both in the experimental and in the theoretical arenas, arising from the fact that the new particles are physical states detectable in experiments (as opposed to the heavy case, where the new particles appear only as virtual particles in internal lines of Feynman diagrams). In general, we can distinguish several kinds of experimental signatures [ Ferber:2022rsf , Bruggisser:2023npd , Alda:2025uwo ] :"},{"type":"para","text":"Off-shell processes where the corresponding on-shell process is forbidden by Lorentz invariance (e.g. leptonic and radiative meson decays M → X ∗ → ℓ + ℓ − , γ γ M\\to X^{*}\\to\\ell^{+}\\ell^{-},\\gamma\\gamma and neutral meson mixing M → X ∗ → M ¯ M\\to X^{*}\\to\\overline{M} ). These processes are able to impose bounds for all values of mass and lifetime of the new particle, although they are most relevant in the resonant region m X = m M m_{X}=m_{M} ."},{"type":"para","text":"Prompt decays: the particle is produced in a reaction A → B X A\\to BX and subsequently decays into X → C X\\to C (where A A , B B , C C are states that may contain multiple physical particles), and the experiment is not able to resolve the production and decay vertices. The experimental signature is therefore an excess in the rate of A → B C A\\to BC as compared to the SM prediction. If the experiment is able to resolve a distance L min L_{\\mathrm{min}} , then the probability for the new particle to travel less than L min L_{\\mathrm{min}} and decay promptly is"},{"type":"para","text":"where c τ X = c ℏ / Γ X c\\tau_{X}=c\\hbar/\\Gamma_{X} is the proper decay length of the particle, and β X \\beta_{X} and γ X \\gamma_{X} its velocity and Lorentz boost in the lab frame. If the Narrow Width Approximation (NWA) is valid Γ X ≪ m X \\Gamma_{X}\\ll m_{X} , then the NP prediction for this process is given by"},{"type":"para","text":"Displaced vertex decays: the particle is produced in A → B X A\\to BX and decays as X → C X\\to C , but the experiment is able to resolve the production and decay vertices. This is the case of experiments with excellent vertex resolution, like LHCb, Belle II and BESIII. The dedicated displaced vertex analyses of these experiments provide bounds directly on the product σ ( A → B X ) × Γ ( X → C ) \\sigma(A\\to BX)\\times\\Gamma(X\\to C) as a function of both m X m_{X} and c τ X c\\tau_{X} ."},{"type":"para","text":"Invisible decays: The new particle is produced in A → B X A\\to BX processes inside the detector, but it is very long-lived and decays outside the detector. The probability for a decay outside the detector is"},{"type":"para","text":"where L max L_{\\mathrm{max}} is the maximum distance that the experiment is able to resolve. The experimental signature is a process with missing energy and momentum, which usually is interpreted as an excess in a process with neutrinos in the final state,"},{"type":"para","text":"Another possible way to obtain an invisible decay, that we will not explore in this work, happens if the new particle acts as a portal to an invisible sector and decays predominantly to non-interacting dark matter particles."},{"type":"para","text":"From a purely computational point of view, the first difficulty that we encounter is the dependence of the predictions on c τ X c\\tau_{X} , that forces us to calculate every decay channel of X X even for channels that are not directly observed. The exponential functions also induce non-Gaussianities in the likelihood function. Moreover, since some observables are only defined for certain values of the mass and/or the proper decay length of the particle, the likelihood function will not even be continuous at some points of the parameter space. All of this, together with the large number of model parameters and experimental observations at our disposal, make ML-based exploration particularly compelling."},{"type":"heading","level":3,"text":"3.2.1 Axion-like particles"},{"type":"para","text":"One of the most popular and theoretically motivated examples of SM extensions featuring new light, feebly interacting particles are axions and axion-like particles (ALPs). Originally, axions were proposed as an elegant solution to the strong CP problem [ Peccei:1977hh , Weinberg:1977ma , Wilczek:1977pj ] , where a pseudo-scalar particle a a of mass m a m_{a} and decay constant f a f_{a} , acting as the pseudo-Nambu Goldstone boson (pNGB) of the Peccei-Quinn (PQ) U ( 1 ) PQ U(1)_{\\mathrm{PQ}} symmetry, dynamically drives the value of the electric dipole moment of the neutron to zero. However, the axion solution requires a correlation between its mass and decay constant, namely [ Weinberg:1977ma , GrillidiCortona:2015jxo ]"},{"type":"para","text":"which severely constrains the available parameter space. Alternatively, if one chooses to consider m a m_{a} and f a f_{a} as independent parameters, we are instead in the ALP paradigm. While an ALP might not necessarily solve the strong CP problem, it has other interesting applications [ Albertus:2026fbe ] , like serving as a dark matter candidate [ Abbott:1982af , Dine:1982ah , Preskill:1982cy , Co:2019jts , Chang:2019tvx , Co:2020dya ] , explaining the flavour puzzle (axiflavon/flaxion [ Ema:2016ops , Calibbi:2016hwq ] ), neutrino masses (majoron [ Chikashige:1980qk , Chikashige:1980ui , Gelmini:1980re , Rothstein:1992rh ] ), or being generic predictions of string theory (axiverse [ Arvanitaki:2009fg , Gendler:2023kjt ] )."},{"type":"para","text":"From a purely bottom-up perspective, we can study ALPs in the framework of EFTs [ Georgi:1986df ] , as an expansion in powers of 1 / f a 1/f_{a} . The first terms of this expansion are the renormalisable operators for the kinetic and mass terms, and dimension-5 operators describing the lowest-order interactions with SM particles: derivative interactions with fermions that respect the shift-symmetry a → a + constant a\\to a+\\mathrm{constant} , and anomalous interactions with gauge bosons:"},{"type":"para","text":"where V ~ μ ν = 1 2 ϵ μ ν ρ σ V ρ σ \\widetilde{V}^{\\mu\\nu}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\rho\\sigma}V_{\\rho\\sigma} is the dual tensor in the convention ϵ 0123 = + 1 \\epsilon^{0123}=+1 , q q and ℓ \\ell are the S U ( 2 ) L SU(2)_{L} doublets of left-handed quarks and leptons, and u u , d d and e e are the S U ( 2 ) L SU(2)_{L} singlets of right-handed quarks and charged leptons. In general, the couplings to fermions c f i j c_{f}^{ij} are hermitian 3 × 3 3\\times 3 matrices in flavour space."},{"type":"para","text":"From the model building perspective, we can distinguish two very broad categories of UV-complete models, depending on which fermions are charged under the PQ symmetry. If only new heavy fermions are charged under PQ, as is the case in the KSVZ model [ Kim:1979if , Shifman:1979if ] and its generalizations [ DiLuzio:2020wdo , Plakkot:2021xyx , DiLuzio:2024xnt ] , the ALP at the matching scale Λ \\Lambda will only interact with the SM gauge bosons as dictated by the gauge anomalies, unless the heavy and SM fermions mix. On the other hand, if the SM fermions f i f_{i} have PQ charges 𝒳 f i \\mathcal{X}_{f}^{i} , then the ALP at the matching scale will feature diagonal (but not necessarily universal) couplings to fermions c f i i ( Λ ) = − 𝒳 f i c_{f}^{ii}(\\Lambda)=-\\mathcal{X}_{f}^{i} , and couplings to gauge bosons fixed by the anomalies [ DiLuzio:2020wdo ] ,"},{"type":"para","text":"Note that in specific realizations, like DFSZ model [ Zhitnitsky:1980tq , Dine:1981rt ] and its generalizations [ DiLuzio:2020wdo , Diehl:2023uui , Cox:2023squ ] , or flaxion models, the scalar sector will impose constraints to combinations of the PQ charges."},{"type":"para","text":"Once the ALP couplings have been determined at the matching scale Λ ∼ 4 π f a \\Lambda\\sim 4\\pi f_{a} , we must solve the Renormalization Group Equations (RGEs) and integrate out the heavy SM particles ( t t , h h , Z Z , W ± W^{\\pm} ) in order to obtain the values of the coefficients at the physical scale of each observable [ Choi:2017gpf , MartinCamalich:2020dfe , Chala:2020wvs , Bauer:2020jbp , Bonilla:2021ufe , DasBakshi:2023lca , Bresciani:2024shu ] . One consequence of the running and matching is that in models with UV couplings to top quarks and/or W ± W^{\\pm} , sizable off-diagonal couplings in the down sector will be generated."},{"type":"heading","level":3,"text":"3.2.2 ALPs as a solution to the Belle II anomaly"},{"type":"para","text":"The Belle II experiment has found an excess in the branching ratio of B + → K + ν ν ¯ B^{+}\\to K^{+}\\nu\\bar{\\nu} of 2.8 σ 2.8\\,\\sigma with respect to the SM prediction [ Belle-II:2023esi ] . Several analyses of the q 2 q^{2} distribution [ Altmannshofer:2023hkn , Fridell:2023ssf , Bolton:2025fsq , Abumusabh:2025zsr , Gartner:2026clx ] have shown that the excess is compatible with the production of one single particle of mass around 2 GeV 2\\,\\mathrm{GeV} that does not decay inside the detector. Furthermore, by combining also the upper limit for the process B → K ∗ ν ν ¯ B\\to K^{*}\\nu\\bar{\\nu} at BaBar [ BaBar:2013npw ] , with different detector size and Lorentz boost, Ref. [ Alda:2025uwo ] found a lower bound for the proper decay length of the particle, c τ ≥ 80 cm ( 95 % C . L . ) c\\tau\\geq 80\\,\\mathrm{cm}\\ (95\\%\\ \\mathrm{C.L.}) ."},{"type":"para","text":"The identification of this presumptive 2 GeV 2\\,\\mathrm{GeV} particle with an ALP proves to be challenging when we consider scenarios motivated by UV-complete models. On the one hand, the flavour-violating coupling | c s b V | / f a |c_{sb}^{V}|/f_{a} that enters in the ALP production B + → K + a B^{+}\\to K^{+}a needs to be large enough to explain the excess, but on the other hand, ALP decay channels such as a → μ + μ − a\\to\\mu^{+}\\mu^{-} and a → η π π a\\to\\eta\\pi\\pi need to be suppressed in order to respect the constraint on the ALP lifetime."},{"type":"para","text":"Ref. [ Alda:2025uwo ] identified a model that fulfills all the requirements, based on the astrophobic axions of Refs. [ DiLuzio:2017ogq , Bjorkeroth:2019jtx , DiLuzio:2022tyc ] . The PQ charges of the SM fermions are"},{"type":"para","text":"In this construction, the coupling c s b V c_{sb}^{V} is generated by top loops, and it is largely independent of β \\beta . Meanwhile, the coupling to muons receives a tree level contribution c μ μ A ( 0 ) = cos 2 β c_{\\mu\\mu}^{A(0)}=\\cos^{2}\\beta which is partially canceled by the running of the coupling to τ \\tau leptons c τ τ A = cos 2 β − 1 = − sin 2 β c_{\\tau\\tau}^{A}=\\cos^{2}\\beta-1=-\\sin^{2}\\beta , and consequently it is minimized for intermediate β \\beta angles."},{"type":"heading","level":3,"text":"3.2.3 Fit parameters and results"},{"type":"para","text":"The model in [ Alda:2025uwo ] is rather predictive, as it only depends on two free parameters, β \\beta and f a f_{a} . Here we aim to extend it in a more descriptive direction, exploring a larger region of the parameter space by lifting the dependence relations between the PQ charges. The free parameters of our new model will be the PQ charges of the third-generation left-handed fermions 𝒳 q L 3 \\mathcal{X}_{q_{L}}^{3} and 𝒳 ℓ L 3 \\mathcal{X}_{\\ell_{L}}^{3} and the universal PQ charges of the right-handed fermions 𝒳 u R \\mathcal{X}_{u_{R}} , 𝒳 d R \\mathcal{X}_{d_{R}} and 𝒳 e R \\mathcal{X}_{e_{R}} , all of them allowed in the range 𝒳 f ∈ [ − 1 , 1 ] \\mathcal{X}_{f}\\in[-1,1] , as well as the energy scale f a f_{a} in the range f a ∈ [ 10 5 , 10 8 ] GeV f_{a}\\in[10^{5},10^{8}]\\,\\mathrm{GeV} (while fixing Λ = 4 π f a \\Lambda=4\\pi f_{a} ) and the ALP mass in the range m a ∈ [ 1.7 , 2.2 ] GeV m_{a}\\in[1.7,2.2]\\,\\mathrm{GeV} ."},{"type":"para","text":"We perform the fit of this model using the likelihood function implemented by the ALP-aca library [ Alda:2025nsz ] . This likelihood function includes more than 100 measurements, but not all of them are operative depending on the ALP mass and proper lifetime. In our setup, in addition to B → K ( ∗ ) a ( → inv ) B\\to K^{(*)}a(\\to\\mathrm{inv}) processes, we have to consider off-shell ALP processes (leptonic and radiative meson decays, neutral meson mixing), and in the limit of short lifetime, prompt/displaced vertex processes like B → K ( ∗ ) a ( → μ + μ − ) B\\to K^{(*)}a(\\to\\mu^{+}\\mu^{-}) decays."},{"type":"para","text":"The training dataset is generated following an active learning programme in order to improve the coverage of the region of large likelihood. A small sample (800 points) generated by LHS is used as a starting seed. Then, at each iteration of the active learning algorithm, a Gaussian Process (GP), the infinite-dimensional limit of a multivariate Gaussian, is trained on the dataset up to that point. A set of candidate points are proposed, for which the GP predicts its χ 2 \\chi^{2} , and more importantly, the uncertainty of said prediction. The candidate points are scored according to their modified Expected Improvement"},{"type":"para","text":"where μ \\mu and σ \\sigma are the predicted central value and uncertainty of the χ 2 \\chi^{2} for the point, χ ∗ 2 \\chi^{2}_{*} is the minimum χ 2 \\chi^{2} observed so far, and Φ \\Phi and ϕ \\phi are the cumulative distribution function and probability distribution function of the Gaussian distribution, respectively. The parameter ξ \\xi regulates the trade-off between exploration and exploitation: if ξ \\xi is large, the best score will be achieved by points where the GP is very uncertain, while for small ξ \\xi , points with small predicted χ 2 \\chi^{2} are selected instead. At each iteration of the algorithm, the point with the best score is selected, its true χ 2 \\chi^{2} is computed and added to the training dataset, and ξ \\xi is slightly decreased. The total size of the dataset, including both the initial seeds and the points selected by the GP, is 2500 points. The training dataset is depicted in Figure 5 , showing that the highest density is achieved in the region of low χ 2 \\chi^{2} ."},{"type":"para","text":"In order to fully capture the details of the χ 2 \\chi^{2} function in the region around its minimum, we propose a two-step strategy: we first train a XGBoost instance on the sigmoid of the displaced χ 2 \\chi^{2} function,"},{"type":"para","text":"where χ c 2 = χ b . f . 2 + 10 \\chi^{2}_{c}=\\chi^{2}_{\\mathrm{b.f.}}+10 is the cutoff: this first stage acts as a classifier, retaining only the points below the cutoff. The second stage is a XGBoost instance trained only on those points below the cutoff, thus focusing on the relevant region of the parameter space."},{"type":"para","text":"The performance of the classifier stage is characterized by its Receiver Operating Characteristic (ROC) curve in Figure 6 . It presents the True Positive Rate (proportion of the points below the cutoff that are correctly identified by the classifier) and False Positive Rate (proportion of the points above the cutoff that are misidentified) for various thresholds of σ ( χ 2 ) \\sigma(\\chi^{2}) . A random classifier would lie on the diagonal of the ROC, while a perfect classifier would be situated in the upper left corner. By setting the threshold for the classifier at σ ( χ 2 ) = 0.5 \\sigma(\\chi^{2})=0.5 , we achieve TPR = 0.96 \\mathrm{TPR}=0.96 and FPR = 0.16 \\mathrm{FPR}=0.16 on the validation dataset. The Area Under the Curve (AUC) of the ROC plot is AUC = 0.974 \\mathrm{AUC}=0.974 , also indicating a great classifying performance. In the regressor stage, the surrogate achieves a Pearson correlation coefficient r = 0.929 r=0.929 , as shown in Figure 6 (Right)."},{"type":"para","text":"As in the B B -physics case, the decisive advantage of the surrogate is its evaluation speed, quantified in Table 2 . On a single CPU core, one evaluation of the exact ALP-aca likelihood over the seven-dimensional parameter space takes ∼ 1.9 \\sim\\!1.9 s (it must solve the RGEs, perform the matching and recompute more than a hundred observables), whereas the trained XGBoost ensemble predicts the same point in ∼ 3.5 μ \\sim\\!3.5\\,\\mu s, a speed-up of about 5.3 × 10 5 5.3\\times 10^{5} . The gain is most tangible in the construction of the training set: generating the 8 × 10 3 8\\times 10^{3} exact points costs about 4.1 4.1 core-hours of ALP-aca computation, which the emulator re-evaluates in ∼ 0.03 \\sim\\!0.03 s. As in the flavour benchmark, all times are measured on a single core, so the comparison is reproducible and does not depend on the number of cores or on the specific machine."},{"type":"para","text":"The SHAP values help us again to interpret the results of the fit. In Figure 7 (left), we have displayed the mean of absolute values of SHAPs in the dataset, which corresponds to the expected importance of each parameter. The most important parameter is m a m_{a} , since it determines if a solution to the Belle II anomaly is even kinematically allowed in the first place. After that, the two most important parameters are 𝒳 q L 3 \\mathcal{X}_{q_{L}}^{3} and f a f_{a} , which control the top loops and the overall impact of ALP effects. The impact of the PQ charges of right-handed fermions is more limited. In the case of m a m_{a} , we can see in Figure 7 (right) that larger ALP masses, which are less compatible with the q 2 q^{2} distribution of the recast to two-body decay kinematics, result in larger SHAP values, that is, larger contribution to the χ 2 \\chi^{2} function."},{"type":"para","text":"Finally, using the surrogate of the likelihood function and a Markov chain Monte Carlo (MCMC) implemented by the emcee library, we can efficiently sample the posterior distributions of both the parameters of the model and of derived quantities. In Figure 8 , we have obtained the posterior distributions for the input parameters m a m_{a} and f a f_{a} , for the low-energy couplings | c s b V | |c_{sb}^{V}| , | c μ μ A | |c_{\\mu\\mu}^{A}| and | c G | |c_{G}| (all of them at the physical scale μ ∼ m a \\mu\\sim m_{a} ) and the relevant parameters for the Belle II anomaly: c τ a c\\tau_{a} and BR ( B → K a ) \\mathrm{BR}(B\\to Ka) . The posterior distribution correctly reflects functional dependencies, such as c τ a ∝ f a 2 c\\tau_{a}\\propto f_{a}^{2} and BR ( B + → K + a ) ∝ | c s b V | 2 \\mathrm{BR}(B^{+}\\to K^{+}a)\\propto|c_{sb}^{V}|^{2} . Clearly, results favor an ALP mass around 1.8 1.8 GeV, in agreement with previous analyses of the Belle II excess based on the kinematic reconstruction of the missing-energy signal. The preferred values of f a f_{a} lie in the 10 6 − 10 7 10^{6}-10^{7} GeV range, balancing the need for sufficiently large flavour-changing interactions to explain the anomaly against the suppression required by constraints included in the fit. The posterior distributions of the effective couplings show that | c s b V | 2 |c_{sb}^{V}|^{2} must be large enough to generate an observable BR ( B + → K + a ) \\mathrm{BR}(B^{+}\\to K^{+}a) , while | c μ μ A | |c_{\\mu\\mu}^{A}| remains comparatively small in order to suppress visible ALP decays and ensure compatibility with searches for displaced and prompt dimuon resonances."},{"type":"para","text":"In Figure 9 , we have shown again the posterior obtained by the surrogate and MCMC, this time fixing m a = 1.8 GeV m_{a}=1.8\\,\\mathrm{GeV} , for c τ a c\\tau_{a} and BR ( B + → K + a ) \\mathrm{BR}(B^{+}\\to K^{+}a) (in purple), compared to the fit using only the B → K + inv B\\to K+\\mathrm{inv} data alone. The BaBar and Belle II data was only able to set a lower bound for the decay length of the ALP, but the combination with other observables in the fit provides complementary information that further restricts the proper decay length. As it was the case in the SMEFT analysis, we once more underline the importance of global explorations of the parameter space, which hugely benefit from the ML techniques like the ones we have described."},{"type":"heading","level":2,"text":"3.3 Cosmology"},{"type":"para","text":"The last decades have seen cosmology become precision physics. And in this path, the methods used in order to do parameter inference and data analysis have become more and more similar to to the ones used in high energy physics. There are different types of cosmological probes when doing cosmological parameter inference. The space-time geometry can be determined through the distance-redshift relation through time-domain probes or through the study of the large-scale structure of the Universe. The use of these probes has allowed us to define a baseline cosmological standard model, the flat Λ \\Lambda CDM, but some tensions between observations may indicate the need for NP [ CosmoVerseNetwork:2025alb ] . In all cases, the likelihood is again built from a χ 2 \\chi^{2} comparing a handful of cosmological parameters against hundreds of measurements through theoretical predictions that require numerical distance integrals, so the same obstacles encountered in the flavour fits (expensive forward evaluations and curved parameter degeneracies) reappear here and make the ML emulation strategy of Section 2 equally attractive. The main cosmological probes used in this work are the distance–redshift relation, measured with Type Ia supernovae as standard candles and with baryon acoustic oscillations as a standard ruler, complemented by the Cosmic Microwave Background (CMB) as a prior on cosmological parameters."},{"type":"heading","level":3,"text":"3.3.1 Cosmological probes and the likelihood"},{"type":"para","text":"The baseline model for cosmological analysis is the flat Λ \\Lambda CDM model where the accelerated expansion of the Universe is explained by the cosmological Λ \\Lambda with constant density and spatial flatness at each moment of cosmological time. The expansion history of the Universe is encoded in the dimensionless Friedmann equation: E ( z ) ≡ H ( z ) / H 0 E(z)\\equiv H(z)/H_{0} ,"},{"type":"para","text":"where spatial flatness fixes the dark-energy density parameter to 1 − Ω m 1-\\Omega_{m} and radiation is negligible at the redshifts probed. The dynamics are driven by the parameters: H 0 H_{0} is the Hubble constant, Ω m = Ω b + Ω c d m \\Omega_{m}=\\Omega_{b}+\\Omega_{cdm} is the density parameter of matter in the Universe, comprised by ordinary matter ( Ω b \\Omega_{b} ) and cold dark matter ( Ω c d m \\Omega_{cdm} ). The dynamical nature of dark energy is enclosed in f D E ( z ) f_{DE}(z) . For the Λ \\Lambda CDM model, as the energy density of the cosmological constant component does not evolve with time ( ρ Λ ∼ c o n s t . \\rho_{\\Lambda}\\sim const. ), f D E = 1 f_{DE}=1 ."},{"type":"para","text":"In the last decade, different cosmological probes, old and new, seem to converge in that the most plausible model for cosmology was Λ \\Lambda CDM. But recently, supernovae and standard ruler measurements have indicated a possible dynamical dark energy component. The evidence for dynamical dark energy is based upon the flat w 0 w a w_{0}w_{a} CDM dark-energy model, in which the dark-energy equation of state is allowed to evolve with redshift according to the Chevallier–Polarski–Linder (CPL) parameterisation [ Chevallier:2000qy , Linder:2002et ] ,"},{"type":"para","text":"which reduces to a cosmological constant ( Λ \\Lambda CDM) for w 0 = − 1 w_{0}=-1 , w a = 0 w_{a}=0 . In this particular case, the energy density of dynamical dark energy component evolves according to:"},{"type":"para","text":"All geometric observables follow from common definitions. The transverse distance between two objects at a given redshift is the comoving (transverse) distance, D M ( z ) D_{M}(z) , while the radial geometric distance relation with redshift is the Hubble distance, D H ( z ) D_{H}(z) , and finally we can capture the isotropic nature of a spherical ruler with the spherically averaged distance, D V ( z ) D_{V}(z) , [ SDSS:2005xqv ]"},{"type":"para","text":"so that the model depends on the four parameters 𝜽 = ( Ω m , H 0 , w 0 , w a ) \\bm{\\theta}=(\\Omega_{m},H_{0},w_{0},w_{a}) (reduced to ( Ω m , H 0 ) (\\Omega_{m},H_{0}) in the Λ \\Lambda CDM baseline below). The three probes enter the global likelihood as follows:"},{"type":"para","text":"Type Ia Supernovae are standardisable candles: after light-curve correction their peak brightness is fixed, so the observed apparent magnitude m B m_{B} measures the luminosity distance through the distance modulus"},{"type":"para","text":"with M M the absolute magnitude. Because M M is fully degenerate with H 0 H_{0} , it is analytically marginalised, so the supernovae constrain the shape of the distance–redshift relation rather than its absolute scale. The χ 2 \\chi^{2} uses the full statistical + + systematic covariance matrix 𝒞 \\mathcal{C} , χ SN 2 = Δ 𝖳 𝒞 − 1 Δ \\chi^{2}_{\\rm SN}=\\Delta^{\\mathsf{T}}\\mathcal{C}^{-1}\\Delta with Δ i = μ i obs − μ ( z i ) \\Delta_{i}=\\mu_{i}^{\\rm obs}-\\mu(z_{i}) . We use the Pantheon + + / SH0ES compilation [ Scolnic:2021amr , Brout:2022vxf ] (1657 supernovae, with the Cepheid calibrators excluded so that M M is marginalised) and, as an independent sample, the Dark Energy Survey five-year SN Ia data set (DES-SN5YR), in both its 2024 release (1829 supernovae) [ DES:2024tys ] and its 2025 update [ DES:2025sn5yr ] (1820 supernovae); for DES the magnitude offset is marginalised analytically, including the corresponding normalisation term."},{"type":"para","text":"Baryon Acoustic Oscillations imprint a fixed comoving scale, the sound horizon at the baryon drag epoch r d r_{d} , that acts as a standard ruler in the galaxy distribution. The sound horizon is given by: r d = ∫ z d ∞ c s ( z ) H ( z ) 𝑑 z r_{d}=\\int_{z_{d}}^{\\infty}{\\frac{c_{s}(z)}{H(z)}dz} , where z d z_{d} is the redshift at decoupling and the speed of sound prior to recombination, c s ( z ) c_{s}(z) , is set by the baryon, ρ b \\rho_{b} and radiation, ρ γ \\rho_{\\gamma} , densities as c s ( z ) = c [ 3 ( 1 + 3 ρ b ( z ) 4 ρ γ ( z ) ) ] − 1 c_{s}(z)=c\\left[\\sqrt{3\\left(1+\\frac{3\\rho_{b}(z)}{4\\rho_{\\gamma}(z)}\\right)}\\right]^{-1} . Surveys measure the ratios D M / r d D_{M}/r_{d} , D H / r d D_{H}/r_{d} and the isotropic combination D V / r d D_{V}/r_{d} at several effective redshifts. We use the 13 measurements of the DESI Data Release 2 [ DESI:2025zgx ] with their Gaussian likelihood χ BAO 2 = Δ 𝖳 𝒞 BAO − 1 Δ \\chi^{2}_{\\rm BAO}=\\Delta^{\\mathsf{T}}\\mathcal{C}^{-1}_{\\rm BAO}\\Delta , fixing r d r_{d} to its Planck fiducial value."},{"type":"para","text":"Cosmic Microwave Background. Rather than recomputing the full CMB power spectra at every point, we summarise the Planck constraint as a Gaussian prior on the late-time parameters. The Planck PR4 + + lensing chain in the CPL model obtained in the re-analysis of the latest Planck data [ Tristram:2023haj ] (superseding the official PR3 [ Planck:2018vyg ] ) made by [ 2026PDU....5202283A ] provides a covariance matrix for its full set of about 25 25 parameters; of these we keep only the five that map onto our fit, the Hubble constant H 0 H_{0} , the two physical densities of baryons ω b \\omega_{b} and cold dark matter ω cdm \\omega_{\\rm cdm} (with ω x ≡ Ω x h 2 \\omega_{x}\\equiv\\Omega_{x}h^{2} and h = H 0 / 100 h=H_{0}/100 ), and the dark-energy parameters w 0 w_{0} and w a w_{a} , discarding the remaining nuisance columns. Because our sampling variable is the total matter density rather than its two components, we combine them into ω m = ω b + ω cdm \\omega_{m}=\\omega_{b}+\\omega_{\\rm cdm} by linear error propagation, which collapses the 3 × 3 3\\times 3 block of ( H 0 , ω b , ω cdm ) (H_{0},\\omega_{b},\\omega_{\\rm cdm}) into a correlated 2 × 2 2\\times 2 covariance for ( H 0 , ω m ) (H_{0},\\omega_{m}) ; the dark-energy parameters are carried with their Planck variances. The result is a 4 × 4 4\\times 4 covariance Σ CMB \\Sigma_{\\rm CMB} for ( H 0 , ω m , w 0 , w a ) (H_{0},\\omega_{m},w_{0},w_{a}) in which H 0 H_{0} and ω m \\omega_{m} stay correlated, while w 0 w_{0} and w a w_{a} enter as independent Gaussian constraints. Centred on the Planck Λ \\Lambda CDM best fit ( w 0 = − 1 w_{0}=-1 , w a = 0 w_{a}=0 ), it contributes"},{"type":"para","text":"where ω m = Ω m ( H 0 / 100 ) 2 \\omega_{m}=\\Omega_{m}(H_{0}/100)^{2} is evaluated from the sampled parameters and the starred quantities are the Planck central values. Combining the two densities by propagating their covariance is the exact marginalisation of the joint Gaussian onto ( H 0 , ω m ) (H_{0},\\omega_{m}) , so for observables that depend only on the total matter density it is equivalent to carrying ω b \\omega_{b} and ω cdm \\omega_{\\rm cdm} as explicit parameters; the correlated ( H 0 , ω m ) (H_{0},\\omega_{m}) block is retained, while in this particular covariance w 0 w_{0} and w a w_{a} happen to be uncorrelated with the rest and enter as independent Gaussians of width σ ( w 0 ) = 0.02 \\sigma(w_{0})=0.02 , σ ( w a ) = 0.05 \\sigma(w_{a})=0.05 . The same prior is used in every CMB combination of Sections 3.3.4 and 3.3.5 ."},{"type":"para","text":"Combined data sets are fit by summing the individual χ 2 \\chi^{2} functions, χ tot 2 ( 𝜽 ) = ∑ k χ k 2 ( 𝜽 ) \\chi^{2}_{\\rm tot}(\\bm{\\theta})=\\sum_{k}\\chi^{2}_{k}(\\bm{\\theta}) , exactly as in the flavour global fit."},{"type":"heading","level":3,"text":"3.3.2 Surrogate construction and the choice of training target"},{"type":"para","text":"For each data combination we train an XGBoost surrogate of χ tot 2 ( 𝜽 ) \\chi^{2}_{\\rm tot}(\\bm{\\theta}) following the methodology of Section 2 . The training design combines two-dimensional slices through the best-fit point, a uniform space-filling draw over the full prior box ( Ω m ∈ [ 0.1 , 0.9 ] \\Omega_{m}\\!\\in\\![0.1,0.9] , H 0 ∈ [ 20 , 100 ] H_{0}\\!\\in\\![20,100] , w 0 ∈ [ − 3 , 0.2 ] w_{0}\\!\\in\\![-3,0.2] , w a ∈ [ − 3 , 2 ] w_{a}\\!\\in\\![-3,2] ), and a Gaussian cloud whose covariance is the Hessian of the fit, so that the narrow, anisotropic w 0 w_{0} – w a w_{a} degeneracy is sampled densely where it matters most. Each design contains 2.5 2.5 – 5 × 10 5 5\\times 10^{5} points, with the exact distance integrals of Eqs. ( 25 )–( 26 ) evaluated in parallel across CPU cores."},{"type":"para","text":"A central, problem-specific ingredient is the choice of training target. The cosmological χ 2 \\chi^{2} spans an enormous dynamic range, from χ min 2 ∼ 10 3 \\chi^{2}_{\\rm min}\\!\\sim\\!10^{3} at the best fit to ≳ 10 6 \\gtrsim\\!10^{6} at the edges of the prior box, while the physically relevant structure, the Δ χ 2 ∼ 2 \\Delta\\chi^{2}\\!\\sim\\!2 – 6 6 band that defines the 1 σ 1\\,\\sigma and 2 σ 2\\,\\sigma confidence regions, lives in a tiny sliver near the minimum. Crucially, the absolute scale of χ min 2 \\chi^{2}_{\\rm min} differs by two orders of magnitude between the data combinations, from χ min 2 ≈ 1.7 × 10 3 \\chi^{2}_{\\rm min}\\!\\approx\\!1.7\\times 10^{3} for the supernova + + BAO fits (more than 1600 1600 data points) down to χ min 2 ≈ 5.6 \\chi^{2}_{\\rm min}\\!\\approx\\!5.6 for the BAO-only fit (13 points), and this is what makes the two naive targets inadequate in a data-dependent way:"},{"type":"para","text":"Training directly on χ 2 \\chi^{2} lets the loss be dominated by the high- χ 2 \\chi^{2} tail: the surrogate fits the outer wall of the likelihood well but is blind to the 𝒪 ( few ) \\mathcal{O}(\\text{few}) variations near the minimum. This happens for every combination, irrespective of χ min 2 \\chi^{2}_{\\rm min} , and makes the recovered contours unusable."},{"type":"para","text":"Training on a plain logarithm, log 10 χ 2 \\log_{10}\\chi^{2} , compresses the tail, but whether it works now depends on where the minimum falls on the logarithmic curve, which is precisely why it succeeds for some data sets and fails for others. For the supernova + + BAO and + + CMB fits the minimum is large, so it sits on the locally flat part of the logarithm and the decisive 1 σ 1\\,\\sigma – 2 σ 2\\,\\sigma band maps onto a negligible change of the target ( log 10 1706 − log 10 1700 ≈ 1.5 × 10 − 3 \\log_{10}1706-\\log_{10}1700\\approx 1.5\\times 10^{-3} ): the contours remain unresolved. For the BAO-only fit the minimum is small, so it sits on the steep part of the logarithm ( log 10 12 − log 10 5.6 ≈ 0.33 \\log_{10}12-\\log_{10}5.6\\approx 0.33 ) and the plain transform does resolve the contours. No single un-shifted logarithm can therefore serve all combinations at once."},{"type":"para","text":"Both problems are solved by a shifted- log 10 \\log_{10} transform, which is the target adopted throughout this work:"},{"type":"para","text":"where χ min 2 \\chi^{2}_{\\rm min} is the minimum over the training set. The shift maps the best fit to y = 0 y=0 regardless of the absolute value of χ min 2 \\chi^{2}_{\\rm min} , so that the minimum of every likelihood, whether χ min 2 ≈ 5 \\chi^{2}_{\\rm min}\\!\\approx\\!5 or ≈ 1700 \\approx\\!1700 , is placed on the same steep, well-resolved part of the transform; this is what removes the data-dependence of the plain logarithm discussed above. The unit offset additionally guarantees a strictly positive argument ( ≥ 1 \\geq 1 ), removing the log ( 0 ) \\log(0) singularity that a naive shift χ 2 − χ min 2 \\chi^{2}-\\chi^{2}_{\\rm min} would introduce exactly at the densely sampled best-fit point. The transform compresses the 10 6 10^{6} tail and stretches precisely the near-minimum band that encodes the confidence regions, amplifying it by about three orders of magnitude in target space. The surrogate is trained on y y and its predictions are mapped back to linear χ 2 \\chi^{2} through the exact inverse in Eq. ( 29 ), so that downstream contour and SHAP computations operate on physical χ 2 \\chi^{2} values. With this target the emulators reach a coefficient of determination R 2 = 0.997 R^{2}=0.997 – 0.9999 0.9999 on a held-out validation set across all data combinations."},{"type":"para","text":"Posterior distributions are then mapped by running the parallel Random-Walk Metropolis–Hastings sampler of Section 2 (1024 chains, with the proposal covariance set to the Hessian of the fit) directly on the emulated likelihood, and rendered as marginalised contours. Because each likelihood call is a single forward pass through the tree ensemble rather than a set of numerical distance integrals, the cost of the posterior exploration is reduced by more than an order of magnitude, as quantified in Section 3.3.5 ."},{"type":"heading","level":3,"text":"3.3.3 Baseline: the Λ \\Lambda CDM fit"},{"type":"para","text":"As a simple, fully controlled example we first emulate the two-parameter Λ \\Lambda CDM likelihood of Pantheon + + combined with DESI BAO, fitting only ( Ω m , H 0 ) (\\Omega_{m},H_{0}) . The surrogate reproduces the exact χ 2 \\chi^{2} with R 2 = 0.99988 R^{2}=0.99988 , and the emulator MCMC recovers"},{"type":"para","text":"at χ min 2 = 1703.6 \\chi^{2}_{\\rm min}=1703.6 . Figure 10 shows the resulting Ω m \\Omega_{m} – H 0 H_{0} posterior: a single, closed, mildly anti-correlated ellipse with no secondary modes or open directions. These values reproduce the standard concordance picture, Ω m = 0.310 \\Omega_{m}=0.310 matching the Planck and DESI determinations to within 1 σ 1\\,\\sigma and the supernova-only Pantheon + + value Ω m ≈ 0.33 \\Omega_{m}\\!\\approx\\!0.33 . Because the supernova absolute magnitude is marginalised, the fit carries no SH0ES distance-ladder anchor, so H 0 H_{0} is set by the BAO scale with the Planck sound horizon r d r_{d} ; it therefore lands on the Planck-like value H 0 ≈ 68 H_{0}\\!\\approx\\!68 rather than the higher local-distance-ladder measurement, exactly as this dataset combination behaves in the DESI and Pantheon + + analyses. This baseline confirms that the pipeline reproduces the established result before turning to the more demanding, and more interesting, four-parameter dark-energy case."},{"type":"heading","level":3,"text":"3.3.4 Dynamical dark energy: w 0 w a w_{0}w_{a} CDM with and without CMB"},{"type":"para","text":"We now turn to the full w 0 w a w_{0}w_{a} CDM model and study how the dark-energy constraints depend on the supernova sample and on the inclusion of the CMB prior. We consider three supernova compilations (Pantheon + + , DES-SN5YR 2024 and 2025), each combined with DESI BAO, and contrast the fit without and with the Planck prior. The best-fit parameters of the six emulators are collected in Table 3 , and the corresponding posteriors are compared pairwise in Figure 11 ."},{"type":"para","text":"A consistent pattern emerges. Without the CMB prior, all three supernova + + BAO combinations prefer a dynamical dark energy with w 0 > − 1 w_{0}>-1 and w a < 0 w_{a}<0 , i.e. an equation of state that crosses the phantom divide as it evolves, the same hint of evolving dark energy recently reported by DESI. The associated contours are broad, with σ ( w a ) ≈ 0.4 \\sigma(w_{a})\\approx 0.4 , reflecting the well-known w 0 w_{0} – w a w_{a} geometric degeneracy that the surrogate captures as an elongated, curved ridge. The deviation is strongest for Pantheon + + and DES-2024 and milder for the DES-2025 update, which sits between the two. Adding the Planck prior pulls every fit back towards Λ \\Lambda CDM ( w 0 ≈ − 0.96 w_{0}\\approx-0.96 to − 0.97 -0.97 , w a ≈ 0.02 w_{a}\\approx 0.02 ) and tightens the constraints dramatically: the uncertainty on w a w_{a} shrinks by about a factor of ten (from ∼ 0.42 \\sim 0.42 to ∼ 0.045 \\sim 0.045 ), on w 0 w_{0} by a factor of five, and on Ω m \\Omega_{m} by a factor of three. Figure 11 makes this contraction visually explicit for the three samples."},{"type":"para","text":"Reading the Pantheon + + panel directly (the DES panels are qualitatively identical), the supernova + + BAO posterior (blue, filled) is a wide, tilted ellipse in the w 0 w_{0} – w a w_{a} plane, elongated along the anti-correlation direction and displaced from the Λ \\Lambda CDM point (dashed lines) towards w 0 > − 1 w_{0}>-1 , w a < 0 w_{a}<0 , whereas the + + CMB posterior (red) collapses to a small blob centred on ( w 0 , w a ) = ( − 1 , 0 ) (w_{0},w_{a})=(-1,0) ; the Ω m \\Omega_{m} and H 0 H_{0} marginals, already well determined, barely move. The supernova + + BAO best fits, for instance ( w 0 , w a ) ≈ ( − 0.78 , − 0.73 ) (w_{0},w_{a})\\approx(-0.78,-0.73) for DES-2024, fall in the same region and direction of the w 0 w_{0} – w a w_{a} plane as the dynamical-dark-energy preference reported by the DESI DR2 analysis for the analogous DESI + + supernova combinations, so the emulator reproduces that published result."},{"type":"para","text":"The CMB enters here not as a forward computation of the temperature and polarisation spectra, but as a compressed Gaussian constraint built from the Planck PR4 + + lensing covariance matrix (Eq. 28 ), a standard and economical way to fold the CMB into a late-time fit. It carries the Planck constraints on all four parameters, with the H 0 H_{0} – ω m \\omega_{m} correlation retained, and in particular the tight bounds on the dark-energy sector that the supernovae and BAO leave nearly free; this is what allows it to tighten w 0 w_{0} and w a w_{a} by up to an order of magnitude and to draw the joint fit into agreement with Λ \\Lambda CDM. The relevant point for the present work is that the surrogate reproduces this compressed-CMB likelihood as faithfully as the supernova + + BAO one, so the contraction of the contours is a genuine output of the emulated fit and not an artefact of the emulation."},{"type":"heading","level":3,"text":"3.3.5 Computational benchmark: surrogate versus exact likelihood"},{"type":"para","text":"The cosmological application also offers a clean setting to quantify the computational gain of the ML emulation, because here the exact likelihood is cheap enough to be sampled directly for comparison, unlike the flavour case, where a brute-force scan is prohibitive. We benchmark three nested data combinations of increasing constraining power, BAO only, BAO & Pantheon + + and CMB & BAO & Pantheon + + , which illustrate how each probe closes the w 0 w_{0} – w a w_{a} degeneracy: with only the 13 BAO points the four-parameter fit is essentially unconstrained ( χ min 2 = 5.6 \\chi^{2}_{\\rm min}=5.6 , with w a w_{a} railing against the prior edge), adding the supernovae closes the contour, and the CMB prior tightens it onto Λ \\Lambda CDM (Figure 12 )."},{"type":"para","text":"We run the same Random-Walk Metropolis–Hastings sampler (1024 chains, identical settings and seed) with four interchangeable likelihood back-ends: the XGBoost surrogate evaluated on the GPU, and the exact χ 2 \\chi^{2} computed with (i) the astropy -based reference parallelised over a CPU process pool (“Astropy”), (ii) a vectorized NumPy reimplementation of the distance integrals on the CPU (“CPU”), and (iii) a JAX vmap kernel on the GPU (“GPU”) [ jax2018github ] . All four back-ends are run on the same workstation, an 18-core/36-thread Intel Core i9-10980XE CPU ( 3.0 3.0 GHz base, 4.8 4.8 GHz turbo) with 188 188 GB of RAM and an NVIDIA RTX A6000 GPU ( 48 48 GB). The Astropy reference must be evaluated one cosmology at a time and cannot be vectorized across the sampled points; its only parallelism is to spread those points over a process pool (up to 16 workers), which scales poorly and becomes impractical for the large supernova likelihoods (the n/a entries in Table 4 ). This is precisely why the distance integrals are reimplemented directly as the vectorized NumPy (CPU) and JAX (GPU) back-ends, which evaluate the whole batch of points in a single call. Crucially, all four back-ends produce statistically identical posteriors, confirming that the surrogate is accurate and not merely fast. The wall-clock times are summarised in Table 4 : the surrogate completes each run in 4.8 4.8 – 7.4 7.4 s, 3.6 3.6 – 22.4 × 22.4\\times faster than the CPU (vectorized NumPy) likelihood and 7.2 7.2 – 12.1 × 12.1\\times faster than the exact GPU kernel, with the advantage growing for the more expensive Pantheon + + likelihoods."},{"type":"para","text":"The structure of Table 4 reflects two competing effects. The emulator’s per-evaluation cost is essentially independent of the dataset, since a forward pass through the tree ensemble does not depend on the number of observables, whereas the exact likelihood scales with the data; this is why the speed-up over the CPU back-end grows from ∼ 4 × \\sim\\!4\\times for the 13 BAO points to ≳ 20 × \\gtrsim\\!20\\times for the ∼ 1700 \\sim\\!1700 -supernova Pantheon + + fits. The GPU exact kernel behaves oppositely at small data volume: for BAO only its fixed kernel-launch and host–device-transfer overhead dominates the tiny per-step compute, so it is slower than the CPU (vectorized NumPy) code ( 90 90 vs 27 27 s), whereas for the much heavier Pantheon + + likelihood that overhead is amortised and the GPU wins. Finally, because every chain is run to a fixed effective-sample-size target rather than a fixed number of steps, the wall-clock time also tracks how fast each posterior mixes: the tightly CMB-constrained fit converges in fewer steps, so BAO & Pantheon + + & CMB is cheaper than BAO & Pantheon + + despite containing more data."},{"type":"para","text":"The same acceleration benefits the up-front cost of building the training set. Evaluating the 2 × 10 5 2\\times 10^{5} exact χ 2 \\chi^{2} points needed to train a supervised surrogate takes about 4 × 10 3 4\\times 10^{3} s (more than an hour) with the astropy reference on 16 CPU cores for the Pantheon + + combinations, whereas the batched JAX /GPU implementation of the distance integrals produces the same set in a few seconds. The GPU and Astropy evaluations agree to better than one unit in χ 2 \\chi^{2} near the minimum, where the training signal matters; they differ only in the extreme tails ( χ 2 ∼ 10 6 \\chi^{2}\\!\\sim\\!10^{6} ), a harmless discrepancy since those points sit in the compressed part of the shifted-log transform of Eq. ( 29 ). Thus both the construction and the sampling of the emulated likelihood are accelerated, and the one-time training cost is amortised over the whole posterior analysis, after which the surrogate replaces the physics computation entirely."},{"type":"heading","level":3,"text":"3.3.6 Feature importance"},{"type":"para","text":"As in the flavour analysis, SHAP values provide a transparent, model-independent ranking of the parameters by their mean absolute contribution to the emulated χ 2 \\chi^{2} , that is, by how steeply the likelihood responds to each one across the sampled volume, which is a direct proxy for how tightly each parameter is constrained. The rankings (Figure 13 ) follow the expected physics and, tellingly, reorganise when the CMB prior is added."},{"type":"para","text":"In the Λ \\Lambda CDM baseline the two parameters are almost tied, Ω m \\Omega_{m} ( 0.67 0.67 ) just above H 0 H_{0} ( 0.63 0.63 ): both are comparably and tightly determined by the joint supernova + + BAO geometry, with Ω m \\Omega_{m} marginally ahead because it enters every distance through E ( z ) E(z) . In the w 0 w a w_{0}w_{a} CDM supernova + + BAO fit the background parameters dominate, H 0 H_{0} ( 0.47 0.47 ) and Ω m \\Omega_{m} ( 0.44 0.44 ) well above the dark-energy shape parameters w 0 w_{0} ( 0.24 0.24 ) and w a w_{a} ( 0.16 0.16 ). Within the dark-energy sector w 0 w_{0} always ranks above w a w_{a} , and this ordering has a clear physical origin: w 0 w_{0} is the value of the equation of state today, so it controls w ( z ) w(z) at low redshift, exactly where dark energy dominates the energy budget and where the supernovae and BAO are most numerous and constraining, whereas w a w_{a} only sets the evolution through the z / ( 1 + z ) z/(1+z) factor, which vanishes at z = 0 z=0 and grows only towards higher redshift, where dark energy is subdominant to matter and the data are sparser. w 0 w_{0} therefore has far more leverage on the observables than w a w_{a} , as the marginalised errors confirm ( σ ( w 0 ) ≈ 0.066 \\sigma(w_{0})\\approx 0.066 versus σ ( w a ) ≈ 0.42 \\sigma(w_{a})\\approx 0.42 , i.e. w 0 w_{0} about six times better determined), and w a w_{a} , the flattest and most degenerate direction, ranks last; this is the feature-importance counterpart of the elongated w 0 w_{0} – w a w_{a} ridge of Figure 11 . Adding the compressed CMB prior inverts the background-versus-dark-energy hierarchy: w 0 w_{0} jumps to the top ( 0.53 0.53 ) and w a w_{a} to second ( 0.27 0.27 ), while Ω m \\Omega_{m} ( 0.25 0.25 ) and H 0 H_{0} ( 0.24 0.24 ) drop. The CMB prior adds tight, direct constraints on w 0 w_{0} and w a w_{a} , which the supernovae and BAO left almost free, so the χ 2 \\chi^{2} now rises steeply along w 0 w_{0} and w a w_{a} and they take most of the SHAP weight, whereas Ω m \\Omega_{m} and H 0 H_{0} , already well measured and consistent with the prior, contribute proportionally less. Note that w 0 w_{0} stays ahead of w a w_{a} throughout (the same ∼ 2 : 1 \\sim\\!2\\!:\\!1 ratio with and without the prior), for the same low-redshift-leverage reason. The SHAP reordering is thus the feature-level fingerprint of the contour contraction seen in Figure 11 , and the agreement between this data-driven ranking and the expected constraining structure confirms that the surrogate has learned the correct physical dependencies rather than numerical artefacts."},{"type":"para","text":"Beyond this global ranking, SHAP also resolves how each parameter acts point by point across the sampled volume. Figure 14 shows this for w 0 w_{0} in the Pantheon + + & \\& DESI BAO fit: its SHAP value, the signed contribution of w 0 w_{0} to the emulated χ 2 \\chi^{2} at each sampled cosmology, plotted against w 0 w_{0} and coloured by w a w_{a} . The points trace a clear parabola, the data-driven image of the χ 2 \\chi^{2} well along w 0 w_{0} : the contribution is least, and slightly negative, in the densely sampled best-fit region near w 0 ≈ − 0.8 w_{0}\\!\\approx\\!-0.8 , where w 0 w_{0} sits at its optimum and helps lower the χ 2 \\chi^{2} , and grows steeply and positive as w 0 w_{0} departs in either direction and drives the χ 2 \\chi^{2} up. It is the magnitude of this contribution, not its sign, that defines importance: the mean absolute value reported in Figure 13 is dominated by these steep flanks, so the parabola itself is the fingerprint of a tightly constrained parameter, whereas a flat, weakly constrained direction such as w a w_{a} shows small SHAP values throughout and no comparable well. The small contribution in the best-fit cloud therefore does not mean that w 0 w_{0} is unimportant there; it simply reflects that, at the bottom of the well, the likelihood is locally flat in w 0 w_{0} , exactly as expected for a parameter pinned at its optimum. The colour finally encodes the residual w 0 w_{0} – w a w_{a} interaction: at fixed w 0 w_{0} the contribution still varies with w a w_{a} , the feature-level counterpart of the elongated w 0 w_{0} – w a w_{a} degeneracy of Figure 11 . Analogous per-parameter plots for the remaining parameters and data combinations are collected in the public PhysicsML repository (see the Data Availability Statement)."},{"type":"heading","level":3,"text":"3.3.7 What the emulation brings"},{"type":"para","text":"The cosmological application plays a double role in this work. Methodologically it is a validation of the ML strategy: unlike the flavour and ALP likelihoods, the exact cosmological χ 2 \\chi^{2} is cheap enough to be sampled directly, so we could run the same MCMC with four independent back-ends (Section 3.3.5 ) and verify that the surrogate posteriors are statistically identical to the exact ones. This is a check that cannot be carried out where the exact likelihood is prohibitive, and it certifies that the order-of-magnitude speedup of the emulator does not come at the price of accuracy."},{"type":"para","text":"Beyond this cross-check, the cosmological fits make the practical advantages of the approach concrete. First, speed: once trained, the emulator samples each posterior in a few seconds against tens to hundreds of seconds for the exact likelihood, and even the up-front construction of the training set is accelerated on the GPU, so the cost is paid once and amortised over the whole analysis; the advantage grows with the cost of the underlying likelihood, which is precisely why it becomes decisive in the flavour and ALP applications. Second, robustness and adaptability: a single pipeline handles the two-parameter Λ \\Lambda CDM fit and the four-parameter w 0 w a w_{0}w_{a} CDM one, three different supernova samples, the presence or absence of the CMB prior, and likelihoods that are tightly ( χ min 2 ≈ 1700 \\chi^{2}_{\\rm min}\\!\\approx\\!1700 ) or barely ( χ min 2 ≈ 5 \\chi^{2}_{\\rm min}\\!\\approx\\!5 ) constrained, with only the shifted-log target of Eq. ( 29 ) needed to make the surrogate resolve all of them. Third, resolution: the emulator returns smooth, closed confidence regions even for the strongly curved w 0 w_{0} – w a w_{a} degeneracy and the nearly flat BAO-only landscape, where a direct grid would be either too coarse or too expensive. Fourth, transparency: the SHAP analysis recovers the physically expected constraining hierarchy and its reorganisation when the CMB is added, so the gain in speed does not turn the inference into a black box. Taken together, these results show that the framework of Section 2 transfers cleanly from high-energy physics to cosmology, the cosmological case providing the controlled benchmark that certifies the method while the more expensive applications reap the largest computational gains."},{"type":"heading","level":1,"text":"4 Conclusions"},{"type":"para","text":"We have presented a general and interpretable Machine Learning framework for accelerating likelihood-based inference in high-energy physics and cosmology. Using XGBoost surrogate models combined with SHAP interpretability and parallel Markov-Chain Monte Carlo sampling, the framework efficiently explores high-dimensional, non-Gaussian parameter spaces. This demonstrates that the approach is general rather than tied to an underlying physical system."},{"type":"para","text":"The surrogate’s robustness across diverse likelihoods stems from three core methodological choices. First, the training design combines a global space-filling prior sample with a local, Hessian-covariance-scaled best-fit cloud to automatically resolve narrow, anisotropic degeneracies. This design is augmented by an active-learning Gaussian process expected improvement in sparse or sharply structured regions. Second, training targets are dynamically adapted to the local landscape, using a direct log-likelihood for the flavour fit, a two-stage classifier-plus-regressor via a sigmoid transform for the axion-like-particle fit, and a shifted- log 10 \\log_{10} transform to resolve cosmological confidence regions across a χ min 2 \\chi^{2}_{\\rm min} spanning two orders of magnitude. Finally, regression trees partition the parameter space adaptively to capture narrow ridges, sharp transitions, and plateau-like regions without neural-network oversmoothing. The space-filling baseline ensures that emulator evaluations remain strictly within the validated training domain."},{"type":"para","text":"Across the three cases studied, the framework delivered substantial computational gains while preserving statistical accuracy. Likelihood evaluations were accelerated by speed-ups of up to 10 5 10^{5} in flavour and axion-like particle analyses, while posterior sampling in the cosmological application achieved speed-ups ranging from 4 4 to 22 22 over the CPU (vectorized NumPy) implementation and from 7 7 to 12 12 over the JAX/GPU kernel. The advantage grows as the underlying likelihood becomes more expensive. Besides, a key advantage of the framework is its interpretability. SHAP analyses consistently recovered the expected hierarchy of parameter and observable importance, providing a transparent physical interpretation of the surrogate predictions. The cosmological case doubles as a controlled validation: because the exact likelihood could be sampled with four independent back-ends, we could verify directly that the surrogate posteriors are statistically identical to the exact ones."},{"type":"para","text":"The three applications show that the framework reproduces the relevant phenomenology at a fraction of the computational cost. It demonstrates that the almost-perfect R D R_{D} – BR ( B + → K + ν ν ¯ ) \\mathrm{BR}(B^{+}\\to K^{+}\\nu\\bar{\\nu}) correlation found in the previous Scenario II [ Alda:2021rgt ] fit disappears when C 1 C_{1} and C 3 C_{3} vary independently, identifies the axion-like-particle parameter region favored by the Belle II B + → K + ν ν ¯ B^{+}\\to K^{+}\\nu\\bar{\\nu} excess, and recovers the preference for dynamical dark energy in supernova + + BAO data, which reverts to Λ \\Lambda CDM once the compressed CMB prior is included. In all cases, SHAP values provide a transparent identification of the measurements driving the results."},{"type":"para","text":"Overall, our results demonstrate that interpretable Machine Learning surrogates provide a fast, accurate, and scalable solution for likelihood-based inference. The framework is readily transferable to other global-fit problems and offers a promising foundation for future developments, including automated active learning, differentiable emulators, and applications to higher-dimensional parameter spaces."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Physically Consistent Parameter Inference: Transparent Machine Learning Emulation in High Energy Physics and Cosmology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Jorge Alda a,b 1 1 1 jorge.aldagallo@unipd.it , Jacobo Asorey b,c 2 2 2 jasorey@unizar.es , Alejandro Mir b,c 3 3 3 amir@unizar.es , Siannah Peñaranda b,c 4 4 4 siannah@unizar.es\"\n },\n {\n \"type\": \"para\",\n \"text\": \"a Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova & INFN Sezione di Padova, via Marzolo 8, 35131 Padova, Italy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"b Centro de Astropartículas y Física de Altas Energías (CAPA), Pedro Cerbuna 12 50009 Zaragoza, Spain\"\n },\n {\n \"type\": \"para\",\n \"text\": \"c Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite recent progress, the exploration of highly non-Gaussian likelihoods remains computationally challenging, motivating the development of fast and interpretable surrogate models. Contemporary analyses in particle physics and cosmology often involve exploring high-dimensional parameter spaces that are constrained by hundreds of correlated measurements. Examples include flavour anomalies in semileptonic B B meson decays and constraints on light new particles. The resulting likelihood functions frequently exhibit intricate structures, such as strong parameter correlations, non-Gaussian behavior, multiple local minima, and elongated degeneracy directions. At the same time, evaluating the theoretical predictions required for these likelihoods often requires computationally intensive calculations, rendering exhaustive parameter scans impractical.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditional statistical approaches rely on repeated evaluations of the likelihood function using Monte Carlo sampling techniques, such as Markov chain Monte Carlo (MCMC) or nested sampling. However, the computational cost of these approaches increases rapidly with the dimensionality and complexity of the problem. This has motivated increasing interest in Machine Learning (ML) techniques that can emulate expensive likelihood functions while preserving the statistical information necessary for robust inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combining global fits with interpretable ML techniques provides a powerful framework for disentangling complex correlations among observables, identifying the most informative measurements, and improving the physical interpretability of New Physics (NP) constraints in the phenomenology of particle physics and cosmology. Gradient-boosted decision trees have emerged as a particularly attractive ML algorithm due to their flexibility, robustness, and ability to accurately capture nonlinear structures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An important challenge in this type of analysis is understanding the relationship between observables and model parameters. SHAP values provide a systematic approach to quantifying the contribution of individual features to model predictions. This enables transparent and physically meaningful interpretations of ML models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we present a unified and interpretable ML framework for global fits in high energy physics and cosmology, in which an XGBoost gradient-boosted regression ensemble is trained to emulate the (generally non-Gaussian) log-likelihood, while SHAP values keep the resulting inference transparent and physically meaningful. Rather than tailoring the method to a single problem, we demonstrate its generality across three representative global-fit applications of increasing dimensionality and complexity, spanning both particle physics and cosmology. These include the flavour anomalies in semileptonic B B meson decays, light and feebly-interacting new states (axion-like particles addressing the Belle II B + → K + ν ν ¯ B^{+}\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} excess), and dynamical dark-energy fits that drive contemporary precision cosmology. In all three cases, the surrogate reproduces the exact likelihood to high accuracy and accelerates parameter-space exploration by several orders of magnitude. Furthermore, SHAP analysis recovers the physically expected hierarchy of observables and parameters, ensuring that the gain in computational efficiency does not come at the expense of interpretability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The paper is organised as follows. Section 2 introduces the ML algorithm at the heart of the framework: the construction of the XGBoost surrogate of the χ 2 \\\\chi^{2} function, the underlying statistical formalism (likelihood, profile likelihood and confidence regions), the strategies used to build and sample the training set, the choice of training target, the SHAP interpretation of feature importance, and the Markov-Chain Monte Carlo exploration of the emulated posterior together with its closure tests. Section 3 presents three applications of the framework. We first analyse flavour anomalies in semileptonic B B meson decays, construct an XGBoost surrogate model of the SMEFT19 global likelihood, and use it to map high-resolution confidence regions. We also rank the model parameters with SHAP, and study the correlations among low-energy observables that characterise the considered scenario. Then, we consider light NP, where the lightness of the new states induces non-Gaussian and even discontinuous likelihoods. As a concrete example, we explore axion-like particles as a candidate explanation for the Belle II B + → K + ν ν ¯ B^{+}\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} excess, training the surrogate through an active-learning programme and a two-stage classifier-plus-regressor strategy to resolve the relevant region of parameter space. Finally, we address cosmology, fitting the Λ \\\\Lambda CDM and dynamical dark-energy ( w 0 w a w_{0}w_{a} CDM) models to Type Ia supernovae, baryon acoustic oscillations, and a compressed cosmic-microwave-background prior. Because the exact cosmological likelihood is cheap enough to sample directly, this application doubles as a controlled validation of the method, in which the emulated posteriors are shown to be statistically indistinguishable from the exact ones. Our conclusions and perspectives are presented in Section 4 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Machine Learning Algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ML algorithms are powerful tools for global fits both in high energy physics and cosmology. One central feature of the framework is the application of SHAP values to quantify the relative importance of each parameter in constraining the fit. This provides a model-independent ranking of model features based on their impact on specific Wilson coefficients of an effective field theory, and allows a transparent mapping between experimental inputs and the parameters of the effective field theories, going beyond standard χ 2 \\\\chi^{2} -based sensitivity analyses [ Alda:2021rgt , Alda:2020okk , AldaGallo:2021cku , Penaranda:2024ssy , MirRamos:2025ijy , Alda:2026jnc ] . With this type of analysis, we can identify the regions of parameter space allowed by any model compatible with current experimental data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We adopt a ML approach to approximate the log-likelihood function. Specifically, we use the XGBoost algorithm [ Chen:2016btl ] , an ensemble method based on regression trees that can efficiently approximate arbitrary functions. Regression trees divide the parameter space into subsets and assign a numerical value to each leaf. By combining many trees into an ensemble, the model constructs an approximation of the log-likelihood function using supervised learning. The training dataset consists of parameter points and their corresponding precomputed log-likelihood values. Model optimization relies on a loss function that measures prediction accuracy and a regularization term that limits model complexity to avoid overfitting. The ensemble is built iteratively by adding new tree structures while applying shrinkage techniques to improve generalization. Using the machine-learned approximation instead of the exact likelihood significantly reduces computational costs, since Monte Carlo methods require a large number of likelihood evaluations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SHAP values [ Lundberg:2017uca , Lundberg:2018 ] are used to interpret the influence of individual parameters in the approximation. Based on concepts from cooperative game theory [ Shapley:1953 ] , SHAP values quantify the contribution of each feature to the model prediction while satisfying properties such as local accuracy, missingness, and consistency. SHAP analysis evaluates the marginal impact of including each parameter in the model and provides insight into the relative importance of the different fit parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The statistical framework summarised in this section follows Ref. [ Alda:2026jnc ] . The central statistical object in a global fit is the likelihood function ℒ ( θ ) \\\\mathcal{L}(\\\\theta) , defined as the conditional probability of observing the experimental data given the model parameters 𝜽 \\\\bm{\\\\theta} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is convenient to work with χ 2 ( 𝜽 ) ≡ − 2 log ℒ ( 𝜽 ) \\\\chi^{2}(\\\\bm{\\\\theta})\\\\equiv-2\\\\log\\\\mathcal{L}(\\\\bm{\\\\theta}) . For N N statistically independent observables with Gaussian uncertainties,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where y i y_{i} are the experimental measurements, x i ( 𝜽 ) x_{i}(\\\\bm{\\\\theta}) are the theoretical predictions, and σ i 2 = σ y i 2 + σ x i 2 \\\\sigma_{i}^{2}=\\\\sigma_{y_{i}}^{2}+\\\\sigma_{x_{i}}^{2} is the sum of experimental and theoretical variances. When the observables are correlated, the product is replaced by the multivariate Gaussian χ 2 ( 𝜽 ) = ∑ i j ( y i − x i ( 𝜽 ) ) ( 𝒞 − 1 ) i j ( y j − x j ( 𝜽 ) ) \\\\chi^{2}(\\\\bm{\\\\theta})=\\\\sum_{ij}(y_{i}-x_{i}(\\\\bm{\\\\theta}))\\\\,(\\\\mathcal{C}^{-1})_{ij}\\\\,(y_{j}-x_{j}(\\\\bm{\\\\theta})) , with 𝒞 \\\\mathcal{C} the combined experimental and theoretical covariance matrix. The best-fit point is the Maximum Likelihood Estimator (MLE),\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and the goodness of a new-physics hypothesis is quantified by its pull with respect to the Standard Model (SM), Δ χ SM 2 = χ 2 ( 𝜽 SM ) − χ 2 ( 𝜽 ^ ) \\\\Delta\\\\chi^{2}_{\\\\rm SM}=\\\\chi^{2}(\\\\bm{\\\\theta}_{\\\\rm SM})-\\\\chi^{2}(\\\\hat{\\\\bm{\\\\theta}}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Confidence regions are obtained via the profile likelihood. The profile likelihood for the i i -th parameter is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and the statistic test\"\n },\n {\n \"type\": \"para\",\n \"text\": \"follows asymptotically a χ 2 \\\\chi^{2} distribution with one degree of freedom (Wilks’ theorem [ Wilks:1938dza ] ). The k σ k\\\\,\\\\sigma confidence interval on θ i \\\\theta^{i} is therefore the set of values for which λ P ( θ i ) ≤ k 2 \\\\lambda_{P}(\\\\theta^{i})\\\\leq k^{2} . For two-dimensional confidence regions one uses the profile likelihood over the remaining parameters, and the 1 σ 1\\\\,\\\\sigma and 2 σ 2\\\\,\\\\sigma contours correspond to Δ χ 2 = 2.30 \\\\Delta\\\\chi^{2}=2.30 and 6.18 6.18 respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, however, the likelihood in Eq. ( 2 ) is often only a simplified picture. Some applications considered in this work involve likelihoods that are non-Gaussian, depend on a large number of correlated observables, and require expensive theoretical computations for each prediction x i ( 𝜽 ) x_{i}(\\\\bm{\\\\theta}) . Mapping the full χ 2 \\\\chi^{2} surface therefore requires a large number of evaluations, making a brute-force scan computationally prohibitive. We address this by training a ML surrogate to approximate the χ 2 \\\\chi^{2} function [ Alda:2026jnc ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The training dataset 𝒟 = { ( θ i , y i ) } i = 1 N \\\\mathcal{D}=\\\\{(\\\\theta_{i},\\\\,y_{i})\\\\}_{i=1}^{N} is built by evaluating the exact χ 2 \\\\chi^{2} at a set of parameter-space points. To distribute these points efficiently we combine a space-filling design over the full prior volume with a denser draw concentrated around the best-fit region, where the likelihood is largest and the confidence contours are most sensitive. For the space-filling component we use, depending on the application, plain uniform random sampling, Latin Hypercube Sampling (LHS) [ McKay:1979 ] , or a low-discrepancy Sobol sequence [ Sobol:1967 ] . The latter two guarantee substantially more uniform coverage than a purely random sample of the same size, while uniform sampling is adequate whenever a large point budget and the dense best-fit cloud already cover the relevant region. The denser best-fit draw is most efficiently realised as a Gaussian cloud whose covariance is derived from the Hessian of the fit, so that the sampling follows the actual likelihood ellipsoid and automatically resolves narrow, anisotropic degeneracies without a hand-tuned grid. When the relevant region is not known a priori, or when the likelihood develops sharp or narrow structures that a static design samples too coarsely, this base design is refined iteratively through active learning. At each iteration an inexpensive auxiliary model is trained on the data accumulated so far, typically a Gaussian Process (GP), the infinite-dimensional limit of a multivariate Gaussian, which returns both a prediction and an uncertainty at every point. The next points to be evaluated exactly are then chosen by maximising an acquisition function such as the Expected Improvement, which balances exploration of the regions where the surrogate is most uncertain against exploitation of the regions of lowest χ 2 \\\\chi^{2} . The exact χ 2 \\\\chi^{2} is computed at the selected points, they are appended to the training set, and the cycle repeats, progressively concentrating the sampling effort where it most improves the emulated likelihood.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The surrogate model is an XGBoost gradient-boosted ensemble of regression trees [ Chen:2016btl ] . A single regression tree partitions the parameter space into T T disjoint regions (“leaves”) through a sequence of binary cuts and assigns a constant weight to each one, defining a function f ( θ ) = w q ( θ ) f(\\\\theta)=w_{q(\\\\theta)} , where q ( θ ) ∈ { 1 , … , T } q(\\\\theta)\\\\in\\\\{1,\\\\ldots,T\\\\} is the leaf into which θ \\\\theta falls and w q ( θ ) w_{q(\\\\theta)} its weight. Since a single tree is too rigid to reproduce an arbitrary function, the surrogate is built as an ensemble of K K trees,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"whose output ϕ ( 𝜽 ) \\\\phi(\\\\bm{\\\\theta}) is the approximation of the training target (the χ 2 \\\\chi^{2} or the log-likelihood). The trees are obtained by supervised learning on the dataset 𝒟 \\\\mathcal{D} , by minimising the objective\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which balances a loss function against a regularisation term. The loss ℓ \\\\ell measures the discrepancy between the target y i y_{i} and its approximation ϕ ( θ i ) \\\\phi(\\\\theta_{i}) ; we use the mean absolute error, ℓ ( ϕ ( θ i ) , y i ) = | ϕ ( θ i ) − y i | \\\\ell(\\\\phi(\\\\theta_{i}),\\\\,y_{i})=|\\\\phi(\\\\theta_{i})-y_{i}| , which is more robust to the large- χ 2 \\\\chi^{2} outliers than a quadratic loss. The regularisation Ω \\\\Omega penalises the complexity of each tree,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with T T the number of leaves and w w the vector of leaf weights, so that the hyperparameters γ \\\\gamma and λ \\\\lambda introduced below directly control the trade-off between flexibility and overfitting. XGBoost additionally provides an optional L 1 L_{1} penalty α ‖ w ‖ 1 \\\\alpha\\\\,\\\\|w\\\\|_{1} on the leaf weights (the hyperparameter α \\\\alpha below), which we employ in the axion-like-particle fit. The ensemble is built iteratively: starting from a single tree that captures the overall shape of the χ 2 \\\\chi^{2} surface, each subsequent tree provides an incremental correction whose newly added weights are scaled (shrinkage) by a learning rate η < 1 \\\\eta<1 . Training is stopped early as soon as the validation loss stagnates for a fixed number of consecutive rounds. Because regression trees partition the parameter space adaptively rather than assuming global smoothness, they are particularly well suited to likelihoods with narrow ridges, sharp transitions and plateau-like regions, where neural-network surrogates would require significantly larger training sets and careful regularisation to avoid over-smoothing the fine structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The behavior of the model is controlled by a set of hyperparameters:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Number of estimators : number of trees in the ensemble.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Maximum depth : maximum number of levels per tree. Large values increase expressiveness but also the tendency to overfit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Learning rate ( η \\\\eta ): scales the contribution of each tree. Smaller values require more trees but reduce overfitting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Minimum split loss ( γ \\\\gamma ): minimum loss reduction required to create a new partition in a leaf. Larger values yield more conservative trees.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regularisation parameters ( α \\\\alpha , λ \\\\lambda ): penalise large leaf weights, further preventing overfitting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Subsampling fractions ( subsample , colsample_bytree ): proportion of training rows and feature columns used per tree, introducing randomness and reducing variance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These hyperparameters can be tuned automatically using Optuna [ Akiba:2019 ] , a gradient-free optimisation library based on a tree-structured Parzen estimator, although for low-dimensional, smooth landscapes a fixed, conservative configuration (moderate depth, small learning rate, aggressive early stopping) is often more robust. The choice of training target (direct χ 2 \\\\chi^{2} , sigmoid transform, or shifted-log 10 transform) depends on the structure of the likelihood landscape and is discussed in each application section. Because tree ensembles do not extrapolate reliably beyond their training range, the surrogate is trusted only within the sampled domain: the space-filling component of the training design covers the full prior volume, and the sampling density is increased wherever the landscape is most structured, so that the emulated likelihood remains faithful across the whole parameter space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After training, SHAP values [ Lundberg:2017uca , Lundberg:2018 ] are computed to quantify the individual contribution of each parameter to the model output at every point in the parameter space. Denoting by F F the full set of n n features (the fit parameters), the SHAP value of the j j -th feature at a point 𝜽 \\\\bm{\\\\theta} is its average marginal contribution over all subsets S S of the remaining features [ Lundberg:2017uca , Lundberg:2018 ] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where f 𝜽 ( S ) f_{\\\\bm{\\\\theta}}(S) is the prediction obtained using only the features in the subset S S . These attributions satisfy three key properties: local accuracy, missingness, and consistency. Local accuracy means that they decompose the model output exactly,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with ϕ 0 \\\\phi_{0} the base value, equal to the average prediction over the dataset. Missingness dictates that a feature missing from the input space or irrelevant is assigned a null contribution ( ϕ j = 0 \\\\phi_{j}=0 ). Consistency guarantees that if the model changes such that the marginal contribution of a feature increases or remains the same regardless of the other features, its corresponding SHAP value will not decrease.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluating Eq. ( 9 ) directly would require retraining the model on every feature subset, an operation whose computational cost grows exponentially with the number of features. For tree ensembles, however, the exact SHAP values are recovered in polynomial time by the TreeSHAP algorithm [ Lundberg:2018 ] . Aggregated over the training dataset, SHAP values provide a global, model-independent ranking of parameter importance that complements the standard Δ χ 2 \\\\Delta\\\\chi^{2} sensitivity analysis and reveals non-linear dependencies and correlations between parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To map the posterior distribution, the emulated χ 2 \\\\chi^{2} is sampled with a Markov-Chain Monte Carlo (MCMC) algorithm. A tentative point 𝜽 ′ \\\\bm{\\\\theta}^{\\\\prime} is accepted over the current one with the Metropolis criterion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with u u a random number uniformly distributed between 0 and 1 1 ; the proposal uses a Gaussian kernel whose covariance is, after a burn-in phase, set to the Cholesky factor of the fit covariance, substantially improving mixing in correlated parameter spaces. A parallel Random-Walk Metropolis–Hastings variant running n chains n_{\\\\rm chains} independent chains can be used, with sampling stopped automatically once the effective sample size exceeds a predefined target. A useful closure test is the distribution of Δ χ 2 = 2 [ log ℒ ( 𝜽 ^ ) − log ℒ ( 𝜽 ) ] \\\\Delta\\\\chi^{2}=2\\\\,[\\\\log\\\\mathcal{L}(\\\\hat{\\\\bm{\\\\theta}})-\\\\log\\\\mathcal{L}(\\\\bm{\\\\theta})] over the accepted points: in the Gaussian limit, where the log-likelihood is locally quadratic, it follows a χ 2 \\\\chi^{2} distribution with a number of degrees of freedom equal to the number of fit parameters (Wilks’ theorem [ Wilks:1938dza ] ), so that any departure from this reference is itself a quantitative diagnostic of the non-Gaussianity of the likelihood. Because each forward pass through the trained ensemble replaces a full physics-library evaluation, the surrogate reduces the wall-clock time of the posterior exploration by several orders of magnitude.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Applications\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 B B Physics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Flavour physics is an incredibly sensitive testing ground for the Standard Model. Because flavour-changing neutral currents (FCNCs) are highly suppressed within the SM framework, they are uniquely susceptible to the virtual contributions of heavy particles. In recent years, a series of persistent discrepancies between SM predictions and experimental data, known as flavour anomalies, have intensified interest in this sector [ LHCb:2017vlu , LHCb:2021trn , HFLAV:CKM2025 ] . To analyze these anomalies without committing to a single specific ultraviolet theory, effective field theories are a very useful tool [ Alda:2020okk , Alda:2021rgt , AldaGallo:2021cku ] . By systematically integrating out heavy degrees of freedom, such as hypothetical NP mediators, we are left with an effective theory of contact interactions, governed by operators and their associated coupling strengths, known as Wilson Coefficients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One of the most powerful features of flavour physics is that a single NP mediator cannot alter an isolated observable without triggering a cascade of predictable shifts in others. Global fits exploit these correlations [ Alda:2021rgt ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Semileptonic B B meson decays provide some of the most stringent low-energy tests of the SM. Over the last decade a coherent set of deviations, the flavour anomalies, has accumulated in three classes of transitions: the charged-current b → c τ ν b\\\\to c\\\\tau\\\\nu ratios R D R_{D} , R D ∗ R_{D^{*}} and R J / ψ R_{J/\\\\psi} ; the neutral-current b → s ν ν ¯ b\\\\to s\\\\nu\\\\bar{\\\\nu} branching ratios, most notably the 2.8 σ 2.8\\\\,\\\\sigma excess in BR ( B + → K + ν ν ¯ ) \\\\mathrm{BR}(B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu}) reported by Belle II [ Belle-II:2023esi ] ; and the b → s ℓ + ℓ − b\\\\to s\\\\ell^{+}\\\\ell^{-} lepton-flavour-universality ratios R K R_{K} , R K ∗ R_{K^{*}} . Following our previous global analyses [ Alda:2020okk , Alda:2021rgt , AldaGallo:2021cku , Penaranda:2024ssy ] , the short-distance NP is parameterised through the Wilson coefficients of the SMEFT, defined at the scale Λ = 1 TeV \\\\Lambda=1\\\\,\\\\mathrm{TeV} in the Warsaw basis and evolved down to the scale of each observable, and the fit is performed with the SMEFT19 package [ SMEFT19 ] , which builds the global likelihood from more than a hundred observables by interfacing flavio [ Straub:2018kue ] , smelli [ Aebischer:2018iyb ] and wilson [ Aebischer:2018bkb ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We work in Scenario III of the mass-rotation framework introduced in Ref. [ Penaranda:2024ssy ] , in which the singlet and triplet four-fermion operators carry independent Wilson coefficients C 1 ≠ C 3 C_{1}\\\\neq C_{3} and no mixing is allowed in the lepton sector ( α ℓ = β ℓ = 0 \\\\alpha^{\\\\ell}=\\\\beta^{\\\\ell}=0 ). The fit therefore depends on three parameters ( C 1 , C 3 , β q ) (C_{1},\\\\,C_{3},\\\\,\\\\beta^{q}) , with β q \\\\beta^{q} the quark mixing angle between the second and third generations. The relevant transitions probe different combinations of the coefficients: b → s ν ν ¯ b\\\\to s\\\\nu\\\\bar{\\\\nu} receives tree-level contributions proportional to C 1 − C 3 C_{1}-C_{3} , b → c ℓ ν b\\\\to c\\\\ell\\\\nu is governed by C 3 C_{3} , and b → s ℓ + ℓ − b\\\\to s\\\\ell^{+}\\\\ell^{-} depends on C 1 + C 3 C_{1}+C_{3} . Treating C 1 C_{1} and C 3 C_{3} as independent, unlike Scenarios I and II where C 1 = C 3 C_{1}=C_{3} , is precisely what allows the fit to accommodate the enhanced B + → K + ν ν ¯ B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} rate together with the R D ( ∗ ) R_{D^{(*)}} anomalies. The parameters are scanned over C 1 , C 3 ∈ [ − 0.30 , 0.00 ] C_{1},\\\\,C_{3}\\\\in[-0.30,\\\\,0.00] and β q ∈ [ 0.00 , 3.20 ] \\\\beta^{q}\\\\in[0.00,\\\\,3.20] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With respect to Ref. [ Penaranda:2024ssy ] we update the dominant charged- and neutral-current inputs to their most recent determinations. For the b → c τ ν b\\\\to c\\\\tau\\\\nu ratios we use the HFLAV CKM 2025 world averages [ HFLAV:CKM2025 ] , R D = 0.358 ± 0.024 R_{D}=0.358\\\\pm 0.024 and R D ∗ = 0.281 ± 0.011 R_{D^{*}}=0.281\\\\pm 0.011 with a correlation coefficient of − 0.37 -0.37 , supplemented by the two R J / ψ R_{J/\\\\psi} measurements, 0.71 ± 0.17 ± 0.18 0.71\\\\pm 0.17\\\\pm 0.18 from LHCb [ LHCb:2017vlu ] and 0.49 ± 0.26 0.49\\\\pm 0.26 from CMS [ CMS:2025jfx ] , which we include as independent constraints. For b → s ν ν ¯ b\\\\to s\\\\nu\\\\bar{\\\\nu} we include the Belle II 362 fb − 1 362\\\\,\\\\mathrm{fb}^{-1} result BR ( B + → K + ν ν ¯ ) = ( 2.3 ± 0.7 ) × 10 − 5 \\\\mathrm{BR}(B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu})=(2.3\\\\pm 0.7)\\\\times 10^{-5} [ Belle-II:2023esi ] . The b → s ℓ + ℓ − b\\\\to s\\\\ell^{+}\\\\ell^{-} ratios R K R_{K} , R K ∗ R_{K^{*}} , now in agreement with the SM, are kept at their reference values and do not move the fit, since lepton-flavour universality is exact in Scenario III [ Penaranda:2024ssy ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train an XGBoost surrogate of the SMEFT19 global log-likelihood over the three-dimensional space ( C 1 , C 3 , β q ) (C_{1},C_{3},\\\\beta^{q}) , following the methodology of Section 2 and first applied to this problem in Refs. [ Alda:2021rgt , Penaranda:2024ssy ] . The training set comprises a base design of ∼ 1.4 × 10 4 \\\\sim\\\\!1.4\\\\times 10^{4} points: a Sobol space-filling sample over the full domain, a Gaussian cloud drawn with the fit covariance around the best-fit point (which resolves the narrow C 1 C_{1} direction automatically), and three two-dimensional grids used to validate the likelihood maps. This base design is supplemented by a further ∼ 1.2 × 10 4 \\\\sim\\\\!1.2\\\\times 10^{4} exact evaluations concentrated, through an active-learning refinement, in the weakly-constrained high- β q \\\\beta^{q} region where it is sparse, bringing the total to ∼ 2.6 × 10 4 \\\\sim\\\\!2.6\\\\times 10^{4} points. Densifying this region improves the coverage of the extended β q \\\\beta^{q} ridge and mitigates the over-extrapolation of the surrogate where the training data would otherwise be scarce. Trained directly on the log-likelihood with a small learning rate and aggressive early stopping, the surrogate reaches a Pearson correlation coefficient r = 0.99 r=0.99 ( R 2 = 0.98 R^{2}=0.98 ) on a held-out validation set (Figure 1 a) while evaluating each point far faster than the exact likelihood, as quantified below.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decisive advantage of the surrogate is its evaluation speed, quantified in Table 1 . All times quoted here are measured on a single CPU core, so the comparison is reproducible and does not depend on the number of cores or on the specific machine. Once trained, the XGBoost ensemble predicts the log-likelihood in ∼ 34 μ \\\\sim\\\\!34\\\\,\\\\mu s per point, to be compared with ∼ 3 \\\\sim\\\\!3 s for a single evaluation of the exact SMEFT19 global likelihood, which requires running the SMEFT-to-WET renormalization-group evolution and recomputing more than a hundred observables at each point. This is a speed-up of about 1.0 × 10 5 1.0\\\\times 10^{5} per evaluation. The gain is most tangible in the construction of the training set: generating the ∼ 2.6 × 10 4 \\\\sim\\\\!2.6\\\\times 10^{4} exact points that define the emulator costs about 24 24 core-hours of SMEFT19 computation, whereas the trained surrogate re-evaluates the same number of points in ∼ 0.9 \\\\sim\\\\!0.9 s. It is precisely this gain that makes the high-resolution confidence contours and the SHAP sensitivity analysis computationally feasible, while the agreement of the emulated likelihood maps with the exact ones (Figure 2 ) guarantees that the speed-up does not come at the price of accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fit locates the best-fit point at\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with Δ χ SM 2 = 48.8 \\\\Delta\\\\chi^{2}_{\\\\rm SM}=48.8 , corresponding to a pull above 6 σ 6\\\\,\\\\sigma with respect to the SM. The fit clearly breaks the C 1 = C 3 C_{1}=C_{3} degeneracy, confirming the qualitative picture of Scenario III with the updated data. Beyond the pointwise accuracy of Figure 1 a, Figure 1 b shows the distribution of Δ χ 2 = χ bf 2 − χ 2 \\\\Delta\\\\chi^{2}=\\\\chi^{2}_{\\\\rm bf}-\\\\chi^{2} for a Monte Carlo sample drawn from the emulated likelihood, together with the χ 2 \\\\chi^{2} distribution for three degrees of freedom expected in the Gaussian (Wilks) limit. The sample follows the overall shape of this reference but is displaced towards somewhat larger Δ χ 2 \\\\Delta\\\\chi^{2} . This is not a deficiency of the emulator or the sampler but a direct imprint of the non-Gaussian structure of the likelihood: the quark-mixing angle β q \\\\beta^{q} is only weakly constrained and remains nearly degenerate over a broad range. This is visible in the two-dimensional log-likelihood maps of Figure 2 , where β q \\\\beta^{q} develops an extended, curved ridge in the ( C 1 , β q ) (C_{1},\\\\beta^{q}) and ( C 3 , β q ) (C_{3},\\\\beta^{q}) planes, far from the compact Gaussian ellipse that a quadratic log-likelihood would produce. The χ 3 2 \\\\chi^{2}_{3} curve should therefore be read as the Gaussian reference recovered in the asymptotic limit, not as an expectation the distribution is required to match; its controlled departure is precisely the non-Gaussian behavior the emulator is designed to capture. The same maps show excellent agreement between the exact SMEFT19 evaluation and the emulator across the ( C 1 , C 3 ) (C_{1},C_{3}) , ( C 1 , β q ) (C_{1},\\\\beta^{q}) and ( C 3 , β q ) (C_{3},\\\\beta^{q}) planes, confirming that the surrogate reproduces this structure faithfully.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The SHAP analysis (Figure 3 ) ranks the parameters as C 3 > C 1 > β q C_{3}>C_{1}>\\\\beta^{q} . This hierarchy aligns with the underlying physics: the dominant sensitivity of the global fit comes from the charged-current R D ( ∗ ) R_{D^{(*)}} observables (controlled by C 3 C_{3} ), followed by the neutral-current B → K ( ∗ ) ν ν ¯ B\\\\to K^{(*)}\\\\nu\\\\bar{\\\\nu} modes (controlled by C 1 C_{1} ), while the quark-mixing angle β q \\\\beta^{q} modulates both sectors without dominating. The agreement between the data-driven SHAP ranking and the expected operator structure confirms that the surrogate captures the correct EFT dynamics rather than overfitting numerical noise.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using a sample of 1.5 × 10 4 1.5\\\\times 10^{4} points generated with the emulator, we run the SMEFT-to-WET renormalization-group evolution at each point and compute the Pearson correlation coefficients between the resulting low-energy observables (Figure 4 ). Because no NP enters the lepton sector in Scenario III ( α ℓ = β ℓ = 0 \\\\alpha^{\\\\ell}=\\\\beta^{\\\\ell}=0 , so that C 10 μ = 0 C_{10}^{\\\\mu}=0 and R K ( ∗ ) = 1 R_{K^{(*)}}=1 exactly), the b → s ℓ + ℓ − b\\\\to s\\\\ell^{+}\\\\ell^{-} ratios and the B ( s ) → μ + μ − B_{(s)}\\\\to\\\\mu^{+}\\\\mu^{-} branching ratios remain at their SM values and show no variation across the sample; the matrix therefore retains only the observables that genuinely respond to ( C 1 , C 3 , β q ) (C_{1},C_{3},\\\\beta^{q}) . The b → c τ ν b\\\\to c\\\\tau\\\\nu ratios R D R_{D} and R J / ψ R_{J/\\\\psi} are perfectly correlated, reflecting their common origin in the single coefficient C V L ∝ C 3 C_{V_{L}}\\\\propto C_{3} , while the neutral-current modes BR ( B + → K + ν ν ¯ ) \\\\mathrm{BR}(B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu}) and BR ( K + → π + ν ν ¯ ) \\\\mathrm{BR}(K^{+}\\\\!\\\\to\\\\pi^{+}\\\\nu\\\\bar{\\\\nu}) respond to the orthogonal combination C 1 − C 3 C_{1}-C_{3} . The central result concerns the interplay between R D R_{D} and BR ( B + → K + ν ν ¯ ) \\\\mathrm{BR}(B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu}) : the almost-perfect correlation between these two observables found in Scenario II [ Alda:2021rgt ] disappears (it drops to ≈ 0.06 \\\\approx 0.06 ) once C 1 C_{1} and C 3 C_{3} are allowed to vary independently, since BR ( B + → K + ν ν ¯ ) ∝ ( C 1 − C 3 ) \\\\mathrm{BR}(B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu})\\\\propto(C_{1}-C_{3}) while R D ∝ C 3 R_{D}\\\\propto C_{3} . The lack of a fixed correlation between the enhanced B + → K + ν ν ¯ B^{+}\\\\!\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} rate and the b → c τ ν b\\\\to c\\\\tau\\\\nu ratios is a distinctive prediction of Scenario III and a target for future experimental scrutiny.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Light New Physics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If the NP candidate is light and feebly interacting, there is a variety of new challenges, both in the experimental and in the theoretical arenas, arising from the fact that the new particles are physical states detectable in experiments (as opposed to the heavy case, where the new particles appear only as virtual particles in internal lines of Feynman diagrams). In general, we can distinguish several kinds of experimental signatures [ Ferber:2022rsf , Bruggisser:2023npd , Alda:2025uwo ] :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Off-shell processes where the corresponding on-shell process is forbidden by Lorentz invariance (e.g. leptonic and radiative meson decays M → X ∗ → ℓ + ℓ − , γ γ M\\\\to X^{*}\\\\to\\\\ell^{+}\\\\ell^{-},\\\\gamma\\\\gamma and neutral meson mixing M → X ∗ → M ¯ M\\\\to X^{*}\\\\to\\\\overline{M} ). These processes are able to impose bounds for all values of mass and lifetime of the new particle, although they are most relevant in the resonant region m X = m M m_{X}=m_{M} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt decays: the particle is produced in a reaction A → B X A\\\\to BX and subsequently decays into X → C X\\\\to C (where A A , B B , C C are states that may contain multiple physical particles), and the experiment is not able to resolve the production and decay vertices. The experimental signature is therefore an excess in the rate of A → B C A\\\\to BC as compared to the SM prediction. If the experiment is able to resolve a distance L min L_{\\\\mathrm{min}} , then the probability for the new particle to travel less than L min L_{\\\\mathrm{min}} and decay promptly is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where c τ X = c ℏ / Γ X c\\\\tau_{X}=c\\\\hbar/\\\\Gamma_{X} is the proper decay length of the particle, and β X \\\\beta_{X} and γ X \\\\gamma_{X} its velocity and Lorentz boost in the lab frame. If the Narrow Width Approximation (NWA) is valid Γ X ≪ m X \\\\Gamma_{X}\\\\ll m_{X} , then the NP prediction for this process is given by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Displaced vertex decays: the particle is produced in A → B X A\\\\to BX and decays as X → C X\\\\to C , but the experiment is able to resolve the production and decay vertices. This is the case of experiments with excellent vertex resolution, like LHCb, Belle II and BESIII. The dedicated displaced vertex analyses of these experiments provide bounds directly on the product σ ( A → B X ) × Γ ( X → C ) \\\\sigma(A\\\\to BX)\\\\times\\\\Gamma(X\\\\to C) as a function of both m X m_{X} and c τ X c\\\\tau_{X} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Invisible decays: The new particle is produced in A → B X A\\\\to BX processes inside the detector, but it is very long-lived and decays outside the detector. The probability for a decay outside the detector is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where L max L_{\\\\mathrm{max}} is the maximum distance that the experiment is able to resolve. The experimental signature is a process with missing energy and momentum, which usually is interpreted as an excess in a process with neutrinos in the final state,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Another possible way to obtain an invisible decay, that we will not explore in this work, happens if the new particle acts as a portal to an invisible sector and decays predominantly to non-interacting dark matter particles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From a purely computational point of view, the first difficulty that we encounter is the dependence of the predictions on c τ X c\\\\tau_{X} , that forces us to calculate every decay channel of X X even for channels that are not directly observed. The exponential functions also induce non-Gaussianities in the likelihood function. Moreover, since some observables are only defined for certain values of the mass and/or the proper decay length of the particle, the likelihood function will not even be continuous at some points of the parameter space. All of this, together with the large number of model parameters and experimental observations at our disposal, make ML-based exploration particularly compelling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.1 Axion-like particles\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One of the most popular and theoretically motivated examples of SM extensions featuring new light, feebly interacting particles are axions and axion-like particles (ALPs). Originally, axions were proposed as an elegant solution to the strong CP problem [ Peccei:1977hh , Weinberg:1977ma , Wilczek:1977pj ] , where a pseudo-scalar particle a a of mass m a m_{a} and decay constant f a f_{a} , acting as the pseudo-Nambu Goldstone boson (pNGB) of the Peccei-Quinn (PQ) U ( 1 ) PQ U(1)_{\\\\mathrm{PQ}} symmetry, dynamically drives the value of the electric dipole moment of the neutron to zero. However, the axion solution requires a correlation between its mass and decay constant, namely [ Weinberg:1977ma , GrillidiCortona:2015jxo ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which severely constrains the available parameter space. Alternatively, if one chooses to consider m a m_{a} and f a f_{a} as independent parameters, we are instead in the ALP paradigm. While an ALP might not necessarily solve the strong CP problem, it has other interesting applications [ Albertus:2026fbe ] , like serving as a dark matter candidate [ Abbott:1982af , Dine:1982ah , Preskill:1982cy , Co:2019jts , Chang:2019tvx , Co:2020dya ] , explaining the flavour puzzle (axiflavon/flaxion [ Ema:2016ops , Calibbi:2016hwq ] ), neutrino masses (majoron [ Chikashige:1980qk , Chikashige:1980ui , Gelmini:1980re , Rothstein:1992rh ] ), or being generic predictions of string theory (axiverse [ Arvanitaki:2009fg , Gendler:2023kjt ] ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From a purely bottom-up perspective, we can study ALPs in the framework of EFTs [ Georgi:1986df ] , as an expansion in powers of 1 / f a 1/f_{a} . The first terms of this expansion are the renormalisable operators for the kinetic and mass terms, and dimension-5 operators describing the lowest-order interactions with SM particles: derivative interactions with fermions that respect the shift-symmetry a → a + constant a\\\\to a+\\\\mathrm{constant} , and anomalous interactions with gauge bosons:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where V ~ μ ν = 1 2 ϵ μ ν ρ σ V ρ σ \\\\widetilde{V}^{\\\\mu\\\\nu}=\\\\frac{1}{2}\\\\epsilon^{\\\\mu\\\\nu\\\\rho\\\\sigma}V_{\\\\rho\\\\sigma} is the dual tensor in the convention ϵ 0123 = + 1 \\\\epsilon^{0123}=+1 , q q and ℓ \\\\ell are the S U ( 2 ) L SU(2)_{L} doublets of left-handed quarks and leptons, and u u , d d and e e are the S U ( 2 ) L SU(2)_{L} singlets of right-handed quarks and charged leptons. In general, the couplings to fermions c f i j c_{f}^{ij} are hermitian 3 × 3 3\\\\times 3 matrices in flavour space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From the model building perspective, we can distinguish two very broad categories of UV-complete models, depending on which fermions are charged under the PQ symmetry. If only new heavy fermions are charged under PQ, as is the case in the KSVZ model [ Kim:1979if , Shifman:1979if ] and its generalizations [ DiLuzio:2020wdo , Plakkot:2021xyx , DiLuzio:2024xnt ] , the ALP at the matching scale Λ \\\\Lambda will only interact with the SM gauge bosons as dictated by the gauge anomalies, unless the heavy and SM fermions mix. On the other hand, if the SM fermions f i f_{i} have PQ charges 𝒳 f i \\\\mathcal{X}_{f}^{i} , then the ALP at the matching scale will feature diagonal (but not necessarily universal) couplings to fermions c f i i ( Λ ) = − 𝒳 f i c_{f}^{ii}(\\\\Lambda)=-\\\\mathcal{X}_{f}^{i} , and couplings to gauge bosons fixed by the anomalies [ DiLuzio:2020wdo ] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that in specific realizations, like DFSZ model [ Zhitnitsky:1980tq , Dine:1981rt ] and its generalizations [ DiLuzio:2020wdo , Diehl:2023uui , Cox:2023squ ] , or flaxion models, the scalar sector will impose constraints to combinations of the PQ charges.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once the ALP couplings have been determined at the matching scale Λ ∼ 4 π f a \\\\Lambda\\\\sim 4\\\\pi f_{a} , we must solve the Renormalization Group Equations (RGEs) and integrate out the heavy SM particles ( t t , h h , Z Z , W ± W^{\\\\pm} ) in order to obtain the values of the coefficients at the physical scale of each observable [ Choi:2017gpf , MartinCamalich:2020dfe , Chala:2020wvs , Bauer:2020jbp , Bonilla:2021ufe , DasBakshi:2023lca , Bresciani:2024shu ] . One consequence of the running and matching is that in models with UV couplings to top quarks and/or W ± W^{\\\\pm} , sizable off-diagonal couplings in the down sector will be generated.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.2 ALPs as a solution to the Belle II anomaly\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Belle II experiment has found an excess in the branching ratio of B + → K + ν ν ¯ B^{+}\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} of 2.8 σ 2.8\\\\,\\\\sigma with respect to the SM prediction [ Belle-II:2023esi ] . Several analyses of the q 2 q^{2} distribution [ Altmannshofer:2023hkn , Fridell:2023ssf , Bolton:2025fsq , Abumusabh:2025zsr , Gartner:2026clx ] have shown that the excess is compatible with the production of one single particle of mass around 2 GeV 2\\\\,\\\\mathrm{GeV} that does not decay inside the detector. Furthermore, by combining also the upper limit for the process B → K ∗ ν ν ¯ B\\\\to K^{*}\\\\nu\\\\bar{\\\\nu} at BaBar [ BaBar:2013npw ] , with different detector size and Lorentz boost, Ref. [ Alda:2025uwo ] found a lower bound for the proper decay length of the particle, c τ ≥ 80 cm ( 95 % C . L . ) c\\\\tau\\\\geq 80\\\\,\\\\mathrm{cm}\\\\ (95\\\\%\\\\ \\\\mathrm{C.L.}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The identification of this presumptive 2 GeV 2\\\\,\\\\mathrm{GeV} particle with an ALP proves to be challenging when we consider scenarios motivated by UV-complete models. On the one hand, the flavour-violating coupling | c s b V | / f a |c_{sb}^{V}|/f_{a} that enters in the ALP production B + → K + a B^{+}\\\\to K^{+}a needs to be large enough to explain the excess, but on the other hand, ALP decay channels such as a → μ + μ − a\\\\to\\\\mu^{+}\\\\mu^{-} and a → η π π a\\\\to\\\\eta\\\\pi\\\\pi need to be suppressed in order to respect the constraint on the ALP lifetime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ref. [ Alda:2025uwo ] identified a model that fulfills all the requirements, based on the astrophobic axions of Refs. [ DiLuzio:2017ogq , Bjorkeroth:2019jtx , DiLuzio:2022tyc ] . The PQ charges of the SM fermions are\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this construction, the coupling c s b V c_{sb}^{V} is generated by top loops, and it is largely independent of β \\\\beta . Meanwhile, the coupling to muons receives a tree level contribution c μ μ A ( 0 ) = cos 2 β c_{\\\\mu\\\\mu}^{A(0)}=\\\\cos^{2}\\\\beta which is partially canceled by the running of the coupling to τ \\\\tau leptons c τ τ A = cos 2 β − 1 = − sin 2 β c_{\\\\tau\\\\tau}^{A}=\\\\cos^{2}\\\\beta-1=-\\\\sin^{2}\\\\beta , and consequently it is minimized for intermediate β \\\\beta angles.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.2.3 Fit parameters and results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model in [ Alda:2025uwo ] is rather predictive, as it only depends on two free parameters, β \\\\beta and f a f_{a} . Here we aim to extend it in a more descriptive direction, exploring a larger region of the parameter space by lifting the dependence relations between the PQ charges. The free parameters of our new model will be the PQ charges of the third-generation left-handed fermions 𝒳 q L 3 \\\\mathcal{X}_{q_{L}}^{3} and 𝒳 ℓ L 3 \\\\mathcal{X}_{\\\\ell_{L}}^{3} and the universal PQ charges of the right-handed fermions 𝒳 u R \\\\mathcal{X}_{u_{R}} , 𝒳 d R \\\\mathcal{X}_{d_{R}} and 𝒳 e R \\\\mathcal{X}_{e_{R}} , all of them allowed in the range 𝒳 f ∈ [ − 1 , 1 ] \\\\mathcal{X}_{f}\\\\in[-1,1] , as well as the energy scale f a f_{a} in the range f a ∈ [ 10 5 , 10 8 ] GeV f_{a}\\\\in[10^{5},10^{8}]\\\\,\\\\mathrm{GeV} (while fixing Λ = 4 π f a \\\\Lambda=4\\\\pi f_{a} ) and the ALP mass in the range m a ∈ [ 1.7 , 2.2 ] GeV m_{a}\\\\in[1.7,2.2]\\\\,\\\\mathrm{GeV} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We perform the fit of this model using the likelihood function implemented by the ALP-aca library [ Alda:2025nsz ] . This likelihood function includes more than 100 measurements, but not all of them are operative depending on the ALP mass and proper lifetime. In our setup, in addition to B → K ( ∗ ) a ( → inv ) B\\\\to K^{(*)}a(\\\\to\\\\mathrm{inv}) processes, we have to consider off-shell ALP processes (leptonic and radiative meson decays, neutral meson mixing), and in the limit of short lifetime, prompt/displaced vertex processes like B → K ( ∗ ) a ( → μ + μ − ) B\\\\to K^{(*)}a(\\\\to\\\\mu^{+}\\\\mu^{-}) decays.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The training dataset is generated following an active learning programme in order to improve the coverage of the region of large likelihood. A small sample (800 points) generated by LHS is used as a starting seed. Then, at each iteration of the active learning algorithm, a Gaussian Process (GP), the infinite-dimensional limit of a multivariate Gaussian, is trained on the dataset up to that point. A set of candidate points are proposed, for which the GP predicts its χ 2 \\\\chi^{2} , and more importantly, the uncertainty of said prediction. The candidate points are scored according to their modified Expected Improvement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where μ \\\\mu and σ \\\\sigma are the predicted central value and uncertainty of the χ 2 \\\\chi^{2} for the point, χ ∗ 2 \\\\chi^{2}_{*} is the minimum χ 2 \\\\chi^{2} observed so far, and Φ \\\\Phi and ϕ \\\\phi are the cumulative distribution function and probability distribution function of the Gaussian distribution, respectively. The parameter ξ \\\\xi regulates the trade-off between exploration and exploitation: if ξ \\\\xi is large, the best score will be achieved by points where the GP is very uncertain, while for small ξ \\\\xi , points with small predicted χ 2 \\\\chi^{2} are selected instead. At each iteration of the algorithm, the point with the best score is selected, its true χ 2 \\\\chi^{2} is computed and added to the training dataset, and ξ \\\\xi is slightly decreased. The total size of the dataset, including both the initial seeds and the points selected by the GP, is 2500 points. The training dataset is depicted in Figure 5 , showing that the highest density is achieved in the region of low χ 2 \\\\chi^{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In order to fully capture the details of the χ 2 \\\\chi^{2} function in the region around its minimum, we propose a two-step strategy: we first train a XGBoost instance on the sigmoid of the displaced χ 2 \\\\chi^{2} function,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where χ c 2 = χ b . f . 2 + 10 \\\\chi^{2}_{c}=\\\\chi^{2}_{\\\\mathrm{b.f.}}+10 is the cutoff: this first stage acts as a classifier, retaining only the points below the cutoff. The second stage is a XGBoost instance trained only on those points below the cutoff, thus focusing on the relevant region of the parameter space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The performance of the classifier stage is characterized by its Receiver Operating Characteristic (ROC) curve in Figure 6 . It presents the True Positive Rate (proportion of the points below the cutoff that are correctly identified by the classifier) and False Positive Rate (proportion of the points above the cutoff that are misidentified) for various thresholds of σ ( χ 2 ) \\\\sigma(\\\\chi^{2}) . A random classifier would lie on the diagonal of the ROC, while a perfect classifier would be situated in the upper left corner. By setting the threshold for the classifier at σ ( χ 2 ) = 0.5 \\\\sigma(\\\\chi^{2})=0.5 , we achieve TPR = 0.96 \\\\mathrm{TPR}=0.96 and FPR = 0.16 \\\\mathrm{FPR}=0.16 on the validation dataset. The Area Under the Curve (AUC) of the ROC plot is AUC = 0.974 \\\\mathrm{AUC}=0.974 , also indicating a great classifying performance. In the regressor stage, the surrogate achieves a Pearson correlation coefficient r = 0.929 r=0.929 , as shown in Figure 6 (Right).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As in the B B -physics case, the decisive advantage of the surrogate is its evaluation speed, quantified in Table 2 . On a single CPU core, one evaluation of the exact ALP-aca likelihood over the seven-dimensional parameter space takes ∼ 1.9 \\\\sim\\\\!1.9 s (it must solve the RGEs, perform the matching and recompute more than a hundred observables), whereas the trained XGBoost ensemble predicts the same point in ∼ 3.5 μ \\\\sim\\\\!3.5\\\\,\\\\mu s, a speed-up of about 5.3 × 10 5 5.3\\\\times 10^{5} . The gain is most tangible in the construction of the training set: generating the 8 × 10 3 8\\\\times 10^{3} exact points costs about 4.1 4.1 core-hours of ALP-aca computation, which the emulator re-evaluates in ∼ 0.03 \\\\sim\\\\!0.03 s. As in the flavour benchmark, all times are measured on a single core, so the comparison is reproducible and does not depend on the number of cores or on the specific machine.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The SHAP values help us again to interpret the results of the fit. In Figure 7 (left), we have displayed the mean of absolute values of SHAPs in the dataset, which corresponds to the expected importance of each parameter. The most important parameter is m a m_{a} , since it determines if a solution to the Belle II anomaly is even kinematically allowed in the first place. After that, the two most important parameters are 𝒳 q L 3 \\\\mathcal{X}_{q_{L}}^{3} and f a f_{a} , which control the top loops and the overall impact of ALP effects. The impact of the PQ charges of right-handed fermions is more limited. In the case of m a m_{a} , we can see in Figure 7 (right) that larger ALP masses, which are less compatible with the q 2 q^{2} distribution of the recast to two-body decay kinematics, result in larger SHAP values, that is, larger contribution to the χ 2 \\\\chi^{2} function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, using the surrogate of the likelihood function and a Markov chain Monte Carlo (MCMC) implemented by the emcee library, we can efficiently sample the posterior distributions of both the parameters of the model and of derived quantities. In Figure 8 , we have obtained the posterior distributions for the input parameters m a m_{a} and f a f_{a} , for the low-energy couplings | c s b V | |c_{sb}^{V}| , | c μ μ A | |c_{\\\\mu\\\\mu}^{A}| and | c G | |c_{G}| (all of them at the physical scale μ ∼ m a \\\\mu\\\\sim m_{a} ) and the relevant parameters for the Belle II anomaly: c τ a c\\\\tau_{a} and BR ( B → K a ) \\\\mathrm{BR}(B\\\\to Ka) . The posterior distribution correctly reflects functional dependencies, such as c τ a ∝ f a 2 c\\\\tau_{a}\\\\propto f_{a}^{2} and BR ( B + → K + a ) ∝ | c s b V | 2 \\\\mathrm{BR}(B^{+}\\\\to K^{+}a)\\\\propto|c_{sb}^{V}|^{2} . Clearly, results favor an ALP mass around 1.8 1.8 GeV, in agreement with previous analyses of the Belle II excess based on the kinematic reconstruction of the missing-energy signal. The preferred values of f a f_{a} lie in the 10 6 − 10 7 10^{6}-10^{7} GeV range, balancing the need for sufficiently large flavour-changing interactions to explain the anomaly against the suppression required by constraints included in the fit. The posterior distributions of the effective couplings show that | c s b V | 2 |c_{sb}^{V}|^{2} must be large enough to generate an observable BR ( B + → K + a ) \\\\mathrm{BR}(B^{+}\\\\to K^{+}a) , while | c μ μ A | |c_{\\\\mu\\\\mu}^{A}| remains comparatively small in order to suppress visible ALP decays and ensure compatibility with searches for displaced and prompt dimuon resonances.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Figure 9 , we have shown again the posterior obtained by the surrogate and MCMC, this time fixing m a = 1.8 GeV m_{a}=1.8\\\\,\\\\mathrm{GeV} , for c τ a c\\\\tau_{a} and BR ( B + → K + a ) \\\\mathrm{BR}(B^{+}\\\\to K^{+}a) (in purple), compared to the fit using only the B → K + inv B\\\\to K+\\\\mathrm{inv} data alone. The BaBar and Belle II data was only able to set a lower bound for the decay length of the ALP, but the combination with other observables in the fit provides complementary information that further restricts the proper decay length. As it was the case in the SMEFT analysis, we once more underline the importance of global explorations of the parameter space, which hugely benefit from the ML techniques like the ones we have described.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Cosmology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The last decades have seen cosmology become precision physics. And in this path, the methods used in order to do parameter inference and data analysis have become more and more similar to to the ones used in high energy physics. There are different types of cosmological probes when doing cosmological parameter inference. The space-time geometry can be determined through the distance-redshift relation through time-domain probes or through the study of the large-scale structure of the Universe. The use of these probes has allowed us to define a baseline cosmological standard model, the flat Λ \\\\Lambda CDM, but some tensions between observations may indicate the need for NP [ CosmoVerseNetwork:2025alb ] . In all cases, the likelihood is again built from a χ 2 \\\\chi^{2} comparing a handful of cosmological parameters against hundreds of measurements through theoretical predictions that require numerical distance integrals, so the same obstacles encountered in the flavour fits (expensive forward evaluations and curved parameter degeneracies) reappear here and make the ML emulation strategy of Section 2 equally attractive. The main cosmological probes used in this work are the distance–redshift relation, measured with Type Ia supernovae as standard candles and with baryon acoustic oscillations as a standard ruler, complemented by the Cosmic Microwave Background (CMB) as a prior on cosmological parameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.1 Cosmological probes and the likelihood\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline model for cosmological analysis is the flat Λ \\\\Lambda CDM model where the accelerated expansion of the Universe is explained by the cosmological Λ \\\\Lambda with constant density and spatial flatness at each moment of cosmological time. The expansion history of the Universe is encoded in the dimensionless Friedmann equation: E ( z ) ≡ H ( z ) / H 0 E(z)\\\\equiv H(z)/H_{0} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where spatial flatness fixes the dark-energy density parameter to 1 − Ω m 1-\\\\Omega_{m} and radiation is negligible at the redshifts probed. The dynamics are driven by the parameters: H 0 H_{0} is the Hubble constant, Ω m = Ω b + Ω c d m \\\\Omega_{m}=\\\\Omega_{b}+\\\\Omega_{cdm} is the density parameter of matter in the Universe, comprised by ordinary matter ( Ω b \\\\Omega_{b} ) and cold dark matter ( Ω c d m \\\\Omega_{cdm} ). The dynamical nature of dark energy is enclosed in f D E ( z ) f_{DE}(z) . For the Λ \\\\Lambda CDM model, as the energy density of the cosmological constant component does not evolve with time ( ρ Λ ∼ c o n s t . \\\\rho_{\\\\Lambda}\\\\sim const. ), f D E = 1 f_{DE}=1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the last decade, different cosmological probes, old and new, seem to converge in that the most plausible model for cosmology was Λ \\\\Lambda CDM. But recently, supernovae and standard ruler measurements have indicated a possible dynamical dark energy component. The evidence for dynamical dark energy is based upon the flat w 0 w a w_{0}w_{a} CDM dark-energy model, in which the dark-energy equation of state is allowed to evolve with redshift according to the Chevallier–Polarski–Linder (CPL) parameterisation [ Chevallier:2000qy , Linder:2002et ] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which reduces to a cosmological constant ( Λ \\\\Lambda CDM) for w 0 = − 1 w_{0}=-1 , w a = 0 w_{a}=0 . In this particular case, the energy density of dynamical dark energy component evolves according to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All geometric observables follow from common definitions. The transverse distance between two objects at a given redshift is the comoving (transverse) distance, D M ( z ) D_{M}(z) , while the radial geometric distance relation with redshift is the Hubble distance, D H ( z ) D_{H}(z) , and finally we can capture the isotropic nature of a spherical ruler with the spherically averaged distance, D V ( z ) D_{V}(z) , [ SDSS:2005xqv ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so that the model depends on the four parameters 𝜽 = ( Ω m , H 0 , w 0 , w a ) \\\\bm{\\\\theta}=(\\\\Omega_{m},H_{0},w_{0},w_{a}) (reduced to ( Ω m , H 0 ) (\\\\Omega_{m},H_{0}) in the Λ \\\\Lambda CDM baseline below). The three probes enter the global likelihood as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Type Ia Supernovae are standardisable candles: after light-curve correction their peak brightness is fixed, so the observed apparent magnitude m B m_{B} measures the luminosity distance through the distance modulus\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with M M the absolute magnitude. Because M M is fully degenerate with H 0 H_{0} , it is analytically marginalised, so the supernovae constrain the shape of the distance–redshift relation rather than its absolute scale. The χ 2 \\\\chi^{2} uses the full statistical + + systematic covariance matrix 𝒞 \\\\mathcal{C} , χ SN 2 = Δ 𝖳 𝒞 − 1 Δ \\\\chi^{2}_{\\\\rm SN}=\\\\Delta^{\\\\mathsf{T}}\\\\mathcal{C}^{-1}\\\\Delta with Δ i = μ i obs − μ ( z i ) \\\\Delta_{i}=\\\\mu_{i}^{\\\\rm obs}-\\\\mu(z_{i}) . We use the Pantheon + + / SH0ES compilation [ Scolnic:2021amr , Brout:2022vxf ] (1657 supernovae, with the Cepheid calibrators excluded so that M M is marginalised) and, as an independent sample, the Dark Energy Survey five-year SN Ia data set (DES-SN5YR), in both its 2024 release (1829 supernovae) [ DES:2024tys ] and its 2025 update [ DES:2025sn5yr ] (1820 supernovae); for DES the magnitude offset is marginalised analytically, including the corresponding normalisation term.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baryon Acoustic Oscillations imprint a fixed comoving scale, the sound horizon at the baryon drag epoch r d r_{d} , that acts as a standard ruler in the galaxy distribution. The sound horizon is given by: r d = ∫ z d ∞ c s ( z ) H ( z ) 𝑑 z r_{d}=\\\\int_{z_{d}}^{\\\\infty}{\\\\frac{c_{s}(z)}{H(z)}dz} , where z d z_{d} is the redshift at decoupling and the speed of sound prior to recombination, c s ( z ) c_{s}(z) , is set by the baryon, ρ b \\\\rho_{b} and radiation, ρ γ \\\\rho_{\\\\gamma} , densities as c s ( z ) = c [ 3 ( 1 + 3 ρ b ( z ) 4 ρ γ ( z ) ) ] − 1 c_{s}(z)=c\\\\left[\\\\sqrt{3\\\\left(1+\\\\frac{3\\\\rho_{b}(z)}{4\\\\rho_{\\\\gamma}(z)}\\\\right)}\\\\right]^{-1} . Surveys measure the ratios D M / r d D_{M}/r_{d} , D H / r d D_{H}/r_{d} and the isotropic combination D V / r d D_{V}/r_{d} at several effective redshifts. We use the 13 measurements of the DESI Data Release 2 [ DESI:2025zgx ] with their Gaussian likelihood χ BAO 2 = Δ 𝖳 𝒞 BAO − 1 Δ \\\\chi^{2}_{\\\\rm BAO}=\\\\Delta^{\\\\mathsf{T}}\\\\mathcal{C}^{-1}_{\\\\rm BAO}\\\\Delta , fixing r d r_{d} to its Planck fiducial value.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cosmic Microwave Background. Rather than recomputing the full CMB power spectra at every point, we summarise the Planck constraint as a Gaussian prior on the late-time parameters. The Planck PR4 + + lensing chain in the CPL model obtained in the re-analysis of the latest Planck data [ Tristram:2023haj ] (superseding the official PR3 [ Planck:2018vyg ] ) made by [ 2026PDU....5202283A ] provides a covariance matrix for its full set of about 25 25 parameters; of these we keep only the five that map onto our fit, the Hubble constant H 0 H_{0} , the two physical densities of baryons ω b \\\\omega_{b} and cold dark matter ω cdm \\\\omega_{\\\\rm cdm} (with ω x ≡ Ω x h 2 \\\\omega_{x}\\\\equiv\\\\Omega_{x}h^{2} and h = H 0 / 100 h=H_{0}/100 ), and the dark-energy parameters w 0 w_{0} and w a w_{a} , discarding the remaining nuisance columns. Because our sampling variable is the total matter density rather than its two components, we combine them into ω m = ω b + ω cdm \\\\omega_{m}=\\\\omega_{b}+\\\\omega_{\\\\rm cdm} by linear error propagation, which collapses the 3 × 3 3\\\\times 3 block of ( H 0 , ω b , ω cdm ) (H_{0},\\\\omega_{b},\\\\omega_{\\\\rm cdm}) into a correlated 2 × 2 2\\\\times 2 covariance for ( H 0 , ω m ) (H_{0},\\\\omega_{m}) ; the dark-energy parameters are carried with their Planck variances. The result is a 4 × 4 4\\\\times 4 covariance Σ CMB \\\\Sigma_{\\\\rm CMB} for ( H 0 , ω m , w 0 , w a ) (H_{0},\\\\omega_{m},w_{0},w_{a}) in which H 0 H_{0} and ω m \\\\omega_{m} stay correlated, while w 0 w_{0} and w a w_{a} enter as independent Gaussian constraints. Centred on the Planck Λ \\\\Lambda CDM best fit ( w 0 = − 1 w_{0}=-1 , w a = 0 w_{a}=0 ), it contributes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ω m = Ω m ( H 0 / 100 ) 2 \\\\omega_{m}=\\\\Omega_{m}(H_{0}/100)^{2} is evaluated from the sampled parameters and the starred quantities are the Planck central values. Combining the two densities by propagating their covariance is the exact marginalisation of the joint Gaussian onto ( H 0 , ω m ) (H_{0},\\\\omega_{m}) , so for observables that depend only on the total matter density it is equivalent to carrying ω b \\\\omega_{b} and ω cdm \\\\omega_{\\\\rm cdm} as explicit parameters; the correlated ( H 0 , ω m ) (H_{0},\\\\omega_{m}) block is retained, while in this particular covariance w 0 w_{0} and w a w_{a} happen to be uncorrelated with the rest and enter as independent Gaussians of width σ ( w 0 ) = 0.02 \\\\sigma(w_{0})=0.02 , σ ( w a ) = 0.05 \\\\sigma(w_{a})=0.05 . The same prior is used in every CMB combination of Sections 3.3.4 and 3.3.5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combined data sets are fit by summing the individual χ 2 \\\\chi^{2} functions, χ tot 2 ( 𝜽 ) = ∑ k χ k 2 ( 𝜽 ) \\\\chi^{2}_{\\\\rm tot}(\\\\bm{\\\\theta})=\\\\sum_{k}\\\\chi^{2}_{k}(\\\\bm{\\\\theta}) , exactly as in the flavour global fit.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.2 Surrogate construction and the choice of training target\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each data combination we train an XGBoost surrogate of χ tot 2 ( 𝜽 ) \\\\chi^{2}_{\\\\rm tot}(\\\\bm{\\\\theta}) following the methodology of Section 2 . The training design combines two-dimensional slices through the best-fit point, a uniform space-filling draw over the full prior box ( Ω m ∈ [ 0.1 , 0.9 ] \\\\Omega_{m}\\\\!\\\\in\\\\![0.1,0.9] , H 0 ∈ [ 20 , 100 ] H_{0}\\\\!\\\\in\\\\![20,100] , w 0 ∈ [ − 3 , 0.2 ] w_{0}\\\\!\\\\in\\\\![-3,0.2] , w a ∈ [ − 3 , 2 ] w_{a}\\\\!\\\\in\\\\![-3,2] ), and a Gaussian cloud whose covariance is the Hessian of the fit, so that the narrow, anisotropic w 0 w_{0} – w a w_{a} degeneracy is sampled densely where it matters most. Each design contains 2.5 2.5 – 5 × 10 5 5\\\\times 10^{5} points, with the exact distance integrals of Eqs. ( 25 )–( 26 ) evaluated in parallel across CPU cores.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central, problem-specific ingredient is the choice of training target. The cosmological χ 2 \\\\chi^{2} spans an enormous dynamic range, from χ min 2 ∼ 10 3 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\sim\\\\!10^{3} at the best fit to ≳ 10 6 \\\\gtrsim\\\\!10^{6} at the edges of the prior box, while the physically relevant structure, the Δ χ 2 ∼ 2 \\\\Delta\\\\chi^{2}\\\\!\\\\sim\\\\!2 – 6 6 band that defines the 1 σ 1\\\\,\\\\sigma and 2 σ 2\\\\,\\\\sigma confidence regions, lives in a tiny sliver near the minimum. Crucially, the absolute scale of χ min 2 \\\\chi^{2}_{\\\\rm min} differs by two orders of magnitude between the data combinations, from χ min 2 ≈ 1.7 × 10 3 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\approx\\\\!1.7\\\\times 10^{3} for the supernova + + BAO fits (more than 1600 1600 data points) down to χ min 2 ≈ 5.6 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\approx\\\\!5.6 for the BAO-only fit (13 points), and this is what makes the two naive targets inadequate in a data-dependent way:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training directly on χ 2 \\\\chi^{2} lets the loss be dominated by the high- χ 2 \\\\chi^{2} tail: the surrogate fits the outer wall of the likelihood well but is blind to the 𝒪 ( few ) \\\\mathcal{O}(\\\\text{few}) variations near the minimum. This happens for every combination, irrespective of χ min 2 \\\\chi^{2}_{\\\\rm min} , and makes the recovered contours unusable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training on a plain logarithm, log 10 χ 2 \\\\log_{10}\\\\chi^{2} , compresses the tail, but whether it works now depends on where the minimum falls on the logarithmic curve, which is precisely why it succeeds for some data sets and fails for others. For the supernova + + BAO and + + CMB fits the minimum is large, so it sits on the locally flat part of the logarithm and the decisive 1 σ 1\\\\,\\\\sigma – 2 σ 2\\\\,\\\\sigma band maps onto a negligible change of the target ( log 10 1706 − log 10 1700 ≈ 1.5 × 10 − 3 \\\\log_{10}1706-\\\\log_{10}1700\\\\approx 1.5\\\\times 10^{-3} ): the contours remain unresolved. For the BAO-only fit the minimum is small, so it sits on the steep part of the logarithm ( log 10 12 − log 10 5.6 ≈ 0.33 \\\\log_{10}12-\\\\log_{10}5.6\\\\approx 0.33 ) and the plain transform does resolve the contours. No single un-shifted logarithm can therefore serve all combinations at once.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Both problems are solved by a shifted- log 10 \\\\log_{10} transform, which is the target adopted throughout this work:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where χ min 2 \\\\chi^{2}_{\\\\rm min} is the minimum over the training set. The shift maps the best fit to y = 0 y=0 regardless of the absolute value of χ min 2 \\\\chi^{2}_{\\\\rm min} , so that the minimum of every likelihood, whether χ min 2 ≈ 5 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\approx\\\\!5 or ≈ 1700 \\\\approx\\\\!1700 , is placed on the same steep, well-resolved part of the transform; this is what removes the data-dependence of the plain logarithm discussed above. The unit offset additionally guarantees a strictly positive argument ( ≥ 1 \\\\geq 1 ), removing the log ( 0 ) \\\\log(0) singularity that a naive shift χ 2 − χ min 2 \\\\chi^{2}-\\\\chi^{2}_{\\\\rm min} would introduce exactly at the densely sampled best-fit point. The transform compresses the 10 6 10^{6} tail and stretches precisely the near-minimum band that encodes the confidence regions, amplifying it by about three orders of magnitude in target space. The surrogate is trained on y y and its predictions are mapped back to linear χ 2 \\\\chi^{2} through the exact inverse in Eq. ( 29 ), so that downstream contour and SHAP computations operate on physical χ 2 \\\\chi^{2} values. With this target the emulators reach a coefficient of determination R 2 = 0.997 R^{2}=0.997 – 0.9999 0.9999 on a held-out validation set across all data combinations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Posterior distributions are then mapped by running the parallel Random-Walk Metropolis–Hastings sampler of Section 2 (1024 chains, with the proposal covariance set to the Hessian of the fit) directly on the emulated likelihood, and rendered as marginalised contours. Because each likelihood call is a single forward pass through the tree ensemble rather than a set of numerical distance integrals, the cost of the posterior exploration is reduced by more than an order of magnitude, as quantified in Section 3.3.5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.3 Baseline: the Λ \\\\Lambda CDM fit\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As a simple, fully controlled example we first emulate the two-parameter Λ \\\\Lambda CDM likelihood of Pantheon + + combined with DESI BAO, fitting only ( Ω m , H 0 ) (\\\\Omega_{m},H_{0}) . The surrogate reproduces the exact χ 2 \\\\chi^{2} with R 2 = 0.99988 R^{2}=0.99988 , and the emulator MCMC recovers\"\n },\n {\n \"type\": \"para\",\n \"text\": \"at χ min 2 = 1703.6 \\\\chi^{2}_{\\\\rm min}=1703.6 . Figure 10 shows the resulting Ω m \\\\Omega_{m} – H 0 H_{0} posterior: a single, closed, mildly anti-correlated ellipse with no secondary modes or open directions. These values reproduce the standard concordance picture, Ω m = 0.310 \\\\Omega_{m}=0.310 matching the Planck and DESI determinations to within 1 σ 1\\\\,\\\\sigma and the supernova-only Pantheon + + value Ω m ≈ 0.33 \\\\Omega_{m}\\\\!\\\\approx\\\\!0.33 . Because the supernova absolute magnitude is marginalised, the fit carries no SH0ES distance-ladder anchor, so H 0 H_{0} is set by the BAO scale with the Planck sound horizon r d r_{d} ; it therefore lands on the Planck-like value H 0 ≈ 68 H_{0}\\\\!\\\\approx\\\\!68 rather than the higher local-distance-ladder measurement, exactly as this dataset combination behaves in the DESI and Pantheon + + analyses. This baseline confirms that the pipeline reproduces the established result before turning to the more demanding, and more interesting, four-parameter dark-energy case.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.4 Dynamical dark energy: w 0 w a w_{0}w_{a} CDM with and without CMB\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now turn to the full w 0 w a w_{0}w_{a} CDM model and study how the dark-energy constraints depend on the supernova sample and on the inclusion of the CMB prior. We consider three supernova compilations (Pantheon + + , DES-SN5YR 2024 and 2025), each combined with DESI BAO, and contrast the fit without and with the Planck prior. The best-fit parameters of the six emulators are collected in Table 3 , and the corresponding posteriors are compared pairwise in Figure 11 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A consistent pattern emerges. Without the CMB prior, all three supernova + + BAO combinations prefer a dynamical dark energy with w 0 > − 1 w_{0}>-1 and w a < 0 w_{a}<0 , i.e. an equation of state that crosses the phantom divide as it evolves, the same hint of evolving dark energy recently reported by DESI. The associated contours are broad, with σ ( w a ) ≈ 0.4 \\\\sigma(w_{a})\\\\approx 0.4 , reflecting the well-known w 0 w_{0} – w a w_{a} geometric degeneracy that the surrogate captures as an elongated, curved ridge. The deviation is strongest for Pantheon + + and DES-2024 and milder for the DES-2025 update, which sits between the two. Adding the Planck prior pulls every fit back towards Λ \\\\Lambda CDM ( w 0 ≈ − 0.96 w_{0}\\\\approx-0.96 to − 0.97 -0.97 , w a ≈ 0.02 w_{a}\\\\approx 0.02 ) and tightens the constraints dramatically: the uncertainty on w a w_{a} shrinks by about a factor of ten (from ∼ 0.42 \\\\sim 0.42 to ∼ 0.045 \\\\sim 0.045 ), on w 0 w_{0} by a factor of five, and on Ω m \\\\Omega_{m} by a factor of three. Figure 11 makes this contraction visually explicit for the three samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reading the Pantheon + + panel directly (the DES panels are qualitatively identical), the supernova + + BAO posterior (blue, filled) is a wide, tilted ellipse in the w 0 w_{0} – w a w_{a} plane, elongated along the anti-correlation direction and displaced from the Λ \\\\Lambda CDM point (dashed lines) towards w 0 > − 1 w_{0}>-1 , w a < 0 w_{a}<0 , whereas the + + CMB posterior (red) collapses to a small blob centred on ( w 0 , w a ) = ( − 1 , 0 ) (w_{0},w_{a})=(-1,0) ; the Ω m \\\\Omega_{m} and H 0 H_{0} marginals, already well determined, barely move. The supernova + + BAO best fits, for instance ( w 0 , w a ) ≈ ( − 0.78 , − 0.73 ) (w_{0},w_{a})\\\\approx(-0.78,-0.73) for DES-2024, fall in the same region and direction of the w 0 w_{0} – w a w_{a} plane as the dynamical-dark-energy preference reported by the DESI DR2 analysis for the analogous DESI + + supernova combinations, so the emulator reproduces that published result.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The CMB enters here not as a forward computation of the temperature and polarisation spectra, but as a compressed Gaussian constraint built from the Planck PR4 + + lensing covariance matrix (Eq. 28 ), a standard and economical way to fold the CMB into a late-time fit. It carries the Planck constraints on all four parameters, with the H 0 H_{0} – ω m \\\\omega_{m} correlation retained, and in particular the tight bounds on the dark-energy sector that the supernovae and BAO leave nearly free; this is what allows it to tighten w 0 w_{0} and w a w_{a} by up to an order of magnitude and to draw the joint fit into agreement with Λ \\\\Lambda CDM. The relevant point for the present work is that the surrogate reproduces this compressed-CMB likelihood as faithfully as the supernova + + BAO one, so the contraction of the contours is a genuine output of the emulated fit and not an artefact of the emulation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.5 Computational benchmark: surrogate versus exact likelihood\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cosmological application also offers a clean setting to quantify the computational gain of the ML emulation, because here the exact likelihood is cheap enough to be sampled directly for comparison, unlike the flavour case, where a brute-force scan is prohibitive. We benchmark three nested data combinations of increasing constraining power, BAO only, BAO & Pantheon + + and CMB & BAO & Pantheon + + , which illustrate how each probe closes the w 0 w_{0} – w a w_{a} degeneracy: with only the 13 BAO points the four-parameter fit is essentially unconstrained ( χ min 2 = 5.6 \\\\chi^{2}_{\\\\rm min}=5.6 , with w a w_{a} railing against the prior edge), adding the supernovae closes the contour, and the CMB prior tightens it onto Λ \\\\Lambda CDM (Figure 12 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We run the same Random-Walk Metropolis–Hastings sampler (1024 chains, identical settings and seed) with four interchangeable likelihood back-ends: the XGBoost surrogate evaluated on the GPU, and the exact χ 2 \\\\chi^{2} computed with (i) the astropy -based reference parallelised over a CPU process pool (“Astropy”), (ii) a vectorized NumPy reimplementation of the distance integrals on the CPU (“CPU”), and (iii) a JAX vmap kernel on the GPU (“GPU”) [ jax2018github ] . All four back-ends are run on the same workstation, an 18-core/36-thread Intel Core i9-10980XE CPU ( 3.0 3.0 GHz base, 4.8 4.8 GHz turbo) with 188 188 GB of RAM and an NVIDIA RTX A6000 GPU ( 48 48 GB). The Astropy reference must be evaluated one cosmology at a time and cannot be vectorized across the sampled points; its only parallelism is to spread those points over a process pool (up to 16 workers), which scales poorly and becomes impractical for the large supernova likelihoods (the n/a entries in Table 4 ). This is precisely why the distance integrals are reimplemented directly as the vectorized NumPy (CPU) and JAX (GPU) back-ends, which evaluate the whole batch of points in a single call. Crucially, all four back-ends produce statistically identical posteriors, confirming that the surrogate is accurate and not merely fast. The wall-clock times are summarised in Table 4 : the surrogate completes each run in 4.8 4.8 – 7.4 7.4 s, 3.6 3.6 – 22.4 × 22.4\\\\times faster than the CPU (vectorized NumPy) likelihood and 7.2 7.2 – 12.1 × 12.1\\\\times faster than the exact GPU kernel, with the advantage growing for the more expensive Pantheon + + likelihoods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure of Table 4 reflects two competing effects. The emulator’s per-evaluation cost is essentially independent of the dataset, since a forward pass through the tree ensemble does not depend on the number of observables, whereas the exact likelihood scales with the data; this is why the speed-up over the CPU back-end grows from ∼ 4 × \\\\sim\\\\!4\\\\times for the 13 BAO points to ≳ 20 × \\\\gtrsim\\\\!20\\\\times for the ∼ 1700 \\\\sim\\\\!1700 -supernova Pantheon + + fits. The GPU exact kernel behaves oppositely at small data volume: for BAO only its fixed kernel-launch and host–device-transfer overhead dominates the tiny per-step compute, so it is slower than the CPU (vectorized NumPy) code ( 90 90 vs 27 27 s), whereas for the much heavier Pantheon + + likelihood that overhead is amortised and the GPU wins. Finally, because every chain is run to a fixed effective-sample-size target rather than a fixed number of steps, the wall-clock time also tracks how fast each posterior mixes: the tightly CMB-constrained fit converges in fewer steps, so BAO & Pantheon + + & CMB is cheaper than BAO & Pantheon + + despite containing more data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same acceleration benefits the up-front cost of building the training set. Evaluating the 2 × 10 5 2\\\\times 10^{5} exact χ 2 \\\\chi^{2} points needed to train a supervised surrogate takes about 4 × 10 3 4\\\\times 10^{3} s (more than an hour) with the astropy reference on 16 CPU cores for the Pantheon + + combinations, whereas the batched JAX /GPU implementation of the distance integrals produces the same set in a few seconds. The GPU and Astropy evaluations agree to better than one unit in χ 2 \\\\chi^{2} near the minimum, where the training signal matters; they differ only in the extreme tails ( χ 2 ∼ 10 6 \\\\chi^{2}\\\\!\\\\sim\\\\!10^{6} ), a harmless discrepancy since those points sit in the compressed part of the shifted-log transform of Eq. ( 29 ). Thus both the construction and the sampling of the emulated likelihood are accelerated, and the one-time training cost is amortised over the whole posterior analysis, after which the surrogate replaces the physics computation entirely.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.6 Feature importance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As in the flavour analysis, SHAP values provide a transparent, model-independent ranking of the parameters by their mean absolute contribution to the emulated χ 2 \\\\chi^{2} , that is, by how steeply the likelihood responds to each one across the sampled volume, which is a direct proxy for how tightly each parameter is constrained. The rankings (Figure 13 ) follow the expected physics and, tellingly, reorganise when the CMB prior is added.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the Λ \\\\Lambda CDM baseline the two parameters are almost tied, Ω m \\\\Omega_{m} ( 0.67 0.67 ) just above H 0 H_{0} ( 0.63 0.63 ): both are comparably and tightly determined by the joint supernova + + BAO geometry, with Ω m \\\\Omega_{m} marginally ahead because it enters every distance through E ( z ) E(z) . In the w 0 w a w_{0}w_{a} CDM supernova + + BAO fit the background parameters dominate, H 0 H_{0} ( 0.47 0.47 ) and Ω m \\\\Omega_{m} ( 0.44 0.44 ) well above the dark-energy shape parameters w 0 w_{0} ( 0.24 0.24 ) and w a w_{a} ( 0.16 0.16 ). Within the dark-energy sector w 0 w_{0} always ranks above w a w_{a} , and this ordering has a clear physical origin: w 0 w_{0} is the value of the equation of state today, so it controls w ( z ) w(z) at low redshift, exactly where dark energy dominates the energy budget and where the supernovae and BAO are most numerous and constraining, whereas w a w_{a} only sets the evolution through the z / ( 1 + z ) z/(1+z) factor, which vanishes at z = 0 z=0 and grows only towards higher redshift, where dark energy is subdominant to matter and the data are sparser. w 0 w_{0} therefore has far more leverage on the observables than w a w_{a} , as the marginalised errors confirm ( σ ( w 0 ) ≈ 0.066 \\\\sigma(w_{0})\\\\approx 0.066 versus σ ( w a ) ≈ 0.42 \\\\sigma(w_{a})\\\\approx 0.42 , i.e. w 0 w_{0} about six times better determined), and w a w_{a} , the flattest and most degenerate direction, ranks last; this is the feature-importance counterpart of the elongated w 0 w_{0} – w a w_{a} ridge of Figure 11 . Adding the compressed CMB prior inverts the background-versus-dark-energy hierarchy: w 0 w_{0} jumps to the top ( 0.53 0.53 ) and w a w_{a} to second ( 0.27 0.27 ), while Ω m \\\\Omega_{m} ( 0.25 0.25 ) and H 0 H_{0} ( 0.24 0.24 ) drop. The CMB prior adds tight, direct constraints on w 0 w_{0} and w a w_{a} , which the supernovae and BAO left almost free, so the χ 2 \\\\chi^{2} now rises steeply along w 0 w_{0} and w a w_{a} and they take most of the SHAP weight, whereas Ω m \\\\Omega_{m} and H 0 H_{0} , already well measured and consistent with the prior, contribute proportionally less. Note that w 0 w_{0} stays ahead of w a w_{a} throughout (the same ∼ 2 : 1 \\\\sim\\\\!2\\\\!:\\\\!1 ratio with and without the prior), for the same low-redshift-leverage reason. The SHAP reordering is thus the feature-level fingerprint of the contour contraction seen in Figure 11 , and the agreement between this data-driven ranking and the expected constraining structure confirms that the surrogate has learned the correct physical dependencies rather than numerical artefacts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond this global ranking, SHAP also resolves how each parameter acts point by point across the sampled volume. Figure 14 shows this for w 0 w_{0} in the Pantheon + + & \\\\& DESI BAO fit: its SHAP value, the signed contribution of w 0 w_{0} to the emulated χ 2 \\\\chi^{2} at each sampled cosmology, plotted against w 0 w_{0} and coloured by w a w_{a} . The points trace a clear parabola, the data-driven image of the χ 2 \\\\chi^{2} well along w 0 w_{0} : the contribution is least, and slightly negative, in the densely sampled best-fit region near w 0 ≈ − 0.8 w_{0}\\\\!\\\\approx\\\\!-0.8 , where w 0 w_{0} sits at its optimum and helps lower the χ 2 \\\\chi^{2} , and grows steeply and positive as w 0 w_{0} departs in either direction and drives the χ 2 \\\\chi^{2} up. It is the magnitude of this contribution, not its sign, that defines importance: the mean absolute value reported in Figure 13 is dominated by these steep flanks, so the parabola itself is the fingerprint of a tightly constrained parameter, whereas a flat, weakly constrained direction such as w a w_{a} shows small SHAP values throughout and no comparable well. The small contribution in the best-fit cloud therefore does not mean that w 0 w_{0} is unimportant there; it simply reflects that, at the bottom of the well, the likelihood is locally flat in w 0 w_{0} , exactly as expected for a parameter pinned at its optimum. The colour finally encodes the residual w 0 w_{0} – w a w_{a} interaction: at fixed w 0 w_{0} the contribution still varies with w a w_{a} , the feature-level counterpart of the elongated w 0 w_{0} – w a w_{a} degeneracy of Figure 11 . Analogous per-parameter plots for the remaining parameters and data combinations are collected in the public PhysicsML repository (see the Data Availability Statement).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.7 What the emulation brings\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cosmological application plays a double role in this work. Methodologically it is a validation of the ML strategy: unlike the flavour and ALP likelihoods, the exact cosmological χ 2 \\\\chi^{2} is cheap enough to be sampled directly, so we could run the same MCMC with four independent back-ends (Section 3.3.5 ) and verify that the surrogate posteriors are statistically identical to the exact ones. This is a check that cannot be carried out where the exact likelihood is prohibitive, and it certifies that the order-of-magnitude speedup of the emulator does not come at the price of accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond this cross-check, the cosmological fits make the practical advantages of the approach concrete. First, speed: once trained, the emulator samples each posterior in a few seconds against tens to hundreds of seconds for the exact likelihood, and even the up-front construction of the training set is accelerated on the GPU, so the cost is paid once and amortised over the whole analysis; the advantage grows with the cost of the underlying likelihood, which is precisely why it becomes decisive in the flavour and ALP applications. Second, robustness and adaptability: a single pipeline handles the two-parameter Λ \\\\Lambda CDM fit and the four-parameter w 0 w a w_{0}w_{a} CDM one, three different supernova samples, the presence or absence of the CMB prior, and likelihoods that are tightly ( χ min 2 ≈ 1700 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\approx\\\\!1700 ) or barely ( χ min 2 ≈ 5 \\\\chi^{2}_{\\\\rm min}\\\\!\\\\approx\\\\!5 ) constrained, with only the shifted-log target of Eq. ( 29 ) needed to make the surrogate resolve all of them. Third, resolution: the emulator returns smooth, closed confidence regions even for the strongly curved w 0 w_{0} – w a w_{a} degeneracy and the nearly flat BAO-only landscape, where a direct grid would be either too coarse or too expensive. Fourth, transparency: the SHAP analysis recovers the physically expected constraining hierarchy and its reorganisation when the CMB is added, so the gain in speed does not turn the inference into a black box. Taken together, these results show that the framework of Section 2 transfers cleanly from high-energy physics to cosmology, the cosmological case providing the controlled benchmark that certifies the method while the more expensive applications reap the largest computational gains.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Conclusions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We have presented a general and interpretable Machine Learning framework for accelerating likelihood-based inference in high-energy physics and cosmology. Using XGBoost surrogate models combined with SHAP interpretability and parallel Markov-Chain Monte Carlo sampling, the framework efficiently explores high-dimensional, non-Gaussian parameter spaces. This demonstrates that the approach is general rather than tied to an underlying physical system.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The surrogate’s robustness across diverse likelihoods stems from three core methodological choices. First, the training design combines a global space-filling prior sample with a local, Hessian-covariance-scaled best-fit cloud to automatically resolve narrow, anisotropic degeneracies. This design is augmented by an active-learning Gaussian process expected improvement in sparse or sharply structured regions. Second, training targets are dynamically adapted to the local landscape, using a direct log-likelihood for the flavour fit, a two-stage classifier-plus-regressor via a sigmoid transform for the axion-like-particle fit, and a shifted- log 10 \\\\log_{10} transform to resolve cosmological confidence regions across a χ min 2 \\\\chi^{2}_{\\\\rm min} spanning two orders of magnitude. Finally, regression trees partition the parameter space adaptively to capture narrow ridges, sharp transitions, and plateau-like regions without neural-network oversmoothing. The space-filling baseline ensures that emulator evaluations remain strictly within the validated training domain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across the three cases studied, the framework delivered substantial computational gains while preserving statistical accuracy. Likelihood evaluations were accelerated by speed-ups of up to 10 5 10^{5} in flavour and axion-like particle analyses, while posterior sampling in the cosmological application achieved speed-ups ranging from 4 4 to 22 22 over the CPU (vectorized NumPy) implementation and from 7 7 to 12 12 over the JAX/GPU kernel. The advantage grows as the underlying likelihood becomes more expensive. Besides, a key advantage of the framework is its interpretability. SHAP analyses consistently recovered the expected hierarchy of parameter and observable importance, providing a transparent physical interpretation of the surrogate predictions. The cosmological case doubles as a controlled validation: because the exact likelihood could be sampled with four independent back-ends, we could verify directly that the surrogate posteriors are statistically identical to the exact ones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The three applications show that the framework reproduces the relevant phenomenology at a fraction of the computational cost. It demonstrates that the almost-perfect R D R_{D} – BR ( B + → K + ν ν ¯ ) \\\\mathrm{BR}(B^{+}\\\\to K^{+}\\\\nu\\\\bar{\\\\nu}) correlation found in the previous Scenario II [ Alda:2021rgt ] fit disappears when C 1 C_{1} and C 3 C_{3} vary independently, identifies the axion-like-particle parameter region favored by the Belle II B + → K + ν ν ¯ B^{+}\\\\to K^{+}\\\\nu\\\\bar{\\\\nu} excess, and recovers the preference for dynamical dark energy in supernova + + BAO data, which reverts to Λ \\\\Lambda CDM once the compressed CMB prior is included. In all cases, SHAP values provide a transparent identification of the measurements driving the results.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, our results demonstrate that interpretable Machine Learning surrogates provide a fast, accurate, and scalable solution for likelihood-based inference. The framework is readily transferable to other global-fit problems and offers a promising foundation for future developments, including automated active learning, differentiable emulators, and applications to higher-dimensional parameter spaces.\"\n }\n]","isConversational":false}}},{"rowIdx":289,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12650"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Evidence-Grounded Verified Agentic Reasoning: A Path Toward Eliminating LLM Hallucination in Empirical Inference via Tool-Attested Kernel Proofs"},"authors":{"kind":"list like","value":["Junyu Ren"],"string":"[\n \"Junyu Ren\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12650v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"LLMs hallucinate on empirical facts when making inference, fabricating or reproducing outdated ones (Huang et al., 2025 ) . Tools and external memories were introduced as a mitigation, letting the LLM consult an authoritative source rather than rely on its parametric prior. But tool access alone does not yield verifiable empirical claims: frontier models exhibit strong confirmation bias, dismissing retrieved evidence that conflicts with parametric memory (Xie et al., 2024 ) , and even state-of-the-art models struggle to stay faithful to counterfactual context (Ming et al., 2025 ) ; our own counterfactual stress tests on tables reproduce the pattern at ∼ \\sim 12–15%, robust to an explicit “according to the table” source-anchoring cue ( Section 4.2 )."},{"type":"para","text":"Tool access leaves two structural failures uncovered: accepted outputs need not descend from attested evidence, and accepted deductions need not hold up under formal scrutiny. EG-VAR targets each with a structural contract: every Verified output must descend from an attested tool call ( Theorem 3.1 ), and every accepted proof step must type-check in the Lean kernel under the declared axiom set ( Theorem 3.2 ). Verified means attested evidence plus kernel-checked derivation; everything else surfaces as honest Abstain with a replayable audit trail. We call this combination a verified empirical claim : a claim any auditor can later check without re-running the model."},{"type":"para","text":"Concretely, given a claim like “Brazil scored on average x x more goals in the 2018 opener than Italy” against a spreadsheet, EG-VAR compiles the claim with indeterminate answer into a typed goal in Lean 4. An LLM uses tool calls to fetch attested values and compute statistics from the table, reasons and infers an answer value derived from the attested evidence, and proposes a Lean tactic encoding this derivation as a proof; the Lean kernel type-checks the tactic. Accepted: publish the answer with a machine-checkable proof; rejected: abstain. A third party re-checks offline by re-running Lean; the LLM need not be re-run."},{"type":"heading","level":4,"text":"Our contribution."},{"type":"para","text":"We present EG-VAR (Evidence-Grounded Verified Agentic Reasoning), a source-general architecture for empirical claims grounded in attested external data, with tables as the first end-to-end instantiation. To our knowledge this is the first proof-assistant-verified architecture in Lean 4 for this setting. The LLM is one untrusted layer in a four-layer stack: a deterministic tool layer (L1, attested storage queries), a per-source formalization layer (L2, audited semantic lifts from storage to world facts), a Lean 4 kernel that mints Verified world claims solely via attestation-derived rules, and the solver LLM that chooses the answer witness, substitutes it into the claim’s dependent-sum goal, and proposes a proof the kernel checks. The key architectural commitment is that mkVerified (the sole constructor for Evidence Verified w w ) requires an Attested_T from the runtime, so every verified output structurally descends from a tool call. The certified answer is extracted as the witness of a dependent sum ( Martin-Löf, 1984 ) in the Curry–Howard tradition (Howard, 1980 ) : a question “which entity has property P P ?” becomes a goal of type Σ ( e : String ) ( v : Value ) , Evidence Verified ( … e v ) \\Sigma\\,(e:\\mathrm{String})\\,(v:\\mathrm{Value}),\\mathrm{Evidence~Verified}\\,(\\ldots\\,e\\,v) , and the surface answer is the first projection of the kernel-checked witness. Verified outputs come with replayable proof traces; unverified claims abstain rather than guess. Unlike symbolic factual verifiers, execution-attestation systems, and math theorem provers ( Section 2 ), EG-VAR combines proof-assistant checking with attested empirical grounding."},{"type":"heading","level":4,"text":"Why a proof assistant for empirical grounding?"},{"type":"para","text":"The standard objection is that proof assistants are overkill for empirical reasoning over external sources. We disagree: existing LLM-driven theorem-proving systems (AlphaProof (Google DeepMind, 2024 ) , Aristotle (Achim et al., 2025 ) , DeepSeek-Prover (Xin et al., 2025 ) , Hilbert (Varambally et al., 2025 ) ) already show that the substrate is not the bottleneck; the bottleneck is audited per-source semantic lifting, which our formalization protocol addresses ( Section 3.5 ; instantiated here on tables). This cost is amortized: each per-source lift is audited once and reused across all subsequent claims grounded in that source, so formalization never enters the per-query inference loop."},{"type":"heading","level":4,"text":"Scope of the empirical work."},{"type":"para","text":"All empirical results in this paper use the tabular instantiation and evaluate on TableBench (Wu et al., 2025 ) . This is the first end-to-end demonstration of the L1/L2/lift pattern on a single substrate, not the conceptual boundary of the framework; cross-substrate validation (SQL views, typed APIs, knowledge graphs) is future work."},{"type":"heading","level":4,"text":"Empirical findings."},{"type":"para","text":"On TableBench unambiguous claims (Tier 1 gold-goal benchmark, n = 120 n{=}120 ), EG-VAR attains 120/120 (100%); the same-tool same-model baseline plateaus at 114/120 (95.0%) and the table-only single-shot baseline at 107/120 (89.2%). On a 5-domain counterfactual stress panel (Tier 1.5 binary-compare and direction claims, n = 20 n{=}20 per panel, two models, two injection regimes), EG-VAR remains 100% source-faithful in every cell; on Sonnet the same-tool baseline drops to 80–90% and the no-tool baseline to 50–80%, depending on injection magnitude. Under full end-to-end deployment with the LLM itself as the formalizer (Tier 2, n = 120 n{=}120 ), the pure formalizer-error rate is 3.3% on Sonnet and 1.7% on Opus, with the remaining disagreements split into logged ambiguities (4.2%) and surfaced benchmark-artifact issues (1.7%) that are auditable rather than silent ( Section 4.3 ). The discriminator is robust to model strength, source cue, and flip magnitude."},{"type":"heading","level":4,"text":"A path toward eliminating LLM hallucination."},{"type":"para","text":"The title claim has a precise structural meaning ( Theorem 3.1 ): no Verified output can appear without an attested tool leaf, under any formalizer. Semantic mis-formalization remains non-zero at current LLM capability (3.3% on Sonnet, 1.7% on Opus), but this accuracy axis is addressable by fine-tuning on the curator-grade (claim, gold goal) pairs the paper releases (code supplement in Appendix G ; curator pipeline summary in Section 3.6 with full proposer/critic/kernel enumerate in Section G.2 ; case studies in Section I.2 ) without weakening the structural guarantee."},{"type":"heading","level":4,"text":"Formalization flywheel."},{"type":"para","text":"The long-run bet is not that users will manually write Lean. It is that empirical communication can gradually acquire formal sidecars . In the near term, EG-VAR’s curator corpus supplies supervised and verifier-guided training data for better NL-to-formal translation. In the medium term, many sources already move along a substrate spectrum from CSV to SQL views, OpenAPI schemas, knowledge graphs, and typed registries, reducing the per-source lift burden ( Section K.3 ). In the long term, LLM-generated answers and documents can themselves carry formal copies of key claims alongside prose. Data media already carry informal instructions of use (schemas, READMEs); EG-VAR just requires them formal. When upstream providers themselves run EG-VAR-like pipelines, formalization becomes mechanized rather than handcrafted, and data arrives already bearing kernel-checked Verified instructions. The formalization burden moves out of the per-query inference loop into reusable source-side infrastructure; the kernel boundary continues to enforce that only attested claims are labeled Verified ."},{"type":"heading","level":4,"text":"Governance translation."},{"type":"para","text":"The two safety theorems ( Section 3.6 ), combined with the no-upcast grade discipline, yield three structural governance properties: (i) tool- attestation-as-axiom enables independent audit of every verified output; (ii) grade discipline makes uncertainty typed and queryable rather than a hidden confidence number; (iii) honest abstention (a kernel rejection becomes the user-visible outcome) is a measurable property, not a learned behaviour. We unpack each in Section 5.2 ."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"We compare EG-VAR against four buckets of prior work ( Table 1 ): empirical studies of how LLMs trade off parametric priors against external context; symbolic factual verifiers; LLM-driven theorem provers; and execution-attestation systems. Each contributes part of the trust pipeline, but none combines a proof-assistant kernel, tool-attestation as a typed axiom, and honest abstention as a structural property for empirical claims."},{"type":"para","text":"The closest empirical prior is ClashEval (Wu et al., 2024 ) (prior-vs-context conflict on frontier models across six domains; our counterfactual stress tests in Section 4.2 reproduce its prior-retention ranking). Adjacent substrates: symbolic factual verification (Aly & Vlachos, 2024 ; Krishna et al., 2022 ; Pei et al., 2025 ) with learned proof-construction components and mapping-error failure modes; post-hoc fact-checking (Chern et al., 2023 ; Gao et al., 2023 ; Wei et al., 2024 ; Dhuliawala et al., 2024 ) ; commercial policy guardrails (Amazon Web Services, 2024 ) ; LLM-for-math theorem proving (Google DeepMind, 2024 ; Xin et al., 2025 ; Achim et al., 2025 ; Varambally et al., 2025 ) (same trust model, self-contained claims, no external grounding); autoformalization (Zhou et al., 2024 ) ; long-lineage CS verification (Raggi et al., 2016 ; Bundy, 1988 ; Clarke et al., 2003 ; Lahiri et al., 2013 ) ; and hardware-attested AI verification (Schnabl et al., 2025 ; Grigor et al., 2025a , b ) (attests execution provenance, not claim truth; complementary to EG-VAR). Detailed treatment of each thread, including the axes EG-VAR extends over ClashEval and why self-contained-math provers do not cover empirical claims, is in Appendix A ."},{"type":"heading","level":1,"text":"3 EG-VAR"},{"type":"heading","level":2,"text":"3.1 Four-layer stack and trust ledger"},{"type":"para","text":"EG-VAR partitions the agentic-AI loop into four layers with a strict trust ledger ( Figure 1 ; Threat-model summary box below)."},{"type":"para","text":"L1, Tool layer (trusted, deterministic). The tool layer exposes a fixed set of queries against a source: cell lookups, filtered aggregates, and selection operators on tables, but the pattern is source-agnostic. Each query produces an attested payload: the deterministic answer the tool computed, packaged with a runtime Attested_T witness that records query, source identity, and result. The runtime is the only entity that can produce Attested_T witnesses."},{"type":"para","text":"L2, Per-source formalization (audited offline). Tools attest storage facts (e.g., “cell at row r r , column c c , has value v v ”); claims live at the world-ontology level ( HasProperty , ArgmaxWhere , SumWhere , …). The mapping from L1 storage facts to L2 world facts is a curator’s interpretive commitment (in this paper, an audited proposer–critic–kernel pipeline; Section G.2 ), encoded as a per-source list of typed lifts that the kernel imports as axioms ( Section 3.5 ). The L1/L2 split keeps the formal language describing the world rather than the storage vehicle, the same separation Codd ( 1970 ) drew between a relational schema and the predicates it serves."},{"type":"para","text":"L3, Lean 4 kernel (trusted, formal). Lean type-checks every proof step. The kernel is the only entity authorized to mint Evidence Verified w w for any world-claim w w . There is exactly one mint rule, mkVerified , which requires an Attested_T hypothesis ( Section 3.4 )."},{"type":"para","text":"L4, Solver LLM (untrusted, stochastic). The LLM receives a typed goal and proposes proof tactics. The kernel rejects invalid tactics and the rejection messages drive revision. The solver cannot synthesize Attested_T terms; it can only consume those introduced by tool calls during the loop."},{"type":"heading","level":4,"text":"Trust ledger."},{"type":"para","text":"The kernel, the L1 vocabulary, the per-source lifts, and the tool adapter are auditable artifacts. The solver is type-checked, not trusted: an arbitrarily wrong solver can never produce a verified-but-unsupported output. This is a structural guarantee, not a statistical one. Each verified output is accompanied by the generated Lean proof file, a self-contained replay artifact that any third party with the kernel and the per-source lifts can re-typecheck."},{"type":"heading","level":2,"text":"3.2 Tables as formal objects"},{"type":"para","text":"The deeper move in EG-VAR is on the source side. Empirical verification fails not because claims are unformalized, but because the source’s semantics are implicit in the tool adapter, hidden in fuzzy retrieval, or assumed in prompt scaffolding. The source is a co-equal trust artifact: from the kernel’s perspective a table is an object that needs formalization on the same footing as a claim."},{"type":"para","text":"Concretely, three things are formalized per source: (i) an L1 storage vocabulary ( CellAt , ColumnContains , RowMatches , …) over which tools emit attested payloads; (ii) an L2 world ontology ( HasProperty , ArgmaxWhere , SumWhere , …) over which claims are stated; (iii) a per-source lift catalog ( Section 3.5 ) encoding the curator’s interpretive commitment about which L1 facts attest which L2 claims under this table’s chosen semantics. The trust artifact a third party must inspect is the lift catalog; tools carry L1 facts deterministically, the kernel reasons only over L2 evidence. This separates source-side curation from claim-side reasoning, making both auditable independently."},{"type":"para","text":"A dataset admits multiple consistent formalizations (e.g., a matchup table as rank-2 curried vs rank-1 with composite-domain); these are type-isomorphic in any Cartesian closed setting and the kernel admits any consistent one. The curator picks an interpretation and audits the lift menu against the table; pluralism is architectural. Full gauge-freedom argument in Section K.2 ."},{"type":"para","text":"We trade internal completeness for soundness, decidability, and proof economy: L2 predicate equivalences (e.g. MeanWhere ⇔ SumWhere / CountWhere \\textsf{MeanWhere}\\Leftrightarrow\\textsf{SumWhere}/\\textsf{CountWhere} ) are localized in per-source lifts rather than added as kernel lemmas, keeping proof search decidable and traces inspectable. This mirrors SMT theory combination, description logics, and decidable FOL fragments; detailed rationale in Appendix N ."},{"type":"heading","level":2,"text":"3.3 Evidence grades and downcast-only"},{"type":"para","text":"Evidence carries an explicit grade g ∈ { Verified , Supported , Plausible , Speculative } g\\in\\{\\textsc{Verified},\\textsc{Supported},\\textsc{Plausible},\\textsc{Speculative}\\} ordered by reliability. The kernel admits downcasting only ( g 1 ⪰ g 2 g_{1}\\succeq g_{2} and e : Evidence g 1 w ⇒ downcast e : Evidence g 2 w e:\\texttt{Evidence}~g_{1}~w\\Rightarrow\\texttt{downcast}~e:\\texttt{Evidence}~g_{2}~w ). No upcast rule exists, so conclusions cannot outrank their weakest premise. The only constructor for Evidence Verified _ \\_ is mkVerified , described next; lower grades arise from explicit downgrades (e.g., staleness, source-conflict) or LLM-proposed heuristic claims that the calculus admits at Plausible or Speculative only."},{"type":"heading","level":2,"text":"3.4 The mkVerified rule (sole minter of Verified )"},{"type":"para","text":"Informally: verified evidence exists only if the runtime already supplied an attested payload. The kernel’s mint rule is:"},{"type":"para","text":"The hypothesis att : Attested_T q p is the load-bearing element. Attested_T witnesses are introduced into the proof context only by the runtime in response to a tool call; the solver cannot synthesize them. interp_T q p is the per-tool interpretation function (a small total Lean function) that maps the attested payload to a finite set of L2 world claims; the kernel mints Verified for any w w in that set. There is no other path to Evidence Verified _ \\_ ."},{"type":"heading","level":2,"text":"3.5 Per-source formalization as the curator’s interpretive commitment"},{"type":"para","text":"The interesting design problem is the L1 → \\to L2 bridge. A tool attests “cell at row r r , column “state”, has value “alabama” ” (an L1 storage fact); the claim under verification is “Alabama has the highest HIV incidence rate” (an L2 world fact). The bridge is a per-source list of typed lifts, audited offline by the source’s curator, that the kernel imports as axioms."},{"type":"para","text":"Per-source lifts operationalize the L1 → \\to L2 bridge: typed composition of an attested L1 fact with a audited lift axiom mints an Evidence Verified witness for the matching L2 world-property. Absence of a matching lift is a feature: the solver has no L2 proof path, the kernel rejects, and the system abstains rather than silently guess. The L1/L2 separation is source-agnostic (SQL views, knowledge graphs, and sensor streams admit the same pattern; Section 5.2 ); a complete worked example (HIV argmax via state projection, showing the lift, attested payload, and Sigma-witness extraction) is in Section H.1 ."},{"type":"heading","level":2,"text":"3.6 Operational proof object and safety theorems"},{"type":"para","text":"The safety guarantees are stated about the runtime’s operational proof object , not about Lean’s type theory alone. A proof object is the tuple of artifacts that the Python implementation at src/python/runtime/pipeline.py actually returns from run_pipeline as the fields of PipelineResult :"},{"type":"para","text":"where leanFile is the generated Lean module text, evidence is the time-ordered list of tool-call payloads lifted to observation-leaf axioms, and steps is the history of solver actions and kernel-verbatim feedback. Figure 2 gives a concrete end-to-end instance for one claim. A standalone replay-audit script ( src/python/runtime/verify_axioms.py , Appendix G ) takes leanFile as input and recomputes the whitelist check post-hoc; the script is not invoked in the run_pipeline loop. The primary safety gate at generation time is the Lean kernel invocation lake env lean inside pipeline.py ."},{"type":"heading","level":4,"text":"Trust zones."},{"type":"para","text":"Four roles participate in producing the proof object. Untrusted proposers : the LLM analyst, formalizer, and solver; none can introduce new top-level axioms because their output is constrained to dictionary entries, a proposed goal type, and a tactic body inserted inside a pre-written Sigma type declaration. Audited runtime-orchestrator : the deterministic Python layer that executes tools, emits observation-leaf axioms, assembles the Lean module, invokes the kernel, and records evidence and steps . Trusted checker : the Lean 4 kernel. Trusted domain assumptions : the fixed EG-VAR prelude ( EGVar/Basic.lean , EGVar/TableL1.lean , EGVar/WorldOntology.lean , and EGVar/Tactics.lean ) and the audited per-source lifts Λ ( s ) \\Lambda(s) from data/tier1_table_formalizations.json , committed at curation time per Section 3.6 ."},{"type":"heading","level":4,"text":"Whitelist-compliant module (summary)."},{"type":"para","text":"We call leanFile whitelist-compliant for source s s iff every axiom in the dependency closure of the claim proof belongs to one of four buckets: the fixed EG-VAR prelude ( B1 ), standard Lean 4 core axioms ( B2 ), runtime-emitted observation leaves ( B3 ), or per-source lifts matching the regex lift_[0-9a-f]+_.+ drawn from Λ ( s ) \\Lambda(s) ( B4 ); the closure contains no sorry variant and at least one trust-artifact witness. Full bucket schemas, the replay-audit mechanics, and guarantee-boundary caveats are in Section P.1 ."},{"type":"para","text":"If RuntimeVerify ( s , c s,c ) returns ( leanFile , evidence , steps ) (\\textit{leanFile},\\textit{evidence},\\textit{steps}) with verdict Verified , and leanFile is whitelist-compliant for s s , then every runtime-emitted observation-leaf axiom in the dependency closure of the claim proof is backed by a prior tool execution recorded in evidence, and every remaining axiom dependency lies in the fixed EG-VAR prelude, in the standard Lean 4 core axioms, or in Λ ( s ) \\Lambda(s) ."},{"type":"para","text":"Every inference step that the system accepts as part of a Verified output type-checks in Lean 4’s kernel under the declared axiom set."},{"type":"para","text":"Full proofs (checker-fidelity, emission-discipline, and reserved-name lemmas; complete proofs of Theorems 3.1 and 3.2 ) plus the emission-site table are in Appendices P and P.1 . The kernel call inside run_pipeline ( lake env lean ) is the primary safety gate at generation time; the standalone verify_axioms.py is a post-hoc replay audit a third-party can re-run offline ( Appendices G and P.1 ). Worked examples of attested proofs and per-source lifts are in Appendices H and I ."},{"type":"heading","level":4,"text":"Formalization provenance."},{"type":"para","text":"Tier 1 and Tier 1.5 load fixture-committed gold types τ gold \\tau^{\\mathrm{gold}} and audited per-source lifts Λ ( s ) \\Lambda(s) from a frozen fixture (proposer–critic–kernel pipeline, no formalizer invoked at evaluation); Tier 2 invokes an LLM formalizer with access to Λ ( s ) \\Lambda(s) but no gold type. Full provenance (subagent pipeline, human meta-supervision scope, curator yield, collusion firewall) is in Section G.2 ."},{"type":"heading","level":1,"text":"4 Empirical Evaluation"},{"type":"para","text":"We report three evaluations. The first two load fixture-committed gold goal types, so they isolate the kernelized trust architecture under ideal formalization: Section 4.1 (Tier 1) establishes competitiveness with same-tool baselines on unambiguous claims; Section 4.2 (Tier 1.5) establishes source-faithfulness under counterfactual prior conflict, where the same-tool baseline on Sonnet drops to 80–90% (no-tool 50–80%) while EG-VAR remains 100% across both flip-magnitude regimes and both model strengths. The third evaluation ( Section 4.3 , Tier 2) removes the gold-fixture bypass and tests the deployment path where the LLM itself proposes the formal goal."},{"type":"heading","level":4,"text":"Two axes."},{"type":"para","text":"We separate formal safety (the Verified label is kernel-accepted against the committed trust artifacts; Theorem 3.1 ) from semantic faithfulness (the formalized goal τ \\tau matches the natural-language claim; measured by answer_match against frozen gold answers). Tiers 1 and 1.5 load τ = τ gold \\tau=\\tau^{\\mathrm{gold}} from the frozen fixture, so semantic faithfulness holds by construction; Tier 2 invokes the LLM formalizer, and semantic faithfulness becomes the primary empirical question."},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"Tier 1 and Tier 1.5 share four rungs ( Table 2 ): Table-only (single-shot, no tools); Tools-open (LLM with full runtime tool surface, no per-source menu enforcement); Tools-curated (per-source menu enforced; same tool surface as EG-VAR); EG-VAR (kernel-checked, fixed-goal mode with the curator’s typed goal). All rungs use the same model, prompt structure, and answer judge ( answers_match ); EG-VAR additionally requires that the kernel mints a Verified witness whose Sigma extract agrees with the gold answer. We evaluate two models: claude-sonnet-4-6 and claude-haiku-4-5 (a stronger and a weaker model from the same family). All runs use temperature 0; EG-VAR is deterministic at temperature 0 and we report 1-rep smoke evaluations as canonical for EG-VAR with corroborating multi-rep partial runs (provenance manifest in Appendix G )."},{"type":"heading","level":2,"text":"4.1 Tier 1 gold-goal benchmark: baseline ladder"},{"type":"para","text":"Our LLM-based curator pipeline selects n = 120 n{=}120 unambiguous TableBench claims spanning argmax, sum, count, top- k k , and aggregation shapes; for each claim it produces the typed Lean goal and the per-source lifts (the trust artifact EG-VAR consumes; Section G.2 ). All four rungs are evaluated on Sonnet at temperature 0. Table 3 reports the headline ladder. EG-VAR attains 120/120 (100%) joint formal safety and semantic faithfulness on the gold-formalization path: every claim’s kernel proof passes the whitelist replay ( Appendix G , formal safety axis) and the extracted Sigma witness matches the curator gold (semantic-faithfulness axis). The same-tool same-model baseline (Tools-curated) plateaus at 114/120 (95%) on the latter alone (no formal-safety surface: baselines cannot abstain); removing the per-source menu costs one further claim (Tools-open 113/120); single-shot table-only floor is 107/120 (89.2%)."},{"type":"heading","level":2,"text":"4.2 Tier 1.5 counterfactual benchmark: source-faithfulness under prior conflict"},{"type":"para","text":"Tier 1.5 isolates a qualitatively different failure mode the Tier 1 ladder under-counts. We construct synthetic counterfactual table variants: real TableBench tables with one or two cell values overwritten to contradict the model’s world-knowledge prior. For each variant we ask a binary-comparison or aggregation claim and classify the rung’s output as source-faithful (matches the table), prior-override (matches the un-injected real-world value), or wrong-other . Tier 1.5’s source-faithfulness column is the semantic-faithfulness axis under counterfactual-injection pressure; EG-VAR’s 100% result combines this with formal-safety (kernel replay) that the baselines lack."},{"type":"heading","level":4,"text":"Two flip-magnitude regimes."},{"type":"para","text":"The extreme-flip panel uses magnitude-10-plus reversals (e.g., Italy GDP → \\to{} $50, Patriots → \\to 0–16, Sinopec → \\to{} $50) across 5 domains; the subtle-flip panel uses 10–25% reversals (Mont Blanc → \\to{} 4500 m vs Monte Rosa 4634 m, Madrid GDP 90 vs Berlin 95) on a panel shifted to stable-scalar priors (country populations, EU country area, European city GDP, Forbes top companies, Alpine peak elevations). Each panel is 5 domains × \\times 2 entity-pair injections × \\times 3 claim shapes = = 30 claims per panel (20 binary-compare + direction, 10 cell-lookup; the cell-lookup set doesn’t trigger prior-override and is excluded here). Table 4 reports the n = 20 n{=}20 binary-compare + direction cells; both models, 1 rep (temp 0, deterministic)."},{"type":"para","text":"The extreme-flip panel narrows the residual same-tool gap to the single most entrenched prior family (country populations). The subtle-flip panel generalizes prior-override across stable-scalar domains; tools rescue most of it, and the kernel closes the rest."},{"type":"para","text":"Per-claim robustness axes (authority-cue, model strength, domain coverage) and the same-tool failure-mode taxonomy are in Section E.1 ."},{"type":"heading","level":2,"text":"4.3 Tier 2 end-to-end evaluation: LLM formalizer"},{"type":"para","text":"Tier 1 and Tier 1.5 bypass the NL-to-Lean formalizer step by loading the fixture-committed goal type from a frozen fixture ( Section 3.6 ). Tier 2 removes that bypass: at deployment time, the runtime invokes the LLM formalizer ( Section 3.6 ) to propose lean_goal_type per claim, then proceeds with the same analyst-formalizer-solver-kernel loop. This is the fully automated regime; no curator interaction at runtime."},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"n = 120 n{=}120 claims, same TableBench curated fixture. claude-sonnet-4-6 at temperature 0. Pipeline: analyst → \\to formalizer → \\to solver. The audited per-source formalization Λ ( s ) \\Lambda(s) is still loaded; only the claim-level goal type is LLM-proposed. We classify every output into one of six mutually-exclusive reporting categories, based on three observables: type-equality with the curator’s gold goal type (kernel- rfl on string-normalized forms), kernel acceptance , and answer-match against the gold answer ( Table 5 )."},{"type":"heading","level":4,"text":"Formal safety versus semantic faithfulness."},{"type":"para","text":"Tier 2 measures two distinct properties. Formal safety is structural and model-independent: every Verified proof still depends on an attested tool leaf, regardless of which LLM proposed the goal type ( Theorem 3.1 ). Semantic faithfulness is operational and model-dependent: it measures whether the LLM-proposed goal matches the frozen reading of the claim. Table 5 reports the semantic-faithfulness axis per model; the Semantic formalizer error row is the pure formalizer-error rate."},{"type":"heading","level":4,"text":"Tier 2 under counterfactual pressure."},{"type":"para","text":"To check that the LLM-as-formalizer pipeline is not just a curator-fixture artifact, we re-run Tier 2 Sonnet on the Tier 1.5 extreme-flip panel, restricted to binary-compare and direction claims ( n = 20 n{=}20 , prior-override-injected tables). Every claim produces a source-faithful Verified output ( 20/20 SF ), with the extracted Sigma witness matching the table-injected answer rather than the model’s world-knowledge prior. The Tier 1.5 same-tool baseline drops to 16/20 on the same extreme-flip tables ( Section 4.2 ); Tier 2 with end-to-end LLM formalization matches EG-VAR’s gold-formalization rung on this subset."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"heading","level":2,"text":"5.1 Limitations"},{"type":"para","text":"Three scope limitations bound the empirical claims. (i) Tier 1/1.5 bypass the formalizer by loading fixture-committed gold types from a frozen fixture ( Section 3.6 ); Tier 2 removes that bypass. (ii) Tools, per-source lifts, and the L1 adapter are trusted; a semantically wrong audited lift can certify a wrong formalized claim ( Section 3.5 ). (iii) All evaluations are on single TableBench tables; multi-source, KG, and streaming sources are out of scope. Detailed scope limits are in Appendix M ."},{"type":"heading","level":2,"text":"5.2 Governance translation"},{"type":"para","text":"Theorems 3.1 and 3.2 and the no-upcast grade discipline yield three governance-relevant structural properties."},{"type":"para","text":"(G1) Auditable evidence trails. Every Verified output carries the generated Lean proof file as its replay artifact: a third-party auditor with the kernel, per-source lifts, and tool adapter can re-typecheck the claim from the attested payload without re-running the LLM. The proof term is a truthmaker in Fine ( 2017 ) ’s sense: the specific attested state of affairs that grounds the claim. Composition of evidence across multiple lifts carries the algebraic structure of a provenance semiring (Green et al., 2007 ) . The trust artifact is enumerable (L1 vocabulary, per-source lifts, tool adapter, kernel); audit cost is amortized , each per-source lift audited once at curation time and reused across all subsequent claims."},{"type":"para","text":"(G2) Typed uncertainty. Grade discipline (no-upcast, minimum-of-premises) makes uncertainty inspectable rather than a hidden confidence number. The graded Evidence g g w w object is operationally close to a justification term (Artemov, 2008 ) , where evidence is explicit rather than folded into a model’s posterior. A Plausible output cannot be silently re-introduced downstream as Verified ; an auditor can query which claims arrived at which grade and why."},{"type":"para","text":"(G3) Honest abstention. A kernel rejection surfaces as a user-visible Abstain , not a silent fallback guess. Refusal thresholds are externally specifiable (a downstream policy can require Verified and reject Abstain , or admit Abstain as a flagged state, without re-implementing the trust boundary); the abstention rate is measurable per-source; the failure mode is auditable rather than learned."},{"type":"heading","level":4,"text":"Portability beyond tables."},{"type":"para","text":"The L1/L2/lift pattern is source-agnostic: SQL views, knowledge graphs, and sensor streams each admit the same per-source-formalization treatment under a matching attestation discipline. The empirical cross-substrate panel is itemized in Section 5.3 ; the broader substrate- adoption argument (formalization gap shrinks as sources become formal; Section K.3 )."},{"type":"heading","level":2,"text":"5.3 Future work scoped to governance reach"},{"type":"para","text":"Three extensions: (1) fine-tuning the NL formalizer on the curator-provided (claim, gold type) corpus with kernel acceptance as the reward signal, an accuracy intervention that tightens the Tier 2 residual rate without weakening Theorem 3.1 ; (2) multi-source attestation with conflict-downgrade rules ( Verified → \\to Supported on detected disagreement); (3) cross-substrate panel (KG-grounded, SQL-grounded) to test the L1/L2 portability claim ( Section 5.2 ). Detail in Appendix M ."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"EG-VAR is an architecture for verified empirical claims grounded in attested external data, instantiated on tables as the first end-to-end substrate. Tool access alone does not suffice: frontier LLMs prior-override counterfactual tables on a measurable fraction of claims. EG-VAR addresses this architecturally: the Lean 4 kernel mints Verified claims only under tool attestation and type-checks every accepted proof step. TableBench Tier 1 ( n = 120 n{=}120 ) closes the same-tool gap to zero; Tier 1.5 counterfactuals ( n = 20 n{=}20 per panel, two flip regimes, two models) stay 100% source-faithful while same-tool drops to 80–90% (no-tool 50–80%); end-to-end Tier 2 with LLM formalization bounds formalizer error at 3.3% (Sonnet), 1.7% (Opus)."},{"type":"para","text":"The core contribution is structural. Theorem 3.1 rules out unsupported Verified outputs model-independently; Theorem 3.2 guarantees that every accepted deduction type-checks under the declared axiom set. Remaining failures are not silent hallucinations but measurable, auditable formalization failures, logged ambiguities, or honest abstentions. This is a constructive path toward eliminating LLM hallucination in empirical inference: the safety floor is architectural today; the semantic-formalization residual is an explicit accuracy target addressable by formalizer fine-tuning on the curator corpus, consensus formalization, and longer-term typed sidecars on sources and LLM-generated documents."},{"type":"para","text":"For AI governance, three properties become enforceable at the system boundary rather than learned behaviours: auditable evidence trails via replayable per-claim Lean proofs, typed uncertainty under no-upcast grade discipline, and externally specifiable refusal thresholds via honest abstention. Verifiable grounding generalises as an architectural protocol rather than a table-specific trick: the same L1/L2/lift discipline can in principle apply to SQL views, knowledge graphs, typed registries, and other attested sources. Validating beyond tables is the immediate empirical next step; the longer-run bet is that as sources and documents acquire typed sidecars, more of the formalization burden moves from inference loops into reusable infrastructure."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLMs hallucinate on empirical facts when making inference, fabricating or reproducing outdated ones (Huang et al., 2025 ) . Tools and external memories were introduced as a mitigation, letting the LLM consult an authoritative source rather than rely on its parametric prior. But tool access alone does not yield verifiable empirical claims: frontier models exhibit strong confirmation bias, dismissing retrieved evidence that conflicts with parametric memory (Xie et al., 2024 ) , and even state-of-the-art models struggle to stay faithful to counterfactual context (Ming et al., 2025 ) ; our own counterfactual stress tests on tables reproduce the pattern at ∼ \\\\sim 12–15%, robust to an explicit “according to the table” source-anchoring cue ( Section 4.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tool access leaves two structural failures uncovered: accepted outputs need not descend from attested evidence, and accepted deductions need not hold up under formal scrutiny. EG-VAR targets each with a structural contract: every Verified output must descend from an attested tool call ( Theorem 3.1 ), and every accepted proof step must type-check in the Lean kernel under the declared axiom set ( Theorem 3.2 ). Verified means attested evidence plus kernel-checked derivation; everything else surfaces as honest Abstain with a replayable audit trail. We call this combination a verified empirical claim : a claim any auditor can later check without re-running the model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Concretely, given a claim like “Brazil scored on average x x more goals in the 2018 opener than Italy” against a spreadsheet, EG-VAR compiles the claim with indeterminate answer into a typed goal in Lean 4. An LLM uses tool calls to fetch attested values and compute statistics from the table, reasons and infers an answer value derived from the attested evidence, and proposes a Lean tactic encoding this derivation as a proof; the Lean kernel type-checks the tactic. Accepted: publish the answer with a machine-checkable proof; rejected: abstain. A third party re-checks offline by re-running Lean; the LLM need not be re-run.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Our contribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present EG-VAR (Evidence-Grounded Verified Agentic Reasoning), a source-general architecture for empirical claims grounded in attested external data, with tables as the first end-to-end instantiation. To our knowledge this is the first proof-assistant-verified architecture in Lean 4 for this setting. The LLM is one untrusted layer in a four-layer stack: a deterministic tool layer (L1, attested storage queries), a per-source formalization layer (L2, audited semantic lifts from storage to world facts), a Lean 4 kernel that mints Verified world claims solely via attestation-derived rules, and the solver LLM that chooses the answer witness, substitutes it into the claim’s dependent-sum goal, and proposes a proof the kernel checks. The key architectural commitment is that mkVerified (the sole constructor for Evidence Verified w w ) requires an Attested_T from the runtime, so every verified output structurally descends from a tool call. The certified answer is extracted as the witness of a dependent sum ( Martin-Löf, 1984 ) in the Curry–Howard tradition (Howard, 1980 ) : a question “which entity has property P P ?” becomes a goal of type Σ ( e : String ) ( v : Value ) , Evidence Verified ( … e v ) \\\\Sigma\\\\,(e:\\\\mathrm{String})\\\\,(v:\\\\mathrm{Value}),\\\\mathrm{Evidence~Verified}\\\\,(\\\\ldots\\\\,e\\\\,v) , and the surface answer is the first projection of the kernel-checked witness. Verified outputs come with replayable proof traces; unverified claims abstain rather than guess. Unlike symbolic factual verifiers, execution-attestation systems, and math theorem provers ( Section 2 ), EG-VAR combines proof-assistant checking with attested empirical grounding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why a proof assistant for empirical grounding?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The standard objection is that proof assistants are overkill for empirical reasoning over external sources. We disagree: existing LLM-driven theorem-proving systems (AlphaProof (Google DeepMind, 2024 ) , Aristotle (Achim et al., 2025 ) , DeepSeek-Prover (Xin et al., 2025 ) , Hilbert (Varambally et al., 2025 ) ) already show that the substrate is not the bottleneck; the bottleneck is audited per-source semantic lifting, which our formalization protocol addresses ( Section 3.5 ; instantiated here on tables). This cost is amortized: each per-source lift is audited once and reused across all subsequent claims grounded in that source, so formalization never enters the per-query inference loop.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scope of the empirical work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All empirical results in this paper use the tabular instantiation and evaluate on TableBench (Wu et al., 2025 ) . This is the first end-to-end demonstration of the L1/L2/lift pattern on a single substrate, not the conceptual boundary of the framework; cross-substrate validation (SQL views, typed APIs, knowledge graphs) is future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Empirical findings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On TableBench unambiguous claims (Tier 1 gold-goal benchmark, n = 120 n{=}120 ), EG-VAR attains 120/120 (100%); the same-tool same-model baseline plateaus at 114/120 (95.0%) and the table-only single-shot baseline at 107/120 (89.2%). On a 5-domain counterfactual stress panel (Tier 1.5 binary-compare and direction claims, n = 20 n{=}20 per panel, two models, two injection regimes), EG-VAR remains 100% source-faithful in every cell; on Sonnet the same-tool baseline drops to 80–90% and the no-tool baseline to 50–80%, depending on injection magnitude. Under full end-to-end deployment with the LLM itself as the formalizer (Tier 2, n = 120 n{=}120 ), the pure formalizer-error rate is 3.3% on Sonnet and 1.7% on Opus, with the remaining disagreements split into logged ambiguities (4.2%) and surfaced benchmark-artifact issues (1.7%) that are auditable rather than silent ( Section 4.3 ). The discriminator is robust to model strength, source cue, and flip magnitude.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A path toward eliminating LLM hallucination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The title claim has a precise structural meaning ( Theorem 3.1 ): no Verified output can appear without an attested tool leaf, under any formalizer. Semantic mis-formalization remains non-zero at current LLM capability (3.3% on Sonnet, 1.7% on Opus), but this accuracy axis is addressable by fine-tuning on the curator-grade (claim, gold goal) pairs the paper releases (code supplement in Appendix G ; curator pipeline summary in Section 3.6 with full proposer/critic/kernel enumerate in Section G.2 ; case studies in Section I.2 ) without weakening the structural guarantee.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Formalization flywheel.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The long-run bet is not that users will manually write Lean. It is that empirical communication can gradually acquire formal sidecars . In the near term, EG-VAR’s curator corpus supplies supervised and verifier-guided training data for better NL-to-formal translation. In the medium term, many sources already move along a substrate spectrum from CSV to SQL views, OpenAPI schemas, knowledge graphs, and typed registries, reducing the per-source lift burden ( Section K.3 ). In the long term, LLM-generated answers and documents can themselves carry formal copies of key claims alongside prose. Data media already carry informal instructions of use (schemas, READMEs); EG-VAR just requires them formal. When upstream providers themselves run EG-VAR-like pipelines, formalization becomes mechanized rather than handcrafted, and data arrives already bearing kernel-checked Verified instructions. The formalization burden moves out of the per-query inference loop into reusable source-side infrastructure; the kernel boundary continues to enforce that only attested claims are labeled Verified .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Governance translation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two safety theorems ( Section 3.6 ), combined with the no-upcast grade discipline, yield three structural governance properties: (i) tool- attestation-as-axiom enables independent audit of every verified output; (ii) grade discipline makes uncertainty typed and queryable rather than a hidden confidence number; (iii) honest abstention (a kernel rejection becomes the user-visible outcome) is a measurable property, not a learned behaviour. We unpack each in Section 5.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare EG-VAR against four buckets of prior work ( Table 1 ): empirical studies of how LLMs trade off parametric priors against external context; symbolic factual verifiers; LLM-driven theorem provers; and execution-attestation systems. Each contributes part of the trust pipeline, but none combines a proof-assistant kernel, tool-attestation as a typed axiom, and honest abstention as a structural property for empirical claims.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The closest empirical prior is ClashEval (Wu et al., 2024 ) (prior-vs-context conflict on frontier models across six domains; our counterfactual stress tests in Section 4.2 reproduce its prior-retention ranking). Adjacent substrates: symbolic factual verification (Aly & Vlachos, 2024 ; Krishna et al., 2022 ; Pei et al., 2025 ) with learned proof-construction components and mapping-error failure modes; post-hoc fact-checking (Chern et al., 2023 ; Gao et al., 2023 ; Wei et al., 2024 ; Dhuliawala et al., 2024 ) ; commercial policy guardrails (Amazon Web Services, 2024 ) ; LLM-for-math theorem proving (Google DeepMind, 2024 ; Xin et al., 2025 ; Achim et al., 2025 ; Varambally et al., 2025 ) (same trust model, self-contained claims, no external grounding); autoformalization (Zhou et al., 2024 ) ; long-lineage CS verification (Raggi et al., 2016 ; Bundy, 1988 ; Clarke et al., 2003 ; Lahiri et al., 2013 ) ; and hardware-attested AI verification (Schnabl et al., 2025 ; Grigor et al., 2025a , b ) (attests execution provenance, not claim truth; complementary to EG-VAR). Detailed treatment of each thread, including the axes EG-VAR extends over ClashEval and why self-contained-math provers do not cover empirical claims, is in Appendix A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 EG-VAR\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Four-layer stack and trust ledger\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EG-VAR partitions the agentic-AI loop into four layers with a strict trust ledger ( Figure 1 ; Threat-model summary box below).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"L1, Tool layer (trusted, deterministic). The tool layer exposes a fixed set of queries against a source: cell lookups, filtered aggregates, and selection operators on tables, but the pattern is source-agnostic. Each query produces an attested payload: the deterministic answer the tool computed, packaged with a runtime Attested_T witness that records query, source identity, and result. The runtime is the only entity that can produce Attested_T witnesses.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"L2, Per-source formalization (audited offline). Tools attest storage facts (e.g., “cell at row r r , column c c , has value v v ”); claims live at the world-ontology level ( HasProperty , ArgmaxWhere , SumWhere , …). The mapping from L1 storage facts to L2 world facts is a curator’s interpretive commitment (in this paper, an audited proposer–critic–kernel pipeline; Section G.2 ), encoded as a per-source list of typed lifts that the kernel imports as axioms ( Section 3.5 ). The L1/L2 split keeps the formal language describing the world rather than the storage vehicle, the same separation Codd ( 1970 ) drew between a relational schema and the predicates it serves.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"L3, Lean 4 kernel (trusted, formal). Lean type-checks every proof step. The kernel is the only entity authorized to mint Evidence Verified w w for any world-claim w w . There is exactly one mint rule, mkVerified , which requires an Attested_T hypothesis ( Section 3.4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"L4, Solver LLM (untrusted, stochastic). The LLM receives a typed goal and proposes proof tactics. The kernel rejects invalid tactics and the rejection messages drive revision. The solver cannot synthesize Attested_T terms; it can only consume those introduced by tool calls during the loop.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Trust ledger.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The kernel, the L1 vocabulary, the per-source lifts, and the tool adapter are auditable artifacts. The solver is type-checked, not trusted: an arbitrarily wrong solver can never produce a verified-but-unsupported output. This is a structural guarantee, not a statistical one. Each verified output is accompanied by the generated Lean proof file, a self-contained replay artifact that any third party with the kernel and the per-source lifts can re-typecheck.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Tables as formal objects\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The deeper move in EG-VAR is on the source side. Empirical verification fails not because claims are unformalized, but because the source’s semantics are implicit in the tool adapter, hidden in fuzzy retrieval, or assumed in prompt scaffolding. The source is a co-equal trust artifact: from the kernel’s perspective a table is an object that needs formalization on the same footing as a claim.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Concretely, three things are formalized per source: (i) an L1 storage vocabulary ( CellAt , ColumnContains , RowMatches , …) over which tools emit attested payloads; (ii) an L2 world ontology ( HasProperty , ArgmaxWhere , SumWhere , …) over which claims are stated; (iii) a per-source lift catalog ( Section 3.5 ) encoding the curator’s interpretive commitment about which L1 facts attest which L2 claims under this table’s chosen semantics. The trust artifact a third party must inspect is the lift catalog; tools carry L1 facts deterministically, the kernel reasons only over L2 evidence. This separates source-side curation from claim-side reasoning, making both auditable independently.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A dataset admits multiple consistent formalizations (e.g., a matchup table as rank-2 curried vs rank-1 with composite-domain); these are type-isomorphic in any Cartesian closed setting and the kernel admits any consistent one. The curator picks an interpretation and audits the lift menu against the table; pluralism is architectural. Full gauge-freedom argument in Section K.2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We trade internal completeness for soundness, decidability, and proof economy: L2 predicate equivalences (e.g. MeanWhere ⇔ SumWhere / CountWhere \\\\textsf{MeanWhere}\\\\Leftrightarrow\\\\textsf{SumWhere}/\\\\textsf{CountWhere} ) are localized in per-source lifts rather than added as kernel lemmas, keeping proof search decidable and traces inspectable. This mirrors SMT theory combination, description logics, and decidable FOL fragments; detailed rationale in Appendix N .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Evidence grades and downcast-only\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evidence carries an explicit grade g ∈ { Verified , Supported , Plausible , Speculative } g\\\\in\\\\{\\\\textsc{Verified},\\\\textsc{Supported},\\\\textsc{Plausible},\\\\textsc{Speculative}\\\\} ordered by reliability. The kernel admits downcasting only ( g 1 ⪰ g 2 g_{1}\\\\succeq g_{2} and e : Evidence g 1 w ⇒ downcast e : Evidence g 2 w e:\\\\texttt{Evidence}~g_{1}~w\\\\Rightarrow\\\\texttt{downcast}~e:\\\\texttt{Evidence}~g_{2}~w ). No upcast rule exists, so conclusions cannot outrank their weakest premise. The only constructor for Evidence Verified _ \\\\_ is mkVerified , described next; lower grades arise from explicit downgrades (e.g., staleness, source-conflict) or LLM-proposed heuristic claims that the calculus admits at Plausible or Speculative only.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 The mkVerified rule (sole minter of Verified )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Informally: verified evidence exists only if the runtime already supplied an attested payload. The kernel’s mint rule is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The hypothesis att : Attested_T q p is the load-bearing element. Attested_T witnesses are introduced into the proof context only by the runtime in response to a tool call; the solver cannot synthesize them. interp_T q p is the per-tool interpretation function (a small total Lean function) that maps the attested payload to a finite set of L2 world claims; the kernel mints Verified for any w w in that set. There is no other path to Evidence Verified _ \\\\_ .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Per-source formalization as the curator’s interpretive commitment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interesting design problem is the L1 → \\\\to L2 bridge. A tool attests “cell at row r r , column “state”, has value “alabama” ” (an L1 storage fact); the claim under verification is “Alabama has the highest HIV incidence rate” (an L2 world fact). The bridge is a per-source list of typed lifts, audited offline by the source’s curator, that the kernel imports as axioms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Per-source lifts operationalize the L1 → \\\\to L2 bridge: typed composition of an attested L1 fact with a audited lift axiom mints an Evidence Verified witness for the matching L2 world-property. Absence of a matching lift is a feature: the solver has no L2 proof path, the kernel rejects, and the system abstains rather than silently guess. The L1/L2 separation is source-agnostic (SQL views, knowledge graphs, and sensor streams admit the same pattern; Section 5.2 ); a complete worked example (HIV argmax via state projection, showing the lift, attested payload, and Sigma-witness extraction) is in Section H.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.6 Operational proof object and safety theorems\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The safety guarantees are stated about the runtime’s operational proof object , not about Lean’s type theory alone. A proof object is the tuple of artifacts that the Python implementation at src/python/runtime/pipeline.py actually returns from run_pipeline as the fields of PipelineResult :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where leanFile is the generated Lean module text, evidence is the time-ordered list of tool-call payloads lifted to observation-leaf axioms, and steps is the history of solver actions and kernel-verbatim feedback. Figure 2 gives a concrete end-to-end instance for one claim. A standalone replay-audit script ( src/python/runtime/verify_axioms.py , Appendix G ) takes leanFile as input and recomputes the whitelist check post-hoc; the script is not invoked in the run_pipeline loop. The primary safety gate at generation time is the Lean kernel invocation lake env lean inside pipeline.py .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Trust zones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four roles participate in producing the proof object. Untrusted proposers : the LLM analyst, formalizer, and solver; none can introduce new top-level axioms because their output is constrained to dictionary entries, a proposed goal type, and a tactic body inserted inside a pre-written Sigma type declaration. Audited runtime-orchestrator : the deterministic Python layer that executes tools, emits observation-leaf axioms, assembles the Lean module, invokes the kernel, and records evidence and steps . Trusted checker : the Lean 4 kernel. Trusted domain assumptions : the fixed EG-VAR prelude ( EGVar/Basic.lean , EGVar/TableL1.lean , EGVar/WorldOntology.lean , and EGVar/Tactics.lean ) and the audited per-source lifts Λ ( s ) \\\\Lambda(s) from data/tier1_table_formalizations.json , committed at curation time per Section 3.6 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Whitelist-compliant module (summary).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We call leanFile whitelist-compliant for source s s iff every axiom in the dependency closure of the claim proof belongs to one of four buckets: the fixed EG-VAR prelude ( B1 ), standard Lean 4 core axioms ( B2 ), runtime-emitted observation leaves ( B3 ), or per-source lifts matching the regex lift_[0-9a-f]+_.+ drawn from Λ ( s ) \\\\Lambda(s) ( B4 ); the closure contains no sorry variant and at least one trust-artifact witness. Full bucket schemas, the replay-audit mechanics, and guarantee-boundary caveats are in Section P.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If RuntimeVerify ( s , c s,c ) returns ( leanFile , evidence , steps ) (\\\\textit{leanFile},\\\\textit{evidence},\\\\textit{steps}) with verdict Verified , and leanFile is whitelist-compliant for s s , then every runtime-emitted observation-leaf axiom in the dependency closure of the claim proof is backed by a prior tool execution recorded in evidence, and every remaining axiom dependency lies in the fixed EG-VAR prelude, in the standard Lean 4 core axioms, or in Λ ( s ) \\\\Lambda(s) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every inference step that the system accepts as part of a Verified output type-checks in Lean 4’s kernel under the declared axiom set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Full proofs (checker-fidelity, emission-discipline, and reserved-name lemmas; complete proofs of Theorems 3.1 and 3.2 ) plus the emission-site table are in Appendices P and P.1 . The kernel call inside run_pipeline ( lake env lean ) is the primary safety gate at generation time; the standalone verify_axioms.py is a post-hoc replay audit a third-party can re-run offline ( Appendices G and P.1 ). Worked examples of attested proofs and per-source lifts are in Appendices H and I .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Formalization provenance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tier 1 and Tier 1.5 load fixture-committed gold types τ gold \\\\tau^{\\\\mathrm{gold}} and audited per-source lifts Λ ( s ) \\\\Lambda(s) from a frozen fixture (proposer–critic–kernel pipeline, no formalizer invoked at evaluation); Tier 2 invokes an LLM formalizer with access to Λ ( s ) \\\\Lambda(s) but no gold type. Full provenance (subagent pipeline, human meta-supervision scope, curator yield, collusion firewall) is in Section G.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Empirical Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report three evaluations. The first two load fixture-committed gold goal types, so they isolate the kernelized trust architecture under ideal formalization: Section 4.1 (Tier 1) establishes competitiveness with same-tool baselines on unambiguous claims; Section 4.2 (Tier 1.5) establishes source-faithfulness under counterfactual prior conflict, where the same-tool baseline on Sonnet drops to 80–90% (no-tool 50–80%) while EG-VAR remains 100% across both flip-magnitude regimes and both model strengths. The third evaluation ( Section 4.3 , Tier 2) removes the gold-fixture bypass and tests the deployment path where the LLM itself proposes the formal goal.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Two axes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We separate formal safety (the Verified label is kernel-accepted against the committed trust artifacts; Theorem 3.1 ) from semantic faithfulness (the formalized goal τ \\\\tau matches the natural-language claim; measured by answer_match against frozen gold answers). Tiers 1 and 1.5 load τ = τ gold \\\\tau=\\\\tau^{\\\\mathrm{gold}} from the frozen fixture, so semantic faithfulness holds by construction; Tier 2 invokes the LLM formalizer, and semantic faithfulness becomes the primary empirical question.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tier 1 and Tier 1.5 share four rungs ( Table 2 ): Table-only (single-shot, no tools); Tools-open (LLM with full runtime tool surface, no per-source menu enforcement); Tools-curated (per-source menu enforced; same tool surface as EG-VAR); EG-VAR (kernel-checked, fixed-goal mode with the curator’s typed goal). All rungs use the same model, prompt structure, and answer judge ( answers_match ); EG-VAR additionally requires that the kernel mints a Verified witness whose Sigma extract agrees with the gold answer. We evaluate two models: claude-sonnet-4-6 and claude-haiku-4-5 (a stronger and a weaker model from the same family). All runs use temperature 0; EG-VAR is deterministic at temperature 0 and we report 1-rep smoke evaluations as canonical for EG-VAR with corroborating multi-rep partial runs (provenance manifest in Appendix G ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Tier 1 gold-goal benchmark: baseline ladder\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our LLM-based curator pipeline selects n = 120 n{=}120 unambiguous TableBench claims spanning argmax, sum, count, top- k k , and aggregation shapes; for each claim it produces the typed Lean goal and the per-source lifts (the trust artifact EG-VAR consumes; Section G.2 ). All four rungs are evaluated on Sonnet at temperature 0. Table 3 reports the headline ladder. EG-VAR attains 120/120 (100%) joint formal safety and semantic faithfulness on the gold-formalization path: every claim’s kernel proof passes the whitelist replay ( Appendix G , formal safety axis) and the extracted Sigma witness matches the curator gold (semantic-faithfulness axis). The same-tool same-model baseline (Tools-curated) plateaus at 114/120 (95%) on the latter alone (no formal-safety surface: baselines cannot abstain); removing the per-source menu costs one further claim (Tools-open 113/120); single-shot table-only floor is 107/120 (89.2%).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Tier 1.5 counterfactual benchmark: source-faithfulness under prior conflict\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tier 1.5 isolates a qualitatively different failure mode the Tier 1 ladder under-counts. We construct synthetic counterfactual table variants: real TableBench tables with one or two cell values overwritten to contradict the model’s world-knowledge prior. For each variant we ask a binary-comparison or aggregation claim and classify the rung’s output as source-faithful (matches the table), prior-override (matches the un-injected real-world value), or wrong-other . Tier 1.5’s source-faithfulness column is the semantic-faithfulness axis under counterfactual-injection pressure; EG-VAR’s 100% result combines this with formal-safety (kernel replay) that the baselines lack.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Two flip-magnitude regimes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The extreme-flip panel uses magnitude-10-plus reversals (e.g., Italy GDP → \\\\to{} $50, Patriots → \\\\to 0–16, Sinopec → \\\\to{} $50) across 5 domains; the subtle-flip panel uses 10–25% reversals (Mont Blanc → \\\\to{} 4500 m vs Monte Rosa 4634 m, Madrid GDP 90 vs Berlin 95) on a panel shifted to stable-scalar priors (country populations, EU country area, European city GDP, Forbes top companies, Alpine peak elevations). Each panel is 5 domains × \\\\times 2 entity-pair injections × \\\\times 3 claim shapes = = 30 claims per panel (20 binary-compare + direction, 10 cell-lookup; the cell-lookup set doesn’t trigger prior-override and is excluded here). Table 4 reports the n = 20 n{=}20 binary-compare + direction cells; both models, 1 rep (temp 0, deterministic).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The extreme-flip panel narrows the residual same-tool gap to the single most entrenched prior family (country populations). The subtle-flip panel generalizes prior-override across stable-scalar domains; tools rescue most of it, and the kernel closes the rest.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Per-claim robustness axes (authority-cue, model strength, domain coverage) and the same-tool failure-mode taxonomy are in Section E.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Tier 2 end-to-end evaluation: LLM formalizer\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tier 1 and Tier 1.5 bypass the NL-to-Lean formalizer step by loading the fixture-committed goal type from a frozen fixture ( Section 3.6 ). Tier 2 removes that bypass: at deployment time, the runtime invokes the LLM formalizer ( Section 3.6 ) to propose lean_goal_type per claim, then proceeds with the same analyst-formalizer-solver-kernel loop. This is the fully automated regime; no curator interaction at runtime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"n = 120 n{=}120 claims, same TableBench curated fixture. claude-sonnet-4-6 at temperature 0. Pipeline: analyst → \\\\to formalizer → \\\\to solver. The audited per-source formalization Λ ( s ) \\\\Lambda(s) is still loaded; only the claim-level goal type is LLM-proposed. We classify every output into one of six mutually-exclusive reporting categories, based on three observables: type-equality with the curator’s gold goal type (kernel- rfl on string-normalized forms), kernel acceptance , and answer-match against the gold answer ( Table 5 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Formal safety versus semantic faithfulness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tier 2 measures two distinct properties. Formal safety is structural and model-independent: every Verified proof still depends on an attested tool leaf, regardless of which LLM proposed the goal type ( Theorem 3.1 ). Semantic faithfulness is operational and model-dependent: it measures whether the LLM-proposed goal matches the frozen reading of the claim. Table 5 reports the semantic-faithfulness axis per model; the Semantic formalizer error row is the pure formalizer-error rate.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Tier 2 under counterfactual pressure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To check that the LLM-as-formalizer pipeline is not just a curator-fixture artifact, we re-run Tier 2 Sonnet on the Tier 1.5 extreme-flip panel, restricted to binary-compare and direction claims ( n = 20 n{=}20 , prior-override-injected tables). Every claim produces a source-faithful Verified output ( 20/20 SF ), with the extracted Sigma witness matching the table-injected answer rather than the model’s world-knowledge prior. The Tier 1.5 same-tool baseline drops to 16/20 on the same extreme-flip tables ( Section 4.2 ); Tier 2 with end-to-end LLM formalization matches EG-VAR’s gold-formalization rung on this subset.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three scope limitations bound the empirical claims. (i) Tier 1/1.5 bypass the formalizer by loading fixture-committed gold types from a frozen fixture ( Section 3.6 ); Tier 2 removes that bypass. (ii) Tools, per-source lifts, and the L1 adapter are trusted; a semantically wrong audited lift can certify a wrong formalized claim ( Section 3.5 ). (iii) All evaluations are on single TableBench tables; multi-source, KG, and streaming sources are out of scope. Detailed scope limits are in Appendix M .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Governance translation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorems 3.1 and 3.2 and the no-upcast grade discipline yield three governance-relevant structural properties.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(G1) Auditable evidence trails. Every Verified output carries the generated Lean proof file as its replay artifact: a third-party auditor with the kernel, per-source lifts, and tool adapter can re-typecheck the claim from the attested payload without re-running the LLM. The proof term is a truthmaker in Fine ( 2017 ) ’s sense: the specific attested state of affairs that grounds the claim. Composition of evidence across multiple lifts carries the algebraic structure of a provenance semiring (Green et al., 2007 ) . The trust artifact is enumerable (L1 vocabulary, per-source lifts, tool adapter, kernel); audit cost is amortized , each per-source lift audited once at curation time and reused across all subsequent claims.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(G2) Typed uncertainty. Grade discipline (no-upcast, minimum-of-premises) makes uncertainty inspectable rather than a hidden confidence number. The graded Evidence g g w w object is operationally close to a justification term (Artemov, 2008 ) , where evidence is explicit rather than folded into a model’s posterior. A Plausible output cannot be silently re-introduced downstream as Verified ; an auditor can query which claims arrived at which grade and why.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(G3) Honest abstention. A kernel rejection surfaces as a user-visible Abstain , not a silent fallback guess. Refusal thresholds are externally specifiable (a downstream policy can require Verified and reject Abstain , or admit Abstain as a flagged state, without re-implementing the trust boundary); the abstention rate is measurable per-source; the failure mode is auditable rather than learned.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Portability beyond tables.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The L1/L2/lift pattern is source-agnostic: SQL views, knowledge graphs, and sensor streams each admit the same per-source-formalization treatment under a matching attestation discipline. The empirical cross-substrate panel is itemized in Section 5.3 ; the broader substrate- adoption argument (formalization gap shrinks as sources become formal; Section K.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Future work scoped to governance reach\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three extensions: (1) fine-tuning the NL formalizer on the curator-provided (claim, gold type) corpus with kernel acceptance as the reward signal, an accuracy intervention that tightens the Tier 2 residual rate without weakening Theorem 3.1 ; (2) multi-source attestation with conflict-downgrade rules ( Verified → \\\\to Supported on detected disagreement); (3) cross-substrate panel (KG-grounded, SQL-grounded) to test the L1/L2 portability claim ( Section 5.2 ). Detail in Appendix M .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EG-VAR is an architecture for verified empirical claims grounded in attested external data, instantiated on tables as the first end-to-end substrate. Tool access alone does not suffice: frontier LLMs prior-override counterfactual tables on a measurable fraction of claims. EG-VAR addresses this architecturally: the Lean 4 kernel mints Verified claims only under tool attestation and type-checks every accepted proof step. TableBench Tier 1 ( n = 120 n{=}120 ) closes the same-tool gap to zero; Tier 1.5 counterfactuals ( n = 20 n{=}20 per panel, two flip regimes, two models) stay 100% source-faithful while same-tool drops to 80–90% (no-tool 50–80%); end-to-end Tier 2 with LLM formalization bounds formalizer error at 3.3% (Sonnet), 1.7% (Opus).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The core contribution is structural. Theorem 3.1 rules out unsupported Verified outputs model-independently; Theorem 3.2 guarantees that every accepted deduction type-checks under the declared axiom set. Remaining failures are not silent hallucinations but measurable, auditable formalization failures, logged ambiguities, or honest abstentions. This is a constructive path toward eliminating LLM hallucination in empirical inference: the safety floor is architectural today; the semantic-formalization residual is an explicit accuracy target addressable by formalizer fine-tuning on the curator corpus, consensus formalization, and longer-term typed sidecars on sources and LLM-generated documents.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For AI governance, three properties become enforceable at the system boundary rather than learned behaviours: auditable evidence trails via replayable per-claim Lean proofs, typed uncertainty under no-upcast grade discipline, and externally specifiable refusal thresholds via honest abstention. Verifiable grounding generalises as an architectural protocol rather than a table-specific trick: the same L1/L2/lift discipline can in principle apply to SQL views, knowledge graphs, typed registries, and other attested sources. Validating beyond tables is the immediate empirical next step; the longer-run bet is that as sources and documents acquire typed sidecars, more of the formalization burden moves from inference loops into reusable infrastructure.\"\n }\n]","isConversational":false}}},{"rowIdx":290,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12645"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"AdaPCLA: Adaptive Prior-Calibrated Logit Adjustment for Long-Tailed Longitudinal EHR Generation"},"authors":{"kind":"list like","value":["Shuai Cui","Chen Wenxuan","Wenjie Du","Jian Lou","Dan Li","Wenjie Feng"],"string":"[\n \"Shuai Cui\",\n \"Chen Wenxuan\",\n \"Wenjie Du\",\n \"Jian Lou\",\n \"Dan Li\",\n \"Wenjie Feng\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12645v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Longitudinal Electronic Health Records (EHR) provide a rich view of patient trajectories, capturing diagnoses, treatments, medications, and clinical events over time. They are central to a wide range of clinical applications, including risk prediction [ 31 , 28 , 1 ] , medication recommendation [ 27 , 23 ] , and decision support systems [ 29 , 16 ] . However, data sensitivity and privacy regulations hinder the sharing of real-world EHR data, limiting research and development in related intelligent technologies. High-quality synthetic EHR generation is therefore crucial for privacy-preserving research [ 39 ] ."},{"type":"para","text":"Longitudinal EHRs are challenging to model due to their high dimensionality, temporal sparsity, and heterogeneity. Moreover, a few common clinical patterns coexist alongside numerous rare but clinically significant conditions, and they usually exhibit a long-tailed pattern in the empirical marginal distribution of medical codes for symptoms, indicating that few codes occur frequently while most clinically relevant codes are rare. During the training process of generative models, such rare codes provide limited positive evidence and weak stable learning signals for their temporal, contextual, and co-occurrence dependencies. If a model captures only the dominant head patterns while underrepresenting rare pathological dependencies, the resulting synthetic data may appear realistic on average but become unreliable for the very subpopulations that matter most clinically. Therefore, the key challenge for synthetic EHR generation is twofold: generated records should match the overall real EHR distribution, while also preserving the temporal and co-occurrence dependencies involving rare clinical events to facilitate downstream clinical modeling and cross-population adaptation."},{"type":"para","text":"Existing generative models attempt to solve this problem from different aspects, but none fully resolves the long-tail difficulty. GAN and VAE-based methods [ 9 , 38 ] improved privacy-preserving generation but often struggled to preserve longitudinal dependency structure. Autoregressive and hierarchical models [ 34 ] capture sequential patterns better, while diffusion-style approaches [ 44 , 13 ] improve sample quality and distributional coverage. Nevertheless, across these families, a common difficulty remains: optimization is dominated by frequent and easy-to-learn clinical patterns, whereas rare but clinically important dependencies receive weak and unstable learning signals [ 33 , 22 ] . As a result, synthetic EHR generators may fit common clinical patterns yet fail to preserve rare-event plausibility, downstream utility, and controllable cross-population adaptation."},{"type":"para","text":"To this end, we propose AdaPCLA , a curriculum prior-calibrated framework for long-tailed EHR generation. AdaPCLA applies prior-aware logit adjustment early in training to maintain effective rare-code updates, then gradually anneals this correction so that rare-code temporal and co-occurrence dependencies are internalized into the model parameters. This design supports clean inference without an external bias after training, and it also enables zero-shot distribution control by applying target-prior corrections without retraining. Our theoretical analysis gives two messages: the label-wise empirical NTK characterizes how prior-induced residuals update rare-code intrinsic logits, while the prior-internalization bound identifies annealing speed and NTK conditioning as theoretical factors for retaining prior signals after scaffold removal."},{"type":"para","text":"We conduct a comprehensive evaluation over large real-world datasets. Experimental results show that rather than only improving aggregate scores, AdaPCLA achieves the largest gains on tail structure: compared with HALO, it improves TailPairSeen by 114.2% on MIMIC-III and 65.1% on MIMIC-IV. For the zero-shot generation setting based on MIMIC-IV training to MIMIC-III inference, AdaPCLA reaches AUPRC of 0.922 0.922 without retraining, outperforming GPT-style generation by 3.5% F1, which verifies the effectiveness of the learned structure for cross-population adaptation. In summary, this paper makes the following contributions:"},{"type":"para","text":"We propose AdaPCLA , a prior-guided curriculum framework that strengthens rare-code learning while supporting bias-free inference and zero-shot distribution control."},{"type":"para","text":"We analyze AdaPCLA with label-wise empirical NTK dynamics and govern the prior-internalization bound by annealing speed and the empirical NTK."},{"type":"para","text":"AdaPCLA achieves significant gains in tail plausibility, downstream utility, and zero-shot cross-population adaptation tasks based on the experiments on real-world datasets."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Longitudinal EHR generation."},{"type":"para","text":"Synthetic EHR generation has progressed from GAN-, VAE-, and correlation-aware models for discrete or tabular medical records [ 9 , 38 , 35 ] to temporal structured simulators and longitudinal mixed-type generators [ 43 , 22 , 21 ] . Recent work further improves high-dimensional visit modeling, privacy-preserving synthesis, multimodal diffusion, and LLM-based generation without direct patient-level access [ 34 , 39 , 44 , 13 ] . These models improve temporal dependency modeling and sample realism. However, because these generators are usually trained to fit the overall distribution of the majority of diseases, rare disease codes contribute sparse learning signals, motivating a generation objective that gives tail codes stronger optimization support without weakening longitudinal dependency modeling."},{"type":"heading","level":4,"text":"Long-tailed learning and zero-shot control."},{"type":"para","text":"Long-tailed learning methods such as resampling, focal loss, class-balanced loss, margin-based objectives, decoupled training, and logit adjustment reshape training under imbalanced labels [ 5 , 25 , 10 , 4 , 20 , 30 , 33 ] . These methods are effective for discriminative classification, but direct use in longitudinal multi-label EHR generation is less straightforward: resampling can disrupt patient trajectories, fixed reweighting can destabilize sparse multi-label optimization, and static prior biases can leave the generator dependent on external correction at inference time. Meanwhile, clinical code frequencies vary across hospitals and cohorts [ 32 ] , motivating generation methods that can adapt to target priors without retraining while preserving longitudinal dependencies and co-occurrence structure. Although label-shift correction and controllable tabular/EHR synthesis study related forms of occurrence frequency adaptation [ 26 , 12 , 38 , 17 ] , they do not directly address zero-shot control for long-tailed longitudinal EHR generators."},{"type":"para","text":"A detailed discussion about closely related work is deferred to Appendix A ."},{"type":"heading","level":1,"text":"3 Problem Formulation"},{"type":"heading","level":2,"text":"3.1 Preliminaries"},{"type":"heading","level":4,"text":"Notions."},{"type":"para","text":"Consider a structured EHR as a sequence of patient visits. Let 𝒱 \\mathcal{V} be a | 𝒱 | |\\mathcal{V}| -size vocabulary of clinical events, i.e., medical codes. A patient’s T T -visit record forms a chronological trajectory X = ( 𝐱 1 , … , 𝐱 T ) X=(\\mathbf{x}_{1},\\ldots,\\mathbf{x}_{T}) , where each visit 𝐱 t ∈ { 0 , 1 } | 𝒱 | \\mathbf{x}_{t}\\in\\{0,1\\}^{|\\mathcal{V}|} is a multi-hot vector indicating the presence of codes at step t t ; 𝐱 < t \\mathbf{x}_{<t} denotes the trajectory history of this patient before time step t t . We assume that observed trajectories are independent and identically distributed samples drawn from an unknown underlying distribution 𝒫 ( X ) \\mathcal{P}(X) , which we aim to approximate."},{"type":"heading","level":4,"text":"Logit Adjustment (LA)."},{"type":"para","text":"Logit adjustment can be used to guide the model’s learning under class imbalance by modifying the relative margins between a target class and its competing classes [ 30 , 40 , 41 ] . For a given sample ( x , y ) (x,y) with y ∈ { 1 , … , K } y\\in\\{1,\\ldots,K\\} , consider a multiclass classifier with logits C k ( x ) C_{k}(x) for k = 1 , … , K k=1,\\ldots,K . The logit-adjusted softmax loss can be written as"},{"type":"para","text":"where C y ′ ( x ) C_{y^{\\prime}}(x) is the logit of a competing class y ′ y^{\\prime} , and δ ( y , y ′ ) \\delta(y,y^{\\prime}) is a prior-dependent adjustment weight. Larger δ ( y , y ′ ) \\delta(y,y^{\\prime}) increases the penalty when the prediction value for the competing class y ′ y^{\\prime} approaches or exceeds the target class y y , thereby changing the optimization pressure across imbalanced classes. In prior logit-adjustment methods for imbalanced classification, δ ( y , y ′ ) \\delta(y,y^{\\prime}) is typically chosen as a function of empirical class priors, yielding a prior-calibrated decision boundary."},{"type":"para","text":"In the longitudinal EHR generation, the prediction target at each visit corresponds to a multi-hot vector for all medical codes. Let y ∈ { 0 , 1 } V y\\in\\{0,1\\}^{V} denote the multi-label target over 𝒱 \\mathcal{V} and y c = 1 y_{c}=1 when event c c occurs in the target visit; let π c = ℙ ( y c = 1 ) \\pi_{c}=\\mathbb{P}(y_{c}=1) be the empirical frequency of c c at the visit level. Given the intrinsic model logit z c ( x ) z_{c}(x) , we denote the effective logit after adjustment by z ~ c ( x ) \\tilde{z}_{c}(x) . A natural multi-label analogue applies a prior-dependent shift element-wise, z ~ c ( x ) = z c ( x ) + b c \\tilde{z}_{c}(x)=z_{c}(x)+b_{c} , where b c b_{c} is a bias term determined by the empirical frequency of event c c ."},{"type":"para","text":"However, a fixed prior-dependent bias can make the generator depend on this external correction and prevent it from capturing the rare-event temporal and co-occurrence structure and internalizing them in its own parameters. AdaPCLA therefore treats the prior bias as a temporary optimization scaffold: it preserves rare-event learning signals early in training and is gradually annealed away, so that inference relies on model-encoded rare-event dependencies rather than an external correction."},{"type":"heading","level":2,"text":"3.2 Problem Formulation and Requirements"},{"type":"para","text":"In general, longitudinal EHR generation is a sequential generative modeling task. Its goal is to learn a parametric generative model M ( ⋅ ; θ ) M(\\cdot;\\theta) that approximates 𝒫 ( X ) \\mathcal{P}(X) . For a training set of N N patient trajectories 𝒟 = { X i } i = 1 N \\mathcal{D}=\\{X^{i}\\}_{i=1}^{N} , the maximum-likelihood objective can be written as"},{"type":"para","text":"However, real-world EHRs data are high-dimensional and sparse, and most medical codes in 𝒱 \\mathcal{V} form an extreme long-tailed distribution. This poses a substantial challenge for learning a model M ( ⋅ ; θ ) M(\\cdot;\\theta) in Eq. ( 1 ) that faithfully characterizes the real data distribution 𝒫 ( X ) \\mathcal{P}(X) . In particular, under the binary cross-entropy loss, the gradient contributions associated with tail codes are severely overwhelmed by the large number of head samples. As a result, optimization becomes biased toward a head-dominated solution: for tail codes, gradients from the many non-occurrence labels ( y c = 0 y_{c}=0 ) can dominate the sparse gradients from occurrence labels ( y c = 1 y_{c}=1 ), pushing their logits downward and suppressing their generation probabilities. Such behavior leads the model to disregard the complex conditional dependencies of rare pathologies, ultimately fails to capture the intrinsic structure of the clinical data and to generate realistic EHR records faithfully."},{"type":"para","text":"Consequently, an optimal model M ( ⋅ ; θ ∗ ) M(\\cdot;\\theta^{*}) should accurately approximate 𝒫 ( X ) \\mathcal{P}(X) over the support of the data distribution, encompassing a comprehensive spectrum of medical codes and faithfully generating high-fidelity long-tailed patterns at inference time. In practice, particularly in low-resource and rare-disease settings, inadequate long-tail modeling can cause synthetic generators to miss clinically important rare-event patterns, reducing the usefulness of synthetic EHRs for downstream prediction and adapting generation to populations with different code frequencies."},{"type":"para","text":"Moreover, owing to the privacy constraints and the inherent difficulty of accessing and sharing large-scale real EHR datasets, the learned model is expected to be readily transferable to a new, typically small-scale dataset 𝒟 ^ \\hat{\\mathcal{D}} , which may be drawn from a distinct distribution 𝒫 ′ ( X ) \\mathcal{P}^{\\prime}(X) , thereby enabling the efficient construction of generative models that are well adapted to the target data."},{"type":"heading","level":1,"text":"4 Proposed Model"},{"type":"para","text":"In this section, we propose AdaPCLA , an Adaptive Prior-Calibrated Logit Adjustment method designed to capture the full-spectrum medical code distribution, generate realistic EHR data, and can adapt to new datasets in a zero-shot manner. It treats empirical label priors as a temporary optimization scaffold and encourages the model to internalize this structural information by gradually removing the scaffold through annealing. Figure 1 illustrates the overall framework of AdaPCLA ."},{"type":"para","text":"AdaPCLA consists of three core modules: (i) an EHR Generator Backbone , an autoregressive Transformer that captures longitudinal clinical dependencies; (ii) Statistical Prior Modeling , which converts long-tailed empirical prevalences into a label-wise logit scaffold 𝐛 stat \\mathbf{b}_{\\text{stat}} ; and (iii) Curriculum Prior Injection , which anneals this scaffold so that early rare-event updates are protected while the final generator can operate without an external bias."},{"type":"heading","level":2,"text":"4.1 EHR Generator Backbone"},{"type":"para","text":"AdaPCLA uses an autoregressive Transformer backbone to model longitudinal EHR trajectories. Given historical visits x < t x_{<t} , the backbone embeds the visit events and processes the visit sequence with autoregressive masked self-attention, so that the hidden state used for predicting t t -th step visit depends only on previous records and respects the temporal order:"},{"type":"para","text":"where z t z_{t} denotes the intrinsic logits over the clinical-event vocabulary for the next visit. These logits are produced by the backbone parameters alone and constitute the logit space on which AdaPCLA performs prior-calibrated training during the annealing process."},{"type":"para","text":"Based on the shared representation H t − 1 H_{t-1} , the next visit is modeled with Bernoulli output heads:"},{"type":"para","text":"Equivalently, the visit-level training objective is the Binary Cross-Entropy loss over clinical events:"},{"type":"para","text":"The factorization is imposed only at the output layer; temporal and cross-event dependencies are captured by the shared Transformer representation."},{"type":"heading","level":2,"text":"4.2 Optimization Scaffold: Statistical Prior"},{"type":"para","text":"For the long-tailed EHR generation, a main difficulty lies in rare medical codes, which represent clinical events such as diagnoses, procedures, and medications, contribute far fewer positive updates than the common codes. Even when a positive occurrence of a rare code produces a large local BCE residual, its aggregate effect can be overwhelmed by gradients from high-frequency codes and from the many non-occurrence labels, i.e., y c = 0 y_{c}=0 , induced by the large code vocabulary. To make early training easier, we build a label-wise bias from empirical marginal frequencies."},{"type":"para","text":"Considering numerical stability, we use the smoothed marginal probability π ^ ϵ ( c ) = π ^ ( c ) + ϵ 1 + 2 ϵ \\hat{\\pi}_{\\epsilon}(c)=\\frac{\\hat{\\pi}(c)+\\epsilon}{1+2\\epsilon} for event c c , then we define the statistical bias used in the scaffold as"},{"type":"para","text":"At the start of training, the model logit for c c is close to its empirical log-odds, z 0 , c ( x ) ≈ logit ( π ^ ϵ ( c ) ) z_{0,c}(x)\\approx\\mathrm{logit}\\!\\big(\\hat{\\pi}_{\\epsilon}(c)\\big) . After adding b stat ( c ) b_{\\text{stat}}(c) , the effective logit becomes z ~ 0 , c ( x ) = z 0 , c ( x ) + b stat ( c ) ≈ 0 \\tilde{z}_{0,c}(x)=z_{0,c}(x)+b_{\\text{stat}}(c)\\approx 0 . Thus, when the annealing coefficient is large early in training, the prior bias shifts the effective logit toward zero."},{"type":"para","text":"Since σ ′ ( u ) = σ ( u ) ( 1 − σ ( u ) ) \\sigma^{\\prime}(u)=\\sigma(u)(1-\\sigma(u)) is maximized at u = 0 u=0 , centering the early effective logit z ~ 0 , c \\tilde{z}_{0,c} near zero places rare-code predictions in the high-sensitivity region of the sigmoid. When the logit is close to zero, a small parameter update can change the predicted probability more easily. When the logit has a large magnitude, the sigmoid becomes saturated and its derivative becomes small. Therefore, b stat b_{\\text{stat}} is used to make early optimization more sensitive, especially for rare codes."},{"type":"para","text":"We use b stat b_{\\text{stat}} only during training, rather than in the inference phase. At the beginning of training, many rare codes have low logits and receive logit-increasing gradients only from sparse occurrence labels ( y c = 1 ) (y_{c}=1) . Adding b stat b_{\\text{stat}} moves the effective logits closer to zero and amplifies the influence of these sparse rare-event observations relative to the dominant head-code updates. By reducing b stat b_{\\text{stat}} to zero gradually, the model learns to capture rare-event dependencies from the patient history rather than relying on a fixed external bias when rare events should appear. At inference time, the model will use its own logits alone for data generation."},{"type":"heading","level":2,"text":"4.3 Curriculum Prior Injection"},{"type":"para","text":"Retaining a fixed source-prior bias during the inference phase causes generation to remain perpetually dependent on an external correction term, which will introduce computational inefficiencies and can miscalibrate outputs when the target population has a different code distribution. Therefore, AdaPCLA utilizes the prior bias only as a training scaffold and anneals to zero gradually, encouraging rare-code dependencies to be encoded in the model parameters for bias-free inference."},{"type":"para","text":"During training, we use the modified logits z ~ τ ( x ) = z τ ( x ) + α ( τ ) b stat , \\tilde{z}_{\\tau}(x)=z_{\\tau}(x)+\\alpha(\\tau)b_{\\text{stat}}, where z τ ( x ) z_{\\tau}(x) is the intrinsic logit produced by the backbone, b stat b_{\\text{stat}} is the statistical prior bias, and α ( τ ) ∈ [ 0 , 1 ] \\alpha(\\tau)\\in[0,1] controls the proportion of the prior bias used at step τ \\tau . The schedule starts from α ( 0 ) = 1 \\alpha(0)=1 and ends at α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 ."},{"type":"para","text":"The training loss is the BCE loss computed with the effective logits:"},{"type":"para","text":"At the beginning, α ( τ ) \\alpha(\\tau) is large, so rare-event observations exert a stronger effect on the update direction. During training, α ( τ ) \\alpha(\\tau) becomes smaller, the model then has to rely more on the patient history and less on the external bias. At the end of training, α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 , so the generator uses only its own logits, z ~ 𝒯 ( x ) = z 𝒯 ( x ) \\tilde{z}_{\\mathcal{T}}(x)=z_{\\mathcal{T}}(x) . Thus, AdaPCLA uses the prior bias as a temporary training aid, but inference is performed without any external bias."},{"type":"heading","level":2,"text":"4.4 Zero-Shot Cross-Population Adaptation"},{"type":"para","text":"Data from different scenarios, such as different hospitals, populations, or regions, exhibits varying degrees of statistical differences and biases. Therefore, a model trained on one patient population would be used to generate data for another population with different code occurrence frequency. AdaPCLA adaptively supports such cross-population adaptation by introducing a target-prior correction difference at inference time, without retraining."},{"type":"para","text":"Let π ^ ( c ) \\hat{\\pi}(c) be source prevalence used during training and π target ( c ) \\pi_{\\mathrm{target}}(c) be target occurrence frequency for event c c . We define the logit difference between source and target population distribution as"},{"type":"para","text":"and modify the logits as z t ′ = z t ( θ ) + λ Δ b z^{\\prime}_{t}=z_{t}(\\theta)+\\lambda\\Delta b for inference, where λ ∈ [ 0 , 1 ] \\lambda\\in[0,1] controls the strength of the adjustment. λ = 1 \\lambda=1 applies the full prior correction and λ = 0 \\lambda=0 recovers the original model."},{"type":"para","text":"This correction follows the standard label-wise prior-shift rule: if the conditional distribution of patient context given event c c is unchanged across the source and target populations, i.e., ℙ S ( x ∣ y c ) = ℙ T ( x ∣ y c ) \\mathbb{P}_{S}(x\\mid y_{c})=\\mathbb{P}_{T}(x\\mid y_{c}) , then the Bayes-optimal logit changes by"},{"type":"para","text":"See Appendix D for the detailed proof. This correction is applied independently to each code at each next-visit prediction step. Under the label-wise prior-shift assumption, it steers label-wise occurrence probabilities toward the target population while reusing the longitudinal and co-occurrence structure learned by the backbone, without requiring target-domain retraining."},{"type":"heading","level":1,"text":"5 Theoretical Analysis"},{"type":"para","text":"In this section, we analyze the training dynamics of AdaPCLA and give theoretical insights into why it can retain rare-code learning signals after the prior scaffold is digested."},{"type":"para","text":"First, the prior scaffold changes the BCE residuals, so the SGD update induced by these residuals is written into the intrinsic logits through the shared Transformer backbone. This happens because the scaffold is fixed and independent of the backbone parameters, i.e., ∇ θ ( α k b stat ) = 0 \\nabla_{\\theta}(\\alpha_{k}b_{\\text{stat}})=0 , and therefore ∇ θ z ~ k ( x ) = ∇ θ z k ( x ) \\nabla_{\\theta}\\tilde{z}_{k}(x)=\\nabla_{\\theta}z_{k}(x) . This yields a label-wise gradient-transfer relationship: the residual of one code can influence the intrinsic logit of another code through the corresponding entry of the label-wise empirical NTK. Second, under standard lazy-training assumptions [ 7 ] , we analyze how the effect of the annealed scaffold α ( τ ) b stat \\alpha(\\tau)b_{\\text{stat}} can be absorbed into the intrinsic logits as α ( τ ) \\alpha(\\tau) decays to zero. Together, these results justify that the prior bias can be injected to create an early rare-code learning signal, and the annealing schedule controls whether this signal is retained in the model parameters after the external bias is digested."},{"type":"heading","level":2,"text":"5.1 Gradient Transfer through the Shared Backbone"},{"type":"para","text":"We identify how the prior scaffold affects rare-code intrinsic-logit updates through the label-wise empirical NTK. At training step k k , the effective logits are z ~ k ( x ) = z k ( x ) + α k b stat \\tilde{z}_{k}(x)=z_{k}(x)+\\alpha_{k}b_{\\text{stat}} , and the BCE residual is 𝒢 Ada ( k ) ( x ) = σ ( z ~ k ( x ) ) − y \\mathcal{G}^{(k)}_{\\mathrm{Ada}}(x)=\\sigma(\\tilde{z}_{k}(x))-y which affects the SGD update direction as the scaffold. The additive term α k b stat \\alpha_{k}b_{\\text{stat}} is fixed with respect to the backbone parameters, so ∇ θ ( α k b stat ) = 0 \\nabla_{\\theta}(\\alpha_{k}b_{\\text{stat}})=0 and ∇ θ z ~ k ( x ) = ∇ θ z k ( x ) \\nabla_{\\theta}\\tilde{z}_{k}(x)=\\nabla_{\\theta}z_{k}(x) . The adjusted logits z ~ k ( x ) \\tilde{z}_{k}(x) determine the BCE residuals, and the fixed-scaffold property ensures that the corresponding parameter update changes the intrinsic logits z k ( x ) z_{k}(x) . For a one-step SGD update with step size η > 0 \\eta>0 , the first-order intrinsic-logit update for code c c is"},{"type":"para","text":"where [ 𝒦 ( k ) ] i , j = ⟨ ∇ θ z k ( i ; x ) , ∇ θ z k ( j ; x ) ⟩ . [\\mathcal{K}^{(k)}]_{i,j}=\\left\\langle\\nabla_{\\theta}z_{k}(i;x),\\nabla_{\\theta}z_{k}(j;x)\\right\\rangle. 𝒦 ( k ) \\mathcal{K}^{(k)} is the label-wise empirical NTK for context x x ."},{"type":"para","text":"Therefore, the residual of code j j changes the intrinsic-logit update of code c c through the corresponding entry [ 𝒦 ( k ) ] c , j [\\mathcal{K}^{(k)}]_{c,j} of the label-wise empirical NTK. This relationship shows that the prior scaffold affects intrinsic logits by reshaping BCE residuals, whose effects are then transferred through backbone parameter updates. For each code c c , the update is determined by BCE residuals weighted by the corresponding row of 𝒦 ( k ) \\mathcal{K}^{(k)} . Thus, AdaPCLA strengthens early rare-code learning signals only when those residuals can be transferred through the empirical-NTK structure."},{"type":"heading","level":2,"text":"5.2 Prior-Internalization Bound"},{"type":"para","text":"For a fixed context x x , define the expected BCE risk as"},{"type":"para","text":"Let z ~ ⋆ ( x ) \\tilde{z}^{\\star}(x) be the effective-logit minimizer of the conditional expected BCE risk for the fixed context x x . Because the effective logits are z + α ( τ ) b stat z+\\alpha(\\tau)b_{\\text{stat}} , the corresponding target in intrinsic-logit coordinates is z τ ⋆ ( x ) = z ~ ⋆ ( x ) − α ( τ ) b stat z_{\\tau}^{\\star}(x)=\\tilde{z}^{\\star}(x)-\\alpha(\\tau)b_{\\text{stat}} . Since AdaPCLA removes the scaffold at inference time, prior internalization requires the learned intrinsic logits z τ ( x ) z_{\\tau}(x) to follow the target as α ( τ ) \\alpha(\\tau) decays to zero."},{"type":"para","text":"Under the assumptions stated the internalization error e τ ( x ) = z τ ( x ) − z τ ⋆ ( x ) e_{\\tau}(x)=z_{\\tau}(x)-z_{\\tau}^{\\star}(x) satisfies"},{"type":"para","text":"This prior-internalization bound gives two design implications. First, smoother annealing reduces sup s | α ˙ ( s ) | \\sup_{s}|\\dot{\\alpha}(s)| , which lowers the internalization error caused by a moving scaffold. Second, a larger minimum eigenvalue λ min ( 𝒦 ) \\lambda_{\\min}(\\mathcal{K}) of the label-wise empirical NTK reduces the bound, meaning that the intrinsic logits can stay closer to the moving target. Because α ( 0 ) = 1 \\alpha(0)=1 and α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 , the moving target in intrinsic-logit coordinates shifts by z 𝒯 ⋆ ( x ) − z 0 ⋆ ( x ) = b stat z_{\\mathcal{T}}^{\\star}(x)-z_{0}^{\\star}(x)=b_{\\text{stat}} . When the internalization error e τ ( x ) e_{\\tau}(x) remains small along the annealing path, the learned intrinsic logits inherit this shift, so z 𝒯 ( x ) − z 0 ( x ) ≈ b stat z_{\\mathcal{T}}(x)-z_{0}(x)\\approx b_{\\text{stat}} . This supports the intended internalization property of AdaPCLA : the prior signal introduced by the training scaffold is absorbed into the model’s own logits through the annealing trajectory, enabling inference without an external prior correction."},{"type":"heading","level":1,"text":"6 Experiments"},{"type":"heading","level":2,"text":"6.1 Experimental Setting"},{"type":"heading","level":4,"text":"Datasets and preprocessing."},{"type":"para","text":"We evaluate on MIMIC-III [ 19 ] and MIMIC-IV [ 18 ] . Each patient is represented as a sequence of visits, where each visit is a deduplicated set of International Classification of Diseases (ICD) diagnosis codes. We use patient-level train/validation/test splits and apply the same preprocessing and vocabulary mapping to all methods."},{"type":"heading","level":4,"text":"Baselines."},{"type":"para","text":"We compare AdaPCLA with representative EHR generators, including LSTM [ 8 ] , EVA [ 22 ] , SynTEG [ 43 ] , GPT-style [ 13 ] , and HALO (hierarchical autoregressive EHR generator) [ 34 ] . For each baseline, we follow the model architecture and training recipe described in the original paper, and use the same training cohort and evaluation protocol as AdaPCLA ."},{"type":"heading","level":4,"text":"Metrics."},{"type":"para","text":"We evaluate the quanlity of synthetic EHR data. For distributional fidelity, we examine occurrence frequency and visit-level code co-occurrence statistics. For tail clinical plausibility, we report PairSeen/TailPairSeen, TailCtxJSD, and TailTopKJac. For downstream utility, we use the Train-on-Synthetic, Test-on-Real (TSTR) protocol and report Accuracy, F1, and AUPRC."},{"type":"heading","level":2,"text":"6.2 Experimental Results"},{"type":"heading","level":4,"text":"Tail clinical plausibility."},{"type":"para","text":"Table 1 reports tail clinical plausibility on MIMIC-III/-IV. AdaPCLA gives the strongest overall tail-focused performance across the two datasets. On MIMIC-IV, AdaPCLA ranks first on all four metrics: PairSeen, TailPairSeen, TailCtxJSD, and TailTopKJac. In particular, it improves TailPairSeen from 0.0498 0.0498 with HALO to 0.0822 0.0822 , a relative gain of 65.1 % 65.1\\% ; and also reduces TailCtxJSD from 0.6652 0.6652 to 0.6313 0.6313 , showing better tail-context consistency."},{"type":"para","text":"On MIMIC-III, AdaPCLA achieves the best TailPairSeen and TailCtxJSD. Compared with HALO, it improves TailPairSeen from 0.0520 0.0520 to 0.1114 0.1114 , and reduces TailCtxJSD from 0.6558 0.6558 to 0.5981 0.5981 . Although HALO has a higher overall PairSeen and LSTM has a higher TailTopKJac, AdaPCLA provides a better balance between tail support and tail-context consistency. Overall, these results show that AdaPCLA better preserves rare-event co-occurrence structure than other baselines."},{"type":"para","text":"EVA and SynTEG show unstable tail plausibility. On MIMIC-III, their PairSeen and TailPairSeen scores are close to zero, which means that many generated intra-visit pairs are not supported by the training corpus. A likely reason is that VAE- and GAN-based generators often produce dense multi-hot visits after thresholding. When a visit contains too many events, the number of intra-visit pairs grows quickly, and this increases the chance of unseen co-occurrences. This also hurts tail-context metrics because the local neighborhood of tail events can deviate from the real data distribution."},{"type":"para","text":"AdaPCLA uses conditional code probabilities together with sampling-time control over the number of generated codes in each visit. This design better preserves the natural sparsity of clinical visits and reduces implausible tail-code co-occurrences."},{"type":"heading","level":2,"text":"6.3 Model Analysis"},{"type":"para","text":"We evaluate the properties and performance of AdaPCLA based on the following aspects: internalization ablation study, downstream utility, and zero-shot distribution control adaptations."},{"type":"heading","level":4,"text":"Ablation study."},{"type":"para","text":"Table 3 (a) compares different variants of AdaPCLA and HALO based on how prior-bias is used for the generative model training. HALO is the backbone generator without any prior bias. Static-PCLA (Consistent) uses the fixed prior bias both during training and inference, which tests whether a permanent bias can improve generation if it is always available. Static-PCLA (Train-only) uses the same fixed bias during training but removes it at inference, which tests whether the model still work after the external bias is taken away. AdaPCLA uses the bias only during training and gradually anneals to zero, and inference is performed without any external bias."},{"type":"para","text":"The results show a clear difference between using a fixed bias and annealing it. Static-PCLA (Consistent) improves over HALO on several tail metrics, which means that the prior bias is useful when it is kept at inference. However, Static-PCLA (Train-only) fails after the bias is removed: TailPairSeen and TailCtxJSD cannot be computed, and TailTopKJac drops to 0.0000 0.0000 . This means that a fixed training bias alone does not make the model learn the rare-event structure."},{"type":"para","text":"AdaPCLA does not use an external bias at inference, but still achieves the best metrics in this ablation. It obtains the highest TailPairSeen 0.1239 0.1239 and TailTopKJac 0.0356 0.0356 , and the lowest TailCtxJSD 0.5940 0.5940 , outperforming all the baselines. This suggests that dynamic annealing helps the backbone learn useful prior information in its own logits, instead of depending on a fixed external correction."},{"type":"para","text":"Figure 2 further visualizes this annealing process by tracking how selected probe-code probabilities change over training checkpoints. A probe code is a candidate code evaluated under a fixed patient-history context, and the probes are grouped into related rare, unrelated rare, and wrong codes. The similar trajectory patterns under 30 and 100 contexts suggest that annealing produces stable internalization dynamics as the probing set is enlarged."},{"type":"heading","level":4,"text":"Downstream predictive utility."},{"type":"para","text":"We use the TSTR protocol to test whether synthetic data supports real downstream prediction. For each generator, we train a 25-label diagnosis classifier only on synthetic data and evaluate it on the held-out real test set. As shown in Table 2 , AdaPCLA achieves the best Accuracy, F1, and AUPRC on both MIMIC-III and MIMIC-IV. These results indicate that AdaPCLA preserves predictive signals that transfer from synthetic data to real-data."},{"type":"heading","level":4,"text":"Zero-shot distribution control."},{"type":"para","text":"We also test whether AdaPCLA can adapt to cross-population generation without retraining. The model is trained on the source population and then applied to the target population using only the target code occurrence rates through the delta-bias correction in Sec. 4.4 . Table 3 (b) reports the zero-shot transfer results from MIMIC-IV to MIMIC-III. These results show that AdaPCLA can use the target prior to adjust generated code frequencies, while retaining longitudinal dependencies and co-occurrence patterns learned from the source population."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"In this paper, we proposed AdaPCLA , a curriculum prior-guided framework for long-tailed longitudinal EHR generation. By using empirical priors as a temporary training scaffold and gradually annealing them away, AdaPCLA strengthens rare-code learning during optimization without requiring an external bias during inference. Experiments on real-world dataset show that AdaPCLA improves tail plausibility, downstream utility, and zero-shot distribution control over strong EHR generation baselines. Finally, AdaPCLA supports zero-shot generation adaptation for cross-population setting in a training-free manner."},{"type":"para","text":"Limitations. The current evaluation limited to diagnosis-code sequences from two public EHR benchmarks. More validation for AdaPCLA over diverse hospitals and richer EHR modalities beyond diagnosis codes is expected as future work."},{"type":"heading","level":2,"text":"A.1 Generative Modeling for Electronic Health Records"},{"type":"para","text":"Synthetic EHR generation builds on a broader clinical sequence modeling literature, where recurrent and temporal point-process models show that diagnoses, procedures, medications, and other clinical events cannot be treated as independent observations [ 8 , 11 , 37 ] . This temporal view is especially important for longitudinal synthesis: a realistic generator should preserve visit-level ordering, delayed comorbidities, and clinically meaningful code co-occurrences, rather than only matching aggregate code counts."},{"type":"para","text":"Synthetic EHR generation has progressed from GAN-, VAE-, and correlation-aware models for discrete or tabular medical records [ 9 , 38 , 35 ] to temporal structured simulators and longitudinal mixed-type generators [ 43 , 22 , 21 ] . Hierarchical autoregressive frameworks further improve high-dimensional visit modeling by directly modeling long-span patient trajectories [ 34 ] . Recent systems also study privacy-preserving synthesis, multimodal diffusion, and LLM-based generation without direct patient-level access [ 39 , 44 , 13 ] , improving temporal dependency modeling and sample realism."},{"type":"para","text":"Despite this progress, most existing generators are still trained to fit the overall distribution of the majority of diseases. In long-tailed EHR data, rare disease codes contribute sparse and unstable learning signals for their temporal, contextual, and co-occurrence dependencies. As a result, a generator may improve average realism while underrepresenting rare-event plausibility and rare-code context consistency. AdaPCLA follows the motivation stated in the main text: it gives tail codes stronger optimization support through a temporary prior scaffold, while preserving longitudinal dependency modeling and enabling bias-free inference after annealing."},{"type":"heading","level":2,"text":"A.2 Long-Tailed Learning in EHR"},{"type":"para","text":"Long-tailed learning methods reshape training under imbalanced labels, with most canonical methods developed for discriminative classification. Resampling methods rebalance the empirical distribution by constructing or selecting minority examples [ 5 ] . Loss-level methods, including focal loss [ 25 ] and class-balanced loss [ 10 ] , change the gradient contribution of different labels. Margin-based objectives enlarge rare-class margins [ 4 ] , decoupled training separates representation learning from classifier calibration [ 20 ] , and logit adjustment adds label-frequency terms to logits to obtain a prior-calibrated decision boundary [ 30 , 33 ] . Later variants further study dynamic or learnable adjustments [ 40 ] ."},{"type":"para","text":"These methods are effective for classification, but direct use in longitudinal multi-label EHR generation is less straightforward. Resampling can disrupt patient trajectories, fixed reweighting can destabilize sparse multi-label optimization, and static prior biases can leave the generator dependent on external correction at inference time. Multi-label long-tail methods [ 36 , 24 ] further show that label co-occurrence and label-frequency imbalance interact, which is precisely the setting faced by longitudinal EHR synthesis."},{"type":"para","text":"Recent studies also analyze long-tailed learning through learning dynamics and optimization trajectories. Neural-collapse-based analyses study how imbalanced training affects decision boundaries [ 14 ] , while probing-based methods use training dynamics to guide data pruning or bias mitigation [ 6 ] . However, these interventions can rely on heuristic data dropping, which may discard informative tail samples, or maintain a permanent dependency on external priors."},{"type":"para","text":"In contrast, AdaPCLA uses a curriculum-learning strategy [ 3 ] . By treating the statistical prior as a temporary optimization scaffold and annealing it over time, AdaPCLA strengthens rare-code learning signals while encouraging the backbone to encode prior information into its own logits. This improves tail fidelity without discarding patient trajectories or requiring a fixed inference-time bias."},{"type":"heading","level":2,"text":"A.3 Distribution Shift and Zero-Shot Adaptation"},{"type":"para","text":"Clinical code frequencies vary across hospitals, populations, and regions due to dataset shift [ 32 ] . Standard approaches for handling such shifts typically rely on domain adaptation or fine-tuning, which require target-domain data access and additional retraining. This motivates generation methods that can adapt to target priors without retraining while preserving longitudinal dependencies and co-occurrence structure."},{"type":"para","text":"One relevant setting is label shift or target-prior shift, where label prevalences change while the conditional structure is assumed to be comparatively stable. Classical and modern work estimates or corrects such shifts using black-box predictors, regularized target-prior estimation, or unified label-shift estimators [ 26 , 2 , 12 ] . Related domain-adaptation work studies target and conditional shift [ 42 ] , and recent time-series adaptation considers feature and label shifts in sequential data [ 15 ] . These studies motivate a label-wise prior-shift view for clinical synthesis: when target code occurrence frequencies change, a generator should adjust label-wise occurrence probabilities while retaining useful conditional dependencies."},{"type":"para","text":"Controllable tabular and EHR synthesis provides a related perspective. Conditional tabular generators [ 38 ] and synthetic EHR frameworks [ 39 , 13 ] can condition generation on available attributes or external summaries. However, they are not primarily designed for zero-shot control of long-tailed longitudinal EHR generators through target-prior correction over thousands of sparse medical codes. Recent work on structural fidelity further emphasizes that preserving causal or conditional dependencies is different from matching marginal distributions [ 17 ] . This distinction is important for cross-population generation, where occurrence frequencies may shift while clinically meaningful co-occurrence structure should remain reusable."},{"type":"para","text":"AdaPCLA follows this direction by separating the learned longitudinal structure from marginal occurrence frequency. After the source prior is absorbed during training, the target-prior correction in Sec. 4.4 provides a lightweight mechanism for zero-shot cross-population adaptation: the generator steers label-wise occurrence probabilities toward the target population without retraining the backbone. The intended benefit is controlled occurrence frequency adaptation while reusing the longitudinal and co-occurrence structure learned from source EHR data."},{"type":"heading","level":2,"text":"B.1 Model Analysis"},{"type":"para","text":"In this section, we conduct a deep-dive analysis to examine the core theoretical claims of AdaPCLA : (1) the internalization of prior knowledge, (2) the translation of tail fidelity into downstream utility, and (3) the disentanglement of structure and occurrence frequency for zero-shot control."},{"type":"heading","level":3,"text":"B.1.1 Prior-Internalization Ablation"},{"type":"para","text":"To examine whether the prior bias is absorbed into the model or remains an external correction, we compare dynamic annealing against static-bias variants in Table 3 . This analysis follows the curriculum construction in Sec. 4.3 and focuses on whether the backbone can preserve rare-event structure during bias-free inference."},{"type":"heading","level":4,"text":"Effect of Removing a Fixed Bias"},{"type":"para","text":"The ablation shows a clear difference between using a fixed bias and annealing it. The Static-PCLA (Train-only) variant applies the fixed bias during training but removes it at inference. After this bias is removed, TailPairSeen and TailCtxJSD cannot be computed, and TailTopKJac drops to 0.0000 0.0000 . This negative control indicates that a fixed training bias alone does not make the backbone encode rare-event structure into its own logits. Instead, the model remains dependent on the fixed correction and fails to preserve tail plausibility once the correction is removed at inference time."},{"type":"heading","level":4,"text":"Evidence of Prior Internalization"},{"type":"para","text":"AdaPCLA does not use an external bias at inference, but still achieves the best tail metrics in this ablation. It obtains the highest TailPairSeen 0.1239 and TailTopKJac 0.0356 , and the lowest TailCtxJSD 0.5940 . This result is consistent with the gradient dynamics analyzed in Sec. 4.3 : the prior scaffold changes BCE residuals during training, and the resulting updates are written into the intrinsic logits through the shared Transformer backbone. It suggests that dynamic annealing helps the backbone learn useful prior information in its own logits, instead of depending on a fixed external correction during sampling."},{"type":"heading","level":4,"text":"Probability Dynamics during Annealing"},{"type":"para","text":"Beyond the ablation table, we further examine how selected code probabilities change as the annealing coefficient α ( τ ) \\alpha(\\tau) decays. Figure 3 reports four complementary views, from global tail statistics to a single clinical context. As Fig. 3 (a) shows, the average last-step log-probability of a broad set of tail codes steadily increases across checkpoints, indicating that the curriculum increases tail-code probabilities as a whole. On the micro-probing set, Fig. 3 (b) shows systematic differences across probe types (related_rare / unrelated_rare / wrong), while Fig. 3 (c) is consistent with the label-wise empirical NTK analysis in Sec. 4.3 : probability gains Δ log p \\Delta\\log p grow with the initial gradient similarity between a disease and its context bundle. Finally, Fig. 3 (d) evaluates the context of Case A (cardiovascular patient, Sec. B.2 ). Anticoagulant (V5861), the clinically correct tail medication for Atrial Fibrillation, rises and converges to the Oracle line, whereas Traffic accident (E8192), the hallucination produced by HALO, remains at the bottom. This is consistent with the Case Study: AdaPCLA later generates Anticoagulant because its probability is selectively elevated during annealing. Additional robustness checks for micro-probing are provided in Appendix J ."},{"type":"heading","level":3,"text":"B.1.2 Annealing Schedule Effects on Related-Rare Probes"},{"type":"para","text":"We compare different annealing schedules by analyzing probability trajectories on tail-relevant probe codes. We do not use a global logit-difference norm such as ‖ z 𝒯 − z 0 − b stat ‖ \\|z_{\\mathcal{T}}-z_{0}-b_{\\text{stat}}\\| as the main empirical measure. First, the fixed scaffold b stat b_{\\text{stat}} has a much larger norm than the logit changes induced by training, making schedule-dependent changes difficult to isolate. Second, the norm aggregates over the entire vocabulary and does not focus on rare-code dimensions that are relevant to a fixed patient-history context."},{"type":"para","text":"To measure these schedule effects, we use related_rare gain and AUC summaries based on micro-probe trajectories. Following the micro-probe setting in the main text, we focus on pairs ( x , c ) (x,c) labeled as related_rare, where c c is a tail code clinically related to the patient-history context x x . For each pair, we record the last-step log-probability trajectory { log p t } t = 0 T \\{\\log p_{t}\\}_{t=0}^{T} over 100 100 micro-probe checkpoints and define three summaries:"},{"type":"para","text":"Final gain : Δ log p = log p T − log p 0 \\Delta\\log p=\\log p_{T}-\\log p_{0} , measuring the net log-probability change at the end of annealing for each related_rare code."},{"type":"para","text":"AUC :"},{"type":"para","text":"which sums the positive log-probability changes along the annealing trajectory."},{"type":"para","text":"Normalized AUC :"},{"type":"para","text":"which rescales AUC to approximately [ 0 , 1 ] [0,1] and summarizes the distribution of log-probability gains across checkpoints relative to the final checkpoint."},{"type":"para","text":"These summaries depend only on increments of log p t \\log p_{t} and do not explicitly involve the global bias b stat b_{\\text{stat}} . They measure the magnitude and consistency of log-probability increases for tail-relevant related_rare codes along the annealing trajectory."},{"type":"para","text":"On the related_rare micro-probes, we compute final gain, AUC, and normalized AUC for four annealing schedules: slow , base , cosine , and fast . For final gain Δ log p \\Delta\\log p , the mean values follow"},{"type":"para","text":"but the differences are small compared to the standard deviations (gain_std ≈ 1.0 \\approx 1.0 ). We therefore do not interpret final gain alone as providing a robust ordering across schedules."},{"type":"para","text":"AUC and normalized AUC show a clearer ordering. For both raw AUC and normalized AUC, we observe"},{"type":"para","text":"where the fast schedule has the largest related_rare AUC, while the slow schedule has the smallest AUC. The normalized AUC values follow the same ordering, indicating that fast annealing increases related_rare log-probabilities earlier and more steadily than slow annealing. We therefore use AUC and normalized AUC as the main schedule-dependent summaries for Experiment 3, and use final gain as a supporting quantity."},{"type":"para","text":"These related_rare gain and AUC summaries should not be interpreted as direct numerical estimates of the theoretical prior-internalization error ‖ e τ ( x ) ‖ 𝒦 − 1 \\|e_{\\tau}(x)\\|_{\\mathcal{K}^{-1}} . They summarize log-probability changes for selected related_rare codes, rather than reconstructing the label-wise empirical NTK flow or the moving target z τ ⋆ ( x ) z_{\\tau}^{\\star}(x) . Accordingly, this experiment provides an empirical analysis of schedule-dependent probability trajectories on tail-relevant codes."},{"type":"para","text":"Taken together, the results show that annealing speed affects the timing and magnitude of related_rare probability increases during training. Fast annealing produces earlier and more sustained log-probability gains, whereas slow annealing produces later and weaker accumulated gains. We use these findings as the schedule-dependent mechanism analysis for Experiment 3, not as a strict numerical validation of the prior-internalization bound."},{"type":"heading","level":2,"text":"B.2 Qualitative Analysis: Clinical Case Study"},{"type":"para","text":"To move beyond aggregate statistics, we examine the Clinical Coherence of synthetic patient trajectories. We present two complementary case studies on MIMIC-IV to illustrate how AdaPCLA captures complex dependencies that baselines miss."},{"type":"heading","level":4,"text":"Case A: Longitudinal Trajectory Consistency."},{"type":"para","text":"We select a real patient with a complex history of Cardiovascular and Metabolic disorders . As detailed in Table 5 , given the context of Visits 1–2 (containing Hyperlipidemia, Atrial Fibrillation, and Hypertension), we compare the generated third visit across models."},{"type":"para","text":"Ground Truth Logic: The patient’s condition progresses logically: chronic cardiovascular issues persist, requiring continued medication (Anticoagulants) and monitoring."},{"type":"para","text":"Baseline Failure (Hallucination): The HALO baseline captures generic codes (Hypertension) but suffers from severe semantic hallucination . It predicts \"Traffic accident (E8192)\" , a completely irrelevant external cause disconnected from the patient’s medical history. This typifies the \"Gradient Domination\" failure: without the scaffold, the model’s attention is scattered by noise. LSTM and GPT similarly fail to recall the specific comorbidity structure (e.g., predicting unrelated HIV or Pneumonia codes)."},{"type":"para","text":"AdaPCLA Success (Coherence): In contrast, AdaPCLA correctly recovers the sophisticated treatment logic. It successfully predicts \"Long-term use of anticoagulants (V5861)\" , a critical medication strictly tied to the history of Atrial Fibrillation (42731) . Furthermore, it generates \"Chest pain\" , a clinically plausible symptom given the cardiac history. This illustrates that AdaPCLA has internalized the causal link: Atrial Fib → \\rightarrow Anticoagulant Use , a dependency that requires looking past marginal frequencies."},{"type":"heading","level":4,"text":"Case B: Tail Co-occurrence Topology"},{"type":"para","text":"In Table 6 (Appendix), we investigate the local neighborhood of a specific tail code: Squamous cell carcinoma of skin (17322) . We examine the top-10 co-occurring codes generated by each model."},{"type":"para","text":"Manifold Reconstruction: AdaPCLA successfully reconstructs the correct disease manifold. It identifies 5 out of the top 10 true comorbidities, including specific procedures like Aortocoronary bypass (V4581) and chronic conditions like Hyperlipidemia ."},{"type":"para","text":"Baseline Scattering: HALO’s predictions are scattered and generic (e.g., Dizziness, Chronic pain), lacking disease-specific correlation. LSTM and GPT fail to generate any meaningful co-occurrences for this rare code, indicating Mode Collapse in the tail region."},{"type":"para","text":"These cases qualitatively illustrate that AdaPCLA ’s superior quantitative metrics (e.g., TailPairSeen) translate into tangible clinical validity."},{"type":"heading","level":2,"text":"B.3 Prediction Reliability and Internalization Consistency"},{"type":"para","text":"Beyond transferability and controllability, we investigate two complementary aspects: (1) the reliability of predicted probabilities, especially in the long tail; and (2) the consistency between observed curriculum dynamics and the theory in Sec. 4.3 ."},{"type":"heading","level":4,"text":"Prediction Confidence Reliability"},{"type":"para","text":"We evaluate predicted probabilities on a held-out real test set with ECE , Brier Score , and NLL , reporting both overall and tail-only subsets. AdaPCLA consistently outperforms GPT-style and HALO in both settings; complete numbers and reliability diagrams are reported in the detailed results section."},{"type":"heading","level":4,"text":"Expanded Probing Analysis"},{"type":"para","text":"We further enlarge the probing set from 30 to 100 contexts and recompute related_rare / unrelated_rare / wrong probe trajectories. Table 8 and Figure 2 show that the trajectory scale and type separation remain qualitatively consistent under the larger probe set."},{"type":"heading","level":4,"text":"Summary of Outputs"},{"type":"para","text":"Together, confidence metrics and expanded probing provide empirical support by examining probability reliability and curriculum dynamics. Exploratory λ min ( 𝒦 ) \\lambda_{\\min}(\\mathcal{K}) correlation analysis is reported in Appendix G and treated as non-conclusive evidence."},{"type":"para","text":"We analyze the curriculum-induced logit dynamics for a fixed input context x x . Recall the effective logits z ~ τ ( x ) = z τ ( x ) + α ( τ ) b stat \\tilde{z}_{\\tau}(x)=z_{\\tau}(x)+\\alpha(\\tau)b_{\\text{stat}} , where α : [ 0 , 𝒯 ] → [ 0 , 1 ] \\alpha:[0,\\mathcal{T}]\\to[0,1] is continuously differentiable with α ( 0 ) = 1 \\alpha(0)=1 and α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 ."},{"type":"para","text":"Why we use an expected-risk surrogate A pointwise BCE loss with a single hard multi-label target y ∈ { 0 , 1 } V y\\in\\{0,1\\}^{V} generally does not admit a finite minimizer in logit space, because the minimizer is attained only in the limit z c → ± ∞ z_{c}\\to\\pm\\infty . To obtain a well-posed moving optimum, we work with the conditional expected BCE risk at a fixed context x x :"},{"type":"para","text":"We suppress the explicit dependence on x x in what follows."},{"type":"para","text":"Under the lazy-training approximation, we assume that the label-wise empirical NTK remains approximately constant along the trajectory. Because we analyze the dynamics of different labels for the same context x x , the kernel here is the label-wise coupling matrix"},{"type":"para","text":"rather than the usual sample–sample NTK. On the active label subspace we assume 𝒦 \\mathcal{K} is positive definite, so that 𝒦 − 1 \\mathcal{K}^{-1} is well-defined there. We write ‖ v ‖ 𝒦 − 1 := v ⊤ 𝒦 − 1 v \\|v\\|_{\\mathcal{K}^{-1}}:=\\sqrt{v^{\\top}\\mathcal{K}^{-1}v} ."},{"type":"para","text":"The kernel-induced gradient flow in intrinsic-logit coordinates is z ˙ τ = − 𝒦 ∇ F τ ( z τ ) \\dot{z}_{\\tau}=-\\mathcal{K}\\nabla F_{\\tau}(z_{\\tau}) ."},{"type":"para","text":"Assumptions"},{"type":"para","text":"A1 (Local strong monotonicity / local strong convexity type condition). There exists a convex neighborhood 𝒟 ⊂ ℝ V \\mathcal{D}\\subset\\mathbb{R}^{V} and a constant μ > 0 \\mu>0 such that, for all τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] and all u , v ∈ 𝒟 u,v\\in\\mathcal{D} ,"},{"type":"para","text":"This is the standard first-order formulation of strong monotonicity of ∇ F τ \\nabla F_{\\tau} on 𝒟 \\mathcal{D} ; when F τ F_{\\tau} is twice differentiable, it is implied by the local Hessian lower bound ∇ 2 F τ ( z ) ⪰ μ I \\nabla^{2}F_{\\tau}(z)\\succeq\\mu I on 𝒟 \\mathcal{D} ."},{"type":"para","text":"A2 (Smoothness of the objective–curriculum pair and endpoint schedule). The joint map ( τ , z ) ↦ ∇ F τ ( z ) (\\tau,z)\\mapsto\\nabla F_{\\tau}(z) is continuously differentiable on [ 0 , 𝒯 ] × 𝒟 [0,\\mathcal{T}]\\times\\mathcal{D} , and α : [ 0 , 𝒯 ] → [ 0 , 1 ] \\alpha:[0,\\mathcal{T}]\\to[0,1] is continuously differentiable with the endpoint conditions α ( 0 ) = 1 \\alpha(0)=1 and α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 (matching the curriculum construction in the main text)."},{"type":"para","text":"A3 (Constant, nondegenerate kernel on the active subspace). The label-wise kernel 𝒦 \\mathcal{K} is constant and symmetric on the active label subspace, and is positive definite there with λ min ( 𝒦 ) > 0 \\lambda_{\\min}(\\mathcal{K})>0 . In particular, the 𝒦 − 1 \\mathcal{K}^{-1} –inner product and the induced norm ∥ ⋅ ∥ 𝒦 − 1 \\|\\cdot\\|_{\\mathcal{K}^{-1}} used below are well-defined on that subspace."},{"type":"para","text":"Lemma A.1 (Regularity of the moving minimizer) Suppose that for each τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] , there exists a unique point z τ ⋆ ∈ 𝒟 z_{\\tau}^{\\star}\\in\\mathcal{D} such that ∇ F τ ( z τ ⋆ ) = 0 \\nabla F_{\\tau}(z_{\\tau}^{\\star})=0 . Under Assumptions A1–A2 and the existence/uniqueness hypothesis above, the path τ ↦ z τ ⋆ \\tau\\mapsto z_{\\tau}^{\\star} is continuously differentiable. Moreover, there exists a finite constant B ⋆ > 0 B_{\\star}>0 such that"},{"type":"para","text":"Proof. Under A1–A2, the Hessian ∇ 2 F τ ( z τ ⋆ ) \\nabla^{2}F_{\\tau}(z_{\\tau}^{\\star}) is nonsingular at each minimizer (as in the standard link between strong monotonicity and positive definiteness of the Jacobian of the gradient). The implicit function theorem therefore yields a local C 1 C^{1} branch of solutions to ∇ F τ ( z ) = 0 \\nabla F_{\\tau}(z)=0 near each ( τ , z τ ⋆ ) (\\tau,z_{\\tau}^{\\star}) . Together with the assumed uniqueness of z τ ⋆ z_{\\tau}^{\\star} in 𝒟 \\mathcal{D} for every τ \\tau , these local branches glue to a globally defined C 1 C^{1} path τ ↦ z τ ⋆ \\tau\\mapsto z_{\\tau}^{\\star} on [ 0 , 𝒯 ] [0,\\mathcal{T}] ."},{"type":"para","text":"Differentiating the optimality condition ∇ F τ ( z τ ⋆ ) = 0 \\nabla F_{\\tau}(z_{\\tau}^{\\star})=0 along this path gives"},{"type":"para","text":"Hence"},{"type":"para","text":"Because the τ \\tau -dependence enters through α ( τ ) \\alpha(\\tau) , ∂ τ ∇ F τ ( z τ ⋆ ) = α ˙ ( τ ) ∂ α ∇ F τ ( z τ ⋆ ) \\partial_{\\tau}\\nabla F_{\\tau}(z_{\\tau}^{\\star})=\\dot{\\alpha}(\\tau)\\,\\partial_{\\alpha}\\nabla F_{\\tau}(z_{\\tau}^{\\star}) . Therefore"},{"type":"para","text":"The map"},{"type":"para","text":"is continuous on [ 0 , 𝒯 ] [0,\\mathcal{T}] (smoothness of the data along the minimizer path), hence bounded on this compact interval. Taking the ∥ ⋅ ∥ 𝒦 − 1 \\|\\cdot\\|_{\\mathcal{K}^{-1}} –operator norm of the displayed expression and absorbing | α ˙ ( τ ) | \\lvert\\dot{\\alpha}(\\tau)\\rvert yields a finite uniform constant B ⋆ B_{\\star} such that Eq. ( 3 ) holds for all τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] . ■ \\blacksquare"},{"type":"para","text":"Theorem A.2 (Prior-internalization error in the 𝒦 − 1 \\mathcal{K}^{-1} -norm). Define the internalization error e τ = z τ − z τ ⋆ e_{\\tau}=z_{\\tau}-z_{\\tau}^{\\star} and r τ = ‖ e τ ‖ 𝒦 − 1 r_{\\tau}=\\|e_{\\tau}\\|_{\\mathcal{K}^{-1}} . Assume additionally that z τ , z τ ⋆ ∈ 𝒟 z_{\\tau},z_{\\tau}^{\\star}\\in\\mathcal{D} for all τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] . Under Assumptions A1–A3 together with this invariant-neighborhood condition, for all τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] ,"},{"type":"para","text":"Proof We follow three steps: (i) define the internalization error and a Lyapunov function; (ii) derive a scalar differential inequality for the radius r τ r_{\\tau} ; (iii) apply Grönwall to close the bound."},{"type":"heading","level":4,"text":"Step 1: internalization error and Lyapunov function."},{"type":"para","text":"Because z ˙ τ = − 𝒦 ∇ F τ ( z τ ) \\dot{z}_{\\tau}=-\\mathcal{K}\\nabla F_{\\tau}(z_{\\tau}) and ∇ F τ ( z τ ⋆ ) = 0 \\nabla F_{\\tau}(z_{\\tau}^{\\star})=0 , we have"},{"type":"para","text":"Define the Lyapunov function V τ = 1 2 ‖ e τ ‖ 𝒦 − 1 2 = 1 2 e τ ⊤ 𝒦 − 1 e τ V_{\\tau}=\\frac{1}{2}\\|e_{\\tau}\\|_{\\mathcal{K}^{-1}}^{2}=\\frac{1}{2}e_{\\tau}^{\\top}\\mathcal{K}^{-1}e_{\\tau} . Since 𝒦 \\mathcal{K} is constant and symmetric,"},{"type":"heading","level":4,"text":"Step 2: Lyapunov bound and the radius reduction (Lyapunov-to- r r bridge)"},{"type":"para","text":"Since z τ , z τ ⋆ ∈ 𝒟 z_{\\tau},z_{\\tau}^{\\star}\\in\\mathcal{D} for all τ ∈ [ 0 , 𝒯 ] \\tau\\in[0,\\mathcal{T}] , Assumption A1 with u = z τ u=z_{\\tau} and v = z τ ⋆ v=z_{\\tau}^{\\star} gives"},{"type":"para","text":"Moreover, since 𝒦 ⪰ λ min ( 𝒦 ) I \\mathcal{K}\\succeq\\lambda_{\\min}(\\mathcal{K})I on the active subspace,"},{"type":"para","text":"Using Cauchy–Schwarz in the 𝒦 − 1 \\mathcal{K}^{-1} inner product together with Lemma A.1,"},{"type":"para","text":"Combining the above inequalities yields the Lyapunov inequality"},{"type":"para","text":"We now isolate the key reduction from V ˙ τ \\dot{V}_{\\tau} to a scalar inequality for r τ r_{\\tau} . Because V τ = 1 2 r τ 2 V_{\\tau}=\\frac{1}{2}r_{\\tau}^{2} and r τ ≥ 0 r_{\\tau}\\geq 0 , the chain rule gives V ˙ τ = r τ r ˙ τ \\dot{V}_{\\tau}=r_{\\tau}\\dot{r}_{\\tau} at points where r τ > 0 r_{\\tau}>0 ; substituting into ( 5 ) and dividing by r τ r_{\\tau} yields"},{"type":"para","text":"(At instants with r τ = 0 r_{\\tau}=0 , the same inequality follows by continuity of the underlying quantities along the trajectory.) We refer to the passage from ( 5 ) to ( 6 ) as the radius reduction step ."},{"type":"heading","level":4,"text":"Step 3: Grönwall closure"},{"type":"para","text":"Applying Grönwall’s inequality to Eq. ( 6 ) gives"},{"type":"para","text":"Finally,"},{"type":"para","text":"which yields Eq. ( 4 ). ■ \\blacksquare"},{"type":"para","text":"Corollary (Prior internalization: structural identity vs. realized endpoint deviation) Structural assumption (ideal moving optimum) Assume additionally that the effective-risk minimizer z ~ ⋆ ( x ) \\tilde{z}^{\\star}(x) depends only on the conditional distribution ℙ ( Y ∣ x ) \\mathbb{P}(Y\\mid x) and is therefore independent of α ( τ ) \\alpha(\\tau) . Then the quasi-static intrinsic minimizer admits the affine representation z τ ⋆ = z ~ ⋆ − α ( τ ) b stat z_{\\tau}^{\\star}=\\tilde{z}^{\\star}-\\alpha(\\tau)b_{\\text{stat}} , and, together with the schedule endpoints α ( 0 ) = 1 \\alpha(0)=1 and α ( 𝒯 ) = 0 \\alpha(\\mathcal{T})=0 , the ideal endpoint shift identity z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\mathcal{T}}^{\\star}-z_{0}^{\\star}=b_{\\text{stat}} holds. Without this structural assumption, one should not treat z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\mathcal{T}}^{\\star}-z_{0}^{\\star}=b_{\\text{stat}} as automatic."},{"type":"para","text":"Realized logits vs. internalization error. Under the ideal endpoint identity z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\mathcal{T}}^{\\star}-z_{0}^{\\star}=b_{\\text{stat}} , the exact algebraic decomposition z 𝒯 − z 0 − b stat = e 𝒯 − e 0 z_{\\mathcal{T}}-z_{0}-b_{\\text{stat}}=e_{\\mathcal{T}}-e_{0} rewrites the realized endpoint deviation from b stat b_{\\text{stat}} into a difference of internalization errors at τ = 𝒯 \\tau=\\mathcal{T} and τ = 0 \\tau=0 ."},{"type":"para","text":"Using Theorem A.2 to control ‖ e τ ‖ 𝒦 − 1 \\|e_{\\tau}\\|_{\\mathcal{K}^{-1}} and the triangle inequality ‖ e 𝒯 − e 0 ‖ 𝒦 − 1 ≤ ‖ e 𝒯 ‖ 𝒦 − 1 + ‖ e 0 ‖ 𝒦 − 1 \\|e_{\\mathcal{T}}-e_{0}\\|_{\\mathcal{K}^{-1}}\\leq\\|e_{\\mathcal{T}}\\|_{\\mathcal{K}^{-1}}+\\|e_{0}\\|_{\\mathcal{K}^{-1}} , with a = μ λ min ( 𝒦 ) a=\\mu\\lambda_{\\min}(\\mathcal{K}) ,"},{"type":"para","text":"Thus the endpoint identity is exact up to a transient initialization term and an annealing-induced internalization term. In particular, if e 0 = 0 e_{0}=0 or ‖ e 0 ‖ 𝒦 − 1 = 𝒪 ( sup s | α ˙ ( s ) | ) \\|e_{0}\\|_{\\mathcal{K}^{-1}}=\\mathcal{O}(\\sup_{s}|\\dot{\\alpha}(s)|) , then"},{"type":"para","text":"Fix a prediction step and a code c c . Let 𝒟 S \\mathcal{D}_{S} and 𝒟 T \\mathcal{D}_{T} denote the source and target domains. Under the label-wise prior-shift assumption, ℙ S ( x ∣ y c ) = ℙ T ( x ∣ y c ) \\mathbb{P}_{S}(x\\mid y_{c})=\\mathbb{P}_{T}(x\\mid y_{c}) , while only the class prior changes from π S ( c ) \\pi_{S}(c) to π T ( c ) \\pi_{T}(c) ."},{"type":"para","text":"The Bayes-optimal per-code logit in domain 𝒟 ∈ { 𝒟 S , 𝒟 T } \\mathcal{D}\\in\\{\\mathcal{D}_{S},\\mathcal{D}_{T}\\} is"},{"type":"para","text":"Applying Bayes’ rule gives"},{"type":"para","text":"The first term is invariant across domains by assumption, so subtracting the source and target expressions yields"},{"type":"para","text":"Hence the exact label-wise Bayes correction is obtained by adding the delta bias Δ b ( c ) = logit ( π T ( c ) ) − logit ( π S ( c ) ) \\Delta b(c)=\\mathrm{logit}\\!\\big(\\pi_{T}(c)\\big)-\\mathrm{logit}\\!\\big(\\pi_{S}(c)\\big) . This proves the claim. The main-text interpolation parameter λ \\lambda is exact when λ = 1 \\lambda=1 ; values λ ∈ [ 0 , 1 ) \\lambda\\in[0,1) correspond to a practical interpolation heuristic rather than to the exact Bayes identity. ■ \\blacksquare ∎"},{"type":"para","text":"Remark. This observation is deliberately label-wise and step-wise. It justifies marginal prior correction for each code at a fixed prediction step, but it does not by itself imply exact matching of the full autoregressive joint distribution after rollout."},{"type":"para","text":"In this section, we provide supporting proofs for the claims used in the main text."},{"type":"heading","level":2,"text":"E.1 Proof of the Static Centering Property"},{"type":"para","text":"We prove a local centering statement rather than a global optimality claim."},{"type":"para","text":"Bayesian decomposition For a fixed medical code c c , the Bayes-optimal effective logit under BCE is the posterior log-odds z ~ c ⋆ ( x ) = log ℙ ( y c = 1 ∣ x ) ℙ ( y c = 0 ∣ x ) \\tilde{z}_{c}^{\\star}(x)=\\log\\frac{\\mathbb{P}(y_{c}=1\\mid x)}{\\mathbb{P}(y_{c}=0\\mid x)} . Applying Bayes’ rule gives the standard decomposition"},{"type":"para","text":"Here 𝒮 c ( x ) \\mathcal{S}_{c}(x) is the structural term, while logit ( π ( c ) ) \\mathrm{logit}(\\pi(c)) is the prevalence term."},{"type":"para","text":"Prior-centered initialization Let π ^ ϵ ( c ) = π ^ ( c ) + ϵ 1 + 2 ϵ \\hat{\\pi}_{\\epsilon}(c)=\\frac{\\hat{\\pi}(c)+\\epsilon}{1+2\\epsilon} and b stat ( c ) = − logit ( π ^ ϵ ( c ) ) b_{\\text{stat}}(c)=-\\mathrm{logit}\\!\\big(\\hat{\\pi}_{\\epsilon}(c)\\big) . Assume that, near initialization, the intrinsic logits satisfy z 0 , c ( x ) = logit ( π ^ ϵ ( c ) ) + r c ( x ) z_{0,c}(x)=\\mathrm{logit}\\!\\big(\\hat{\\pi}_{\\epsilon}(c)\\big)+r_{c}(x) , with | r c ( x ) | ≪ 1 |r_{c}(x)|\\ll 1 . Then for any static additive shift b ( c ) b(c) the initial effective logit is z ~ 0 , c ( x ) = z 0 , c ( x ) + b ( c ) = logit ( π ^ ϵ ( c ) ) + r c ( x ) + b ( c ) \\tilde{z}_{0,c}(x)=z_{0,c}(x)+b(c)=\\mathrm{logit}\\!\\big(\\hat{\\pi}_{\\epsilon}(c)\\big)+r_{c}(x)+b(c) . The unique shift that centers the nominal operating point (the case r c ( x ) = 0 r_{c}(x)=0 ) at zero is b ( c ) = − logit ( π ^ ϵ ( c ) ) = b stat ( c ) b(c)=-\\mathrm{logit}\\!\\big(\\hat{\\pi}_{\\epsilon}(c)\\big)=b_{\\text{stat}}(c) , for which z ~ 0 , c ( x ) = r c ( x ) ≈ 0 \\tilde{z}_{0,c}(x)=r_{c}(x)\\approx 0 ."},{"type":"para","text":"Local sensitivity maximization For BCE with sigmoid link, the local logit-to-probability sensitivity and the pointwise curvature are both σ ′ ( u ) = σ ( u ) ( 1 − σ ( u ) ) \\sigma^{\\prime}(u)=\\sigma(u)(1-\\sigma(u)) . This quantity is uniquely maximized at u = 0 u=0 . Therefore, among additive shifts depending only on π ^ ϵ ( c ) \\hat{\\pi}_{\\epsilon}(c) , the choice b stat ( c ) b_{\\text{stat}}(c) is the unique shift that centers the local operating point at the maximal-sensitivity region of the sigmoid. This establishes the centering claim used in the main text."},{"type":"para","text":"Finally, because the scaffold is an x x -independent additive term, it translates the logit coordinate without changing the structural component 𝒮 c ( x ) \\mathcal{S}_{c}(x) in Eq. ( 7 ). Thus it acts as a local optimization scaffold rather than as a modification of the underlying structure. ■ \\blacksquare"},{"type":"heading","level":2,"text":"E.2 Lemma: NTK Gradient Flow under BCE Loss"},{"type":"para","text":"Before analyzing the curriculum dynamics, we formalize the continuous-time gradient flow of neural network outputs under Binary Cross-Entropy (BCE) loss."},{"type":"para","text":"Let 𝒟 = { ( x i , y i ) } i = 1 n \\mathcal{D}=\\{(x_{i},y_{i})\\}_{i=1}^{n} be the training dataset. Let 𝐟 ( t ) = [ f θ ( t ) ( x 1 ) , … , f θ ( t ) ( x n ) ] ⊤ ∈ ℝ n \\mathbf{f}(t)=[f_{\\theta(t)}(x_{1}),\\dots,f_{\\theta(t)}(x_{n})]^{\\top}\\in\\mathbb{R}^{n} denote the intrinsic logits of the neural network at time t t , and 𝐲 = [ y 1 , … , y n ] ⊤ ∈ { 0 , 1 } n \\mathbf{y}=[y_{1},\\dots,y_{n}]^{\\top}\\in\\{0,1\\}^{n} be the corresponding labels. Under continuous-time gradient descent on the BCE loss, the evolution of the network outputs follows the non-linear Ordinary Differential Equation (ODE)"},{"type":"para","text":"where H ( t ) ∈ ℝ n × n H(t)\\in\\mathbb{R}^{n\\times n} is the empirical NTK matrix with entries"},{"type":"para","text":"The parameter trajectory under gradient flow satisfies"},{"type":"para","text":"Applying the chain rule gives"},{"type":"para","text":"For any x k x_{k} ,"},{"type":"para","text":"Stacking these equations over k = 1 , … , n k=1,\\dots,n yields the claimed matrix form. ■ \\blacksquare ∎"},{"type":"para","text":"Remark. Under AdaPCLA , the same identity applies to the effective logits f ~ θ ( t ) ( x ) = f θ ( t ) ( x ) + α ( t ) b stat \\tilde{f}_{\\theta(t)}(x)=f_{\\theta(t)}(x)+\\alpha(t)b_{\\text{stat}} , because the scaffold is independent of θ \\theta and therefore ∇ θ f ~ θ ( t ) ( x ) = ∇ θ f θ ( t ) ( x ) \\nabla_{\\theta}\\tilde{f}_{\\theta(t)}(x)=\\nabla_{\\theta}f_{\\theta(t)}(x) ."},{"type":"para","text":"We now write the discrete first-order update under the actual AdaPCLA training objective. Fix one SGD step and condition on the current annealing value α t \\alpha_{t} as constant during that step. Define the effective logits z ~ t ( x ) = z t ( x ) + α t b stat \\tilde{z}^{t}(x)=z^{t}(x)+\\alpha_{t}b_{\\text{stat}} , where z t ( x ) = f θ t ( x ) ∈ ℝ V z^{t}(x)=f_{\\theta^{t}}(x)\\in\\mathbb{R}^{V} denotes the intrinsic logits."},{"type":"para","text":"Let x o x_{o} be an observing example and ( x u , y u ) (x_{u},y_{u}) an updating example. We study the change in the log-probability vector log σ ( z ~ t ( x o ) ) \\log\\sigma(\\tilde{z}^{t}(x_{o})) . A first-order Taylor expansion gives"},{"type":"para","text":"Therefore,"},{"type":"para","text":"Under SGD on the AdaPCLA BCE loss,"},{"type":"para","text":"Because α t b stat \\alpha_{t}b_{\\text{stat}} is independent of θ \\theta within this step, ∇ θ z ~ t ( x ) = ∇ θ z t ( x ) \\nabla_{\\theta}\\tilde{z}^{t}(x)=\\nabla_{\\theta}z^{t}(x) . Applying the chain rule on both factors in Eq. ( 8 ) yields the following compact decomposition. For readability, write"},{"type":"para","text":"Then"},{"type":"para","text":"where"},{"type":"para","text":"and"},{"type":"para","text":"Substituting Eq. ( 9 ) into Eq. ( 8 ) gives"},{"type":"para","text":"The matrix 𝒜 Ada t ( x o ) \\mathcal{A}_{\\mathrm{Ada}}^{t}(x_{o}) is diagonal because sigmoid factorizes across labels:"},{"type":"para","text":"Therefore, as in independent BCE, the output nonlinearity does not introduce explicit cross-label terms. Any cross-label effect is determined by the shared-backbone kernel 𝒦 t ( x o , x u ) \\mathcal{K}^{t}(x_{o},x_{u}) ."},{"type":"para","text":"Finally, because z ~ t ( x ) = z t ( x ) + α t b stat \\tilde{z}^{t}(x)=z^{t}(x)+\\alpha_{t}b_{\\text{stat}} differs from z t ( x ) z^{t}(x) by a θ \\theta -independent constant, the intrinsic-logit increment obeys"},{"type":"para","text":"which is the form used in the main text. The deterministic effect induced by changes in α t \\alpha_{t} across optimization steps is analyzed separately in Appendix C . ■ \\blacksquare"},{"type":"para","text":"The prior-internalization bound in Appendix C and Sec. 4.3 identifies the minimum eigenvalue λ min ( 𝒦 ) \\lambda_{\\min}(\\mathcal{K}) of the label-wise empirical NTK as a factor that reduces the internalization-error bound. As an exploratory analysis, we compute λ min ( 𝒦 x ) \\lambda_{\\min}(\\mathcal{K}_{x}) at the initial checkpoint for each probing context and compare it with the empirical related-rare AUC and final gain. The label-wise empirical NTK is restricted to the related-rare codes of the same context."},{"type":"para","text":"30-context results: On the 30-context micro-probe set, the Spearman correlations are positive but weak: Spearman( λ min ( 𝒦 init ) \\lambda_{\\min}(\\mathcal{K}_{\\mathrm{init}}) , related-rare AUC) ≈ 0.30 \\approx 0.30 ( p ≈ 0.11 p\\approx 0.11 ) and Spearman( λ min ( 𝒦 init ) \\lambda_{\\min}(\\mathcal{K}_{\\mathrm{init}}) , related-rare gain) ≈ 0.16 \\approx 0.16 ( p ≈ 0.40 p\\approx 0.40 ). The signs are consistent with the bound, but the correlations do not reach conventional significance."},{"type":"para","text":"100-context robustness check: We extend the analysis to 100 probing contexts. The correlations become close to zero: Spearman( λ min ( 𝒦 init ) \\lambda_{\\min}(\\mathcal{K}_{\\mathrm{init}}) , AUC) ≈ 0.02 \\approx 0.02 ( p ≈ 0.86 p\\approx 0.86 ) and Spearman( λ min ( 𝒦 init ) \\lambda_{\\min}(\\mathcal{K}_{\\mathrm{init}}) , gain) ≈ − 0.06 \\approx-0.06 ( p ≈ 0.53 p\\approx 0.53 ). The final-gain correlation is negative under this larger probing set."},{"type":"para","text":"Conclusion: We do not treat the λ min ( 𝒦 ) \\lambda_{\\min}(\\mathcal{K}) and related-rare AUC/gain analysis as empirical validation of the bound. The 30-context result is statistically modest, and the 100-context extension shows that the correlation is not robust. We report this analysis for completeness as non-conclusive evidence."},{"type":"para","text":"This section provides detailed prediction-reliability statistics for the downstream 25-label DiagnosisModel evaluated on the real MIMIC-IV test split. We report ECE, Brier Score, and Negative Log-Likelihood (NLL) for AdaPCLA , HALO, and GPT-style under two scopes: (i) overall (all labels), and (ii) tail-only (bottom-1/3 labels by training-set frequency)."},{"type":"para","text":"Figure 4 shows the corresponding reliability diagrams for both the overall and tail-only scopes. Across confidence bins, AdaPCLA has the smallest deviation from the identity line, especially in the high-confidence region. GPT-style and HALO show larger confidence errors, particularly on rare labels."},{"type":"para","text":"This section reports quantitative summaries for the micro-probing experiments with 30 and 100 patient-history contexts. For each context and probe type (related_rare / unrelated_rare / wrong), we compute the AUC of Δ log p ( τ ) \\Delta\\log p(\\tau) over annealing checkpoints and the final Δ log p \\Delta\\log p at the end of annealing. We then report the mean and standard deviation across contexts."},{"type":"para","text":"Figure 2 in the main text tracks selected probe-code probabilities under 30 and 100 probing contexts. Table 8 reports the corresponding AUC of Δ log p ( τ ) \\Delta\\log p(\\tau) and final Δ log p \\Delta\\log p for each probe type."},{"type":"para","text":"All experiments reported in this paper were run in the sft_lab virtual environment on a server with 8 × \\times NVIDIA RTX 4090 GPUs. Table 9 lists the main software and dependency versions used for training and evaluation. We recommend using compatible versions for reproduction."},{"type":"para","text":"This section reports frequency–rank plots for MIMIC-III and MIMIC-IV, comparing Real training data with synthetic data generated by HALO, LSTM, GPT-style, and AdaPCLA . These plots summarize the ranked marginal occurrence-frequency distribution of medical codes. They do not evaluate code-set support, tail clinical plausibility, rare-code context consistency, or downstream predictive utility. Therefore, we use them only as supplementary distributional-fidelity views. Model comparison should rely on Table 1 for PairSeen, TailPairSeen, TailCtxJSD, and TailTopKJac, Table 2 for downstream utility, and Sec. B.2 for qualitative clinical examples."},{"type":"para","text":"This section reports per-code marginal occurrence-frequency views that complement the frequency–rank plots in Sec. L . Each point corresponds to one medical code present in the Real training set."},{"type":"para","text":"Frequency–frequency scatter: For each code c c , the horizontal axis is its visit-level frequency in Real and the vertical axis is its visit-level frequency in a given model (one subplot per model: HALO, LSTM, GPT-style, AdaPCLA ). Both axes use a symmetric log scale. Points near the diagonal y = x y=x indicate that the model’s marginal frequency for that code is close to Real; points above or below the diagonal indicate higher or lower generated frequency than Real. These plots evaluate code-level marginal occurrence frequencies, but they do not measure tail clinical plausibility or rare-event co-occurrence structure."},{"type":"para","text":"Rank–rank scatter: For each code c c , the horizontal axis is its rank in Real (1 = most frequent) and the vertical axis is its rank in the model’s synthetic data; both axes are log-scaled. Points near the diagonal indicate agreement between the Real code rank and the synthetic code rank. Codes never generated by a model are assigned a rank beyond the maximum. Together, the frequency–frequency and rank–rank figures report per-code marginal alignment with the training distribution."},{"type":"para","text":"This section reports code–code co-occurrence matrices for the top-200 most frequent medical codes on MIMIC-IV. We compute visit-level co-occurrence counts for Real, HALO, LSTM, and AdaPCLA , and then apply a PPMI (Positive Pointwise Mutual Information) transform. To keep the panels comparable, we perform hierarchical clustering on the Real PPMI matrix using average linkage and apply the resulting code order to all models. The visualization compares how well each generator preserves frequent-code co-occurrence patterns learned from the Real training data."},{"type":"para","text":"This section reports the full P ( code ∣ label ) P(\\text{code}\\mid\\text{label}) heatmaps referenced in Sec. 6 . We partition medical codes into three frequency buckets: head (most frequent), mid, and tail (long tail). For each bucket, we compute the conditional distribution of codes given the bucket label. The heatmaps use a 3 × 4 3\\times 4 layout: each row corresponds to one bucket, and each column corresponds to one data source (Real training data, HALO, LSTM, or AdaPCLA ) on MIMIC-IV. These plots provide supplementary distributional-fidelity views for code-level probability structure within each frequency regime. AdaPCLA shows closer alignment with Real across the three buckets than HALO and LSTM, which is consistent with the tail plausibility and co-occurrence results in Table 1 ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Longitudinal Electronic Health Records (EHR) provide a rich view of patient trajectories, capturing diagnoses, treatments, medications, and clinical events over time. They are central to a wide range of clinical applications, including risk prediction [ 31 , 28 , 1 ] , medication recommendation [ 27 , 23 ] , and decision support systems [ 29 , 16 ] . However, data sensitivity and privacy regulations hinder the sharing of real-world EHR data, limiting research and development in related intelligent technologies. High-quality synthetic EHR generation is therefore crucial for privacy-preserving research [ 39 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Longitudinal EHRs are challenging to model due to their high dimensionality, temporal sparsity, and heterogeneity. Moreover, a few common clinical patterns coexist alongside numerous rare but clinically significant conditions, and they usually exhibit a long-tailed pattern in the empirical marginal distribution of medical codes for symptoms, indicating that few codes occur frequently while most clinically relevant codes are rare. During the training process of generative models, such rare codes provide limited positive evidence and weak stable learning signals for their temporal, contextual, and co-occurrence dependencies. If a model captures only the dominant head patterns while underrepresenting rare pathological dependencies, the resulting synthetic data may appear realistic on average but become unreliable for the very subpopulations that matter most clinically. Therefore, the key challenge for synthetic EHR generation is twofold: generated records should match the overall real EHR distribution, while also preserving the temporal and co-occurrence dependencies involving rare clinical events to facilitate downstream clinical modeling and cross-population adaptation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Existing generative models attempt to solve this problem from different aspects, but none fully resolves the long-tail difficulty. GAN and VAE-based methods [ 9 , 38 ] improved privacy-preserving generation but often struggled to preserve longitudinal dependency structure. Autoregressive and hierarchical models [ 34 ] capture sequential patterns better, while diffusion-style approaches [ 44 , 13 ] improve sample quality and distributional coverage. Nevertheless, across these families, a common difficulty remains: optimization is dominated by frequent and easy-to-learn clinical patterns, whereas rare but clinically important dependencies receive weak and unstable learning signals [ 33 , 22 ] . As a result, synthetic EHR generators may fit common clinical patterns yet fail to preserve rare-event plausibility, downstream utility, and controllable cross-population adaptation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To this end, we propose AdaPCLA , a curriculum prior-calibrated framework for long-tailed EHR generation. AdaPCLA applies prior-aware logit adjustment early in training to maintain effective rare-code updates, then gradually anneals this correction so that rare-code temporal and co-occurrence dependencies are internalized into the model parameters. This design supports clean inference without an external bias after training, and it also enables zero-shot distribution control by applying target-prior corrections without retraining. Our theoretical analysis gives two messages: the label-wise empirical NTK characterizes how prior-induced residuals update rare-code intrinsic logits, while the prior-internalization bound identifies annealing speed and NTK conditioning as theoretical factors for retaining prior signals after scaffold removal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct a comprehensive evaluation over large real-world datasets. Experimental results show that rather than only improving aggregate scores, AdaPCLA achieves the largest gains on tail structure: compared with HALO, it improves TailPairSeen by 114.2% on MIMIC-III and 65.1% on MIMIC-IV. For the zero-shot generation setting based on MIMIC-IV training to MIMIC-III inference, AdaPCLA reaches AUPRC of 0.922 0.922 without retraining, outperforming GPT-style generation by 3.5% F1, which verifies the effectiveness of the learned structure for cross-population adaptation. In summary, this paper makes the following contributions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose AdaPCLA , a prior-guided curriculum framework that strengthens rare-code learning while supporting bias-free inference and zero-shot distribution control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We analyze AdaPCLA with label-wise empirical NTK dynamics and govern the prior-internalization bound by annealing speed and the empirical NTK.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA achieves significant gains in tail plausibility, downstream utility, and zero-shot cross-population adaptation tasks based on the experiments on real-world datasets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Longitudinal EHR generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Synthetic EHR generation has progressed from GAN-, VAE-, and correlation-aware models for discrete or tabular medical records [ 9 , 38 , 35 ] to temporal structured simulators and longitudinal mixed-type generators [ 43 , 22 , 21 ] . Recent work further improves high-dimensional visit modeling, privacy-preserving synthesis, multimodal diffusion, and LLM-based generation without direct patient-level access [ 34 , 39 , 44 , 13 ] . These models improve temporal dependency modeling and sample realism. However, because these generators are usually trained to fit the overall distribution of the majority of diseases, rare disease codes contribute sparse learning signals, motivating a generation objective that gives tail codes stronger optimization support without weakening longitudinal dependency modeling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Long-tailed learning and zero-shot control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Long-tailed learning methods such as resampling, focal loss, class-balanced loss, margin-based objectives, decoupled training, and logit adjustment reshape training under imbalanced labels [ 5 , 25 , 10 , 4 , 20 , 30 , 33 ] . These methods are effective for discriminative classification, but direct use in longitudinal multi-label EHR generation is less straightforward: resampling can disrupt patient trajectories, fixed reweighting can destabilize sparse multi-label optimization, and static prior biases can leave the generator dependent on external correction at inference time. Meanwhile, clinical code frequencies vary across hospitals and cohorts [ 32 ] , motivating generation methods that can adapt to target priors without retraining while preserving longitudinal dependencies and co-occurrence structure. Although label-shift correction and controllable tabular/EHR synthesis study related forms of occurrence frequency adaptation [ 26 , 12 , 38 , 17 ] , they do not directly address zero-shot control for long-tailed longitudinal EHR generators.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A detailed discussion about closely related work is deferred to Appendix A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Problem Formulation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Preliminaries\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Notions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a structured EHR as a sequence of patient visits. Let 𝒱 \\\\mathcal{V} be a | 𝒱 | |\\\\mathcal{V}| -size vocabulary of clinical events, i.e., medical codes. A patient’s T T -visit record forms a chronological trajectory X = ( 𝐱 1 , … , 𝐱 T ) X=(\\\\mathbf{x}_{1},\\\\ldots,\\\\mathbf{x}_{T}) , where each visit 𝐱 t ∈ { 0 , 1 } | 𝒱 | \\\\mathbf{x}_{t}\\\\in\\\\{0,1\\\\}^{|\\\\mathcal{V}|} is a multi-hot vector indicating the presence of codes at step t t ; 𝐱 < t \\\\mathbf{x}_{<t} denotes the trajectory history of this patient before time step t t . We assume that observed trajectories are independent and identically distributed samples drawn from an unknown underlying distribution 𝒫 ( X ) \\\\mathcal{P}(X) , which we aim to approximate.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Logit Adjustment (LA).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Logit adjustment can be used to guide the model’s learning under class imbalance by modifying the relative margins between a target class and its competing classes [ 30 , 40 , 41 ] . For a given sample ( x , y ) (x,y) with y ∈ { 1 , … , K } y\\\\in\\\\{1,\\\\ldots,K\\\\} , consider a multiclass classifier with logits C k ( x ) C_{k}(x) for k = 1 , … , K k=1,\\\\ldots,K . The logit-adjusted softmax loss can be written as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where C y ′ ( x ) C_{y^{\\\\prime}}(x) is the logit of a competing class y ′ y^{\\\\prime} , and δ ( y , y ′ ) \\\\delta(y,y^{\\\\prime}) is a prior-dependent adjustment weight. Larger δ ( y , y ′ ) \\\\delta(y,y^{\\\\prime}) increases the penalty when the prediction value for the competing class y ′ y^{\\\\prime} approaches or exceeds the target class y y , thereby changing the optimization pressure across imbalanced classes. In prior logit-adjustment methods for imbalanced classification, δ ( y , y ′ ) \\\\delta(y,y^{\\\\prime}) is typically chosen as a function of empirical class priors, yielding a prior-calibrated decision boundary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the longitudinal EHR generation, the prediction target at each visit corresponds to a multi-hot vector for all medical codes. Let y ∈ { 0 , 1 } V y\\\\in\\\\{0,1\\\\}^{V} denote the multi-label target over 𝒱 \\\\mathcal{V} and y c = 1 y_{c}=1 when event c c occurs in the target visit; let π c = ℙ ( y c = 1 ) \\\\pi_{c}=\\\\mathbb{P}(y_{c}=1) be the empirical frequency of c c at the visit level. Given the intrinsic model logit z c ( x ) z_{c}(x) , we denote the effective logit after adjustment by z ~ c ( x ) \\\\tilde{z}_{c}(x) . A natural multi-label analogue applies a prior-dependent shift element-wise, z ~ c ( x ) = z c ( x ) + b c \\\\tilde{z}_{c}(x)=z_{c}(x)+b_{c} , where b c b_{c} is a bias term determined by the empirical frequency of event c c .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, a fixed prior-dependent bias can make the generator depend on this external correction and prevent it from capturing the rare-event temporal and co-occurrence structure and internalizing them in its own parameters. AdaPCLA therefore treats the prior bias as a temporary optimization scaffold: it preserves rare-event learning signals early in training and is gradually annealed away, so that inference relies on model-encoded rare-event dependencies rather than an external correction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Problem Formulation and Requirements\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In general, longitudinal EHR generation is a sequential generative modeling task. Its goal is to learn a parametric generative model M ( ⋅ ; θ ) M(\\\\cdot;\\\\theta) that approximates 𝒫 ( X ) \\\\mathcal{P}(X) . For a training set of N N patient trajectories 𝒟 = { X i } i = 1 N \\\\mathcal{D}=\\\\{X^{i}\\\\}_{i=1}^{N} , the maximum-likelihood objective can be written as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, real-world EHRs data are high-dimensional and sparse, and most medical codes in 𝒱 \\\\mathcal{V} form an extreme long-tailed distribution. This poses a substantial challenge for learning a model M ( ⋅ ; θ ) M(\\\\cdot;\\\\theta) in Eq. ( 1 ) that faithfully characterizes the real data distribution 𝒫 ( X ) \\\\mathcal{P}(X) . In particular, under the binary cross-entropy loss, the gradient contributions associated with tail codes are severely overwhelmed by the large number of head samples. As a result, optimization becomes biased toward a head-dominated solution: for tail codes, gradients from the many non-occurrence labels ( y c = 0 y_{c}=0 ) can dominate the sparse gradients from occurrence labels ( y c = 1 y_{c}=1 ), pushing their logits downward and suppressing their generation probabilities. Such behavior leads the model to disregard the complex conditional dependencies of rare pathologies, ultimately fails to capture the intrinsic structure of the clinical data and to generate realistic EHR records faithfully.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequently, an optimal model M ( ⋅ ; θ ∗ ) M(\\\\cdot;\\\\theta^{*}) should accurately approximate 𝒫 ( X ) \\\\mathcal{P}(X) over the support of the data distribution, encompassing a comprehensive spectrum of medical codes and faithfully generating high-fidelity long-tailed patterns at inference time. In practice, particularly in low-resource and rare-disease settings, inadequate long-tail modeling can cause synthetic generators to miss clinically important rare-event patterns, reducing the usefulness of synthetic EHRs for downstream prediction and adapting generation to populations with different code frequencies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, owing to the privacy constraints and the inherent difficulty of accessing and sharing large-scale real EHR datasets, the learned model is expected to be readily transferable to a new, typically small-scale dataset 𝒟 ^ \\\\hat{\\\\mathcal{D}} , which may be drawn from a distinct distribution 𝒫 ′ ( X ) \\\\mathcal{P}^{\\\\prime}(X) , thereby enabling the efficient construction of generative models that are well adapted to the target data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Proposed Model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we propose AdaPCLA , an Adaptive Prior-Calibrated Logit Adjustment method designed to capture the full-spectrum medical code distribution, generate realistic EHR data, and can adapt to new datasets in a zero-shot manner. It treats empirical label priors as a temporary optimization scaffold and encourages the model to internalize this structural information by gradually removing the scaffold through annealing. Figure 1 illustrates the overall framework of AdaPCLA .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA consists of three core modules: (i) an EHR Generator Backbone , an autoregressive Transformer that captures longitudinal clinical dependencies; (ii) Statistical Prior Modeling , which converts long-tailed empirical prevalences into a label-wise logit scaffold 𝐛 stat \\\\mathbf{b}_{\\\\text{stat}} ; and (iii) Curriculum Prior Injection , which anneals this scaffold so that early rare-event updates are protected while the final generator can operate without an external bias.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 EHR Generator Backbone\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA uses an autoregressive Transformer backbone to model longitudinal EHR trajectories. Given historical visits x < t x_{<t} , the backbone embeds the visit events and processes the visit sequence with autoregressive masked self-attention, so that the hidden state used for predicting t t -th step visit depends only on previous records and respects the temporal order:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where z t z_{t} denotes the intrinsic logits over the clinical-event vocabulary for the next visit. These logits are produced by the backbone parameters alone and constitute the logit space on which AdaPCLA performs prior-calibrated training during the annealing process.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Based on the shared representation H t − 1 H_{t-1} , the next visit is modeled with Bernoulli output heads:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equivalently, the visit-level training objective is the Binary Cross-Entropy loss over clinical events:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The factorization is imposed only at the output layer; temporal and cross-event dependencies are captured by the shared Transformer representation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Optimization Scaffold: Statistical Prior\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the long-tailed EHR generation, a main difficulty lies in rare medical codes, which represent clinical events such as diagnoses, procedures, and medications, contribute far fewer positive updates than the common codes. Even when a positive occurrence of a rare code produces a large local BCE residual, its aggregate effect can be overwhelmed by gradients from high-frequency codes and from the many non-occurrence labels, i.e., y c = 0 y_{c}=0 , induced by the large code vocabulary. To make early training easier, we build a label-wise bias from empirical marginal frequencies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Considering numerical stability, we use the smoothed marginal probability π ^ ϵ ( c ) = π ^ ( c ) + ϵ 1 + 2 ϵ \\\\hat{\\\\pi}_{\\\\epsilon}(c)=\\\\frac{\\\\hat{\\\\pi}(c)+\\\\epsilon}{1+2\\\\epsilon} for event c c , then we define the statistical bias used in the scaffold as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the start of training, the model logit for c c is close to its empirical log-odds, z 0 , c ( x ) ≈ logit ( π ^ ϵ ( c ) ) z_{0,c}(x)\\\\approx\\\\mathrm{logit}\\\\!\\\\big(\\\\hat{\\\\pi}_{\\\\epsilon}(c)\\\\big) . After adding b stat ( c ) b_{\\\\text{stat}}(c) , the effective logit becomes z ~ 0 , c ( x ) = z 0 , c ( x ) + b stat ( c ) ≈ 0 \\\\tilde{z}_{0,c}(x)=z_{0,c}(x)+b_{\\\\text{stat}}(c)\\\\approx 0 . Thus, when the annealing coefficient is large early in training, the prior bias shifts the effective logit toward zero.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since σ ′ ( u ) = σ ( u ) ( 1 − σ ( u ) ) \\\\sigma^{\\\\prime}(u)=\\\\sigma(u)(1-\\\\sigma(u)) is maximized at u = 0 u=0 , centering the early effective logit z ~ 0 , c \\\\tilde{z}_{0,c} near zero places rare-code predictions in the high-sensitivity region of the sigmoid. When the logit is close to zero, a small parameter update can change the predicted probability more easily. When the logit has a large magnitude, the sigmoid becomes saturated and its derivative becomes small. Therefore, b stat b_{\\\\text{stat}} is used to make early optimization more sensitive, especially for rare codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use b stat b_{\\\\text{stat}} only during training, rather than in the inference phase. At the beginning of training, many rare codes have low logits and receive logit-increasing gradients only from sparse occurrence labels ( y c = 1 ) (y_{c}=1) . Adding b stat b_{\\\\text{stat}} moves the effective logits closer to zero and amplifies the influence of these sparse rare-event observations relative to the dominant head-code updates. By reducing b stat b_{\\\\text{stat}} to zero gradually, the model learns to capture rare-event dependencies from the patient history rather than relying on a fixed external bias when rare events should appear. At inference time, the model will use its own logits alone for data generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Curriculum Prior Injection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Retaining a fixed source-prior bias during the inference phase causes generation to remain perpetually dependent on an external correction term, which will introduce computational inefficiencies and can miscalibrate outputs when the target population has a different code distribution. Therefore, AdaPCLA utilizes the prior bias only as a training scaffold and anneals to zero gradually, encouraging rare-code dependencies to be encoded in the model parameters for bias-free inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"During training, we use the modified logits z ~ τ ( x ) = z τ ( x ) + α ( τ ) b stat , \\\\tilde{z}_{\\\\tau}(x)=z_{\\\\tau}(x)+\\\\alpha(\\\\tau)b_{\\\\text{stat}}, where z τ ( x ) z_{\\\\tau}(x) is the intrinsic logit produced by the backbone, b stat b_{\\\\text{stat}} is the statistical prior bias, and α ( τ ) ∈ [ 0 , 1 ] \\\\alpha(\\\\tau)\\\\in[0,1] controls the proportion of the prior bias used at step τ \\\\tau . The schedule starts from α ( 0 ) = 1 \\\\alpha(0)=1 and ends at α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The training loss is the BCE loss computed with the effective logits:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the beginning, α ( τ ) \\\\alpha(\\\\tau) is large, so rare-event observations exert a stronger effect on the update direction. During training, α ( τ ) \\\\alpha(\\\\tau) becomes smaller, the model then has to rely more on the patient history and less on the external bias. At the end of training, α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 , so the generator uses only its own logits, z ~ 𝒯 ( x ) = z 𝒯 ( x ) \\\\tilde{z}_{\\\\mathcal{T}}(x)=z_{\\\\mathcal{T}}(x) . Thus, AdaPCLA uses the prior bias as a temporary training aid, but inference is performed without any external bias.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Zero-Shot Cross-Population Adaptation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data from different scenarios, such as different hospitals, populations, or regions, exhibits varying degrees of statistical differences and biases. Therefore, a model trained on one patient population would be used to generate data for another population with different code occurrence frequency. AdaPCLA adaptively supports such cross-population adaptation by introducing a target-prior correction difference at inference time, without retraining.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let π ^ ( c ) \\\\hat{\\\\pi}(c) be source prevalence used during training and π target ( c ) \\\\pi_{\\\\mathrm{target}}(c) be target occurrence frequency for event c c . We define the logit difference between source and target population distribution as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and modify the logits as z t ′ = z t ( θ ) + λ Δ b z^{\\\\prime}_{t}=z_{t}(\\\\theta)+\\\\lambda\\\\Delta b for inference, where λ ∈ [ 0 , 1 ] \\\\lambda\\\\in[0,1] controls the strength of the adjustment. λ = 1 \\\\lambda=1 applies the full prior correction and λ = 0 \\\\lambda=0 recovers the original model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This correction follows the standard label-wise prior-shift rule: if the conditional distribution of patient context given event c c is unchanged across the source and target populations, i.e., ℙ S ( x ∣ y c ) = ℙ T ( x ∣ y c ) \\\\mathbb{P}_{S}(x\\\\mid y_{c})=\\\\mathbb{P}_{T}(x\\\\mid y_{c}) , then the Bayes-optimal logit changes by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"See Appendix D for the detailed proof. This correction is applied independently to each code at each next-visit prediction step. Under the label-wise prior-shift assumption, it steers label-wise occurrence probabilities toward the target population while reusing the longitudinal and co-occurrence structure learned by the backbone, without requiring target-domain retraining.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Theoretical Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we analyze the training dynamics of AdaPCLA and give theoretical insights into why it can retain rare-code learning signals after the prior scaffold is digested.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, the prior scaffold changes the BCE residuals, so the SGD update induced by these residuals is written into the intrinsic logits through the shared Transformer backbone. This happens because the scaffold is fixed and independent of the backbone parameters, i.e., ∇ θ ( α k b stat ) = 0 \\\\nabla_{\\\\theta}(\\\\alpha_{k}b_{\\\\text{stat}})=0 , and therefore ∇ θ z ~ k ( x ) = ∇ θ z k ( x ) \\\\nabla_{\\\\theta}\\\\tilde{z}_{k}(x)=\\\\nabla_{\\\\theta}z_{k}(x) . This yields a label-wise gradient-transfer relationship: the residual of one code can influence the intrinsic logit of another code through the corresponding entry of the label-wise empirical NTK. Second, under standard lazy-training assumptions [ 7 ] , we analyze how the effect of the annealed scaffold α ( τ ) b stat \\\\alpha(\\\\tau)b_{\\\\text{stat}} can be absorbed into the intrinsic logits as α ( τ ) \\\\alpha(\\\\tau) decays to zero. Together, these results justify that the prior bias can be injected to create an early rare-code learning signal, and the annealing schedule controls whether this signal is retained in the model parameters after the external bias is digested.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Gradient Transfer through the Shared Backbone\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We identify how the prior scaffold affects rare-code intrinsic-logit updates through the label-wise empirical NTK. At training step k k , the effective logits are z ~ k ( x ) = z k ( x ) + α k b stat \\\\tilde{z}_{k}(x)=z_{k}(x)+\\\\alpha_{k}b_{\\\\text{stat}} , and the BCE residual is 𝒢 Ada ( k ) ( x ) = σ ( z ~ k ( x ) ) − y \\\\mathcal{G}^{(k)}_{\\\\mathrm{Ada}}(x)=\\\\sigma(\\\\tilde{z}_{k}(x))-y which affects the SGD update direction as the scaffold. The additive term α k b stat \\\\alpha_{k}b_{\\\\text{stat}} is fixed with respect to the backbone parameters, so ∇ θ ( α k b stat ) = 0 \\\\nabla_{\\\\theta}(\\\\alpha_{k}b_{\\\\text{stat}})=0 and ∇ θ z ~ k ( x ) = ∇ θ z k ( x ) \\\\nabla_{\\\\theta}\\\\tilde{z}_{k}(x)=\\\\nabla_{\\\\theta}z_{k}(x) . The adjusted logits z ~ k ( x ) \\\\tilde{z}_{k}(x) determine the BCE residuals, and the fixed-scaffold property ensures that the corresponding parameter update changes the intrinsic logits z k ( x ) z_{k}(x) . For a one-step SGD update with step size η > 0 \\\\eta>0 , the first-order intrinsic-logit update for code c c is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where [ 𝒦 ( k ) ] i , j = ⟨ ∇ θ z k ( i ; x ) , ∇ θ z k ( j ; x ) ⟩ . [\\\\mathcal{K}^{(k)}]_{i,j}=\\\\left\\\\langle\\\\nabla_{\\\\theta}z_{k}(i;x),\\\\nabla_{\\\\theta}z_{k}(j;x)\\\\right\\\\rangle. 𝒦 ( k ) \\\\mathcal{K}^{(k)} is the label-wise empirical NTK for context x x .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, the residual of code j j changes the intrinsic-logit update of code c c through the corresponding entry [ 𝒦 ( k ) ] c , j [\\\\mathcal{K}^{(k)}]_{c,j} of the label-wise empirical NTK. This relationship shows that the prior scaffold affects intrinsic logits by reshaping BCE residuals, whose effects are then transferred through backbone parameter updates. For each code c c , the update is determined by BCE residuals weighted by the corresponding row of 𝒦 ( k ) \\\\mathcal{K}^{(k)} . Thus, AdaPCLA strengthens early rare-code learning signals only when those residuals can be transferred through the empirical-NTK structure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Prior-Internalization Bound\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a fixed context x x , define the expected BCE risk as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let z ~ ⋆ ( x ) \\\\tilde{z}^{\\\\star}(x) be the effective-logit minimizer of the conditional expected BCE risk for the fixed context x x . Because the effective logits are z + α ( τ ) b stat z+\\\\alpha(\\\\tau)b_{\\\\text{stat}} , the corresponding target in intrinsic-logit coordinates is z τ ⋆ ( x ) = z ~ ⋆ ( x ) − α ( τ ) b stat z_{\\\\tau}^{\\\\star}(x)=\\\\tilde{z}^{\\\\star}(x)-\\\\alpha(\\\\tau)b_{\\\\text{stat}} . Since AdaPCLA removes the scaffold at inference time, prior internalization requires the learned intrinsic logits z τ ( x ) z_{\\\\tau}(x) to follow the target as α ( τ ) \\\\alpha(\\\\tau) decays to zero.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the assumptions stated the internalization error e τ ( x ) = z τ ( x ) − z τ ⋆ ( x ) e_{\\\\tau}(x)=z_{\\\\tau}(x)-z_{\\\\tau}^{\\\\star}(x) satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This prior-internalization bound gives two design implications. First, smoother annealing reduces sup s | α ˙ ( s ) | \\\\sup_{s}|\\\\dot{\\\\alpha}(s)| , which lowers the internalization error caused by a moving scaffold. Second, a larger minimum eigenvalue λ min ( 𝒦 ) \\\\lambda_{\\\\min}(\\\\mathcal{K}) of the label-wise empirical NTK reduces the bound, meaning that the intrinsic logits can stay closer to the moving target. Because α ( 0 ) = 1 \\\\alpha(0)=1 and α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 , the moving target in intrinsic-logit coordinates shifts by z 𝒯 ⋆ ( x ) − z 0 ⋆ ( x ) = b stat z_{\\\\mathcal{T}}^{\\\\star}(x)-z_{0}^{\\\\star}(x)=b_{\\\\text{stat}} . When the internalization error e τ ( x ) e_{\\\\tau}(x) remains small along the annealing path, the learned intrinsic logits inherit this shift, so z 𝒯 ( x ) − z 0 ( x ) ≈ b stat z_{\\\\mathcal{T}}(x)-z_{0}(x)\\\\approx b_{\\\\text{stat}} . This supports the intended internalization property of AdaPCLA : the prior signal introduced by the training scaffold is absorbed into the model’s own logits through the annealing trajectory, enabling inference without an external prior correction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1 Experimental Setting\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Datasets and preprocessing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on MIMIC-III [ 19 ] and MIMIC-IV [ 18 ] . Each patient is represented as a sequence of visits, where each visit is a deduplicated set of International Classification of Diseases (ICD) diagnosis codes. We use patient-level train/validation/test splits and apply the same preprocessing and vocabulary mapping to all methods.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare AdaPCLA with representative EHR generators, including LSTM [ 8 ] , EVA [ 22 ] , SynTEG [ 43 ] , GPT-style [ 13 ] , and HALO (hierarchical autoregressive EHR generator) [ 34 ] . For each baseline, we follow the model architecture and training recipe described in the original paper, and use the same training cohort and evaluation protocol as AdaPCLA .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the quanlity of synthetic EHR data. For distributional fidelity, we examine occurrence frequency and visit-level code co-occurrence statistics. For tail clinical plausibility, we report PairSeen/TailPairSeen, TailCtxJSD, and TailTopKJac. For downstream utility, we use the Train-on-Synthetic, Test-on-Real (TSTR) protocol and report Accuracy, F1, and AUPRC.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2 Experimental Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Tail clinical plausibility.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 reports tail clinical plausibility on MIMIC-III/-IV. AdaPCLA gives the strongest overall tail-focused performance across the two datasets. On MIMIC-IV, AdaPCLA ranks first on all four metrics: PairSeen, TailPairSeen, TailCtxJSD, and TailTopKJac. In particular, it improves TailPairSeen from 0.0498 0.0498 with HALO to 0.0822 0.0822 , a relative gain of 65.1 % 65.1\\\\% ; and also reduces TailCtxJSD from 0.6652 0.6652 to 0.6313 0.6313 , showing better tail-context consistency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On MIMIC-III, AdaPCLA achieves the best TailPairSeen and TailCtxJSD. Compared with HALO, it improves TailPairSeen from 0.0520 0.0520 to 0.1114 0.1114 , and reduces TailCtxJSD from 0.6558 0.6558 to 0.5981 0.5981 . Although HALO has a higher overall PairSeen and LSTM has a higher TailTopKJac, AdaPCLA provides a better balance between tail support and tail-context consistency. Overall, these results show that AdaPCLA better preserves rare-event co-occurrence structure than other baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EVA and SynTEG show unstable tail plausibility. On MIMIC-III, their PairSeen and TailPairSeen scores are close to zero, which means that many generated intra-visit pairs are not supported by the training corpus. A likely reason is that VAE- and GAN-based generators often produce dense multi-hot visits after thresholding. When a visit contains too many events, the number of intra-visit pairs grows quickly, and this increases the chance of unseen co-occurrences. This also hurts tail-context metrics because the local neighborhood of tail events can deviate from the real data distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA uses conditional code probabilities together with sampling-time control over the number of generated codes in each visit. This design better preserves the natural sparsity of clinical visits and reduces implausible tail-code co-occurrences.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.3 Model Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the properties and performance of AdaPCLA based on the following aspects: internalization ablation study, downstream utility, and zero-shot distribution control adaptations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Ablation study.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 (a) compares different variants of AdaPCLA and HALO based on how prior-bias is used for the generative model training. HALO is the backbone generator without any prior bias. Static-PCLA (Consistent) uses the fixed prior bias both during training and inference, which tests whether a permanent bias can improve generation if it is always available. Static-PCLA (Train-only) uses the same fixed bias during training but removes it at inference, which tests whether the model still work after the external bias is taken away. AdaPCLA uses the bias only during training and gradually anneals to zero, and inference is performed without any external bias.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results show a clear difference between using a fixed bias and annealing it. Static-PCLA (Consistent) improves over HALO on several tail metrics, which means that the prior bias is useful when it is kept at inference. However, Static-PCLA (Train-only) fails after the bias is removed: TailPairSeen and TailCtxJSD cannot be computed, and TailTopKJac drops to 0.0000 0.0000 . This means that a fixed training bias alone does not make the model learn the rare-event structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA does not use an external bias at inference, but still achieves the best metrics in this ablation. It obtains the highest TailPairSeen 0.1239 0.1239 and TailTopKJac 0.0356 0.0356 , and the lowest TailCtxJSD 0.5940 0.5940 , outperforming all the baselines. This suggests that dynamic annealing helps the backbone learn useful prior information in its own logits, instead of depending on a fixed external correction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 further visualizes this annealing process by tracking how selected probe-code probabilities change over training checkpoints. A probe code is a candidate code evaluated under a fixed patient-history context, and the probes are grouped into related rare, unrelated rare, and wrong codes. The similar trajectory patterns under 30 and 100 contexts suggest that annealing produces stable internalization dynamics as the probing set is enlarged.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Downstream predictive utility.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the TSTR protocol to test whether synthetic data supports real downstream prediction. For each generator, we train a 25-label diagnosis classifier only on synthetic data and evaluate it on the held-out real test set. As shown in Table 2 , AdaPCLA achieves the best Accuracy, F1, and AUPRC on both MIMIC-III and MIMIC-IV. These results indicate that AdaPCLA preserves predictive signals that transfer from synthetic data to real-data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Zero-shot distribution control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also test whether AdaPCLA can adapt to cross-population generation without retraining. The model is trained on the source population and then applied to the target population using only the target code occurrence rates through the delta-bias correction in Sec. 4.4 . Table 3 (b) reports the zero-shot transfer results from MIMIC-IV to MIMIC-III. These results show that AdaPCLA can use the target prior to adjust generated code frequencies, while retaining longitudinal dependencies and co-occurrence patterns learned from the source population.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we proposed AdaPCLA , a curriculum prior-guided framework for long-tailed longitudinal EHR generation. By using empirical priors as a temporary training scaffold and gradually annealing them away, AdaPCLA strengthens rare-code learning during optimization without requiring an external bias during inference. Experiments on real-world dataset show that AdaPCLA improves tail plausibility, downstream utility, and zero-shot distribution control over strong EHR generation baselines. Finally, AdaPCLA supports zero-shot generation adaptation for cross-population setting in a training-free manner.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limitations. The current evaluation limited to diagnosis-code sequences from two public EHR benchmarks. More validation for AdaPCLA over diverse hospitals and richer EHR modalities beyond diagnosis codes is expected as future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 Generative Modeling for Electronic Health Records\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Synthetic EHR generation builds on a broader clinical sequence modeling literature, where recurrent and temporal point-process models show that diagnoses, procedures, medications, and other clinical events cannot be treated as independent observations [ 8 , 11 , 37 ] . This temporal view is especially important for longitudinal synthesis: a realistic generator should preserve visit-level ordering, delayed comorbidities, and clinically meaningful code co-occurrences, rather than only matching aggregate code counts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Synthetic EHR generation has progressed from GAN-, VAE-, and correlation-aware models for discrete or tabular medical records [ 9 , 38 , 35 ] to temporal structured simulators and longitudinal mixed-type generators [ 43 , 22 , 21 ] . Hierarchical autoregressive frameworks further improve high-dimensional visit modeling by directly modeling long-span patient trajectories [ 34 ] . Recent systems also study privacy-preserving synthesis, multimodal diffusion, and LLM-based generation without direct patient-level access [ 39 , 44 , 13 ] , improving temporal dependency modeling and sample realism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite this progress, most existing generators are still trained to fit the overall distribution of the majority of diseases. In long-tailed EHR data, rare disease codes contribute sparse and unstable learning signals for their temporal, contextual, and co-occurrence dependencies. As a result, a generator may improve average realism while underrepresenting rare-event plausibility and rare-code context consistency. AdaPCLA follows the motivation stated in the main text: it gives tail codes stronger optimization support through a temporary prior scaffold, while preserving longitudinal dependency modeling and enabling bias-free inference after annealing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 Long-Tailed Learning in EHR\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Long-tailed learning methods reshape training under imbalanced labels, with most canonical methods developed for discriminative classification. Resampling methods rebalance the empirical distribution by constructing or selecting minority examples [ 5 ] . Loss-level methods, including focal loss [ 25 ] and class-balanced loss [ 10 ] , change the gradient contribution of different labels. Margin-based objectives enlarge rare-class margins [ 4 ] , decoupled training separates representation learning from classifier calibration [ 20 ] , and logit adjustment adds label-frequency terms to logits to obtain a prior-calibrated decision boundary [ 30 , 33 ] . Later variants further study dynamic or learnable adjustments [ 40 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These methods are effective for classification, but direct use in longitudinal multi-label EHR generation is less straightforward. Resampling can disrupt patient trajectories, fixed reweighting can destabilize sparse multi-label optimization, and static prior biases can leave the generator dependent on external correction at inference time. Multi-label long-tail methods [ 36 , 24 ] further show that label co-occurrence and label-frequency imbalance interact, which is precisely the setting faced by longitudinal EHR synthesis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent studies also analyze long-tailed learning through learning dynamics and optimization trajectories. Neural-collapse-based analyses study how imbalanced training affects decision boundaries [ 14 ] , while probing-based methods use training dynamics to guide data pruning or bias mitigation [ 6 ] . However, these interventions can rely on heuristic data dropping, which may discard informative tail samples, or maintain a permanent dependency on external priors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast, AdaPCLA uses a curriculum-learning strategy [ 3 ] . By treating the statistical prior as a temporary optimization scaffold and annealing it over time, AdaPCLA strengthens rare-code learning signals while encouraging the backbone to encode prior information into its own logits. This improves tail fidelity without discarding patient trajectories or requiring a fixed inference-time bias.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.3 Distribution Shift and Zero-Shot Adaptation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Clinical code frequencies vary across hospitals, populations, and regions due to dataset shift [ 32 ] . Standard approaches for handling such shifts typically rely on domain adaptation or fine-tuning, which require target-domain data access and additional retraining. This motivates generation methods that can adapt to target priors without retraining while preserving longitudinal dependencies and co-occurrence structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One relevant setting is label shift or target-prior shift, where label prevalences change while the conditional structure is assumed to be comparatively stable. Classical and modern work estimates or corrects such shifts using black-box predictors, regularized target-prior estimation, or unified label-shift estimators [ 26 , 2 , 12 ] . Related domain-adaptation work studies target and conditional shift [ 42 ] , and recent time-series adaptation considers feature and label shifts in sequential data [ 15 ] . These studies motivate a label-wise prior-shift view for clinical synthesis: when target code occurrence frequencies change, a generator should adjust label-wise occurrence probabilities while retaining useful conditional dependencies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Controllable tabular and EHR synthesis provides a related perspective. Conditional tabular generators [ 38 ] and synthetic EHR frameworks [ 39 , 13 ] can condition generation on available attributes or external summaries. However, they are not primarily designed for zero-shot control of long-tailed longitudinal EHR generators through target-prior correction over thousands of sparse medical codes. Recent work on structural fidelity further emphasizes that preserving causal or conditional dependencies is different from matching marginal distributions [ 17 ] . This distinction is important for cross-population generation, where occurrence frequencies may shift while clinically meaningful co-occurrence structure should remain reusable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA follows this direction by separating the learned longitudinal structure from marginal occurrence frequency. After the source prior is absorbed during training, the target-prior correction in Sec. 4.4 provides a lightweight mechanism for zero-shot cross-population adaptation: the generator steers label-wise occurrence probabilities toward the target population without retraining the backbone. The intended benefit is controlled occurrence frequency adaptation while reusing the longitudinal and co-occurrence structure learned from source EHR data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Model Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we conduct a deep-dive analysis to examine the core theoretical claims of AdaPCLA : (1) the internalization of prior knowledge, (2) the translation of tail fidelity into downstream utility, and (3) the disentanglement of structure and occurrence frequency for zero-shot control.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"B.1.1 Prior-Internalization Ablation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To examine whether the prior bias is absorbed into the model or remains an external correction, we compare dynamic annealing against static-bias variants in Table 3 . This analysis follows the curriculum construction in Sec. 4.3 and focuses on whether the backbone can preserve rare-event structure during bias-free inference.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Effect of Removing a Fixed Bias\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ablation shows a clear difference between using a fixed bias and annealing it. The Static-PCLA (Train-only) variant applies the fixed bias during training but removes it at inference. After this bias is removed, TailPairSeen and TailCtxJSD cannot be computed, and TailTopKJac drops to 0.0000 0.0000 . This negative control indicates that a fixed training bias alone does not make the backbone encode rare-event structure into its own logits. Instead, the model remains dependent on the fixed correction and fails to preserve tail plausibility once the correction is removed at inference time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Evidence of Prior Internalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA does not use an external bias at inference, but still achieves the best tail metrics in this ablation. It obtains the highest TailPairSeen 0.1239 and TailTopKJac 0.0356 , and the lowest TailCtxJSD 0.5940 . This result is consistent with the gradient dynamics analyzed in Sec. 4.3 : the prior scaffold changes BCE residuals during training, and the resulting updates are written into the intrinsic logits through the shared Transformer backbone. It suggests that dynamic annealing helps the backbone learn useful prior information in its own logits, instead of depending on a fixed external correction during sampling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Probability Dynamics during Annealing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond the ablation table, we further examine how selected code probabilities change as the annealing coefficient α ( τ ) \\\\alpha(\\\\tau) decays. Figure 3 reports four complementary views, from global tail statistics to a single clinical context. As Fig. 3 (a) shows, the average last-step log-probability of a broad set of tail codes steadily increases across checkpoints, indicating that the curriculum increases tail-code probabilities as a whole. On the micro-probing set, Fig. 3 (b) shows systematic differences across probe types (related_rare / unrelated_rare / wrong), while Fig. 3 (c) is consistent with the label-wise empirical NTK analysis in Sec. 4.3 : probability gains Δ log p \\\\Delta\\\\log p grow with the initial gradient similarity between a disease and its context bundle. Finally, Fig. 3 (d) evaluates the context of Case A (cardiovascular patient, Sec. B.2 ). Anticoagulant (V5861), the clinically correct tail medication for Atrial Fibrillation, rises and converges to the Oracle line, whereas Traffic accident (E8192), the hallucination produced by HALO, remains at the bottom. This is consistent with the Case Study: AdaPCLA later generates Anticoagulant because its probability is selectively elevated during annealing. Additional robustness checks for micro-probing are provided in Appendix J .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"B.1.2 Annealing Schedule Effects on Related-Rare Probes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare different annealing schedules by analyzing probability trajectories on tail-relevant probe codes. We do not use a global logit-difference norm such as ‖ z 𝒯 − z 0 − b stat ‖ \\\\|z_{\\\\mathcal{T}}-z_{0}-b_{\\\\text{stat}}\\\\| as the main empirical measure. First, the fixed scaffold b stat b_{\\\\text{stat}} has a much larger norm than the logit changes induced by training, making schedule-dependent changes difficult to isolate. Second, the norm aggregates over the entire vocabulary and does not focus on rare-code dimensions that are relevant to a fixed patient-history context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To measure these schedule effects, we use related_rare gain and AUC summaries based on micro-probe trajectories. Following the micro-probe setting in the main text, we focus on pairs ( x , c ) (x,c) labeled as related_rare, where c c is a tail code clinically related to the patient-history context x x . For each pair, we record the last-step log-probability trajectory { log p t } t = 0 T \\\\{\\\\log p_{t}\\\\}_{t=0}^{T} over 100 100 micro-probe checkpoints and define three summaries:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Final gain : Δ log p = log p T − log p 0 \\\\Delta\\\\log p=\\\\log p_{T}-\\\\log p_{0} , measuring the net log-probability change at the end of annealing for each related_rare code.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AUC :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which sums the positive log-probability changes along the annealing trajectory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Normalized AUC :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which rescales AUC to approximately [ 0 , 1 ] [0,1] and summarizes the distribution of log-probability gains across checkpoints relative to the final checkpoint.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These summaries depend only on increments of log p t \\\\log p_{t} and do not explicitly involve the global bias b stat b_{\\\\text{stat}} . They measure the magnitude and consistency of log-probability increases for tail-relevant related_rare codes along the annealing trajectory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the related_rare micro-probes, we compute final gain, AUC, and normalized AUC for four annealing schedules: slow , base , cosine , and fast . For final gain Δ log p \\\\Delta\\\\log p , the mean values follow\"\n },\n {\n \"type\": \"para\",\n \"text\": \"but the differences are small compared to the standard deviations (gain_std ≈ 1.0 \\\\approx 1.0 ). We therefore do not interpret final gain alone as providing a robust ordering across schedules.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AUC and normalized AUC show a clearer ordering. For both raw AUC and normalized AUC, we observe\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the fast schedule has the largest related_rare AUC, while the slow schedule has the smallest AUC. The normalized AUC values follow the same ordering, indicating that fast annealing increases related_rare log-probabilities earlier and more steadily than slow annealing. We therefore use AUC and normalized AUC as the main schedule-dependent summaries for Experiment 3, and use final gain as a supporting quantity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These related_rare gain and AUC summaries should not be interpreted as direct numerical estimates of the theoretical prior-internalization error ‖ e τ ( x ) ‖ 𝒦 − 1 \\\\|e_{\\\\tau}(x)\\\\|_{\\\\mathcal{K}^{-1}} . They summarize log-probability changes for selected related_rare codes, rather than reconstructing the label-wise empirical NTK flow or the moving target z τ ⋆ ( x ) z_{\\\\tau}^{\\\\star}(x) . Accordingly, this experiment provides an empirical analysis of schedule-dependent probability trajectories on tail-relevant codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taken together, the results show that annealing speed affects the timing and magnitude of related_rare probability increases during training. Fast annealing produces earlier and more sustained log-probability gains, whereas slow annealing produces later and weaker accumulated gains. We use these findings as the schedule-dependent mechanism analysis for Experiment 3, not as a strict numerical validation of the prior-internalization bound.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Qualitative Analysis: Clinical Case Study\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To move beyond aggregate statistics, we examine the Clinical Coherence of synthetic patient trajectories. We present two complementary case studies on MIMIC-IV to illustrate how AdaPCLA captures complex dependencies that baselines miss.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case A: Longitudinal Trajectory Consistency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We select a real patient with a complex history of Cardiovascular and Metabolic disorders . As detailed in Table 5 , given the context of Visits 1–2 (containing Hyperlipidemia, Atrial Fibrillation, and Hypertension), we compare the generated third visit across models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Logic: The patient’s condition progresses logically: chronic cardiovascular issues persist, requiring continued medication (Anticoagulants) and monitoring.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Failure (Hallucination): The HALO baseline captures generic codes (Hypertension) but suffers from severe semantic hallucination . It predicts \\\"Traffic accident (E8192)\\\" , a completely irrelevant external cause disconnected from the patient’s medical history. This typifies the \\\"Gradient Domination\\\" failure: without the scaffold, the model’s attention is scattered by noise. LSTM and GPT similarly fail to recall the specific comorbidity structure (e.g., predicting unrelated HIV or Pneumonia codes).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPCLA Success (Coherence): In contrast, AdaPCLA correctly recovers the sophisticated treatment logic. It successfully predicts \\\"Long-term use of anticoagulants (V5861)\\\" , a critical medication strictly tied to the history of Atrial Fibrillation (42731) . Furthermore, it generates \\\"Chest pain\\\" , a clinically plausible symptom given the cardiac history. This illustrates that AdaPCLA has internalized the causal link: Atrial Fib → \\\\rightarrow Anticoagulant Use , a dependency that requires looking past marginal frequencies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case B: Tail Co-occurrence Topology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Table 6 (Appendix), we investigate the local neighborhood of a specific tail code: Squamous cell carcinoma of skin (17322) . We examine the top-10 co-occurring codes generated by each model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Manifold Reconstruction: AdaPCLA successfully reconstructs the correct disease manifold. It identifies 5 out of the top 10 true comorbidities, including specific procedures like Aortocoronary bypass (V4581) and chronic conditions like Hyperlipidemia .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Scattering: HALO’s predictions are scattered and generic (e.g., Dizziness, Chronic pain), lacking disease-specific correlation. LSTM and GPT fail to generate any meaningful co-occurrences for this rare code, indicating Mode Collapse in the tail region.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These cases qualitatively illustrate that AdaPCLA ’s superior quantitative metrics (e.g., TailPairSeen) translate into tangible clinical validity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 Prediction Reliability and Internalization Consistency\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond transferability and controllability, we investigate two complementary aspects: (1) the reliability of predicted probabilities, especially in the long tail; and (2) the consistency between observed curriculum dynamics and the theory in Sec. 4.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Prediction Confidence Reliability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate predicted probabilities on a held-out real test set with ECE , Brier Score , and NLL , reporting both overall and tail-only subsets. AdaPCLA consistently outperforms GPT-style and HALO in both settings; complete numbers and reliability diagrams are reported in the detailed results section.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Expanded Probing Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further enlarge the probing set from 30 to 100 contexts and recompute related_rare / unrelated_rare / wrong probe trajectories. Table 8 and Figure 2 show that the trajectory scale and type separation remain qualitatively consistent under the larger probe set.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Summary of Outputs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Together, confidence metrics and expanded probing provide empirical support by examining probability reliability and curriculum dynamics. Exploratory λ min ( 𝒦 ) \\\\lambda_{\\\\min}(\\\\mathcal{K}) correlation analysis is reported in Appendix G and treated as non-conclusive evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We analyze the curriculum-induced logit dynamics for a fixed input context x x . Recall the effective logits z ~ τ ( x ) = z τ ( x ) + α ( τ ) b stat \\\\tilde{z}_{\\\\tau}(x)=z_{\\\\tau}(x)+\\\\alpha(\\\\tau)b_{\\\\text{stat}} , where α : [ 0 , 𝒯 ] → [ 0 , 1 ] \\\\alpha:[0,\\\\mathcal{T}]\\\\to[0,1] is continuously differentiable with α ( 0 ) = 1 \\\\alpha(0)=1 and α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why we use an expected-risk surrogate A pointwise BCE loss with a single hard multi-label target y ∈ { 0 , 1 } V y\\\\in\\\\{0,1\\\\}^{V} generally does not admit a finite minimizer in logit space, because the minimizer is attained only in the limit z c → ± ∞ z_{c}\\\\to\\\\pm\\\\infty . To obtain a well-posed moving optimum, we work with the conditional expected BCE risk at a fixed context x x :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We suppress the explicit dependence on x x in what follows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the lazy-training approximation, we assume that the label-wise empirical NTK remains approximately constant along the trajectory. Because we analyze the dynamics of different labels for the same context x x , the kernel here is the label-wise coupling matrix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"rather than the usual sample–sample NTK. On the active label subspace we assume 𝒦 \\\\mathcal{K} is positive definite, so that 𝒦 − 1 \\\\mathcal{K}^{-1} is well-defined there. We write ‖ v ‖ 𝒦 − 1 := v ⊤ 𝒦 − 1 v \\\\|v\\\\|_{\\\\mathcal{K}^{-1}}:=\\\\sqrt{v^{\\\\top}\\\\mathcal{K}^{-1}v} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The kernel-induced gradient flow in intrinsic-logit coordinates is z ˙ τ = − 𝒦 ∇ F τ ( z τ ) \\\\dot{z}_{\\\\tau}=-\\\\mathcal{K}\\\\nabla F_{\\\\tau}(z_{\\\\tau}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assumptions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A1 (Local strong monotonicity / local strong convexity type condition). There exists a convex neighborhood 𝒟 ⊂ ℝ V \\\\mathcal{D}\\\\subset\\\\mathbb{R}^{V} and a constant μ > 0 \\\\mu>0 such that, for all τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] and all u , v ∈ 𝒟 u,v\\\\in\\\\mathcal{D} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is the standard first-order formulation of strong monotonicity of ∇ F τ \\\\nabla F_{\\\\tau} on 𝒟 \\\\mathcal{D} ; when F τ F_{\\\\tau} is twice differentiable, it is implied by the local Hessian lower bound ∇ 2 F τ ( z ) ⪰ μ I \\\\nabla^{2}F_{\\\\tau}(z)\\\\succeq\\\\mu I on 𝒟 \\\\mathcal{D} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A2 (Smoothness of the objective–curriculum pair and endpoint schedule). The joint map ( τ , z ) ↦ ∇ F τ ( z ) (\\\\tau,z)\\\\mapsto\\\\nabla F_{\\\\tau}(z) is continuously differentiable on [ 0 , 𝒯 ] × 𝒟 [0,\\\\mathcal{T}]\\\\times\\\\mathcal{D} , and α : [ 0 , 𝒯 ] → [ 0 , 1 ] \\\\alpha:[0,\\\\mathcal{T}]\\\\to[0,1] is continuously differentiable with the endpoint conditions α ( 0 ) = 1 \\\\alpha(0)=1 and α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 (matching the curriculum construction in the main text).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A3 (Constant, nondegenerate kernel on the active subspace). The label-wise kernel 𝒦 \\\\mathcal{K} is constant and symmetric on the active label subspace, and is positive definite there with λ min ( 𝒦 ) > 0 \\\\lambda_{\\\\min}(\\\\mathcal{K})>0 . In particular, the 𝒦 − 1 \\\\mathcal{K}^{-1} –inner product and the induced norm ∥ ⋅ ∥ 𝒦 − 1 \\\\|\\\\cdot\\\\|_{\\\\mathcal{K}^{-1}} used below are well-defined on that subspace.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Lemma A.1 (Regularity of the moving minimizer) Suppose that for each τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] , there exists a unique point z τ ⋆ ∈ 𝒟 z_{\\\\tau}^{\\\\star}\\\\in\\\\mathcal{D} such that ∇ F τ ( z τ ⋆ ) = 0 \\\\nabla F_{\\\\tau}(z_{\\\\tau}^{\\\\star})=0 . Under Assumptions A1–A2 and the existence/uniqueness hypothesis above, the path τ ↦ z τ ⋆ \\\\tau\\\\mapsto z_{\\\\tau}^{\\\\star} is continuously differentiable. Moreover, there exists a finite constant B ⋆ > 0 B_{\\\\star}>0 such that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proof. Under A1–A2, the Hessian ∇ 2 F τ ( z τ ⋆ ) \\\\nabla^{2}F_{\\\\tau}(z_{\\\\tau}^{\\\\star}) is nonsingular at each minimizer (as in the standard link between strong monotonicity and positive definiteness of the Jacobian of the gradient). The implicit function theorem therefore yields a local C 1 C^{1} branch of solutions to ∇ F τ ( z ) = 0 \\\\nabla F_{\\\\tau}(z)=0 near each ( τ , z τ ⋆ ) (\\\\tau,z_{\\\\tau}^{\\\\star}) . Together with the assumed uniqueness of z τ ⋆ z_{\\\\tau}^{\\\\star} in 𝒟 \\\\mathcal{D} for every τ \\\\tau , these local branches glue to a globally defined C 1 C^{1} path τ ↦ z τ ⋆ \\\\tau\\\\mapsto z_{\\\\tau}^{\\\\star} on [ 0 , 𝒯 ] [0,\\\\mathcal{T}] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Differentiating the optimality condition ∇ F τ ( z τ ⋆ ) = 0 \\\\nabla F_{\\\\tau}(z_{\\\\tau}^{\\\\star})=0 along this path gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the τ \\\\tau -dependence enters through α ( τ ) \\\\alpha(\\\\tau) , ∂ τ ∇ F τ ( z τ ⋆ ) = α ˙ ( τ ) ∂ α ∇ F τ ( z τ ⋆ ) \\\\partial_{\\\\tau}\\\\nabla F_{\\\\tau}(z_{\\\\tau}^{\\\\star})=\\\\dot{\\\\alpha}(\\\\tau)\\\\,\\\\partial_{\\\\alpha}\\\\nabla F_{\\\\tau}(z_{\\\\tau}^{\\\\star}) . Therefore\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The map\"\n },\n {\n \"type\": \"para\",\n \"text\": \"is continuous on [ 0 , 𝒯 ] [0,\\\\mathcal{T}] (smoothness of the data along the minimizer path), hence bounded on this compact interval. Taking the ∥ ⋅ ∥ 𝒦 − 1 \\\\|\\\\cdot\\\\|_{\\\\mathcal{K}^{-1}} –operator norm of the displayed expression and absorbing | α ˙ ( τ ) | \\\\lvert\\\\dot{\\\\alpha}(\\\\tau)\\\\rvert yields a finite uniform constant B ⋆ B_{\\\\star} such that Eq. ( 3 ) holds for all τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] . ■ \\\\blacksquare\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem A.2 (Prior-internalization error in the 𝒦 − 1 \\\\mathcal{K}^{-1} -norm). Define the internalization error e τ = z τ − z τ ⋆ e_{\\\\tau}=z_{\\\\tau}-z_{\\\\tau}^{\\\\star} and r τ = ‖ e τ ‖ 𝒦 − 1 r_{\\\\tau}=\\\\|e_{\\\\tau}\\\\|_{\\\\mathcal{K}^{-1}} . Assume additionally that z τ , z τ ⋆ ∈ 𝒟 z_{\\\\tau},z_{\\\\tau}^{\\\\star}\\\\in\\\\mathcal{D} for all τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] . Under Assumptions A1–A3 together with this invariant-neighborhood condition, for all τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proof We follow three steps: (i) define the internalization error and a Lyapunov function; (ii) derive a scalar differential inequality for the radius r τ r_{\\\\tau} ; (iii) apply Grönwall to close the bound.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 1: internalization error and Lyapunov function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because z ˙ τ = − 𝒦 ∇ F τ ( z τ ) \\\\dot{z}_{\\\\tau}=-\\\\mathcal{K}\\\\nabla F_{\\\\tau}(z_{\\\\tau}) and ∇ F τ ( z τ ⋆ ) = 0 \\\\nabla F_{\\\\tau}(z_{\\\\tau}^{\\\\star})=0 , we have\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define the Lyapunov function V τ = 1 2 ‖ e τ ‖ 𝒦 − 1 2 = 1 2 e τ ⊤ 𝒦 − 1 e τ V_{\\\\tau}=\\\\frac{1}{2}\\\\|e_{\\\\tau}\\\\|_{\\\\mathcal{K}^{-1}}^{2}=\\\\frac{1}{2}e_{\\\\tau}^{\\\\top}\\\\mathcal{K}^{-1}e_{\\\\tau} . Since 𝒦 \\\\mathcal{K} is constant and symmetric,\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 2: Lyapunov bound and the radius reduction (Lyapunov-to- r r bridge)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since z τ , z τ ⋆ ∈ 𝒟 z_{\\\\tau},z_{\\\\tau}^{\\\\star}\\\\in\\\\mathcal{D} for all τ ∈ [ 0 , 𝒯 ] \\\\tau\\\\in[0,\\\\mathcal{T}] , Assumption A1 with u = z τ u=z_{\\\\tau} and v = z τ ⋆ v=z_{\\\\tau}^{\\\\star} gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, since 𝒦 ⪰ λ min ( 𝒦 ) I \\\\mathcal{K}\\\\succeq\\\\lambda_{\\\\min}(\\\\mathcal{K})I on the active subspace,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using Cauchy–Schwarz in the 𝒦 − 1 \\\\mathcal{K}^{-1} inner product together with Lemma A.1,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combining the above inequalities yields the Lyapunov inequality\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now isolate the key reduction from V ˙ τ \\\\dot{V}_{\\\\tau} to a scalar inequality for r τ r_{\\\\tau} . Because V τ = 1 2 r τ 2 V_{\\\\tau}=\\\\frac{1}{2}r_{\\\\tau}^{2} and r τ ≥ 0 r_{\\\\tau}\\\\geq 0 , the chain rule gives V ˙ τ = r τ r ˙ τ \\\\dot{V}_{\\\\tau}=r_{\\\\tau}\\\\dot{r}_{\\\\tau} at points where r τ > 0 r_{\\\\tau}>0 ; substituting into ( 5 ) and dividing by r τ r_{\\\\tau} yields\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(At instants with r τ = 0 r_{\\\\tau}=0 , the same inequality follows by continuity of the underlying quantities along the trajectory.) We refer to the passage from ( 5 ) to ( 6 ) as the radius reduction step .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 3: Grönwall closure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying Grönwall’s inequality to Eq. ( 6 ) gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which yields Eq. ( 4 ). ■ \\\\blacksquare\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Corollary (Prior internalization: structural identity vs. realized endpoint deviation) Structural assumption (ideal moving optimum) Assume additionally that the effective-risk minimizer z ~ ⋆ ( x ) \\\\tilde{z}^{\\\\star}(x) depends only on the conditional distribution ℙ ( Y ∣ x ) \\\\mathbb{P}(Y\\\\mid x) and is therefore independent of α ( τ ) \\\\alpha(\\\\tau) . Then the quasi-static intrinsic minimizer admits the affine representation z τ ⋆ = z ~ ⋆ − α ( τ ) b stat z_{\\\\tau}^{\\\\star}=\\\\tilde{z}^{\\\\star}-\\\\alpha(\\\\tau)b_{\\\\text{stat}} , and, together with the schedule endpoints α ( 0 ) = 1 \\\\alpha(0)=1 and α ( 𝒯 ) = 0 \\\\alpha(\\\\mathcal{T})=0 , the ideal endpoint shift identity z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\\\mathcal{T}}^{\\\\star}-z_{0}^{\\\\star}=b_{\\\\text{stat}} holds. Without this structural assumption, one should not treat z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\\\mathcal{T}}^{\\\\star}-z_{0}^{\\\\star}=b_{\\\\text{stat}} as automatic.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Realized logits vs. internalization error. Under the ideal endpoint identity z 𝒯 ⋆ − z 0 ⋆ = b stat z_{\\\\mathcal{T}}^{\\\\star}-z_{0}^{\\\\star}=b_{\\\\text{stat}} , the exact algebraic decomposition z 𝒯 − z 0 − b stat = e 𝒯 − e 0 z_{\\\\mathcal{T}}-z_{0}-b_{\\\\text{stat}}=e_{\\\\mathcal{T}}-e_{0} rewrites the realized endpoint deviation from b stat b_{\\\\text{stat}} into a difference of internalization errors at τ = 𝒯 \\\\tau=\\\\mathcal{T} and τ = 0 \\\\tau=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using Theorem A.2 to control ‖ e τ ‖ 𝒦 − 1 \\\\|e_{\\\\tau}\\\\|_{\\\\mathcal{K}^{-1}} and the triangle inequality ‖ e 𝒯 − e 0 ‖ 𝒦 − 1 ≤ ‖ e 𝒯 ‖ 𝒦 − 1 + ‖ e 0 ‖ 𝒦 − 1 \\\\|e_{\\\\mathcal{T}}-e_{0}\\\\|_{\\\\mathcal{K}^{-1}}\\\\leq\\\\|e_{\\\\mathcal{T}}\\\\|_{\\\\mathcal{K}^{-1}}+\\\\|e_{0}\\\\|_{\\\\mathcal{K}^{-1}} , with a = μ λ min ( 𝒦 ) a=\\\\mu\\\\lambda_{\\\\min}(\\\\mathcal{K}) ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus the endpoint identity is exact up to a transient initialization term and an annealing-induced internalization term. In particular, if e 0 = 0 e_{0}=0 or ‖ e 0 ‖ 𝒦 − 1 = 𝒪 ( sup s | α ˙ ( s ) | ) \\\\|e_{0}\\\\|_{\\\\mathcal{K}^{-1}}=\\\\mathcal{O}(\\\\sup_{s}|\\\\dot{\\\\alpha}(s)|) , then\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a prediction step and a code c c . Let 𝒟 S \\\\mathcal{D}_{S} and 𝒟 T \\\\mathcal{D}_{T} denote the source and target domains. Under the label-wise prior-shift assumption, ℙ S ( x ∣ y c ) = ℙ T ( x ∣ y c ) \\\\mathbb{P}_{S}(x\\\\mid y_{c})=\\\\mathbb{P}_{T}(x\\\\mid y_{c}) , while only the class prior changes from π S ( c ) \\\\pi_{S}(c) to π T ( c ) \\\\pi_{T}(c) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Bayes-optimal per-code logit in domain 𝒟 ∈ { 𝒟 S , 𝒟 T } \\\\mathcal{D}\\\\in\\\\{\\\\mathcal{D}_{S},\\\\mathcal{D}_{T}\\\\} is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying Bayes’ rule gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first term is invariant across domains by assumption, so subtracting the source and target expressions yields\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence the exact label-wise Bayes correction is obtained by adding the delta bias Δ b ( c ) = logit ( π T ( c ) ) − logit ( π S ( c ) ) \\\\Delta b(c)=\\\\mathrm{logit}\\\\!\\\\big(\\\\pi_{T}(c)\\\\big)-\\\\mathrm{logit}\\\\!\\\\big(\\\\pi_{S}(c)\\\\big) . This proves the claim. The main-text interpolation parameter λ \\\\lambda is exact when λ = 1 \\\\lambda=1 ; values λ ∈ [ 0 , 1 ) \\\\lambda\\\\in[0,1) correspond to a practical interpolation heuristic rather than to the exact Bayes identity. ■ \\\\blacksquare ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Remark. This observation is deliberately label-wise and step-wise. It justifies marginal prior correction for each code at a fixed prediction step, but it does not by itself imply exact matching of the full autoregressive joint distribution after rollout.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we provide supporting proofs for the claims used in the main text.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.1 Proof of the Static Centering Property\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We prove a local centering statement rather than a global optimality claim.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bayesian decomposition For a fixed medical code c c , the Bayes-optimal effective logit under BCE is the posterior log-odds z ~ c ⋆ ( x ) = log ℙ ( y c = 1 ∣ x ) ℙ ( y c = 0 ∣ x ) \\\\tilde{z}_{c}^{\\\\star}(x)=\\\\log\\\\frac{\\\\mathbb{P}(y_{c}=1\\\\mid x)}{\\\\mathbb{P}(y_{c}=0\\\\mid x)} . Applying Bayes’ rule gives the standard decomposition\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here 𝒮 c ( x ) \\\\mathcal{S}_{c}(x) is the structural term, while logit ( π ( c ) ) \\\\mathrm{logit}(\\\\pi(c)) is the prevalence term.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prior-centered initialization Let π ^ ϵ ( c ) = π ^ ( c ) + ϵ 1 + 2 ϵ \\\\hat{\\\\pi}_{\\\\epsilon}(c)=\\\\frac{\\\\hat{\\\\pi}(c)+\\\\epsilon}{1+2\\\\epsilon} and b stat ( c ) = − logit ( π ^ ϵ ( c ) ) b_{\\\\text{stat}}(c)=-\\\\mathrm{logit}\\\\!\\\\big(\\\\hat{\\\\pi}_{\\\\epsilon}(c)\\\\big) . Assume that, near initialization, the intrinsic logits satisfy z 0 , c ( x ) = logit ( π ^ ϵ ( c ) ) + r c ( x ) z_{0,c}(x)=\\\\mathrm{logit}\\\\!\\\\big(\\\\hat{\\\\pi}_{\\\\epsilon}(c)\\\\big)+r_{c}(x) , with | r c ( x ) | ≪ 1 |r_{c}(x)|\\\\ll 1 . Then for any static additive shift b ( c ) b(c) the initial effective logit is z ~ 0 , c ( x ) = z 0 , c ( x ) + b ( c ) = logit ( π ^ ϵ ( c ) ) + r c ( x ) + b ( c ) \\\\tilde{z}_{0,c}(x)=z_{0,c}(x)+b(c)=\\\\mathrm{logit}\\\\!\\\\big(\\\\hat{\\\\pi}_{\\\\epsilon}(c)\\\\big)+r_{c}(x)+b(c) . The unique shift that centers the nominal operating point (the case r c ( x ) = 0 r_{c}(x)=0 ) at zero is b ( c ) = − logit ( π ^ ϵ ( c ) ) = b stat ( c ) b(c)=-\\\\mathrm{logit}\\\\!\\\\big(\\\\hat{\\\\pi}_{\\\\epsilon}(c)\\\\big)=b_{\\\\text{stat}}(c) , for which z ~ 0 , c ( x ) = r c ( x ) ≈ 0 \\\\tilde{z}_{0,c}(x)=r_{c}(x)\\\\approx 0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Local sensitivity maximization For BCE with sigmoid link, the local logit-to-probability sensitivity and the pointwise curvature are both σ ′ ( u ) = σ ( u ) ( 1 − σ ( u ) ) \\\\sigma^{\\\\prime}(u)=\\\\sigma(u)(1-\\\\sigma(u)) . This quantity is uniquely maximized at u = 0 u=0 . Therefore, among additive shifts depending only on π ^ ϵ ( c ) \\\\hat{\\\\pi}_{\\\\epsilon}(c) , the choice b stat ( c ) b_{\\\\text{stat}}(c) is the unique shift that centers the local operating point at the maximal-sensitivity region of the sigmoid. This establishes the centering claim used in the main text.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, because the scaffold is an x x -independent additive term, it translates the logit coordinate without changing the structural component 𝒮 c ( x ) \\\\mathcal{S}_{c}(x) in Eq. ( 7 ). Thus it acts as a local optimization scaffold rather than as a modification of the underlying structure. ■ \\\\blacksquare\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.2 Lemma: NTK Gradient Flow under BCE Loss\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before analyzing the curriculum dynamics, we formalize the continuous-time gradient flow of neural network outputs under Binary Cross-Entropy (BCE) loss.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒟 = { ( x i , y i ) } i = 1 n \\\\mathcal{D}=\\\\{(x_{i},y_{i})\\\\}_{i=1}^{n} be the training dataset. Let 𝐟 ( t ) = [ f θ ( t ) ( x 1 ) , … , f θ ( t ) ( x n ) ] ⊤ ∈ ℝ n \\\\mathbf{f}(t)=[f_{\\\\theta(t)}(x_{1}),\\\\dots,f_{\\\\theta(t)}(x_{n})]^{\\\\top}\\\\in\\\\mathbb{R}^{n} denote the intrinsic logits of the neural network at time t t , and 𝐲 = [ y 1 , … , y n ] ⊤ ∈ { 0 , 1 } n \\\\mathbf{y}=[y_{1},\\\\dots,y_{n}]^{\\\\top}\\\\in\\\\{0,1\\\\}^{n} be the corresponding labels. Under continuous-time gradient descent on the BCE loss, the evolution of the network outputs follows the non-linear Ordinary Differential Equation (ODE)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where H ( t ) ∈ ℝ n × n H(t)\\\\in\\\\mathbb{R}^{n\\\\times n} is the empirical NTK matrix with entries\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The parameter trajectory under gradient flow satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying the chain rule gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any x k x_{k} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stacking these equations over k = 1 , … , n k=1,\\\\dots,n yields the claimed matrix form. ■ \\\\blacksquare ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Remark. Under AdaPCLA , the same identity applies to the effective logits f ~ θ ( t ) ( x ) = f θ ( t ) ( x ) + α ( t ) b stat \\\\tilde{f}_{\\\\theta(t)}(x)=f_{\\\\theta(t)}(x)+\\\\alpha(t)b_{\\\\text{stat}} , because the scaffold is independent of θ \\\\theta and therefore ∇ θ f ~ θ ( t ) ( x ) = ∇ θ f θ ( t ) ( x ) \\\\nabla_{\\\\theta}\\\\tilde{f}_{\\\\theta(t)}(x)=\\\\nabla_{\\\\theta}f_{\\\\theta(t)}(x) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now write the discrete first-order update under the actual AdaPCLA training objective. Fix one SGD step and condition on the current annealing value α t \\\\alpha_{t} as constant during that step. Define the effective logits z ~ t ( x ) = z t ( x ) + α t b stat \\\\tilde{z}^{t}(x)=z^{t}(x)+\\\\alpha_{t}b_{\\\\text{stat}} , where z t ( x ) = f θ t ( x ) ∈ ℝ V z^{t}(x)=f_{\\\\theta^{t}}(x)\\\\in\\\\mathbb{R}^{V} denotes the intrinsic logits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let x o x_{o} be an observing example and ( x u , y u ) (x_{u},y_{u}) an updating example. We study the change in the log-probability vector log σ ( z ~ t ( x o ) ) \\\\log\\\\sigma(\\\\tilde{z}^{t}(x_{o})) . A first-order Taylor expansion gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under SGD on the AdaPCLA BCE loss,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because α t b stat \\\\alpha_{t}b_{\\\\text{stat}} is independent of θ \\\\theta within this step, ∇ θ z ~ t ( x ) = ∇ θ z t ( x ) \\\\nabla_{\\\\theta}\\\\tilde{z}^{t}(x)=\\\\nabla_{\\\\theta}z^{t}(x) . Applying the chain rule on both factors in Eq. ( 8 ) yields the following compact decomposition. For readability, write\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Substituting Eq. ( 9 ) into Eq. ( 8 ) gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The matrix 𝒜 Ada t ( x o ) \\\\mathcal{A}_{\\\\mathrm{Ada}}^{t}(x_{o}) is diagonal because sigmoid factorizes across labels:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, as in independent BCE, the output nonlinearity does not introduce explicit cross-label terms. Any cross-label effect is determined by the shared-backbone kernel 𝒦 t ( x o , x u ) \\\\mathcal{K}^{t}(x_{o},x_{u}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, because z ~ t ( x ) = z t ( x ) + α t b stat \\\\tilde{z}^{t}(x)=z^{t}(x)+\\\\alpha_{t}b_{\\\\text{stat}} differs from z t ( x ) z^{t}(x) by a θ \\\\theta -independent constant, the intrinsic-logit increment obeys\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which is the form used in the main text. The deterministic effect induced by changes in α t \\\\alpha_{t} across optimization steps is analyzed separately in Appendix C . ■ \\\\blacksquare\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prior-internalization bound in Appendix C and Sec. 4.3 identifies the minimum eigenvalue λ min ( 𝒦 ) \\\\lambda_{\\\\min}(\\\\mathcal{K}) of the label-wise empirical NTK as a factor that reduces the internalization-error bound. As an exploratory analysis, we compute λ min ( 𝒦 x ) \\\\lambda_{\\\\min}(\\\\mathcal{K}_{x}) at the initial checkpoint for each probing context and compare it with the empirical related-rare AUC and final gain. The label-wise empirical NTK is restricted to the related-rare codes of the same context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"30-context results: On the 30-context micro-probe set, the Spearman correlations are positive but weak: Spearman( λ min ( 𝒦 init ) \\\\lambda_{\\\\min}(\\\\mathcal{K}_{\\\\mathrm{init}}) , related-rare AUC) ≈ 0.30 \\\\approx 0.30 ( p ≈ 0.11 p\\\\approx 0.11 ) and Spearman( λ min ( 𝒦 init ) \\\\lambda_{\\\\min}(\\\\mathcal{K}_{\\\\mathrm{init}}) , related-rare gain) ≈ 0.16 \\\\approx 0.16 ( p ≈ 0.40 p\\\\approx 0.40 ). The signs are consistent with the bound, but the correlations do not reach conventional significance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"100-context robustness check: We extend the analysis to 100 probing contexts. The correlations become close to zero: Spearman( λ min ( 𝒦 init ) \\\\lambda_{\\\\min}(\\\\mathcal{K}_{\\\\mathrm{init}}) , AUC) ≈ 0.02 \\\\approx 0.02 ( p ≈ 0.86 p\\\\approx 0.86 ) and Spearman( λ min ( 𝒦 init ) \\\\lambda_{\\\\min}(\\\\mathcal{K}_{\\\\mathrm{init}}) , gain) ≈ − 0.06 \\\\approx-0.06 ( p ≈ 0.53 p\\\\approx 0.53 ). The final-gain correlation is negative under this larger probing set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conclusion: We do not treat the λ min ( 𝒦 ) \\\\lambda_{\\\\min}(\\\\mathcal{K}) and related-rare AUC/gain analysis as empirical validation of the bound. The 30-context result is statistically modest, and the 100-context extension shows that the correlation is not robust. We report this analysis for completeness as non-conclusive evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section provides detailed prediction-reliability statistics for the downstream 25-label DiagnosisModel evaluated on the real MIMIC-IV test split. We report ECE, Brier Score, and Negative Log-Likelihood (NLL) for AdaPCLA , HALO, and GPT-style under two scopes: (i) overall (all labels), and (ii) tail-only (bottom-1/3 labels by training-set frequency).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 shows the corresponding reliability diagrams for both the overall and tail-only scopes. Across confidence bins, AdaPCLA has the smallest deviation from the identity line, especially in the high-confidence region. GPT-style and HALO show larger confidence errors, particularly on rare labels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports quantitative summaries for the micro-probing experiments with 30 and 100 patient-history contexts. For each context and probe type (related_rare / unrelated_rare / wrong), we compute the AUC of Δ log p ( τ ) \\\\Delta\\\\log p(\\\\tau) over annealing checkpoints and the final Δ log p \\\\Delta\\\\log p at the end of annealing. We then report the mean and standard deviation across contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 in the main text tracks selected probe-code probabilities under 30 and 100 probing contexts. Table 8 reports the corresponding AUC of Δ log p ( τ ) \\\\Delta\\\\log p(\\\\tau) and final Δ log p \\\\Delta\\\\log p for each probe type.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments reported in this paper were run in the sft_lab virtual environment on a server with 8 × \\\\times NVIDIA RTX 4090 GPUs. Table 9 lists the main software and dependency versions used for training and evaluation. We recommend using compatible versions for reproduction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports frequency–rank plots for MIMIC-III and MIMIC-IV, comparing Real training data with synthetic data generated by HALO, LSTM, GPT-style, and AdaPCLA . These plots summarize the ranked marginal occurrence-frequency distribution of medical codes. They do not evaluate code-set support, tail clinical plausibility, rare-code context consistency, or downstream predictive utility. Therefore, we use them only as supplementary distributional-fidelity views. Model comparison should rely on Table 1 for PairSeen, TailPairSeen, TailCtxJSD, and TailTopKJac, Table 2 for downstream utility, and Sec. B.2 for qualitative clinical examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports per-code marginal occurrence-frequency views that complement the frequency–rank plots in Sec. L . Each point corresponds to one medical code present in the Real training set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Frequency–frequency scatter: For each code c c , the horizontal axis is its visit-level frequency in Real and the vertical axis is its visit-level frequency in a given model (one subplot per model: HALO, LSTM, GPT-style, AdaPCLA ). Both axes use a symmetric log scale. Points near the diagonal y = x y=x indicate that the model’s marginal frequency for that code is close to Real; points above or below the diagonal indicate higher or lower generated frequency than Real. These plots evaluate code-level marginal occurrence frequencies, but they do not measure tail clinical plausibility or rare-event co-occurrence structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Rank–rank scatter: For each code c c , the horizontal axis is its rank in Real (1 = most frequent) and the vertical axis is its rank in the model’s synthetic data; both axes are log-scaled. Points near the diagonal indicate agreement between the Real code rank and the synthetic code rank. Codes never generated by a model are assigned a rank beyond the maximum. Together, the frequency–frequency and rank–rank figures report per-code marginal alignment with the training distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports code–code co-occurrence matrices for the top-200 most frequent medical codes on MIMIC-IV. We compute visit-level co-occurrence counts for Real, HALO, LSTM, and AdaPCLA , and then apply a PPMI (Positive Pointwise Mutual Information) transform. To keep the panels comparable, we perform hierarchical clustering on the Real PPMI matrix using average linkage and apply the resulting code order to all models. The visualization compares how well each generator preserves frequent-code co-occurrence patterns learned from the Real training data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports the full P ( code ∣ label ) P(\\\\text{code}\\\\mid\\\\text{label}) heatmaps referenced in Sec. 6 . We partition medical codes into three frequency buckets: head (most frequent), mid, and tail (long tail). For each bucket, we compute the conditional distribution of codes given the bucket label. The heatmaps use a 3 × 4 3\\\\times 4 layout: each row corresponds to one bucket, and each column corresponds to one data source (Real training data, HALO, LSTM, or AdaPCLA ) on MIMIC-IV. These plots provide supplementary distributional-fidelity views for code-level probability structure within each frequency regime. AdaPCLA shows closer alignment with Real across the three buckets than HALO and LSTM, which is consistent with the tail plausibility and co-occurrence results in Table 1 .\"\n }\n]","isConversational":false}}},{"rowIdx":291,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12626"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Gradient-free learning of a closed-loop wall controller for turbulent drag reduction"},"authors":{"kind":"list like","value":["Giorgio Maria Cavallazzi","Miguel Pérez Cuadrado","Alfredo Pinelli"],"string":"[\n \"Giorgio Maria Cavallazzi\",\n \"Miguel Pérez Cuadrado\",\n \"Alfredo Pinelli\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12626v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Skin-friction drag in wall-bounded turbulence is generated by quasi-streamwise vortices whose induced wall-normal velocities sustain the Reynolds shear stress above its laminar value (Orlandi & Jiménez, 1994 ) . The vortices are part of a self-sustaining near-wall cycle of low-speed streaks and quasi-streamwise vortices (Kline et al., 1967 ; Jiménez & Moin, 1991 ; Hamilton et al., 1995 ; Waleffe, 1997 ) that persists autonomously within the buffer layer even in the absence of outer-flow forcing (Jiménez & Pinelli, 1999 ) , making that layer the natural target for drag-reduction control."},{"type":"para","text":"In the case of established active flow control methods, like opposition control (Choi et al., 1994 ; Hammond et al., 1998 ; Stroh et al., 2015 ) , spanwise forcing and streamwise-travelling waves of spanwise velocity (Quadrio et al., 2009 ; Gatti & Quadrio, 2016 ; Touber & Leschziner, 2012 ) , the drag-reduction mechanism rests on a specific, physically motivated interaction with the near-wall structures, and the actuation is designed around that interaction from the outset. Controllers discovered by learnt data-driven algorithms carry no such prior; their actuation strategy emerges from the optimisation, and what physical interaction it exploits is not known in advance."},{"type":"para","text":"Reinforcement learning (RL) has been the most productive research area for those controllers. Applied to the classical plane channel flow at low Reynolds numbers, it has produced closed-loop wall policies that report drag reductions competitive with the physics-based benchmarks (Guastoni et al., 2023 ; Sonoda et al., 2023 ; Lee et al., 2023 ; Han & Huang, 2020 ; Zhou et al., 2025 ; Varela et al., 2022 ; Cavallazzi et al., 2025 ) , as surveyed in several reviews (Garnier et al., 2021 ; Brunton et al., 2020 ; Vignon et al., 2023b ) . A common way to keep the formulation tractable is a shared, translationally invariant multi-agent policy (Vignon et al., 2023a ; Guastoni et al., 2023 ) , trained with a centralised critic on a small periodic domain, executed independently at each wall patch, then applied to a larger channel. This template underlies numerous subsequent studies (Sonoda et al., 2023 ; Cavallazzi et al., 2025 ; Wälchli et al., 2024 ; Bae & Koumoutsakos, 2022 ; Suárez et al., 2024 ; Rabault & Kuhnle, 2019 ) . The drag reduction headline figures reported on minimal domains must, however, be read with care: a policy evaluated in a minimal flow unit can drive the flow towards relaminarisation, where the pressure gradient falls to its laminar value and the reported drag reduction saturates at a value that reflects the box rather than the controller (Cavallazzi et al., 2026 ) . A representative comparison therefore requires a domain large enough to keep the flow fully turbulent throughout, the setting adopted here."},{"type":"para","text":"Despite the need for a large evaluation domain, training is in practice confined to a small one by the structure of the multi-agent update rather than by simulation cost alone. A gradient step adjusts the shared policy parameters from a per-patch return whose credit is assigned by a central critic, and that critic must resolve each patch within a joint action space that widens as patches are added, so the estimate deteriorates on larger domains. Two inherent features of the wall-control problem tie this restriction down, one corrupting the credit assignment and the other raising its cost. First, because the actuation must inject no net mass, the joint action is projected onto its zero-mean subspace before reaching the wall. Under this projection, the action applied at any single patch becomes dependent on all others, breaking the separability of per-patch credit (Cavallazzi et al., 2026 ) . Second, every training episode requires advancing a costly direct numerical simulation, so the wall-clock cost of collecting experience grows with the domain. Consequently, training is conventionally restricted to a minimal flow unit, and the policy is transferred only afterwards."},{"type":"para","text":"An evolution strategy removes both constraints. It treats the expected return as a function of the policy parameters. At each generation, a population of perturbed parameter vectors is evaluated over complete flow episodes. Each episode returns a scalar reward. The parameter update is a weighted average of the perturbations, with no backpropagation through the flow and no central critic (Rechenberg, 1973 ; Hansen & Ostermeier, 2001 ; Wierstra et al., 2014 ; Salimans et al., 2017 ) . Because whole episodes are scored, the per-agent credit assignment that the zero-net-mass constraint corrupts does not arise. The population evaluations are independent and parallelise across rollouts. The search therefore runs directly on the large evaluation domain, with no small-box surrogate. To our knowledge, this is the first application of an evolution strategy to the control of a turbulent flow."},{"type":"para","text":"The evolution-strategy controller provides a setting to address a question that has not been settled on a large domain: whether a learnt policy that reaches opposition-level drag reduction does so through the opposition mechanism or through a different reorganisation of the near-wall cycle. Its actuation is smooth and free of the bang-bang saturation that characterises degenerate controllers (Cavallazzi et al., 2026 ) . The controlled flow is therefore interpretable in physical terms. We characterise it with a wall-normal decomposition of the skin friction (Fukagata et al., 2002 ) , Reynolds-stress profiles scaled by each controller’s own friction velocity, the anisotropy invariants of the stress tensor (Lumley & Newman, 1977 ) , and the premultiplied near-wall spectra (del Álamo & Jiménez, 2003 ; del Álamo et al., 2004 ) ."},{"type":"heading","level":1,"text":"2 Problem set-up and methodology"},{"type":"heading","level":2,"text":"2.1 Flow configuration and numerics"},{"type":"para","text":"We perform direct numerical simulations of an incompressible flow in a plane channel at a constant flow rate by using CaNS (Costa, 2018 ; Costa et al., 2021 ) . The solver integrates the unsteady, incompressible Navier-Stokes equations using a low-storage third-order Runge–Kutta advancement within a pressure-projection algorithm. The friction Reynolds number is 𝑅𝑒 τ = H + ≃ 180 \\mathit{Re}_{\\tau}=H^{+}\\simeq 180 . The domain measures ( L x + , L y + , L z + ) ≃ ( 1922 , 576 , 360 ) (L_{x}^{+},L_{y}^{+},L_{z}^{+})\\simeq(1922,576,360) in viscous units, equivalently ( L x / H , L y / H , L z / H ) = ( 10.68 , 3.2 , 2 ) (L_{x}/H,L_{y}/H,L_{z}/H)=(10.68,3.2,2) , with H H the channel half-height. It is discretised on a 256 × 256 × 100 256\\times 256\\times 100 grid with wall-normal points clustered towards the walls. Throughout, u u , v v and w w denote the streamwise, spanwise and wall-normal velocity components, and p p the pressure. Control is applied as wall-normal blowing and suction w w ( x , y , t ) w_{w}(x,y,t) at the wall, Γ w \\Gamma_{w} , with zero net mass flux enforced at every instant, ∫ Γ w w w d x d y = 0 \\int_{\\Gamma_{w}}w_{w}\\,\\mathrm{d}x\\,\\mathrm{d}y=0 . The flow is sensed at a detection plane at z d + ≃ 14 z_{d}^{+}\\simeq 14 . The actuation interval is Δ t + ≃ 5 \\Delta t^{+}\\simeq 5 , longer than the fastest near-wall fluctuations but short compared to the typical streak lifetime."},{"type":"heading","level":2,"text":"2.2 Closed-loop controller and the evolution strategy"},{"type":"para","text":"The controller is a recurrent neural network that maps the velocity field at the detection plane to a wall-normal velocity field at the wall. At each actuation step the hidden state h t h_{t} is updated and a new actuation field a t a_{t} is produced. The zero-net-mass constraint is enforced as the final layer: the projection P : a ↦ a − a ¯ P:a\\mapsto a-\\bar{a} subtracts the spatial mean. The resulting w w ( ⋅ , t ) = P a t w_{w}(\\cdot,t)=Pa_{t} therefore satisfies ∫ Γ w w w d x d y = 0 \\int_{\\Gamma_{w}}w_{w}\\,\\mathrm{d}x\\,\\mathrm{d}y=0 at every instant for any parameter vector θ ∈ ℝ d \\theta\\in\\mathbb{R}^{d} . The same architecture is used by the GRU-MARL controller of Cavallazzi et al. ( 2026 ) , trained by multi-agent gradient descent (Lowe et al., 2017 ) on a minimal flow unit."},{"type":"para","text":"Here, the shared parameters θ \\theta are optimised directly on the large evaluation domain by an evolution strategy (Rechenberg, 1973 ; Wierstra et al., 2014 ; Salimans et al., 2017 ) . The optimisation target is the net energy saving over a complete episode. Let P p = | ⟨ ∂ x p ⟩ | U b P_{p}=|\\langle\\partial_{x}p\\rangle|U_{b} be the mean pumping power and W w = ( L x L y ) − 1 ∫ Γ w w w p d x d y W_{w}=(L_{x}L_{y})^{-1}\\int_{\\Gamma_{w}}w_{w}\\,p\\,\\mathrm{d}x\\,\\mathrm{d}y the net power delivered by the controller at the wall. The total dissipation is ε = P p + W w / H \\varepsilon=P_{p}+W_{w}/H , and the fitness is F ( θ ) = ( ε 0 − ε ( θ ) ) / ε 0 , F(\\theta)=\\left(\\varepsilon_{0}-\\varepsilon(\\theta)\\right)/\\varepsilon_{0}, where ε 0 \\varepsilon_{0} is the uncontrolled value. Two metrics commonly used to score wall controllers omit the wall power W w W_{w} entirely: the bare drag reduction 1 − | ⟨ ∂ x p ⟩ | / | ⟨ ∂ x p ⟩ 0 | 1-|\\langle\\partial_{x}p\\rangle|/|\\langle\\partial_{x}p\\rangle_{0}| , which tracks only the mean pressure gradient, and the cubic surrogate ∼ | w w | 3 \\sim|w_{w}|^{3} , which estimates actuation cost from the actuation amplitude rather than from the power exchanged with the wall. Because neither metric can register the energy a controller adds through the wall, a controller that injects energy can score well on either while lowering F ( θ ) F(\\theta) (Fukagata et al., 2009 ; Marusic et al., 2021 ; Cavallazzi et al., 2026 ) ."},{"type":"para","text":"Unlike gradient-based training, the evolution strategy requires no derivative of F F with respect to θ \\theta . Obtaining one would mean differentiating through both the DNS solver and the recurrent network; instead, each candidate parameter vector is scored by running one complete flow episode and reading off the resulting F F , with the simulation and the network treated as a black box. At each generation, n n pairs of parameter vectors are drawn by adding mirrored Gaussian perturbations to the current estimate m ∈ ℝ d m\\in\\mathbb{R}^{d} , { m ± σ 𝝃 k } k = 1 n \\{m\\pm\\sigma\\bm{\\xi}_{k}\\}_{k=1}^{n} with 𝝃 k ∼ 𝒩 ( 0 , 𝐈 ) \\bm{\\xi}_{k}\\sim\\mathcal{N}(0,\\mathbf{I}) , giving 2 n 2n members evaluated in parallel, each as one DNS episode initialised from the same flow state so that fitness differences reflect θ \\theta rather than the initial condition (Salimans et al., 2017 ) . The mean is then advanced along"},{"type":"para","text":"where υ k ± \\upsilon_{k}^{\\pm} is the centred rank of F ( m ± σ 𝝃 k ) F(m\\pm\\sigma\\bm{\\xi}_{k}) within the generation: each member’s position among the 2 n 2n fitness values, shifted so the ranks average to zero. This makes the update depend only on the relative ordering of the population and not on the numerical scale of F F (Salimans et al., 2017 ) . The parameter mean is then updated from g ^ \\hat{g} using Adam (Kingma & Ba, 2014 ) , a standard adaptive-step-size rule. The 2 n 2n DNS episodes within a generation are mutually independent and can be run concurrently, so given enough parallel resources the wall-clock cost of a generation reduces to that of a single episode."},{"type":"heading","level":2,"text":"2.3 Control cases and diagnostics"},{"type":"para","text":"Four flows are compared: the uncontrolled channel, opposition control, and two learnt controllers that share the architecture of § 2.2 and differ only in how their parameters were obtained. Opposition sets the wall velocity to the negative of the wall-normal velocity sensed at the detection plane, w w = − w ′ ( z d ) w_{w}=-w^{\\prime}(z_{d}) (Choi et al., 1994 ) . The first learnt controller is trained by gradient-based multi-agent learning on a minimal flow unit, within a centralised-training, decentralised-execution framework (Lowe et al., 2017 ; Foerster et al., 2018 ) constrained to keep the actuation physically coherent, and applied to the evaluation domain without further training (Cavallazzi et al., 2026 ) ; it is labelled GRU-MARL. The second is the same policy whose parameters are then optimised directly on the evaluation domain by the evolution strategy of § 2.2 , and is labelled ES. We omit the saturated, memoryless policies whose behaviour motivates a separate study of reward specification (Cavallazzi et al., 2026 ) , since the concern here is the mechanism of controllers that reduce drag at low actuation power."},{"type":"para","text":"Each controlled flow is characterised by its own friction velocity u τ = τ w / ρ u_{\\tau}=\\sqrt{\\tau_{w}/\\rho} (being τ w \\tau_{w} the wall shear stress and ρ \\rho the density), evaluated after the control has acted, and all wall-scaled quantities below are normalised by this flow-specific value."},{"type":"para","text":"The contribution of the Reynolds shear stress to the skin friction is quantified with the identity of Fukagata et al. ( 2002 ) , which decomposes the friction coefficient C f = τ w / ( 1 2 ρ U b 2 ) C_{f}=\\tau_{w}/(\\tfrac{1}{2}\\rho U_{b}^{2}) , with U b = 1 U_{b}=1 constant flow rate, into a laminar contribution C f lam C_{f}^{\\mathrm{lam}} at the same bulk Reynolds number and a wall-normal weighted integral of the Reynolds shear stress,"},{"type":"para","text":"where the integral runs from the wall ( z / H = 0 z/H=0 ) to the centreline ( z / H = 1 z/H=1 ) and the numerical prefactor of the plane-channel identity is c = 12 c=12 (Fukagata et al., 2002 ) . Scale information is taken from the premultiplied spectra of the velocity fluctuations at fixed wall-normal distance (del Álamo et al., 2004 ) ."},{"type":"heading","level":1,"text":"3 Results"},{"type":"para","text":"The comparison begins with the drag reduction itself. Optimised directly on the evaluation domain, the evolution-strategy controller reduces the skin friction by 26.1 % 26.1\\% , above the 17.4 % 17.4\\% obtained by the same architecture trained by gradient-based multi-agent learning on the minimal box and transferred without further training, and marginally above the 22.5 % 22.5\\% of opposition control. The gap between the two learnt controllers is the drag reduction lost when the gradient-based policy is transferred from its minimal training box to the larger channel, and recovered when the evolution strategy optimises on that channel directly. The friction velocities differ between the cases, falling from u τ ≈ 0.064 u_{\\tau}\\approx 0.064 in the uncontrolled flow to 0.056 0.056 for opposition, 0.058 0.058 for GRU-MARL and 0.055 0.055 for the evolution strategy. Reaching comparable drag reduction is the starting point of the comparison; the rest of the paper asks whether the underlying flow states coincide."},{"type":"para","text":"Figure 1 decomposes the skin friction across the channel with the identity ( 2 ), plotted against the wall-normal coordinate z / H z/H and scaled for every case by the single friction velocity of the uncontrolled flow, so that the area between the uncontrolled curve and a controlled one is that controller’s drag reduction. The controlled flows carry a smaller weighted Reynolds stress than the uncontrolled flow across the buffer layer and the region above it, and the cumulative curves show the reduction accruing over the same range of z / H z/H for all three. The relative split between layers is similar across the controllers, the buffer layer carrying about a quarter of the weighted stress in each case, while the turbulent fraction of the mean-energy budget falls from 0.41 0.41 in the uncontrolled flow to about 0.34 0.34 to 0.35 0.35 under control. The decomposition locates the reduction but does not on its own distinguish the controllers; the difference between them lies in how the near-wall stress is organised, which the remaining diagnostics address."},{"type":"heading","level":2,"text":"3.1 Reynolds-stress reorganisation"},{"type":"para","text":"Figure 2 shows the streamwise normal stress ⟨ u ′ u ′ ⟩ + \\left\\langle u^{\\prime}u^{\\prime}\\right\\rangle^{+} and the Reynolds shear stress ⟨ u ′ w ′ ⟩ + \\left\\langle u^{\\prime}w^{\\prime}\\right\\rangle^{+} , each normalised by its own friction velocity. The shear stress, which carries the drag, follows opposition for both learnt controllers, yet their streamwise normal stress differs from that of every other case, opposition included. The learnt controllers therefore reach an opposition-like shear stress through a streamwise fluctuation field that no single benchmark reproduces."},{"type":"para","text":"Figure 3 maps the same information onto the anisotropy-invariant triangle. The trajectories of the controlled flows through the buffer layer occupy different regions of the triangle: a coordinate-free confirmation that the controlled states are distinct and that depart differently from the uncontrolled trajectory. This is consistent with the stress profiles of figure 2 ."},{"type":"heading","level":2,"text":"3.2 Near-wall structure, actuation and scale selectivity"},{"type":"para","text":"Figure 4 shows instantaneous fields of the near-wall velocity and of the wall actuation for each controller. Opposition sets the wall velocity to the negative of the sensed wall-normal velocity fluctuations, so its actuation follows the footprint of the sweeps and ejections at the detection plane by construction. The learnt actuation instead aligns with the streamwise low- and high-speed streaks. Figure 6 makes the relationship explicit: the mean actuation conditioned on the local streamwise and wall-normal fluctuations varies principally with u ′ u^{\\prime} and less strongly with w ′ w^{\\prime} . Across the learnt controllers the actuation correlates with u ′ u^{\\prime} at about 0.5 0.5 to 0.6 0.6 , two to four times stronger in magnitude than its correlation with w ′ w^{\\prime} . The learnt actuation thus tracks the streamwise streaks, and opposition the wall-normal velocity. We describe this spatial relationship only, and do not address the order in which the actuation and the structures influence one another."},{"type":"para","text":"The joint probability density of ( u ′ , w ′ ) (u^{\\prime},w^{\\prime}) at the detection plane, in figure 7 , shows how the two learnt controllers redistribute the momentum-carrying events. In the uncontrolled flow the ejections and sweeps contribute almost equally. Under every controller the sweep contribution exceeds the ejection contribution, so the ejections are weakened preferentially, and the effect is strongest for the evolution-strategy controller, whose ejection contribution to the inner-scaled Reynolds shear stress ⟨ u ′ w ′ ⟩ + \\left\\langle u^{\\prime}w^{\\prime}\\right\\rangle^{+} falls to about − 0.10 -0.10 , from − 0.32 -0.32 in the uncontrolled flow. These are contributions to the shear stress, not the quadrant occupation probabilities annotated in figure 7 ."},{"type":"para","text":"Figure 8 shows the premultiplied spectra of the streamwise and wall-normal velocity fluctuations near the wall. The energy of the controlled flows is modified at the spanwise wavelengths of the near-wall streaks, around λ y + ≃ 100 \\lambda_{y}^{+}\\simeq 100 (del Álamo et al., 2004 ) , and the learnt controllers concentrate their effect there. The streamwise streak energy, peaked near λ x + ≃ 700 \\lambda_{x}^{+}\\simeq 700 in the uncontrolled and GRU-MARL flows, is shifted to shorter wavelengths under opposition and most strongly redistributed towards small scales by the evolution strategy, whose long streaks are broken into shorter fragments (figure 4 )."},{"type":"heading","level":2,"text":"3.3 Smoothness and statistical steadiness of the learnt control"},{"type":"para","text":"The comparison above is meaningful only if the learnt actuation controls the turbulence rather than settling into a fixed pattern set by the box. Figure 6 shows the distribution of wall-actuation values for each controller: it is smooth and stays away from the amplitude bound, with a root-mean-square amplitude of about 0.016 0.016 and a negligible saturated fraction. The streamwise-averaged actuation also remains statistically stationary and does not lock into a standing wave in the spanwise direction. The actuation is, then, an unsaturated response that tracks the evolving flow, not the saturated, frozen forcing that an objective scored on pumping power alone would favour."},{"type":"heading","level":1,"text":"4 Discussion"},{"type":"para","text":"An evolution strategy optimises a closed-loop wall controller directly on the large, fully turbulent channel at 𝑅𝑒 τ ≃ 180 \\mathit{Re}_{\\tau}\\simeq 180 , and recovers the drag reduction that the same architecture loses when a gradient-based multi-agent policy, trained on a minimal box, is transferred to that channel. The gain has a simple reading: the search scores each complete episode by a single energy-aware number and never assigns a separate reward to each wall patch, so the zero-net-mass constraint that links the patches becomes part of how the actuation is formed rather than a source of interference in the optimisation, and the controller is tuned on the same flow it will drive. It reaches a drag reduction of about 26 % 26\\% , above the 17 % 17\\% retained by the transferred minimal-box policy and marginally above the 22 % 22\\% of opposition control. The added cost is a population of rollouts at each generation, which can run in parallel because they are independent."},{"type":"para","text":"The four flows differ substantially in structure, and the three controlled states remain distinct even where their drag reductions are similar. Opposition and the evolution strategy remove comparable amounts of skin friction, yet the diagnostics place their flows in different parts of the buffer layer, with different Reynolds-stress profiles, each scaled by its own friction velocity, and different anisotropy. The evolution-strategy actuation aligns predominantly with the streamwise streaks, whereas opposition acts against the wall-normal velocity (Choi et al., 1994 ; Hammond et al., 1998 ; Stroh et al., 2015 ) ; in redistributing the momentum-carrying events at the detection plane it weakens the ejections preferentially and concentrates its effect near the spanwise streak wavelength λ y + ≃ 100 \\lambda_{y}^{+}\\simeq 100 . Spanwise forcing produces a comparable lateral reorganisation of the streaks (Touber & Leschziner, 2012 ) , though through an open-loop rather than a sensed actuation."},{"type":"para","text":"The clearest separation is in the streaks themselves. The instantaneous fields (figure 4 ) and the streamwise spectra (figure 8 ) show opposition mainly lowering the streak amplitude, whereas the evolution strategy breaks the long streaks into shorter fragments and moves the streamwise energy to smaller scales. That two controllers reaching similar drag reduction act so differently on the coherence of the near-wall cycle suggests different mechanisms, distinguished by the flow topology rather than by the drag figure alone. Because the controller is smooth and was optimised on the same large domain on which it is assessed, this separation reflects the controlled turbulence and not the size of a training box."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Skin-friction drag in wall-bounded turbulence is generated by quasi-streamwise vortices whose induced wall-normal velocities sustain the Reynolds shear stress above its laminar value (Orlandi & Jiménez, 1994 ) . The vortices are part of a self-sustaining near-wall cycle of low-speed streaks and quasi-streamwise vortices (Kline et al., 1967 ; Jiménez & Moin, 1991 ; Hamilton et al., 1995 ; Waleffe, 1997 ) that persists autonomously within the buffer layer even in the absence of outer-flow forcing (Jiménez & Pinelli, 1999 ) , making that layer the natural target for drag-reduction control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the case of established active flow control methods, like opposition control (Choi et al., 1994 ; Hammond et al., 1998 ; Stroh et al., 2015 ) , spanwise forcing and streamwise-travelling waves of spanwise velocity (Quadrio et al., 2009 ; Gatti & Quadrio, 2016 ; Touber & Leschziner, 2012 ) , the drag-reduction mechanism rests on a specific, physically motivated interaction with the near-wall structures, and the actuation is designed around that interaction from the outset. Controllers discovered by learnt data-driven algorithms carry no such prior; their actuation strategy emerges from the optimisation, and what physical interaction it exploits is not known in advance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reinforcement learning (RL) has been the most productive research area for those controllers. Applied to the classical plane channel flow at low Reynolds numbers, it has produced closed-loop wall policies that report drag reductions competitive with the physics-based benchmarks (Guastoni et al., 2023 ; Sonoda et al., 2023 ; Lee et al., 2023 ; Han & Huang, 2020 ; Zhou et al., 2025 ; Varela et al., 2022 ; Cavallazzi et al., 2025 ) , as surveyed in several reviews (Garnier et al., 2021 ; Brunton et al., 2020 ; Vignon et al., 2023b ) . A common way to keep the formulation tractable is a shared, translationally invariant multi-agent policy (Vignon et al., 2023a ; Guastoni et al., 2023 ) , trained with a centralised critic on a small periodic domain, executed independently at each wall patch, then applied to a larger channel. This template underlies numerous subsequent studies (Sonoda et al., 2023 ; Cavallazzi et al., 2025 ; Wälchli et al., 2024 ; Bae & Koumoutsakos, 2022 ; Suárez et al., 2024 ; Rabault & Kuhnle, 2019 ) . The drag reduction headline figures reported on minimal domains must, however, be read with care: a policy evaluated in a minimal flow unit can drive the flow towards relaminarisation, where the pressure gradient falls to its laminar value and the reported drag reduction saturates at a value that reflects the box rather than the controller (Cavallazzi et al., 2026 ) . A representative comparison therefore requires a domain large enough to keep the flow fully turbulent throughout, the setting adopted here.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite the need for a large evaluation domain, training is in practice confined to a small one by the structure of the multi-agent update rather than by simulation cost alone. A gradient step adjusts the shared policy parameters from a per-patch return whose credit is assigned by a central critic, and that critic must resolve each patch within a joint action space that widens as patches are added, so the estimate deteriorates on larger domains. Two inherent features of the wall-control problem tie this restriction down, one corrupting the credit assignment and the other raising its cost. First, because the actuation must inject no net mass, the joint action is projected onto its zero-mean subspace before reaching the wall. Under this projection, the action applied at any single patch becomes dependent on all others, breaking the separability of per-patch credit (Cavallazzi et al., 2026 ) . Second, every training episode requires advancing a costly direct numerical simulation, so the wall-clock cost of collecting experience grows with the domain. Consequently, training is conventionally restricted to a minimal flow unit, and the policy is transferred only afterwards.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An evolution strategy removes both constraints. It treats the expected return as a function of the policy parameters. At each generation, a population of perturbed parameter vectors is evaluated over complete flow episodes. Each episode returns a scalar reward. The parameter update is a weighted average of the perturbations, with no backpropagation through the flow and no central critic (Rechenberg, 1973 ; Hansen & Ostermeier, 2001 ; Wierstra et al., 2014 ; Salimans et al., 2017 ) . Because whole episodes are scored, the per-agent credit assignment that the zero-net-mass constraint corrupts does not arise. The population evaluations are independent and parallelise across rollouts. The search therefore runs directly on the large evaluation domain, with no small-box surrogate. To our knowledge, this is the first application of an evolution strategy to the control of a turbulent flow.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The evolution-strategy controller provides a setting to address a question that has not been settled on a large domain: whether a learnt policy that reaches opposition-level drag reduction does so through the opposition mechanism or through a different reorganisation of the near-wall cycle. Its actuation is smooth and free of the bang-bang saturation that characterises degenerate controllers (Cavallazzi et al., 2026 ) . The controlled flow is therefore interpretable in physical terms. We characterise it with a wall-normal decomposition of the skin friction (Fukagata et al., 2002 ) , Reynolds-stress profiles scaled by each controller’s own friction velocity, the anisotropy invariants of the stress tensor (Lumley & Newman, 1977 ) , and the premultiplied near-wall spectra (del Álamo & Jiménez, 2003 ; del Álamo et al., 2004 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Problem set-up and methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Flow configuration and numerics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We perform direct numerical simulations of an incompressible flow in a plane channel at a constant flow rate by using CaNS (Costa, 2018 ; Costa et al., 2021 ) . The solver integrates the unsteady, incompressible Navier-Stokes equations using a low-storage third-order Runge–Kutta advancement within a pressure-projection algorithm. The friction Reynolds number is 𝑅𝑒 τ = H + ≃ 180 \\\\mathit{Re}_{\\\\tau}=H^{+}\\\\simeq 180 . The domain measures ( L x + , L y + , L z + ) ≃ ( 1922 , 576 , 360 ) (L_{x}^{+},L_{y}^{+},L_{z}^{+})\\\\simeq(1922,576,360) in viscous units, equivalently ( L x / H , L y / H , L z / H ) = ( 10.68 , 3.2 , 2 ) (L_{x}/H,L_{y}/H,L_{z}/H)=(10.68,3.2,2) , with H H the channel half-height. It is discretised on a 256 × 256 × 100 256\\\\times 256\\\\times 100 grid with wall-normal points clustered towards the walls. Throughout, u u , v v and w w denote the streamwise, spanwise and wall-normal velocity components, and p p the pressure. Control is applied as wall-normal blowing and suction w w ( x , y , t ) w_{w}(x,y,t) at the wall, Γ w \\\\Gamma_{w} , with zero net mass flux enforced at every instant, ∫ Γ w w w d x d y = 0 \\\\int_{\\\\Gamma_{w}}w_{w}\\\\,\\\\mathrm{d}x\\\\,\\\\mathrm{d}y=0 . The flow is sensed at a detection plane at z d + ≃ 14 z_{d}^{+}\\\\simeq 14 . The actuation interval is Δ t + ≃ 5 \\\\Delta t^{+}\\\\simeq 5 , longer than the fastest near-wall fluctuations but short compared to the typical streak lifetime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Closed-loop controller and the evolution strategy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The controller is a recurrent neural network that maps the velocity field at the detection plane to a wall-normal velocity field at the wall. At each actuation step the hidden state h t h_{t} is updated and a new actuation field a t a_{t} is produced. The zero-net-mass constraint is enforced as the final layer: the projection P : a ↦ a − a ¯ P:a\\\\mapsto a-\\\\bar{a} subtracts the spatial mean. The resulting w w ( ⋅ , t ) = P a t w_{w}(\\\\cdot,t)=Pa_{t} therefore satisfies ∫ Γ w w w d x d y = 0 \\\\int_{\\\\Gamma_{w}}w_{w}\\\\,\\\\mathrm{d}x\\\\,\\\\mathrm{d}y=0 at every instant for any parameter vector θ ∈ ℝ d \\\\theta\\\\in\\\\mathbb{R}^{d} . The same architecture is used by the GRU-MARL controller of Cavallazzi et al. ( 2026 ) , trained by multi-agent gradient descent (Lowe et al., 2017 ) on a minimal flow unit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, the shared parameters θ \\\\theta are optimised directly on the large evaluation domain by an evolution strategy (Rechenberg, 1973 ; Wierstra et al., 2014 ; Salimans et al., 2017 ) . The optimisation target is the net energy saving over a complete episode. Let P p = | ⟨ ∂ x p ⟩ | U b P_{p}=|\\\\langle\\\\partial_{x}p\\\\rangle|U_{b} be the mean pumping power and W w = ( L x L y ) − 1 ∫ Γ w w w p d x d y W_{w}=(L_{x}L_{y})^{-1}\\\\int_{\\\\Gamma_{w}}w_{w}\\\\,p\\\\,\\\\mathrm{d}x\\\\,\\\\mathrm{d}y the net power delivered by the controller at the wall. The total dissipation is ε = P p + W w / H \\\\varepsilon=P_{p}+W_{w}/H , and the fitness is F ( θ ) = ( ε 0 − ε ( θ ) ) / ε 0 , F(\\\\theta)=\\\\left(\\\\varepsilon_{0}-\\\\varepsilon(\\\\theta)\\\\right)/\\\\varepsilon_{0}, where ε 0 \\\\varepsilon_{0} is the uncontrolled value. Two metrics commonly used to score wall controllers omit the wall power W w W_{w} entirely: the bare drag reduction 1 − | ⟨ ∂ x p ⟩ | / | ⟨ ∂ x p ⟩ 0 | 1-|\\\\langle\\\\partial_{x}p\\\\rangle|/|\\\\langle\\\\partial_{x}p\\\\rangle_{0}| , which tracks only the mean pressure gradient, and the cubic surrogate ∼ | w w | 3 \\\\sim|w_{w}|^{3} , which estimates actuation cost from the actuation amplitude rather than from the power exchanged with the wall. Because neither metric can register the energy a controller adds through the wall, a controller that injects energy can score well on either while lowering F ( θ ) F(\\\\theta) (Fukagata et al., 2009 ; Marusic et al., 2021 ; Cavallazzi et al., 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike gradient-based training, the evolution strategy requires no derivative of F F with respect to θ \\\\theta . Obtaining one would mean differentiating through both the DNS solver and the recurrent network; instead, each candidate parameter vector is scored by running one complete flow episode and reading off the resulting F F , with the simulation and the network treated as a black box. At each generation, n n pairs of parameter vectors are drawn by adding mirrored Gaussian perturbations to the current estimate m ∈ ℝ d m\\\\in\\\\mathbb{R}^{d} , { m ± σ 𝝃 k } k = 1 n \\\\{m\\\\pm\\\\sigma\\\\bm{\\\\xi}_{k}\\\\}_{k=1}^{n} with 𝝃 k ∼ 𝒩 ( 0 , 𝐈 ) \\\\bm{\\\\xi}_{k}\\\\sim\\\\mathcal{N}(0,\\\\mathbf{I}) , giving 2 n 2n members evaluated in parallel, each as one DNS episode initialised from the same flow state so that fitness differences reflect θ \\\\theta rather than the initial condition (Salimans et al., 2017 ) . The mean is then advanced along\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where υ k ± \\\\upsilon_{k}^{\\\\pm} is the centred rank of F ( m ± σ 𝝃 k ) F(m\\\\pm\\\\sigma\\\\bm{\\\\xi}_{k}) within the generation: each member’s position among the 2 n 2n fitness values, shifted so the ranks average to zero. This makes the update depend only on the relative ordering of the population and not on the numerical scale of F F (Salimans et al., 2017 ) . The parameter mean is then updated from g ^ \\\\hat{g} using Adam (Kingma & Ba, 2014 ) , a standard adaptive-step-size rule. The 2 n 2n DNS episodes within a generation are mutually independent and can be run concurrently, so given enough parallel resources the wall-clock cost of a generation reduces to that of a single episode.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Control cases and diagnostics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four flows are compared: the uncontrolled channel, opposition control, and two learnt controllers that share the architecture of § 2.2 and differ only in how their parameters were obtained. Opposition sets the wall velocity to the negative of the wall-normal velocity sensed at the detection plane, w w = − w ′ ( z d ) w_{w}=-w^{\\\\prime}(z_{d}) (Choi et al., 1994 ) . The first learnt controller is trained by gradient-based multi-agent learning on a minimal flow unit, within a centralised-training, decentralised-execution framework (Lowe et al., 2017 ; Foerster et al., 2018 ) constrained to keep the actuation physically coherent, and applied to the evaluation domain without further training (Cavallazzi et al., 2026 ) ; it is labelled GRU-MARL. The second is the same policy whose parameters are then optimised directly on the evaluation domain by the evolution strategy of § 2.2 , and is labelled ES. We omit the saturated, memoryless policies whose behaviour motivates a separate study of reward specification (Cavallazzi et al., 2026 ) , since the concern here is the mechanism of controllers that reduce drag at low actuation power.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each controlled flow is characterised by its own friction velocity u τ = τ w / ρ u_{\\\\tau}=\\\\sqrt{\\\\tau_{w}/\\\\rho} (being τ w \\\\tau_{w} the wall shear stress and ρ \\\\rho the density), evaluated after the control has acted, and all wall-scaled quantities below are normalised by this flow-specific value.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The contribution of the Reynolds shear stress to the skin friction is quantified with the identity of Fukagata et al. ( 2002 ) , which decomposes the friction coefficient C f = τ w / ( 1 2 ρ U b 2 ) C_{f}=\\\\tau_{w}/(\\\\tfrac{1}{2}\\\\rho U_{b}^{2}) , with U b = 1 U_{b}=1 constant flow rate, into a laminar contribution C f lam C_{f}^{\\\\mathrm{lam}} at the same bulk Reynolds number and a wall-normal weighted integral of the Reynolds shear stress,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the integral runs from the wall ( z / H = 0 z/H=0 ) to the centreline ( z / H = 1 z/H=1 ) and the numerical prefactor of the plane-channel identity is c = 12 c=12 (Fukagata et al., 2002 ) . Scale information is taken from the premultiplied spectra of the velocity fluctuations at fixed wall-normal distance (del Álamo et al., 2004 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The comparison begins with the drag reduction itself. Optimised directly on the evaluation domain, the evolution-strategy controller reduces the skin friction by 26.1 % 26.1\\\\% , above the 17.4 % 17.4\\\\% obtained by the same architecture trained by gradient-based multi-agent learning on the minimal box and transferred without further training, and marginally above the 22.5 % 22.5\\\\% of opposition control. The gap between the two learnt controllers is the drag reduction lost when the gradient-based policy is transferred from its minimal training box to the larger channel, and recovered when the evolution strategy optimises on that channel directly. The friction velocities differ between the cases, falling from u τ ≈ 0.064 u_{\\\\tau}\\\\approx 0.064 in the uncontrolled flow to 0.056 0.056 for opposition, 0.058 0.058 for GRU-MARL and 0.055 0.055 for the evolution strategy. Reaching comparable drag reduction is the starting point of the comparison; the rest of the paper asks whether the underlying flow states coincide.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 decomposes the skin friction across the channel with the identity ( 2 ), plotted against the wall-normal coordinate z / H z/H and scaled for every case by the single friction velocity of the uncontrolled flow, so that the area between the uncontrolled curve and a controlled one is that controller’s drag reduction. The controlled flows carry a smaller weighted Reynolds stress than the uncontrolled flow across the buffer layer and the region above it, and the cumulative curves show the reduction accruing over the same range of z / H z/H for all three. The relative split between layers is similar across the controllers, the buffer layer carrying about a quarter of the weighted stress in each case, while the turbulent fraction of the mean-energy budget falls from 0.41 0.41 in the uncontrolled flow to about 0.34 0.34 to 0.35 0.35 under control. The decomposition locates the reduction but does not on its own distinguish the controllers; the difference between them lies in how the near-wall stress is organised, which the remaining diagnostics address.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Reynolds-stress reorganisation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 shows the streamwise normal stress ⟨ u ′ u ′ ⟩ + \\\\left\\\\langle u^{\\\\prime}u^{\\\\prime}\\\\right\\\\rangle^{+} and the Reynolds shear stress ⟨ u ′ w ′ ⟩ + \\\\left\\\\langle u^{\\\\prime}w^{\\\\prime}\\\\right\\\\rangle^{+} , each normalised by its own friction velocity. The shear stress, which carries the drag, follows opposition for both learnt controllers, yet their streamwise normal stress differs from that of every other case, opposition included. The learnt controllers therefore reach an opposition-like shear stress through a streamwise fluctuation field that no single benchmark reproduces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 maps the same information onto the anisotropy-invariant triangle. The trajectories of the controlled flows through the buffer layer occupy different regions of the triangle: a coordinate-free confirmation that the controlled states are distinct and that depart differently from the uncontrolled trajectory. This is consistent with the stress profiles of figure 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Near-wall structure, actuation and scale selectivity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 shows instantaneous fields of the near-wall velocity and of the wall actuation for each controller. Opposition sets the wall velocity to the negative of the sensed wall-normal velocity fluctuations, so its actuation follows the footprint of the sweeps and ejections at the detection plane by construction. The learnt actuation instead aligns with the streamwise low- and high-speed streaks. Figure 6 makes the relationship explicit: the mean actuation conditioned on the local streamwise and wall-normal fluctuations varies principally with u ′ u^{\\\\prime} and less strongly with w ′ w^{\\\\prime} . Across the learnt controllers the actuation correlates with u ′ u^{\\\\prime} at about 0.5 0.5 to 0.6 0.6 , two to four times stronger in magnitude than its correlation with w ′ w^{\\\\prime} . The learnt actuation thus tracks the streamwise streaks, and opposition the wall-normal velocity. We describe this spatial relationship only, and do not address the order in which the actuation and the structures influence one another.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The joint probability density of ( u ′ , w ′ ) (u^{\\\\prime},w^{\\\\prime}) at the detection plane, in figure 7 , shows how the two learnt controllers redistribute the momentum-carrying events. In the uncontrolled flow the ejections and sweeps contribute almost equally. Under every controller the sweep contribution exceeds the ejection contribution, so the ejections are weakened preferentially, and the effect is strongest for the evolution-strategy controller, whose ejection contribution to the inner-scaled Reynolds shear stress ⟨ u ′ w ′ ⟩ + \\\\left\\\\langle u^{\\\\prime}w^{\\\\prime}\\\\right\\\\rangle^{+} falls to about − 0.10 -0.10 , from − 0.32 -0.32 in the uncontrolled flow. These are contributions to the shear stress, not the quadrant occupation probabilities annotated in figure 7 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 8 shows the premultiplied spectra of the streamwise and wall-normal velocity fluctuations near the wall. The energy of the controlled flows is modified at the spanwise wavelengths of the near-wall streaks, around λ y + ≃ 100 \\\\lambda_{y}^{+}\\\\simeq 100 (del Álamo et al., 2004 ) , and the learnt controllers concentrate their effect there. The streamwise streak energy, peaked near λ x + ≃ 700 \\\\lambda_{x}^{+}\\\\simeq 700 in the uncontrolled and GRU-MARL flows, is shifted to shorter wavelengths under opposition and most strongly redistributed towards small scales by the evolution strategy, whose long streaks are broken into shorter fragments (figure 4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Smoothness and statistical steadiness of the learnt control\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The comparison above is meaningful only if the learnt actuation controls the turbulence rather than settling into a fixed pattern set by the box. Figure 6 shows the distribution of wall-actuation values for each controller: it is smooth and stays away from the amplitude bound, with a root-mean-square amplitude of about 0.016 0.016 and a negligible saturated fraction. The streamwise-averaged actuation also remains statistically stationary and does not lock into a standing wave in the spanwise direction. The actuation is, then, an unsaturated response that tracks the evolving flow, not the saturated, frozen forcing that an objective scored on pumping power alone would favour.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An evolution strategy optimises a closed-loop wall controller directly on the large, fully turbulent channel at 𝑅𝑒 τ ≃ 180 \\\\mathit{Re}_{\\\\tau}\\\\simeq 180 , and recovers the drag reduction that the same architecture loses when a gradient-based multi-agent policy, trained on a minimal box, is transferred to that channel. The gain has a simple reading: the search scores each complete episode by a single energy-aware number and never assigns a separate reward to each wall patch, so the zero-net-mass constraint that links the patches becomes part of how the actuation is formed rather than a source of interference in the optimisation, and the controller is tuned on the same flow it will drive. It reaches a drag reduction of about 26 % 26\\\\% , above the 17 % 17\\\\% retained by the transferred minimal-box policy and marginally above the 22 % 22\\\\% of opposition control. The added cost is a population of rollouts at each generation, which can run in parallel because they are independent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The four flows differ substantially in structure, and the three controlled states remain distinct even where their drag reductions are similar. Opposition and the evolution strategy remove comparable amounts of skin friction, yet the diagnostics place their flows in different parts of the buffer layer, with different Reynolds-stress profiles, each scaled by its own friction velocity, and different anisotropy. The evolution-strategy actuation aligns predominantly with the streamwise streaks, whereas opposition acts against the wall-normal velocity (Choi et al., 1994 ; Hammond et al., 1998 ; Stroh et al., 2015 ) ; in redistributing the momentum-carrying events at the detection plane it weakens the ejections preferentially and concentrates its effect near the spanwise streak wavelength λ y + ≃ 100 \\\\lambda_{y}^{+}\\\\simeq 100 . Spanwise forcing produces a comparable lateral reorganisation of the streaks (Touber & Leschziner, 2012 ) , though through an open-loop rather than a sensed actuation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The clearest separation is in the streaks themselves. The instantaneous fields (figure 4 ) and the streamwise spectra (figure 8 ) show opposition mainly lowering the streak amplitude, whereas the evolution strategy breaks the long streaks into shorter fragments and moves the streamwise energy to smaller scales. That two controllers reaching similar drag reduction act so differently on the coherence of the near-wall cycle suggests different mechanisms, distinguished by the flow topology rather than by the drag figure alone. Because the controller is smooth and was optimised on the same large domain on which it is assessed, this separation reflects the controlled turbulence and not the size of a training box.\"\n }\n]","isConversational":false}}},{"rowIdx":292,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.13107"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"DeepCormack: Fermi surface tomography using model-based data-driven algorithms"},"authors":{"kind":"list like","value":["Georg F. B. Lovric","Bryn Drury","Carola-Bibiane Schönlieb","Stephen B. Dugdale","Ander Biguri"],"string":"[\n \"Georg F. B. Lovric\",\n \"Bryn Drury\",\n \"Carola-Bibiane Schönlieb\",\n \"Stephen B. Dugdale\",\n \"Ander Biguri\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.13107v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"The Fermi surface of a material is the surface in reciprocal space (or k -space) separating occupied from unoccupied electronic states at zero temperature ( T = 0 K T=0{\\rm K} ). Its shape is determined entirely by the electronic band structure, i.e. the relationship between electron energy and crystal momentum evaluated at the chemical potential. This shape governs many of the key properties of a metal, including electrical and thermal conductivity, magnetism, and more exotic phenomena such as superconductivity. The Fermi surface is not physical, it is a representation of which electronic states are available for low-energy excitations. Almost all low-energy physics in a metal is controlled by electrons near the Fermi surface: states deeper within it are frozen by the Pauli exclusion principle and cannot participate in transport, magnetic ordering, or Cooper pairing [ 16 , 7 ] . In the free-electron model, the Fermi surface is a simple sphere in k -space, but in real materials it becomes more complex due to the periodic crystal potential leading to electronic band structures that disperse anisotropically. This can lead to Fermi surfaces which comprise multiple interconnected sheets with topologies which are consistent with the underlying crystal symmetry [ 7 ] . These deviations in the Fermi surface topology, in comparison to the free-electron picture, can play a pivotal role in the physical properties of metals, for example, influencing the appearance of superconductivity, magnetic order and charge-density waves [ 7 ] . The geometry of the Fermi surface therefore encodes a great deal about what a metal can do, and having more accurate methods for measuring it is therefore important."},{"type":"para","text":"Angular correlation of annihilation radiation (ACAR) is one such method [ 8 , 18 ] . It is also a method which is bulk sensitive, can be applied at room temperature, and which is suitable for disordered alloys [ 9 , 12 ] . When positrons are implanted into a sample, they thermalise quickly before annihilating with an electron, emitting two photons. As the thermal positrons carry negligible momentum at the point of annihilation, the small angular deviations of the photon pair from exact collinearity (on the order of milliradians) essentially encodes the momentum of the electron immediately prior to annihilation. In a 2D-ACAR experiment, two position-sensitive detectors placed on either side of the sample record the coordinates of the coincident gamma-photons which impinge on each detector, allowing the angular deviations of each detected annihilation event to be recorded. Accumulating on the order of ∼ 10 8 \\sim 10^{8} coincidence events produces a 2D projection of the underlying three-dimensional two-photon momentum density (TPMD). By repeating this at different sample orientations (corresponding to different crystallographic directions), a set of projections is obtained from which the full 3D TPMD can be reconstructed. The features of the Fermi surface geometry can be extracted from the reconstructed density by converting from p -space to k -space via the Lock-Crisp-West (LCW) theorem [ 20 , 8 ] . This LCW procedure sums over reciprocal lattice vector contributions to reduce the extended-zone momentum density into the reduced Brillouin zone, giving us the k -space density which represents the occupation number as seen by the positron . The Fermi surface then appears as a discontinuity in this occupation number, and can be recovered by, for example, computing the gradient. Some further details about the experimental and theoretical background to the measurement of the Fermi surface by the 2D-ACAR method can be found in Refs. [ 14 , 6 ] ."},{"type":"para","text":"Recovering the 3D TPMD from a small number of projections remains a challenging inverse problem. Several reconstruction algorithms have been developed for this purpose, including filtered backprojection, spherical harmonic methods [ 22 , 23 ] , and more recently regularisation-based approaches [ 27 , 28 , 19 ] . In this work, we focus on the planar reconstruction method originally proposed by Cormack [ 4 , 5 , 3 ] and later adapted for Fermi surface studies by Kontrym-Sznajd [ 17 ] . The modified Cormack method (hereafter MCM) is well-suited to 2D-ACAR data because its least-squares properties handle experimental errors through natural truncation of the series, and because it enforces a consistency condition on the projection data that can correct for inconsistencies introduced by noise or normalisation errors. The underlying method uses polar Fourier series, expanding the projection data into sets of orthogonal polynomials (Chebyshev and Zernike), to carry out the reconstruction. The inherent symmetry of the crystals being studied and the relatively smooth nature of the reconstructed densities, make it a particularly powerful technique for recovering information about the Fermi surface."},{"type":"para","text":"Despite the advantages of the MCM, acquiring 2D-ACAR data of sufficient quality for Fermi surface studies is extremely time-consuming in practice. To make measurements with sufficiently high angular resolution (recalling that it is the angle that is proportional to the electron momentum), the joint requirements of high detector efficiency and acceptable position resolution mean that there can be up to 24m between the two detectors. The solid angle subtended by the detectors is therefore very small, resulting in relatively low coincidence count rates in the range of hundreds of counts per second. There is an inherent trade-off between acquisition time and that overall statistical precision of a spectrum, with the collection of more annihilation events reducing noise but requiring longer measurements. With a radioisotope source (typically 22 Na), obtaining a single high-quality data set comprising five to six projections at 200 200 million counts each can takes approximately three to four months. A further complication arises for reconstruction algorithms which operate in a slice-by-slice manner: as the number of recorded counts decreases away from the centre of the projections (corresponding to zero momentum), noise will vary systematically across the reconstructed slices. This results in the reconstructed densities at points away from the centre of the momentum density having much lower quality than ones closer to it. An approach which can simultaneously improve the momentum density reconstruction quality from ACAR data while at the same time lowering the acquisition time (meaning noisier projections) would therefore be desirable."},{"type":"para","text":"Given that 2D-ACAR shares its core measurement geometry with positron emission tomography (PET), there is good reason to expect that machine learning advances in medical image reconstruction could translate productively to Fermi surface tomography, given the significant advances in the field in the last few years, e.g. [ 25 , 11 , 1 , 15 ] . The goal of this work is to investigate whether similar approaches can positively translate to ACAR tomography, using supervised neural networks in conjunction with classical reconstruction techniques to improve the recovered Fermi surface details."},{"type":"para","text":"To this end, we introduce DeepCormack, a hybrid framework combining the Cormack-Kontrym-Sznajd method with supervised deep learning. The network includes a 1D convolutional neural network trained in projection space, a multi-layer perceptron in radial density function space, and a 2D UNet in image space, learning corrections that reduce noise and convolution artefacts while partially compensating for the information loss inherent in the limited angular sampling from 5 5 projections. As reconstruction quality degrades away from the centre of a momentum density, the effect of conditioning the UNet on the statistical precision (characterised by the total number of counts in a spectrum) to make it robust to the variation in noise is also studied."},{"type":"para","text":"To support training and evaluation, we also propose a data-driven simulation framework based on singular value decomposition (SVD) and dynamic mode decomposition (DMD), which enables the generation of realistic, continuously varying synthetic 3D TPMD volumes, based entirely on a DFT-generated reference momentum density for copper [ 10 ] . Copper was chosen for its simple but anisotropic Fermi surface topology, as well as having the same crystal structure (face-centred cubic) as the intermetallic compound ZrZn 2 , for which we have DFT calculations of the TPMD available. This represents an excellent intermediate step between the relatively simple momentum density of copper (with its single Fermi surface sheet) and the challenge of reconstructing from real high quality experimental data (which have been measured by Major et al. [ 21 ] ). With a much larger lattice constant (factor of two) compared to copper, the impact of resolution on the features across the much smaller Brillouin zone is likely to be more significant. Furthermore, with multiple Fermi surface sheets, its momentum density is quite different from copper which provides an excellent test of the training data based on copper. Together, these components form a physically consistent and scalable pipeline for machine learning-assisted ACAR reconstruction under experimentally realistic conditions."},{"type":"para","text":"By investigating the application of modern deep learning tools in ACAR tomography, our hope is to open up the area of Fermiology to the wider inverse problem community. Finding techniques that allow for improved reconstruction while lowering the required statistical precision of the projections can enable faster and more efficient characterization of a material’s properties, and make the study of Fermi surfaces more accessible."},{"type":"para","text":"The paper is organised as follows: Section 2 presents the main algorithm and data generation approach. Section 3 presents experimental results of DeepCormack for different model configurations, across acquisition counts ( 10 − 200 10-200 M), and for an out-of-distribution (OOD) sample of ZrZn 2 to test its ability to generalize. Section 4 discusses these results in depth, while Section 5 summarises the main findings, discusses current limitations, and outlines improvements to the method."},{"type":"heading","level":4,"text":"The main contributions can be summarized as follows:"},{"type":"para","text":"A derivation of the modified Cormack method reconstruction method in operator notation for the field of inverse problems."},{"type":"para","text":"Novel method using SVD and DMD for generating sufficiently realistic 3D-TPMD data to train ML models with;"},{"type":"para","text":"Framework for integrating neural networks into the modified Cormack method for substantially improved Fermi surface reconstruction with ACAR tomography, sufficiently flexible enough to allow for slice-by-slice reconstruction despite varying noise levels across the TPMD volume;"},{"type":"para","text":"Practical demonstrations of DeepCormack’s utility, on both pseudo-real data and DFT-generated TPMD data under simulated experimental conditions."},{"type":"heading","level":1,"text":"2 Methods"},{"type":"para","text":"We propose a family of methods which integrate neural networks into the modified Cormack method (described in section 2.1 ) to improve the two-photon momentum density reconstruction that can be achieved via conventional angular correlation of electron-positron annihilation radiation measurements. This framework, which we have dubbed DeepCormack ( 2.2 ), requires supervised learning. Section 2.3 proposes a data simulation method which allows for paired ground truth and simulated measurement (synthetic) 3D TPMDs to be generated at scale."},{"type":"heading","level":2,"text":"2.1 Fermi Surface Tomography"},{"type":"para","text":"The Fermi surface tomography mathematical model is similar to computed tomography (CT), however given its development happened in parallel to mathematical advances in CT, mathematical descriptions that link both are missing from the literature. This section introduces the Fermi surface tomography forward model, and the one of the most successful and widely applied reconstruction algorithms, namely the Modified Cormack Method (MCM), describing the mathematical foundations and linking it to an operator form of the reconstruction, which can open up the model for further analysis from the inverse problems community (analysis not presented in this work). Thus the description of the MCM presented here differs from the standard one in the Fermiology literature. We recommend the works of [ 4 , 5 , 17 ] for the more standard description of the MCM."},{"type":"para","text":"Firstly, the forward model of the Fermi surface tomography is the shown in Equation 1 as"},{"type":"para","text":"where f ¯ \\bar{f} is the measurement as a vector containing the positive half of our normalized projections as f ¯ ∈ ℝ L {\\bar{f}}\\in\\mathbb{R}^{L} , where L = L p ⋅ Φ L=L_{p}\\cdot\\Phi being the number of measured pixels L p L_{p} in each measured projection Φ \\Phi and ρ ¯ 2 γ ∈ ℝ 2 L P ⋅ 2 L p \\bar{\\rho}^{2\\gamma}\\in\\mathbb{R}^{2L_{P}\\cdot 2L_{p}} is the vectorized reconstructed TPMD image, finally e ~ \\tilde{e} correspond to measurement noise. The operator ℛ \\mathcal{R} is the Radon transform."},{"type":"para","text":"In practice, the measurements are 2D over Φ \\Phi projections at various angles φ \\varphi of a sample’s 3D TPMD. Each projection f φ f_{\\varphi} will have shape f φ ∈ ℝ L p × L p f_{\\varphi}\\in\\mathbb{R}^{L_{p}\\times L_{p}} , but due to mirror symmetry, only one quadrant of the data will be needed, and the other 3 quadrants will be symmetrically folded and averaged to reduce measurement noise, and finally normalized against the first projection. Given the operator is independent per slice of the 2D measurement f φ f_{\\varphi} , we define f f as a single row of f φ f_{\\varphi} , for every φ \\varphi and its vectorized version, f ¯ \\bar{f} . This allows us to define the operators in a more simple way, and highlights that the 3D-TPMD reconstruction process is comprised of a stack of independent 2D problems."},{"type":"para","text":"In tomographic applications where the Radon transform is the operator, the pseudo-inverse ℛ † \\mathcal{R}^{\\dagger} (or reconstruction operator) is often represented by the Filtered Backprojection (FBP) algorithm. However, this is not the case in Fermi surface tomography, as the extremely limited number of projections causes the FBP algorithm to struggle. The MCM, by instead exploiting the inherent known symmetries of the reconstruction, produces superior results and thus has been widely applied."},{"type":"heading","level":3,"text":"2.1.1 The Modified Cormack Method"},{"type":"para","text":"The MCM algorithm can be written it in operator notation, as:"},{"type":"para","text":"where each of the matrices represent a step of the MCM algorithm. Firstly 𝒞 ∈ ℝ M × L \\mathcal{C}\\in\\mathbb{R}^{M\\times L} is a linear interpolator operator, in particular converting each row of the detector from a linear coordinate system sampled from the indices t = cos ( δ ) t=\\cos(\\delta) , with δ \\delta being a vector that linearly samples from [ 0 , π 2 ] [0,\\frac{\\pi}{2}] of size Δ \\Delta , making M = Δ ⋅ Φ M=\\Delta\\cdot\\Phi . This step is often described in the MCM method as a projection of the measurement into the domain of the unit circle with polar coordinates."},{"type":"para","text":"This radial sampling of the measurement allows to expand the data into polar Fourier series, defined by operator 𝒫 ∈ ℝ M × M \\mathcal{P}\\in\\mathbb{R}^{M\\times M} . Let N ( t , φ ) := ( 𝒞 f ) ( t , φ ) N(t,\\varphi):=(\\mathcal{C}f)(t,\\varphi) denote the projection data expressed in the polar coordinates produced by the interpolation step 𝒞 \\mathcal{C} . This operator relies on an important observation, and prior information, of the crystal being measured and its symmetries. G G denotes the reciprocal lattice vector of the crystal being studied, and let | G | |G| denote its order, i.e. the fold of rotational symmetry. For a face-centred cubic (FCC) structure, this corresponds to | G | = 4 |G|=4 — which all of the samples studied in this work have. Because N ( t , φ ) N(t,\\varphi) must be invariant under a rotation of 2 π / | G | 2\\pi/|G| , only harmonics of order n ≡ 0 mod | G | n\\equiv 0\\mod{|G|} can appear in its Fourier expansion, so that"},{"type":"para","text":"In operator form, 𝒫 \\mathcal{P} is thus a symmetry-truncated Fourier transform: it maps 𝒞 f ¯ \\mathcal{C}\\bar{f} onto the coefficients N n ( t ) N_{n}(t) at orders n ≡ 0 mod | G | n\\equiv 0\\mod{|G|} only, discarding all other harmonics a priori. Having obtained the radial harmonic profiles N n ( t ) N_{n}(t) , each is further expanded as a sine series built from Chebyshev polynomials as"},{"type":"para","text":"Equation 4 can be written in matrix form as 𝒮 = I ⊗ 𝒮 φ \\mathcal{S}=I\\otimes\\mathcal{S}^{\\varphi} , where is I ∈ ℝ Φ × Φ I\\in\\mathbb{R}^{\\Phi\\times\\Phi} the identity matrix and 𝒮 φ ∈ ℝ Δ × Δ \\mathcal{S}^{\\varphi}\\in\\mathbb{R}^{\\Delta\\times\\Delta} is 1 1 1 For equation simplicity this works uses the convention of matrix elements starting at index zero, i.e. the first element of a matrix is in index i = 0 , j = 0 i=0,j=0 ."},{"type":"para","text":"We can thus define the process of obtaining Chebyshev coefficients from measured projections as"},{"type":"para","text":"This operator 𝒜 \\mathcal{A} is useful later in this work to generate synthetic data."},{"type":"para","text":"Before the coefficients 𝐚 n \\mathbf{a}_{n} are used to synthesize the density, they are weighted elementwise with diagonal matrix 𝒲 ∈ ℝ M × M \\mathcal{W}\\in\\mathbb{R}^{M\\times M} as:"},{"type":"para","text":"The following step of the MCM reconstruction is to map the Chebyshev coefficients into the radial density functions, ρ ∈ ℝ L × 1 \\rho\\in\\mathbb{R}^{L\\times 1} , using Zernike polynomials, which can be represented as"},{"type":"para","text":"being ω \\omega the radial distance over the unit circle. The construction of the Zernike matrix 𝒵 ∈ ℝ L × M \\mathcal{Z}\\in\\mathbb{R}^{L\\times M} can be summarized as 𝒵 = I ⊗ 𝒵 n \\mathcal{Z}=I\\otimes\\mathcal{Z}^{n} , with an identity matrix I ∈ ℝ Φ × Φ I\\in\\mathbb{R}^{\\Phi\\times\\Phi} and a series of 𝒵 n ∈ ℝ L p × Δ \\mathcal{Z}^{n}\\in\\mathbb{R}^{L_{p}\\times\\Delta} , for n ∈ { r | G | : r ∈ { 0 , 1 , … , Φ } } n\\in\\{r|G|:r\\in\\{0,1,\\dots,\\Phi\\}\\} . Each sub-matrix is then built by computing the Zernike coefficients at the radial density values w i ∈ [ 0 , 1 , … , L ] w_{i}\\in[0,1,\\dots,L] , with a different matrix for each value of n n , as"},{"type":"para","text":"This recovers the operator notation of the Modified Cormack Method:"},{"type":"para","text":"The resulting radial density functions, ρ \\rho , can be converted to the density ρ ¯ 2 γ ( p , θ ) \\bar{\\rho}^{2\\gamma}(p,\\theta) in the polar plane. Each channel ρ n ( p ) \\rho_{n}(p) is modulated by cos ( n θ ) \\cos(n\\theta) and summed:"},{"type":"para","text":"The Equation 11 reconstruction step is illustrated in Appendix A . This recombination step defines the operator 𝒪 \\mathcal{O} , mapping ρ {\\rho} onto ρ ¯ 2 γ \\bar{\\rho}^{2\\gamma} , and gives the full modified Cormack method as originally stated:"},{"type":"para","text":"While in Fermiology this is often the final step, it is of interest to also define the TPMD in Euclidean form ( ρ E 2 γ \\rho_{E}^{2\\gamma} ), as shown in Equation 13 , since the standard CNN architectures used in this work are more naturally suited to a Euclidean grid."},{"type":"para","text":"In operator form, we express this transformation into Euclidean space using operator ℰ \\mathcal{E} as:"},{"type":"heading","level":2,"text":"2.2 DeepCormack"},{"type":"para","text":"This section describes our proposed data-driven method, enhancing the MCM algorithm with deep neural networks. In general, DeepCormack integrates up to three neural networks at different key stages of the modified Cormack method, as shown in Figure 1 . The process works as follows: 2D-ACAR measures projections of a sample’s momentum density at different angles."},{"type":"para","text":"The MCM reconstructs the TPMD at fixed detector row slice-by-slice, and DeepCormack proceeds similarly. However, the normalized projections are first passed through a convolutional neural network (CNN) (details in section 2.2.1 ) in order to denoise and deconvolute them. Equation 10 is then applied to recover the corresponding ∼ 5 \\sim 5 radial density functions, which are subsequently passed through a fully connected multilayer perceptron (MLP) (details in section 2.2.2 ). The MLP-outputted radial density functions are passed through Equation 11 to reconstruct the TPMD ρ 2 γ \\rho^{2\\gamma} . This reconstruction is finally passed through a UNet (details in section 2.2.3 )."},{"type":"para","text":"In this work, we explore several possible configurations of this DeepCormack structure explained in more detail in the Results section."},{"type":"heading","level":3,"text":"2.2.1 1D CNN"},{"type":"para","text":"The first stage, shown in Figure 2 , employs a one-dimensional residual convolutional neural network (CNN) to map noisy measured projections ( f m e a s = f + e ~ ) (f_{meas}=f+\\tilde{e}) to their ideal counterparts ( f G T = f ) ({f}_{GT}=f) . The network accepts as input Φ \\Phi ( 5 5 in our work) angular projections, each of length L p L_{p} ( 256 256 in ours), arranged as a multi-channel 1D signal of shape ( Φ , L p ) (\\Phi,L_{p}) . The architecture consists of a stack of five one dimensional convolutional layers with kernel size 3 3 and unit padding, with the first four layers projecting into 32 32 hidden channels and the final layer projecting back to 5 5 output channels, preserving the input dimensionality. ReLU activations are applied between all intermediate layers. A global residual connection is added from the input directly to the output, such that the network learns a correction Δ \\Delta rather than the full mapping, i.e. f C N N = f meas + CNN ( f meas ) {{{f}}}_{CNN}={{f}}_{\\text{meas}}+\\text{CNN}({{f}}_{\\text{meas}}) . The network is trained by minimising the loss comprising a direct mean-squared error (MSE) between predicted and ideal projections ( ℒ 1 = MSE ( f G T , f C N N ) \\mathcal{L}_{1}=\\text{MSE}\\left({{f}}_{GT},{{f}}_{CNN}\\right) ), and an auxiliary MSE between the ground truth TPMD and CNN-denoised TPMD reconstruction in Euclidian coordinates ( ℒ 2 = MSE ( ℰ ℳ f GT , ℰ ℳ f CNN ) \\mathcal{L}_{2}=\\text{MSE}\\left(\\mathcal{EM}{f}_{\\text{GT}},\\mathcal{EM}{f}_{\\text{CNN}}\\right) ). The full loss term is constructed as ℒ C = α ℒ 1 + ℒ 2 \\mathcal{L}_{\\text{C}}=\\alpha\\mathcal{L}_{1}+\\mathcal{L}_{2} . We found α = 10 \\alpha=10 empirically the best value. This allows the model to denoise the projections, while ensuring the final results are still valid, given that small changes in the measurements can cause large changes in the final image. The network is optimised using the Adam optimiser with a learning rate of 10 − 3 10^{-3} ."},{"type":"heading","level":3,"text":"2.2.2 MLP"},{"type":"para","text":"The second stage employs a fully connected multilayer perceptron (MLP) to refine the radial density function representation of the CNN output, as shown in Figure 3 . The input to the MLP is a tensor of shape ( Φ , L p ) (\\Phi,L_{p}) representing the L p = 256 L_{p}=256 radial grid points each described by Φ = 5 \\Phi=5 Cormack expansion coefficients, which is flattened to a vector of dimension Φ ⋅ L p = 1280 \\Phi\\cdot L_{p}=1280 . The network consists of two hidden layers of width 11 × L p = 2816 11\\times L_{p}=2816 with ReLU activations, followed by a linear output layer projecting back to the input dimensionality. The output is reshaped to ( Φ , L p ) (\\Phi,L_{p}) to match the input. The CNN is frozen, and the MLP receives as input the CNN-denoised projections passed through the MCM to get ρ {\\rho} . The network is supervised entirely in reconstruction space: the loss is the MSE between the Euclidean 2D TPMD from the MLP output and the ground truth ( ℒ M = MSE ( ℰ ρ GT , ℰ ρ MLP ) \\mathcal{L}_{\\text{M}}=\\text{MSE}\\left(\\mathcal{E}{\\rho}_{\\text{GT}},\\mathcal{E}{\\rho}_{\\text{MLP}}\\right) ). This formulation also encourages the MLP to recover features in the TPMD that are lost due to the limited number of projections available in experimental ACAR. The MLP is optimised with the Adam optimiser at a learning rate of 10 − 3 10^{-3} ."},{"type":"heading","level":3,"text":"2.2.3 UNet"},{"type":"para","text":"The third stage is a UNet architecture (Figure 4 ) based on [ 13 ] to learn residual corrections to the two-dimensional TPMD reconstruction. The projections are passed through the CNN, converted to radial density functions ρ {\\rho} , and inputted to the MLP. The output is reconstructed into a quadrant of the 2D image ρ P 2 γ {\\rho}^{2\\gamma}_{P} of shape ( 1 , L p , L p ) (1,L_{p},L_{p}) via the final step of the MCM, and becomes the input of the UNet."},{"type":"para","text":"The encoder comprises three convolutional blocks with channel progressions [ 1 , 64 , 64 , 64 ] [1,64,64,64] , [ 64 , [64, 128 , 128, 128 ] 128] , and [ 128 , 256 , 256 ] [128,256,256] respectively, each followed by 2 × 2 2\\times 2 max-pooling for spatial downsampling. The bottleneck processes features at [ 256 , 512 , 512 ] [256,512,512] channels with a spatial dropout rate of 0.1 0.1 . The decoder mirrors the encoder using transposed convolutions for upsampling, with skip connections concatenating encoder feature maps to the corresponding decoder inputs at each resolution level, and channel progressions [ 512 , 256 , 256 ] [512,256,256] , [ 256 , 128 , 128 ] [256,128,128] , and [ 128 , 64 , 64 ] [128,64,64] . Each convolutional block uses Group Normalisation and ReLU activations. A final 1 × 1 1\\times 1 convolution projects to a single output channel. A global residual connection adds the input image to the network output, enforcing that the UNet learns a residual correction such that ρ 2 γ = ReLU ( ρ pred 2 γ + UNet ( ρ pred 2 γ ) ) {\\rho}^{2\\gamma}=\\text{ReLU}(\\rho^{2\\gamma}_{\\text{pred}}+\\text{UNet}(\\rho^{2\\gamma}_{\\text{pred}})) , where the ReLU enforces physical non-negativity of the momentum distribution. The network is trained by minimising the MSE between the corrected output and the ground-truth TPMD image ( ℒ U = MSE ( ρ 2 γ GT , ρ 2 γ UNet ) \\mathcal{L}_{\\text{U}}=\\text{MSE}\\left({\\rho^{2\\gamma}}_{\\text{GT}},{\\rho^{2\\gamma}}_{\\text{UNet}}\\right) ), using the Adam optimiser at a learning rate of 10 − 4 10^{-4} ."},{"type":"heading","level":3,"text":"2.2.4 FiLM Conditioning"},{"type":"para","text":"A UNet was also tested with and without Feature-wise Linear Modulation (FiLM) [ 24 , 26 ] . FiLM learns functions γ ( z ) \\gamma(z) and β ( z ) \\beta(z) to modulate a neural network’s intermediate feature map 𝐱 \\mathbf{x} via a learned affine transformation:"},{"type":"para","text":"making it possible to condition the network on a signal z z without modifying the weights themselves. In our case, z z is taken as the log-normalised count level averaged across the simulated measurement projections, providing the UNet with an explicit measure of the Poisson noise in the data."},{"type":"para","text":"For the UNet, FiLM is applied to the output of each convolutional block (i.e., after normalisation and activation), rather than between normalisation and the nonlinearity as in the original formulation. z z is first encoded via a sinusoidal embedding and refined by a shared two-layer SiLU MLP, whose output is distributed to a single linear projection per block, producing per-channel ( γ , β ) (\\gamma,\\beta) pairs at each encoder, latent, and decoder stage. FiLM parameters are zero-initialised, such that at the start of training γ = 0 \\gamma=0 , β = 0 \\beta=0 , making the transformation an identity map."},{"type":"heading","level":2,"text":"2.3 Data Simulation"},{"type":"para","text":"As experimental ACAR measurements can take months, constructing 3D TPMD training datasets of the size typically required for supervised deep learning via experimental means alone would be impractical, as it takes 3-4 months to measure a single materials. Computational approaches such as density functional theory (DFT) could in principle provide ground-truth momentum densities from which projections could be taken through them and realistic experimental conditions simulated. However, DFT calculations are themselves rather intensive to compute both in terms of human time (parameter setting, convergence testing etc.) as well as resources (computing times can be of the order of hours, and the densities themselves can require storage at the GB scale). As no publicly available momentum density database currently exists, a key step for making supervised learning feasible requires a data generation method for sufficiently diverse and realistic synthetic 3D TPMDs that continuously evolve slice-by-slice. This would allow the detector resolution to be simulated as a 2D Gaussian convolution, noise via Poisson statistics, and the momentum sampling function (MSF) — a necessary correction due to the ACAR detector geometry, as only photons emitted with perfect collinearity have 100% probability to both hit the detectors — can be applied as a normalized tent function."},{"type":"para","text":"Our proposed data generation approach is applied on a reference momentum density for copper, and consists of two independent steps: first, plausible central slices are generated, using a combination of singular value decomposition (SVD) on the copper Chebyshev coefficients and varying the singular values. Dynamic mode decomposition (DMD) [ 2 ] is then applied, which learns the underlying dynamics of how the copper momentum density evolves continuously from one slice to the next."},{"type":"para","text":"Any number of central slice TPMDs can be generated using SVD, then continuously evolved into a full 3D TPMD using DMD. This will act as our ground truth for the dataset. As outlined in Section 2.3.3 , a corresponding experimental dataset can be simulated by taking 5 5 projections through the ground truth volumes and applying realistic degradation to the data (detector resolution and geometry effects, noise etc)."},{"type":"para","text":"Taking 20 20 ideal projections of the DFT copper momentum density between φ ∈ [ 0 , 45 ] \\varphi\\in[0,45] degrees preserves sufficient information when reconstructed via the MCM (Equation 12 ) to act as an approximation for the ground truth. This allows our data generation method to operate entirely on the extracted Chebyshev coefficients, as defined in Equation 4 , for some projection slice l = p y ∈ [ 0 , L p − 1 ] , l=p_{y}\\in[0,L_{p}-1], L p = 256 , L_{p}=256, that is reconstructed from copper: a ~ n , l m \\tilde{a}_{n\\hskip 1.42271pt,\\hskip 1.42271ptl}^{m} . Here, m ∈ [ 0 , Δ − 1 ] , m\\in[0,\\Delta-1], Δ = 180 \\Delta=180 are the number of angular degrees of freedom when parametrising the projections along the unit circle, and n ∈ [ 0 , Φ − 1 ] n\\in[0,\\Phi-1] denotes the projection index, where Φ = 20 , \\Phi=20, are the number of ideal projections used to approximate the ground truth copper momentum density with the MCM. For some projection Φ , \\Phi, we define the matrix [ 𝐚 ~ 𝚽 ] l m = a ~ Φ , l m \\left[\\mathbf{\\tilde{a}_{\\Phi}}\\right]_{lm}=\\tilde{a}_{\\Phi\\hskip 1.42271pt,\\hskip 1.42271ptl}^{m} , where 𝐚 ~ Φ ∈ ℝ Δ × L p \\mathbf{\\tilde{a}}_{\\Phi}\\in\\mathbb{R}^{\\Delta\\times L_{p}} . When generating the synthetic data, we model each of the Chebyshev coefficients independently via their projection channel by fixing n n , as they are tied to separate ρ \\rho ’s when reconstructed. The label will be omitted hereafter for notational clarity."},{"type":"heading","level":3,"text":"2.3.1 Generating Central Slices"},{"type":"para","text":"The MCM, at the step shown in Equation 4 , decomposes each projection into a set of Chebyshev coefficients to carry out the reconstruction. These are the source values used for our generative process of central slices."},{"type":"para","text":"For a fixed projection, truncated SVD is applied across the angular channel over all unique slice indices, reducing the dimensionality to a level that maintains approximately ∼ 90 % \\sim 90\\% of the variance. This yields an independent lower dimensional latent space per projection channel, each of which is modelled by fitting a multivariate Gaussian to allow sampling new points. When lifted back to coefficient space, the result is a new set of Chebyshev coefficients that can proceed through the rest of the MCM (Equation 12 ), and after some minor tuning will give a realistic central TPMD slice."},{"type":"para","text":"More precisely, for 𝐚 ~ \\mathbf{\\tilde{a}} previously defined in Section 2.3 , let 𝐚 ~ l ∈ ℝ Δ , \\mathbf{\\tilde{a}}_{l}\\in\\mathbb{R}^{\\Delta}, denote the copper Chebyshev coefficient state at a fixed projection n n at a given slice l l . To construct the latent space and sample a new set 𝐚 ^ ∈ ℝ Δ , \\hat{\\mathbf{a}}\\in\\mathbb{R}^{\\Delta}, we begin by constructing the snapshot matrix using the copper values:"},{"type":"para","text":"The temporal 2 2 2 Note that time is not a variable in our work, instead the Chebyshev coefficient variation over slices are the variable. However, for consistency with the DMD and Koopman operator literature, we call this time. mean is defined as"},{"type":"para","text":"allowing us to get the mean-centered data as:"},{"type":"para","text":"where 𝟏 ∈ ℝ L p \\mathbf{1}\\in\\mathbb{R}^{L_{p}} is a vector of ones."},{"type":"para","text":"We compute the singular value decomposition (SVD) of the centered snapshot matrix:"},{"type":"para","text":"where 𝐔 ∈ ℝ Δ × Δ \\mathbf{U}\\in\\mathbb{R}^{\\Delta\\times\\Delta} contains the orthonormal spatial modes, 𝚺 ∈ ℝ Δ × L p \\bm{\\Sigma}\\in\\mathbb{R}^{\\Delta\\times L_{p}} is a diagonal matrix of singular values, and 𝐕 ∈ ℝ L p × L p \\mathbf{V}\\in\\mathbb{R}^{L_{p}\\times L_{p}} contains temporal coefficients. We retain the first K ≪ Δ K\\ll\\Delta dominant modes, yielding the truncated SVD:"},{"type":"para","text":"𝐔 k ∈ ℝ Δ × K \\mathbf{U}_{k}\\in\\mathbb{R}^{\\Delta\\times K} defines a reduced spatial basis. In this step, K = 15 K=15 ( Δ / K = 12 ) (\\Delta/K=12) was found to construct reasonable TPMD features, capturing ∼ 88 % \\sim 88\\% of the total variance across slices. A Gaussian is fitted to the latent coordinates 𝚺 k 𝐕 k ⊤ ∈ ℝ K × L p \\bm{\\Sigma}_{k}\\mathbf{V}_{k}^{\\top}\\in\\mathbb{R}^{K\\times L_{p}} of the L p = 256 L_{p}=256 copper slices, a new latent vector 𝐳 ^ \\hat{\\mathbf{z}} is drawn, and projected back to coefficient space to give a synthetic Chebyshev coefficient 𝐚 ^ ∈ ℝ Δ \\mathbf{\\hat{a}}\\in\\mathbb{R}^{\\Delta} by:"},{"type":"para","text":"where μ k \\mu_{k} and Λ k \\Lambda_{k} are the sample mean and full covariance matrix of the latent coordinates respectively. Repeating this process for all Φ \\Phi channels (with the projection channel n n reinstated) yields the new central-slice coefficient matrix"},{"type":"para","text":"Equation 10 can subsequently be applied to recover the corresponding ρ \\rho and Equation 12 for the reconstructed ρ P 2 γ \\rho^{2\\gamma}_{P} ."},{"type":"para","text":"It was found that applying corrections to the ρ \\rho generated from 𝐀 ^ l = 0 \\hat{\\mathbf{A}}_{\\mathrm{l=0}} improved the overall quality and consistency of the SVD sampling method. First, the isotropic term ρ 0 \\rho_{0} (being the most sensitive and frequently producing poor-quality TPMDs when sampled via SVD) is replaced with the isotropic term drawn from the original copper dataset at a slice index in the range l ∈ [ 0 , 133 ] l\\in[0,133] . 1D Gaussian smoothing is applied along the radial axis of the higher-order density terms ρ n > 0 \\rho_{n>0} to suppress unphysical and overly exotic features that arise from SVD sampling but are not characteristic of real TPMDs. All ρ n \\rho_{n} functions are normalised such that their peak intensities remain within a ∼ 10 % \\sim 10\\% range observed in the corresponding copper reference functions. Finally, a physicality constraint is then enforced: the total radial density ∑ n ρ n \\sum_{n}\\rho_{n} must remain non-negative everywhere. Passing these corrected ρ n \\rho_{n} functions through the final MCM reconstruction step yields the synthetic ground truth TPMD slice to be used in the next stage of the data generation method."},{"type":"para","text":"Since corrections have been applied to the ρ n \\rho_{n} functions, evolving them into full 3D TPMDs will require recomputing the Chebyshev coefficients for each synthetic central slice. This is done by applying the Radon transform at 20 equally spaced projection angles φ ∈ [ 0 ∘ , 45 ∘ ] \\varphi\\in[0^{\\circ},45^{\\circ}] and recovering the corresponding (corrected) Chebyshev coefficients via the MCM (Equation 10 )."},{"type":"heading","level":3,"text":"2.3.2 Evolving the central slice into 3D TPMDs"},{"type":"para","text":"Once feasible generated central slices are created, 3D versions of them need to be derived. In order to extrapolate the central slice in a manner coherent with TPMD structure, we propose a dynamic mode decomposition (DMD) model [ 2 ] . This is trained per-projection channel on the original copper Chebyshev coefficients 𝐚 ~ \\mathbf{\\tilde{a}} . DMD is a particularly powerful method for this: it allows the slice-to-slice non-linear dynamics of the Chebyshev coefficients to be simulated by assuming they evolve linearly in a latent space. It achieves this by computing an approximate Koopman operator for each n n , such that the synthetically generated set of central-slice Chebyshev coefficients 𝐀 ^ l = 0 \\hat{\\mathbf{A}}_{\\mathrm{l=0}} can be evolved into a full set 𝐀 ^ ∈ ℝ L p × Δ × Φ \\hat{\\mathbf{A}}\\in\\mathbb{R}^{L_{p}\\times\\Delta\\times\\Phi} . Passing each of the evolved Chebyshev coefficients through the MCM as 𝒜 f ¯ \\mathcal{A}\\bar{f} in Equation 10 , the result is a continuously evolving 3D TPMD."},{"type":"para","text":"In order to compute the DMD model, we Continue from the truncated SVD for copper in Equation 16 , for a fixed projection channel n n , we now retain the first K = 24 K=24 ( M / K = 7.5 M/K=7.5 ) dominant modes as it was found to give a good trade-off between diversity and stable volume evolution. Each Chebyshev slice snapshot ( 𝐚 ~ l ) (\\tilde{\\mathbf{a}}_{l}) from Equation 15 is projected onto the reduced basis as"},{"type":"para","text":"Allowing us to define the time-shifted latent snapshot matrices:"},{"type":"para","text":"We assume linear evolution in latent space:"},{"type":"para","text":"where 𝒦 ∈ ℝ K × K \\mathcal{K}\\in\\mathbb{R}^{K\\times K} is a finite-dimensional approximation of the Koopman operator, and is obtained by solving the least-squares problem"},{"type":"para","text":"where | | ⋅ | | F ||\\cdot||_{F} is the Frobenius norm. The closed-form solution is"},{"type":"para","text":"with 𝐙 † \\mathbf{Z}^{\\dagger} denoting the Moore–Penrose pseudoinverse."},{"type":"para","text":"Once the Koopman operator 𝒦 \\mathcal{K} is pre-computed in latent space, the generated central slice for the projection channel 𝐚 ^ l \\mathbf{\\hat{a}}_{l} can be rolled out to produce slice a ^ l + 1 \\hat{a}_{l+1} via:"},{"type":"para","text":"resulting in the new Chebyshev snapshot matrix for projection channel n n :"},{"type":"para","text":"Repeating this process for all Φ \\Phi projection channels — each with their own unique Koopman operator 𝒦 n \\mathcal{K}_{n} — will result in a full set of synthetic Chebyshev coefficients:"},{"type":"para","text":"Separately passing each 𝐀 ^ l = [ 𝒜 f ] synthetic \\hat{\\mathbf{A}}_{l}=[\\mathcal{A}f]_{\\text{synthetic}} from l = 0 l=0 to L p − 1 L_{p}-1 into Equation 12 will result in a full 3D TPMD which continuously evolves. Each time a new central slice is constructed with SVD and evolved with DMD, a unique 3D TPMD is created allowing countless samples to be generated for training."},{"type":"para","text":"The effect of taking the copper central slice Chebyshev coefficents and applying DMD to continuously evolve them with the copper-derived Koopman operators is shown in Appendix B , where the Cu ground truth and DMD Fermi surfaces are compared."},{"type":"heading","level":3,"text":"2.3.3 Simulating Experimental Measurement Conditions"},{"type":"para","text":"A final step of generating realistic data for supervised ML is to be able to produce noise similar to the one present in experimental conditions at different count levels. Figure 5 shows our simulated noise pipeline and its effect in MCM reconstructed slices, for both a central slice and a slice 1.1 1.1 au away from it."},{"type":"para","text":"For a synthetic 3D TPMD volume, (i) five 2D projections are computed at angles φ ∈ [ 0 ∘ , 45 ∘ ] \\varphi\\in[0^{\\circ},45^{\\circ}] . (ii) Detector resolution effects ( 0.110 × 0.137 0.110\\times 0.137 a.u. smearing along the projection axis p x p_{x} and p y p_{y} ), (iii) Poisson counting noise and momentum sampling function (MSF) effects are applied to each projection in order to simulate realistic experimental conditions at the desired count rates. The MSF arises as a consequence of the detector geometry, where only electrons emitted with perfect collinearity have a 100 % 100\\% probability of both registering a hit at either detector. This effect can be simulated by multiplying the projections with a normalized tent function. As the synthetic dataset is based on a momentum density for copper, it will retain the sample’s four-fold symmetry, and (iv) each projection’s quadrants can be averaged with one another before removing the MSF contribution. MCM reconstruction is then applied to these degraded projections. This yields matched pairs of ground truth and simulated measurement projections, radial density functions, and 3D TPMD volumes for use in supervised training."},{"type":"heading","level":2,"text":"2.4 Training data for DeepCormack"},{"type":"para","text":"To reduce storage and memory requirements, only a subset of p y p_{y} -slices are retained from each volume. These are drawn randomly from the range p y ∈ [ 0 , 83 ] p_{y}\\in[0,83] out of 256 total slices, a region which accounts for approximately 90% of the total counts in the copper reference data. Appendix C Figure 16 shows three slices taken through different synthetic 3D momentum densities that were generated using our method and used in training."},{"type":"heading","level":1,"text":"3 Results"},{"type":"para","text":"In this section we present experiments of trained DeepCormack models at different scenarios. Before showing model performance, we first validate our noise simulation approach to the simulation of experimental conditions at varying counts by comparing the Fermi surface degradation of ZrZn 2 with that of real experimental data ( 3.1 ). Different DeepCormack model configurations are then tested and compared using three different datasets with simulated measurement conditions at 200 200 M counts: the synthetically generated test data (DMD), the copper TPMD which the synthetic training data used as reference, and ZrZn 2 data ( 3.2 ). This allows for testing the models’ reconstruction capabilities on data that is within the distribution of the training data (the DMD data), data which has features and slice-by-slice structure within distribution of the training data but not directly seen during training (the copper data), and on data out of distribution (OOD) entirely (the ZrZn 2 data)."},{"type":"para","text":"In addition to higher image quality, one goal of DeepCormack is to enable faster scanning time. The next experiment presented in Section 3.3 showcases the model’s performance at lowering the photon counts measured (i.e. reducing acquisition time). This is tested with the same three datasets from the previous experiments, i.e. the DMD, copper, and ZrZn 2 data."},{"type":"heading","level":2,"text":"3.1 Validating the Simulation Approach"},{"type":"para","text":"To validate the DeepCormack framework on real data, the simulated experimental conditions must first be verified against real experimental measurements. This is to confirm that the synthetic data faithfully reproduces the degradation due to noise and smearing of real ACAR data across a range of count levels. To this end, we compare simulated and experimental ACAR measurements of ZrZn 2 across the range 10 − 165 10-165 M counts, using a DFT-calculated momentum density as the ground truth. Computing the gradient of the 3D TPMDs allows calibration quality to be assessed in TPMD space rather than by comparing the reconstructed images directly. The ACAR measurements had simulated measurement conditions applied as outlined in Section 2.3.3 , and repeated for data at 10 10 M, 100 100 M, and 165 165 M counts in order to provide direct comparison with real experimental data."},{"type":"para","text":"Using the nominal experimental parameters gave simulated data that was too smooth and insufficiently noisy, with degradation in Fermi space that was too mild relative to experiment. A better match was obtained by increasing the convolution widths by + 60 % +60\\% to the reference values and using an effective count level of 40 % 40\\% of the experimental total (e.g. 100 100 M experimental counts → \\rightarrow 40 40 M simulated counts). With these adjusted parameters, the degradation matched the experimental reference well in Fermi surface space as shown in Figure 6 , and was similarly consistent in TPMD space (Appendix D Figure 17 ). This mismatch is probably explained by the fact that there are other possible factors producing noise in real measurements that our first order approximation of the noise does not capture."},{"type":"para","text":"This validates the proposed forward model and approximation of the measurement noise, allowing us to generate synthetic measurement data with the correct noise characteristics, a necessary step for the supervised learning proposed in this work, that uses the synthetic TPMDs explained in Section 2.3 . The rest of the experiments use this simulated noise model for training."},{"type":"heading","level":2,"text":"3.2 Model Comparison at Standard Counts"},{"type":"para","text":"Several different DeepCormack model configurations (defined below) were evaluated across three datasets to assess their reconstruction performance and ability to generalize. The DMD test set shares the same distribution as the training data; the copper (Cu) dataset represents a real TPMD, but is still within reasonable distribution of the training data, as it was used as the reference momentum density for the DMD generation; and the ZrZn 2 dataset introduces unique and unseen TPMD features, providing a useful test of OOD generalization. Due to the SNR of the 2D-ACAR technique, a standard experimental measurement will capture around 200 200 M total annihilation events per projection. Therefore, all datasets were simulated with experimental conditions at 200 200 M counts. The DeepCormack configurations explored in this work are the following:"},{"type":"para","text":"CNN : A CNN trained on the projections f f ."},{"type":"para","text":"MLP : An MLP trained on the radial density functions ρ \\rho ."},{"type":"para","text":"UNet : A UNet trained on the TPMD ρ 2 γ \\rho^{2\\gamma} ."},{"type":"para","text":"ETE(CNN + MLP + UNet) : A CNN, MLP, and UNet trained end-to-end."},{"type":"para","text":"CNN PT → \\rightarrow UNet : Pre-trained CNN, where the outputs are used to train a UNet."},{"type":"para","text":"CNN PT + UNet PT : Pre-trained CNN and pre-trained UNet linked together and trained."},{"type":"para","text":"CNN PT + MLP PT + UNet PT : Pre-trained CNN, pre-trained MLP, and pre-trained UNet linked together and trained."},{"type":"para","text":"CNN PT + MLP PT → \\rightarrow UNet : Pre-trained CNN and pre-trained MLP linked together and trained, then their outputs are used to train a new UNet. A skip connection runs from the CNN through the MCM to recover ρ P 2 γ \\rho^{2\\gamma}_{P} , and is concatenated as an additional input into the UNet."},{"type":"para","text":"Results using Peak Signal to Noise Ratio (PSNR) are show in Table 1 , for all the model configurations and the three test datasets. The MLP delivers competitive performance on the DMD and Cu datasets ( 38.24 38.24 and 38.10 38.10 PSNR respectively), but struggles with ZrZn 2 , achieving 30.78 30.78 PSNR, far below the MCM baseline of 33.96 33.96 PSNR. On the other hand, the CNN is modest on DMD ( 33.78 33.78 PSNR) but achieves 36.20 36.20 PSNR on ZrZn 2 — the second highest overall — while retaining the lowest variance of all configurations. The UNet gives the strongest and most consistent results overall, competitive on DMD ( 38.41 38.41 PSNR) and the best on both Cu and ZrZn 2 ( 40.05 40.05 and 37.44 37.44 PSNR respectively). The combination CNN PT + MLP PT → UNet \\text{CNN}_{\\text{PT}}+\\text{MLP}_{\\text{PT}}\\rightarrow\\text{UNet} achieved the highest DMD performance ( 40.69 40.69 PSNR), while also achieving 38.24 38.24 PSNR on the copper data. However, this comes at the cost of OOD generalization, with ZrZn 2 performance falling to 32.51 32.51 PSNR — below even the Modified Cormack baseline."},{"type":"para","text":"These experiments highlight that wile DeepCormack has can have far superior performance than baselines, an appropriately designed training dataset is required to fully exploit its potential. For the rest of the experiments, the following configurations are selected and named for convenience: DC C \\text{DC}_{\\text{C}} is the CNN only configuration (1), DC M \\text{DC}_{\\text{M}} is the MLP only configuration (2), DC U \\text{DC}_{\\text{U}} is the UNet only configuration (3) and finally DC CMU \\text{DC}_{\\text{CMU}} is the configuration that performs best in DMD data, i.e. pretraining the CNN and MLP first, and the training all together with the Unet (8)."},{"type":"para","text":"The goal of ACAR measurements is to recover the Fermi surface, but models output and are trained on TPMDs in p -space. To recover the Fermi surface, the momentum densities for copper were converted to k -space densities using the LCW theorem. Figure 7 shows the full pipeline, where projections are converted to TPMDs, k -space momentum densities, before finally extracting the Fermi surface. As this last step — the Fermi surface — is the variable of interest, the rest of the experiments in this work will show results in this space, even though DeepCormack only reconstructs the first step (from projections to TPMDs)."},{"type":"heading","level":2,"text":"3.3 DeepCormack Performance Across Counts"},{"type":"heading","level":3,"text":"3.3.1 Model Performance across counts on DMD Data"},{"type":"para","text":"To assess how various DeepCormacks perform as data quality degrades, the full framework was evaluated on the DMD test set across count levels ranging from 10 10 M to 200 200 M, and benchmarked against the MCM using both PSNR and Structural Similarity Index Metric (SSIM) as seen in Table 2 and equivalently in Figure 8 ."},{"type":"para","text":"DC CMU \\text{DC}_{\\text{CMU}} outperforms all models at every count level on both metrics. Most notably, its reconstruction at 10 10 M counts ( 38.22 38.22 PSNR) exceeds the MCM at 200 200 M counts ( 32.38 32.38 PSNR), demonstrating that the framework can maintain reconstruction quality at count levels an order of magnitude lower than those currently used in practice. The MCM degrades with decreasing counts, dropping from 32.38 32.38 to 26.12 26.12 PSNR between 200 200 M and 10 10 M, while DC CMU \\text{DC}_{\\text{CMU}} comparatively stable across the same range ( 40.69 40.69 to 38.22 38.22 PSNR)."},{"type":"para","text":"The improvement over the best individual models — the DC M \\text{DC}_{M} and DC U \\text{DC}_{U} , — is more modest, averaging around + 1 − 1.5 +1-1.5 db PSNR across count levels."},{"type":"heading","level":3,"text":"3.3.2 Model Performance across counts on Copper Data"},{"type":"para","text":"Here we test the models on the copper momentum density with simulated measurement conditions applied for counts ranging from 10 − 200 10-200 M. Results can be seen in Table 3 ans Figure 9 . This is the same reference used to generate synthetic training data. Since the copper features are not far to the training distribution, this will test how well the models generalize to real TPMD data rather than purely synthetically generated ones."},{"type":"para","text":"On the Cu data, the DC U \\text{DC}_{\\text{U}} is the strongest performer across all count levels, reaching 40.05 40.05 PSNR at 200 200 M counts and 37.41 37.41 at 10 10 M, outperforming DC CMU \\text{DC}_{\\text{CMU}} , which achieves 38.65 38.65 and 36.04 36.04 dB respectively. DC CMU \\text{DC}_{\\text{CMU}} performs comparably to the DC M \\text{DC}_{\\text{M}} across all counts, around 1.40 1.40 PSNR lower than the DC U \\text{DC}_{\\text{U}} . However, all models again comfortably outperform the MCM at every count level. Notably, DC CMU \\text{DC}_{\\text{CMU}} at 10 10 M counts ( 36.04 36.04 PSNR) still exceeds the MCM at 200 200 M counts ( 31.62 31.62 dB PSNR), consistent with the DMD results."},{"type":"para","text":"When these slices are individually passed through DeepCormack, they need to be renormalized in order to produce coherent 3D TPMDs and Fermi surfaces. This is done using a smoothed version of the measured momentum density data’s maximum intensity across slices."},{"type":"para","text":"Despite the PSNR and SSIM gap between the DC U \\text{DC}_{\\text{U}} and DC CMU \\text{DC}_{\\text{CMU}} in momentum space, the Fermi surface reconstructions show that both produce qualitatively comparable results across the full count range, recovering a greater detail of features along all three slices than the MCM. Notably, the recovered Fermi surface remains largely intact even at 10 10 M counts, with features that are more distinct than the MCM at 200 200 M counts."},{"type":"heading","level":3,"text":"3.3.3 Model Performance on ZrZn 2 Data"},{"type":"para","text":"Here the models are tested on ZrZn 2 data with simulated measurement conditions applied across 10 10 M to 200 200 M counts (Table 4 , Figure 9 ). Unlike the copper data, the ZrZn 2 momentum density has features that are outside the training distribution, making this a strict test of generalization."},{"type":"para","text":"DC CMU \\text{DC}_{\\text{CMU}} s performance is significantly lower on the ZrZn 2 dataset, achieving only 32.82 32.82 PSNR at 200 200 M counts, which is below the MCM ( 33.96 33.96 PSNR) and the DC C \\text{DC}_{\\text{C}} ( 36.20 36.20 PSNR), but greater than that of the DC M \\text{DC}_{\\text{M}} by itself ( 30.78 30.78 PSNR). This performance drop becomes apparent when computing and comparing the Fermi surfaces, as shown in Figure 12 ."},{"type":"para","text":"We found out, however, that conditioning the DC U \\text{DC}_{\\text{U}} with FiLM conditioning has a noticeable impact on the reconstructed Fermi surface for ZrZn 2 , as seen in Figure 13 . Qualitatively, the results suggest that DC U \\text{DC}_{\\text{U}} produces a better Fermi surface than MCM, albeit its unclear if a sufficiently informative one."},{"type":"heading","level":1,"text":"4 Discussion"},{"type":"para","text":"In this work, we proposed a DeepCormack family of models for Fermi surface tomography, together with data generation approaches and testing on a variety of count levels, which is directly proportional to scanning time."},{"type":"para","text":"A large combination of different DeepCormack architecture variations were tested, and results were generally inconclusive. Each configuration carries its own trade-offs. The CNN and MLP are the least intensive to train: the CNN generalizes well but gives weaker in-distribution performance, while the MLP is powerful within the training distribution but struggles on OOD data. The UNet is the most computationally demanding to train, but offers the strongest and most reliable performance overall. We note, however, that the UNet’s advantage likely reflects the limitations of the training data as much as its architectural strengths. There are currently no public databases of experimentally measured or DFT-generated momentum densities for real samples that could support supervised learning, so models were trained entirely on synthetic data derived from a copper reference. With more diverse and representative training data, configurations that perform strongest on the DMD test set — particularly DC CMU \\text{DC}_{\\text{CMU}} , which achieves amongst the highest in-distribution performance while offering considerable architectural flexibility — may prove more competitive. Each component is straightforward to train independently and can subsequently be linked and fine-tuned for a small number of epochs, making this framework practical and adaptable, particularly as richer training data becomes available. Furthermore, if strong reconstruction performance can be maintained at lower count levels, collecting data along more than 5 5 projection directions becomes viable, and the richer angular sampling may allow the full range of model configurations to reach their potential."},{"type":"para","text":"Notably, we saw that the performance gain of the models depend on which data are tested. While pre-training can improve reconstruction quality on data close to the training distribution, it appears to specialize the models towards training-data-like features in a way that does not fully recover with joint training. More broadly, models with sufficient capacity to exploit the structure of the synthetic data tend to do so at the cost of generalization, and the DC U \\text{DC}_{\\text{U}} currently strikes the best balance between the two. It should be noted that these results are obtained on synthetic DMD data drawn from the same distribution as the training data, which naturally favours all models evaluated here. Real experimental data will differ in both noise characteristics and momentum density features, and how well these results carry over depends on how faithfully the simulated data reproduces real measurements — as assessed in Section 3.1 . Training on a broader range of DFT-generated momentum densities, or ideally real experimental data, would likely improve performance and generalization across the board. Testing the models on real experimental data will be explored in future work."},{"type":"para","text":"That said, the lack of generalization to out-of-distribution samples is a limitation, but only a moderate one. As is true of inverse problems in general, learned reconstructions are highly sensitive to the data distribution, meaning they tend to generalize only moderately, particularly when tested on real data. An advantage here is that Fermi surface tomography is a very slow process — one that can rely on precomputed approximations of the ground truth. Currently, ACAR measurements for 200 200 M counts take over 3 3 months of scanning time. However, one can pre-compute a reference momentum density within a day or so. This reference can then be used to generate training data for DeepCormack via DMD, after which one can choose to reduce the measurement time. Our work suggests that this will still produce higher quality Fermi surfaces than the modified Cormack method, in significantly less time. That said, DeepCormack produces significantly better reconstructions than MCM on 200 200 M-count data, so even without the time-reduction possibility, DeepCormack enables much higher quality Fermiology via 2D-ACAR. It is also noteworthy that FiLM conditioning led to some qualitatively better reconstructions than MCM on OOD experimental data, meaning OOD testing cannot be completely discarded."},{"type":"para","text":"In general, for this work it is worth noting that we used the natural image quality metrics PSNR and SSIM to evaluate performance. While, in broad terms, the qualitative evaluations of the results here correlate with an increase in PSNR and SSIM, these metrics may not be the appropriate ones for evaluating fine differences or for choosing a ”best” configuration of DeepCormack. Ideally, the quality of the Fermi surface — or, better still, the quality of quantitative values extracted from the Fermi surface — would be the best metric, as these are what physicists will use, not the images themselves. This is, however, non-trivial, and would require a set of labelled ground truth data, which is not available for this task at the time of writing. Future work should focus on validating DeepCormack with such metrics, evaluating how useful it is for real outcomes with real data. This idea could also be extended to training the models. If MSE/PSNR of the TPMDs that DeepCormack outputs is not the goal, moving the training loss to metrics defined on the Fermi surface would be an improved solution. This is non-trivial, however, as applying the LCW theorem to pseudo-real data is not obvious, and computationally expensive, as it would require storing all TPMD slices during training rather than just a subset."},{"type":"para","text":"All configurations here have been desiged to only 5 5 projections, which constrains the quality of reconstruction regardless of the model used. Given the strong performance of DeepCormack, a natural next step would be to acquire more projections — for example 10-20 — at proportionally lower count levels ( 50 50 — 25 25 % of 200 200 M counts), keeping total acquisition time fixed. DeepCormack could compensate for the lower quality projections, whilst the richer angular sampling could improve reconstruction quality overall. This may also benefit configurations beyond the standalone DC U \\text{DC}_{\\text{U}} , as the CNN and MLP are well-suited to exploiting additional input channels, and modular frameworks such as DC CMU \\text{DC}_{\\text{CMU}} could prove better positioned to take advantage of both denser projection coverage and richer training data."},{"type":"heading","level":1,"text":"5 Conclusions"},{"type":"para","text":"In this work we proposed DeepCormack, a family of data-driven, model-based reconstruction algorithms for Fermi surface tomography. We also propose a data generation model based on Koopman operators to train such models. Results show significant improvement in reconstruction quality compared to the baseline, standard Modified Cormack Method, both in the standard acquisition regime and in significantly reduced ones. This not only allows for much higher quality results for the condensed matter physics community, but also potentially saves weeks of scanning time without any loss (in fact, with an increase) in reconstruction quality."},{"type":"heading","level":4,"text":"Copper Koopman Operator on Copper Central Slice:"},{"type":"para","text":"The copper momentum density is where we derived our per-channel Koopman operators 𝒦 n \\mathcal{K}_{n} from. Here, we evaluate whether evolving the Chebyshev coefficients for the copper central slice via dynamic mode decomposition is sufficient to recover the original structure. To do this comparison, we took the original DFT-generated copper momentum density along with the DMD-evolved copper momentum density, applied LCW-folding to convert from p p -space to k k -space before extracting their Fermi surfaces. Figure 15 compares slices taken through high symmetry planes ( 100 ) (100) , ( 110 ) (110) , and ( 111 ) (111) of the Fermi surfaces."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Fermi surface of a material is the surface in reciprocal space (or k -space) separating occupied from unoccupied electronic states at zero temperature ( T = 0 K T=0{\\\\rm K} ). Its shape is determined entirely by the electronic band structure, i.e. the relationship between electron energy and crystal momentum evaluated at the chemical potential. This shape governs many of the key properties of a metal, including electrical and thermal conductivity, magnetism, and more exotic phenomena such as superconductivity. The Fermi surface is not physical, it is a representation of which electronic states are available for low-energy excitations. Almost all low-energy physics in a metal is controlled by electrons near the Fermi surface: states deeper within it are frozen by the Pauli exclusion principle and cannot participate in transport, magnetic ordering, or Cooper pairing [ 16 , 7 ] . In the free-electron model, the Fermi surface is a simple sphere in k -space, but in real materials it becomes more complex due to the periodic crystal potential leading to electronic band structures that disperse anisotropically. This can lead to Fermi surfaces which comprise multiple interconnected sheets with topologies which are consistent with the underlying crystal symmetry [ 7 ] . These deviations in the Fermi surface topology, in comparison to the free-electron picture, can play a pivotal role in the physical properties of metals, for example, influencing the appearance of superconductivity, magnetic order and charge-density waves [ 7 ] . The geometry of the Fermi surface therefore encodes a great deal about what a metal can do, and having more accurate methods for measuring it is therefore important.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Angular correlation of annihilation radiation (ACAR) is one such method [ 8 , 18 ] . It is also a method which is bulk sensitive, can be applied at room temperature, and which is suitable for disordered alloys [ 9 , 12 ] . When positrons are implanted into a sample, they thermalise quickly before annihilating with an electron, emitting two photons. As the thermal positrons carry negligible momentum at the point of annihilation, the small angular deviations of the photon pair from exact collinearity (on the order of milliradians) essentially encodes the momentum of the electron immediately prior to annihilation. In a 2D-ACAR experiment, two position-sensitive detectors placed on either side of the sample record the coordinates of the coincident gamma-photons which impinge on each detector, allowing the angular deviations of each detected annihilation event to be recorded. Accumulating on the order of ∼ 10 8 \\\\sim 10^{8} coincidence events produces a 2D projection of the underlying three-dimensional two-photon momentum density (TPMD). By repeating this at different sample orientations (corresponding to different crystallographic directions), a set of projections is obtained from which the full 3D TPMD can be reconstructed. The features of the Fermi surface geometry can be extracted from the reconstructed density by converting from p -space to k -space via the Lock-Crisp-West (LCW) theorem [ 20 , 8 ] . This LCW procedure sums over reciprocal lattice vector contributions to reduce the extended-zone momentum density into the reduced Brillouin zone, giving us the k -space density which represents the occupation number as seen by the positron . The Fermi surface then appears as a discontinuity in this occupation number, and can be recovered by, for example, computing the gradient. Some further details about the experimental and theoretical background to the measurement of the Fermi surface by the 2D-ACAR method can be found in Refs. [ 14 , 6 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recovering the 3D TPMD from a small number of projections remains a challenging inverse problem. Several reconstruction algorithms have been developed for this purpose, including filtered backprojection, spherical harmonic methods [ 22 , 23 ] , and more recently regularisation-based approaches [ 27 , 28 , 19 ] . In this work, we focus on the planar reconstruction method originally proposed by Cormack [ 4 , 5 , 3 ] and later adapted for Fermi surface studies by Kontrym-Sznajd [ 17 ] . The modified Cormack method (hereafter MCM) is well-suited to 2D-ACAR data because its least-squares properties handle experimental errors through natural truncation of the series, and because it enforces a consistency condition on the projection data that can correct for inconsistencies introduced by noise or normalisation errors. The underlying method uses polar Fourier series, expanding the projection data into sets of orthogonal polynomials (Chebyshev and Zernike), to carry out the reconstruction. The inherent symmetry of the crystals being studied and the relatively smooth nature of the reconstructed densities, make it a particularly powerful technique for recovering information about the Fermi surface.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite the advantages of the MCM, acquiring 2D-ACAR data of sufficient quality for Fermi surface studies is extremely time-consuming in practice. To make measurements with sufficiently high angular resolution (recalling that it is the angle that is proportional to the electron momentum), the joint requirements of high detector efficiency and acceptable position resolution mean that there can be up to 24m between the two detectors. The solid angle subtended by the detectors is therefore very small, resulting in relatively low coincidence count rates in the range of hundreds of counts per second. There is an inherent trade-off between acquisition time and that overall statistical precision of a spectrum, with the collection of more annihilation events reducing noise but requiring longer measurements. With a radioisotope source (typically 22 Na), obtaining a single high-quality data set comprising five to six projections at 200 200 million counts each can takes approximately three to four months. A further complication arises for reconstruction algorithms which operate in a slice-by-slice manner: as the number of recorded counts decreases away from the centre of the projections (corresponding to zero momentum), noise will vary systematically across the reconstructed slices. This results in the reconstructed densities at points away from the centre of the momentum density having much lower quality than ones closer to it. An approach which can simultaneously improve the momentum density reconstruction quality from ACAR data while at the same time lowering the acquisition time (meaning noisier projections) would therefore be desirable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given that 2D-ACAR shares its core measurement geometry with positron emission tomography (PET), there is good reason to expect that machine learning advances in medical image reconstruction could translate productively to Fermi surface tomography, given the significant advances in the field in the last few years, e.g. [ 25 , 11 , 1 , 15 ] . The goal of this work is to investigate whether similar approaches can positively translate to ACAR tomography, using supervised neural networks in conjunction with classical reconstruction techniques to improve the recovered Fermi surface details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To this end, we introduce DeepCormack, a hybrid framework combining the Cormack-Kontrym-Sznajd method with supervised deep learning. The network includes a 1D convolutional neural network trained in projection space, a multi-layer perceptron in radial density function space, and a 2D UNet in image space, learning corrections that reduce noise and convolution artefacts while partially compensating for the information loss inherent in the limited angular sampling from 5 5 projections. As reconstruction quality degrades away from the centre of a momentum density, the effect of conditioning the UNet on the statistical precision (characterised by the total number of counts in a spectrum) to make it robust to the variation in noise is also studied.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To support training and evaluation, we also propose a data-driven simulation framework based on singular value decomposition (SVD) and dynamic mode decomposition (DMD), which enables the generation of realistic, continuously varying synthetic 3D TPMD volumes, based entirely on a DFT-generated reference momentum density for copper [ 10 ] . Copper was chosen for its simple but anisotropic Fermi surface topology, as well as having the same crystal structure (face-centred cubic) as the intermetallic compound ZrZn 2 , for which we have DFT calculations of the TPMD available. This represents an excellent intermediate step between the relatively simple momentum density of copper (with its single Fermi surface sheet) and the challenge of reconstructing from real high quality experimental data (which have been measured by Major et al. [ 21 ] ). With a much larger lattice constant (factor of two) compared to copper, the impact of resolution on the features across the much smaller Brillouin zone is likely to be more significant. Furthermore, with multiple Fermi surface sheets, its momentum density is quite different from copper which provides an excellent test of the training data based on copper. Together, these components form a physically consistent and scalable pipeline for machine learning-assisted ACAR reconstruction under experimentally realistic conditions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By investigating the application of modern deep learning tools in ACAR tomography, our hope is to open up the area of Fermiology to the wider inverse problem community. Finding techniques that allow for improved reconstruction while lowering the required statistical precision of the projections can enable faster and more efficient characterization of a material’s properties, and make the study of Fermi surfaces more accessible.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The paper is organised as follows: Section 2 presents the main algorithm and data generation approach. Section 3 presents experimental results of DeepCormack for different model configurations, across acquisition counts ( 10 − 200 10-200 M), and for an out-of-distribution (OOD) sample of ZrZn 2 to test its ability to generalize. Section 4 discusses these results in depth, while Section 5 summarises the main findings, discusses current limitations, and outlines improvements to the method.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The main contributions can be summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A derivation of the modified Cormack method reconstruction method in operator notation for the field of inverse problems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Novel method using SVD and DMD for generating sufficiently realistic 3D-TPMD data to train ML models with;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Framework for integrating neural networks into the modified Cormack method for substantially improved Fermi surface reconstruction with ACAR tomography, sufficiently flexible enough to allow for slice-by-slice reconstruction despite varying noise levels across the TPMD volume;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Practical demonstrations of DeepCormack’s utility, on both pseudo-real data and DFT-generated TPMD data under simulated experimental conditions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Methods\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose a family of methods which integrate neural networks into the modified Cormack method (described in section 2.1 ) to improve the two-photon momentum density reconstruction that can be achieved via conventional angular correlation of electron-positron annihilation radiation measurements. This framework, which we have dubbed DeepCormack ( 2.2 ), requires supervised learning. Section 2.3 proposes a data simulation method which allows for paired ground truth and simulated measurement (synthetic) 3D TPMDs to be generated at scale.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Fermi Surface Tomography\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Fermi surface tomography mathematical model is similar to computed tomography (CT), however given its development happened in parallel to mathematical advances in CT, mathematical descriptions that link both are missing from the literature. This section introduces the Fermi surface tomography forward model, and the one of the most successful and widely applied reconstruction algorithms, namely the Modified Cormack Method (MCM), describing the mathematical foundations and linking it to an operator form of the reconstruction, which can open up the model for further analysis from the inverse problems community (analysis not presented in this work). Thus the description of the MCM presented here differs from the standard one in the Fermiology literature. We recommend the works of [ 4 , 5 , 17 ] for the more standard description of the MCM.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Firstly, the forward model of the Fermi surface tomography is the shown in Equation 1 as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where f ¯ \\\\bar{f} is the measurement as a vector containing the positive half of our normalized projections as f ¯ ∈ ℝ L {\\\\bar{f}}\\\\in\\\\mathbb{R}^{L} , where L = L p ⋅ Φ L=L_{p}\\\\cdot\\\\Phi being the number of measured pixels L p L_{p} in each measured projection Φ \\\\Phi and ρ ¯ 2 γ ∈ ℝ 2 L P ⋅ 2 L p \\\\bar{\\\\rho}^{2\\\\gamma}\\\\in\\\\mathbb{R}^{2L_{P}\\\\cdot 2L_{p}} is the vectorized reconstructed TPMD image, finally e ~ \\\\tilde{e} correspond to measurement noise. The operator ℛ \\\\mathcal{R} is the Radon transform.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, the measurements are 2D over Φ \\\\Phi projections at various angles φ \\\\varphi of a sample’s 3D TPMD. Each projection f φ f_{\\\\varphi} will have shape f φ ∈ ℝ L p × L p f_{\\\\varphi}\\\\in\\\\mathbb{R}^{L_{p}\\\\times L_{p}} , but due to mirror symmetry, only one quadrant of the data will be needed, and the other 3 quadrants will be symmetrically folded and averaged to reduce measurement noise, and finally normalized against the first projection. Given the operator is independent per slice of the 2D measurement f φ f_{\\\\varphi} , we define f f as a single row of f φ f_{\\\\varphi} , for every φ \\\\varphi and its vectorized version, f ¯ \\\\bar{f} . This allows us to define the operators in a more simple way, and highlights that the 3D-TPMD reconstruction process is comprised of a stack of independent 2D problems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In tomographic applications where the Radon transform is the operator, the pseudo-inverse ℛ † \\\\mathcal{R}^{\\\\dagger} (or reconstruction operator) is often represented by the Filtered Backprojection (FBP) algorithm. However, this is not the case in Fermi surface tomography, as the extremely limited number of projections causes the FBP algorithm to struggle. The MCM, by instead exploiting the inherent known symmetries of the reconstruction, produces superior results and thus has been widely applied.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.1 The Modified Cormack Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The MCM algorithm can be written it in operator notation, as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where each of the matrices represent a step of the MCM algorithm. Firstly 𝒞 ∈ ℝ M × L \\\\mathcal{C}\\\\in\\\\mathbb{R}^{M\\\\times L} is a linear interpolator operator, in particular converting each row of the detector from a linear coordinate system sampled from the indices t = cos ( δ ) t=\\\\cos(\\\\delta) , with δ \\\\delta being a vector that linearly samples from [ 0 , π 2 ] [0,\\\\frac{\\\\pi}{2}] of size Δ \\\\Delta , making M = Δ ⋅ Φ M=\\\\Delta\\\\cdot\\\\Phi . This step is often described in the MCM method as a projection of the measurement into the domain of the unit circle with polar coordinates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This radial sampling of the measurement allows to expand the data into polar Fourier series, defined by operator 𝒫 ∈ ℝ M × M \\\\mathcal{P}\\\\in\\\\mathbb{R}^{M\\\\times M} . Let N ( t , φ ) := ( 𝒞 f ) ( t , φ ) N(t,\\\\varphi):=(\\\\mathcal{C}f)(t,\\\\varphi) denote the projection data expressed in the polar coordinates produced by the interpolation step 𝒞 \\\\mathcal{C} . This operator relies on an important observation, and prior information, of the crystal being measured and its symmetries. G G denotes the reciprocal lattice vector of the crystal being studied, and let | G | |G| denote its order, i.e. the fold of rotational symmetry. For a face-centred cubic (FCC) structure, this corresponds to | G | = 4 |G|=4 — which all of the samples studied in this work have. Because N ( t , φ ) N(t,\\\\varphi) must be invariant under a rotation of 2 π / | G | 2\\\\pi/|G| , only harmonics of order n ≡ 0 mod | G | n\\\\equiv 0\\\\mod{|G|} can appear in its Fourier expansion, so that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In operator form, 𝒫 \\\\mathcal{P} is thus a symmetry-truncated Fourier transform: it maps 𝒞 f ¯ \\\\mathcal{C}\\\\bar{f} onto the coefficients N n ( t ) N_{n}(t) at orders n ≡ 0 mod | G | n\\\\equiv 0\\\\mod{|G|} only, discarding all other harmonics a priori. Having obtained the radial harmonic profiles N n ( t ) N_{n}(t) , each is further expanded as a sine series built from Chebyshev polynomials as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equation 4 can be written in matrix form as 𝒮 = I ⊗ 𝒮 φ \\\\mathcal{S}=I\\\\otimes\\\\mathcal{S}^{\\\\varphi} , where is I ∈ ℝ Φ × Φ I\\\\in\\\\mathbb{R}^{\\\\Phi\\\\times\\\\Phi} the identity matrix and 𝒮 φ ∈ ℝ Δ × Δ \\\\mathcal{S}^{\\\\varphi}\\\\in\\\\mathbb{R}^{\\\\Delta\\\\times\\\\Delta} is 1 1 1 For equation simplicity this works uses the convention of matrix elements starting at index zero, i.e. the first element of a matrix is in index i = 0 , j = 0 i=0,j=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can thus define the process of obtaining Chebyshev coefficients from measured projections as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This operator 𝒜 \\\\mathcal{A} is useful later in this work to generate synthetic data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before the coefficients 𝐚 n \\\\mathbf{a}_{n} are used to synthesize the density, they are weighted elementwise with diagonal matrix 𝒲 ∈ ℝ M × M \\\\mathcal{W}\\\\in\\\\mathbb{R}^{M\\\\times M} as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The following step of the MCM reconstruction is to map the Chebyshev coefficients into the radial density functions, ρ ∈ ℝ L × 1 \\\\rho\\\\in\\\\mathbb{R}^{L\\\\times 1} , using Zernike polynomials, which can be represented as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"being ω \\\\omega the radial distance over the unit circle. The construction of the Zernike matrix 𝒵 ∈ ℝ L × M \\\\mathcal{Z}\\\\in\\\\mathbb{R}^{L\\\\times M} can be summarized as 𝒵 = I ⊗ 𝒵 n \\\\mathcal{Z}=I\\\\otimes\\\\mathcal{Z}^{n} , with an identity matrix I ∈ ℝ Φ × Φ I\\\\in\\\\mathbb{R}^{\\\\Phi\\\\times\\\\Phi} and a series of 𝒵 n ∈ ℝ L p × Δ \\\\mathcal{Z}^{n}\\\\in\\\\mathbb{R}^{L_{p}\\\\times\\\\Delta} , for n ∈ { r | G | : r ∈ { 0 , 1 , … , Φ } } n\\\\in\\\\{r|G|:r\\\\in\\\\{0,1,\\\\dots,\\\\Phi\\\\}\\\\} . Each sub-matrix is then built by computing the Zernike coefficients at the radial density values w i ∈ [ 0 , 1 , … , L ] w_{i}\\\\in[0,1,\\\\dots,L] , with a different matrix for each value of n n , as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This recovers the operator notation of the Modified Cormack Method:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The resulting radial density functions, ρ \\\\rho , can be converted to the density ρ ¯ 2 γ ( p , θ ) \\\\bar{\\\\rho}^{2\\\\gamma}(p,\\\\theta) in the polar plane. Each channel ρ n ( p ) \\\\rho_{n}(p) is modulated by cos ( n θ ) \\\\cos(n\\\\theta) and summed:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Equation 11 reconstruction step is illustrated in Appendix A . This recombination step defines the operator 𝒪 \\\\mathcal{O} , mapping ρ {\\\\rho} onto ρ ¯ 2 γ \\\\bar{\\\\rho}^{2\\\\gamma} , and gives the full modified Cormack method as originally stated:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While in Fermiology this is often the final step, it is of interest to also define the TPMD in Euclidean form ( ρ E 2 γ \\\\rho_{E}^{2\\\\gamma} ), as shown in Equation 13 , since the standard CNN architectures used in this work are more naturally suited to a Euclidean grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In operator form, we express this transformation into Euclidean space using operator ℰ \\\\mathcal{E} as:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 DeepCormack\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section describes our proposed data-driven method, enhancing the MCM algorithm with deep neural networks. In general, DeepCormack integrates up to three neural networks at different key stages of the modified Cormack method, as shown in Figure 1 . The process works as follows: 2D-ACAR measures projections of a sample’s momentum density at different angles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The MCM reconstructs the TPMD at fixed detector row slice-by-slice, and DeepCormack proceeds similarly. However, the normalized projections are first passed through a convolutional neural network (CNN) (details in section 2.2.1 ) in order to denoise and deconvolute them. Equation 10 is then applied to recover the corresponding ∼ 5 \\\\sim 5 radial density functions, which are subsequently passed through a fully connected multilayer perceptron (MLP) (details in section 2.2.2 ). The MLP-outputted radial density functions are passed through Equation 11 to reconstruct the TPMD ρ 2 γ \\\\rho^{2\\\\gamma} . This reconstruction is finally passed through a UNet (details in section 2.2.3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we explore several possible configurations of this DeepCormack structure explained in more detail in the Results section.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.1 1D CNN\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first stage, shown in Figure 2 , employs a one-dimensional residual convolutional neural network (CNN) to map noisy measured projections ( f m e a s = f + e ~ ) (f_{meas}=f+\\\\tilde{e}) to their ideal counterparts ( f G T = f ) ({f}_{GT}=f) . The network accepts as input Φ \\\\Phi ( 5 5 in our work) angular projections, each of length L p L_{p} ( 256 256 in ours), arranged as a multi-channel 1D signal of shape ( Φ , L p ) (\\\\Phi,L_{p}) . The architecture consists of a stack of five one dimensional convolutional layers with kernel size 3 3 and unit padding, with the first four layers projecting into 32 32 hidden channels and the final layer projecting back to 5 5 output channels, preserving the input dimensionality. ReLU activations are applied between all intermediate layers. A global residual connection is added from the input directly to the output, such that the network learns a correction Δ \\\\Delta rather than the full mapping, i.e. f C N N = f meas + CNN ( f meas ) {{{f}}}_{CNN}={{f}}_{\\\\text{meas}}+\\\\text{CNN}({{f}}_{\\\\text{meas}}) . The network is trained by minimising the loss comprising a direct mean-squared error (MSE) between predicted and ideal projections ( ℒ 1 = MSE ( f G T , f C N N ) \\\\mathcal{L}_{1}=\\\\text{MSE}\\\\left({{f}}_{GT},{{f}}_{CNN}\\\\right) ), and an auxiliary MSE between the ground truth TPMD and CNN-denoised TPMD reconstruction in Euclidian coordinates ( ℒ 2 = MSE ( ℰ ℳ f GT , ℰ ℳ f CNN ) \\\\mathcal{L}_{2}=\\\\text{MSE}\\\\left(\\\\mathcal{EM}{f}_{\\\\text{GT}},\\\\mathcal{EM}{f}_{\\\\text{CNN}}\\\\right) ). The full loss term is constructed as ℒ C = α ℒ 1 + ℒ 2 \\\\mathcal{L}_{\\\\text{C}}=\\\\alpha\\\\mathcal{L}_{1}+\\\\mathcal{L}_{2} . We found α = 10 \\\\alpha=10 empirically the best value. This allows the model to denoise the projections, while ensuring the final results are still valid, given that small changes in the measurements can cause large changes in the final image. The network is optimised using the Adam optimiser with a learning rate of 10 − 3 10^{-3} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.2 MLP\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second stage employs a fully connected multilayer perceptron (MLP) to refine the radial density function representation of the CNN output, as shown in Figure 3 . The input to the MLP is a tensor of shape ( Φ , L p ) (\\\\Phi,L_{p}) representing the L p = 256 L_{p}=256 radial grid points each described by Φ = 5 \\\\Phi=5 Cormack expansion coefficients, which is flattened to a vector of dimension Φ ⋅ L p = 1280 \\\\Phi\\\\cdot L_{p}=1280 . The network consists of two hidden layers of width 11 × L p = 2816 11\\\\times L_{p}=2816 with ReLU activations, followed by a linear output layer projecting back to the input dimensionality. The output is reshaped to ( Φ , L p ) (\\\\Phi,L_{p}) to match the input. The CNN is frozen, and the MLP receives as input the CNN-denoised projections passed through the MCM to get ρ {\\\\rho} . The network is supervised entirely in reconstruction space: the loss is the MSE between the Euclidean 2D TPMD from the MLP output and the ground truth ( ℒ M = MSE ( ℰ ρ GT , ℰ ρ MLP ) \\\\mathcal{L}_{\\\\text{M}}=\\\\text{MSE}\\\\left(\\\\mathcal{E}{\\\\rho}_{\\\\text{GT}},\\\\mathcal{E}{\\\\rho}_{\\\\text{MLP}}\\\\right) ). This formulation also encourages the MLP to recover features in the TPMD that are lost due to the limited number of projections available in experimental ACAR. The MLP is optimised with the Adam optimiser at a learning rate of 10 − 3 10^{-3} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.3 UNet\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The third stage is a UNet architecture (Figure 4 ) based on [ 13 ] to learn residual corrections to the two-dimensional TPMD reconstruction. The projections are passed through the CNN, converted to radial density functions ρ {\\\\rho} , and inputted to the MLP. The output is reconstructed into a quadrant of the 2D image ρ P 2 γ {\\\\rho}^{2\\\\gamma}_{P} of shape ( 1 , L p , L p ) (1,L_{p},L_{p}) via the final step of the MCM, and becomes the input of the UNet.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The encoder comprises three convolutional blocks with channel progressions [ 1 , 64 , 64 , 64 ] [1,64,64,64] , [ 64 , [64, 128 , 128, 128 ] 128] , and [ 128 , 256 , 256 ] [128,256,256] respectively, each followed by 2 × 2 2\\\\times 2 max-pooling for spatial downsampling. The bottleneck processes features at [ 256 , 512 , 512 ] [256,512,512] channels with a spatial dropout rate of 0.1 0.1 . The decoder mirrors the encoder using transposed convolutions for upsampling, with skip connections concatenating encoder feature maps to the corresponding decoder inputs at each resolution level, and channel progressions [ 512 , 256 , 256 ] [512,256,256] , [ 256 , 128 , 128 ] [256,128,128] , and [ 128 , 64 , 64 ] [128,64,64] . Each convolutional block uses Group Normalisation and ReLU activations. A final 1 × 1 1\\\\times 1 convolution projects to a single output channel. A global residual connection adds the input image to the network output, enforcing that the UNet learns a residual correction such that ρ 2 γ = ReLU ( ρ pred 2 γ + UNet ( ρ pred 2 γ ) ) {\\\\rho}^{2\\\\gamma}=\\\\text{ReLU}(\\\\rho^{2\\\\gamma}_{\\\\text{pred}}+\\\\text{UNet}(\\\\rho^{2\\\\gamma}_{\\\\text{pred}})) , where the ReLU enforces physical non-negativity of the momentum distribution. The network is trained by minimising the MSE between the corrected output and the ground-truth TPMD image ( ℒ U = MSE ( ρ 2 γ GT , ρ 2 γ UNet ) \\\\mathcal{L}_{\\\\text{U}}=\\\\text{MSE}\\\\left({\\\\rho^{2\\\\gamma}}_{\\\\text{GT}},{\\\\rho^{2\\\\gamma}}_{\\\\text{UNet}}\\\\right) ), using the Adam optimiser at a learning rate of 10 − 4 10^{-4} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.2.4 FiLM Conditioning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A UNet was also tested with and without Feature-wise Linear Modulation (FiLM) [ 24 , 26 ] . FiLM learns functions γ ( z ) \\\\gamma(z) and β ( z ) \\\\beta(z) to modulate a neural network’s intermediate feature map 𝐱 \\\\mathbf{x} via a learned affine transformation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"making it possible to condition the network on a signal z z without modifying the weights themselves. In our case, z z is taken as the log-normalised count level averaged across the simulated measurement projections, providing the UNet with an explicit measure of the Poisson noise in the data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the UNet, FiLM is applied to the output of each convolutional block (i.e., after normalisation and activation), rather than between normalisation and the nonlinearity as in the original formulation. z z is first encoded via a sinusoidal embedding and refined by a shared two-layer SiLU MLP, whose output is distributed to a single linear projection per block, producing per-channel ( γ , β ) (\\\\gamma,\\\\beta) pairs at each encoder, latent, and decoder stage. FiLM parameters are zero-initialised, such that at the start of training γ = 0 \\\\gamma=0 , β = 0 \\\\beta=0 , making the transformation an identity map.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Data Simulation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As experimental ACAR measurements can take months, constructing 3D TPMD training datasets of the size typically required for supervised deep learning via experimental means alone would be impractical, as it takes 3-4 months to measure a single materials. Computational approaches such as density functional theory (DFT) could in principle provide ground-truth momentum densities from which projections could be taken through them and realistic experimental conditions simulated. However, DFT calculations are themselves rather intensive to compute both in terms of human time (parameter setting, convergence testing etc.) as well as resources (computing times can be of the order of hours, and the densities themselves can require storage at the GB scale). As no publicly available momentum density database currently exists, a key step for making supervised learning feasible requires a data generation method for sufficiently diverse and realistic synthetic 3D TPMDs that continuously evolve slice-by-slice. This would allow the detector resolution to be simulated as a 2D Gaussian convolution, noise via Poisson statistics, and the momentum sampling function (MSF) — a necessary correction due to the ACAR detector geometry, as only photons emitted with perfect collinearity have 100% probability to both hit the detectors — can be applied as a normalized tent function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our proposed data generation approach is applied on a reference momentum density for copper, and consists of two independent steps: first, plausible central slices are generated, using a combination of singular value decomposition (SVD) on the copper Chebyshev coefficients and varying the singular values. Dynamic mode decomposition (DMD) [ 2 ] is then applied, which learns the underlying dynamics of how the copper momentum density evolves continuously from one slice to the next.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Any number of central slice TPMDs can be generated using SVD, then continuously evolved into a full 3D TPMD using DMD. This will act as our ground truth for the dataset. As outlined in Section 2.3.3 , a corresponding experimental dataset can be simulated by taking 5 5 projections through the ground truth volumes and applying realistic degradation to the data (detector resolution and geometry effects, noise etc).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taking 20 20 ideal projections of the DFT copper momentum density between φ ∈ [ 0 , 45 ] \\\\varphi\\\\in[0,45] degrees preserves sufficient information when reconstructed via the MCM (Equation 12 ) to act as an approximation for the ground truth. This allows our data generation method to operate entirely on the extracted Chebyshev coefficients, as defined in Equation 4 , for some projection slice l = p y ∈ [ 0 , L p − 1 ] , l=p_{y}\\\\in[0,L_{p}-1], L p = 256 , L_{p}=256, that is reconstructed from copper: a ~ n , l m \\\\tilde{a}_{n\\\\hskip 1.42271pt,\\\\hskip 1.42271ptl}^{m} . Here, m ∈ [ 0 , Δ − 1 ] , m\\\\in[0,\\\\Delta-1], Δ = 180 \\\\Delta=180 are the number of angular degrees of freedom when parametrising the projections along the unit circle, and n ∈ [ 0 , Φ − 1 ] n\\\\in[0,\\\\Phi-1] denotes the projection index, where Φ = 20 , \\\\Phi=20, are the number of ideal projections used to approximate the ground truth copper momentum density with the MCM. For some projection Φ , \\\\Phi, we define the matrix [ 𝐚 ~ 𝚽 ] l m = a ~ Φ , l m \\\\left[\\\\mathbf{\\\\tilde{a}_{\\\\Phi}}\\\\right]_{lm}=\\\\tilde{a}_{\\\\Phi\\\\hskip 1.42271pt,\\\\hskip 1.42271ptl}^{m} , where 𝐚 ~ Φ ∈ ℝ Δ × L p \\\\mathbf{\\\\tilde{a}}_{\\\\Phi}\\\\in\\\\mathbb{R}^{\\\\Delta\\\\times L_{p}} . When generating the synthetic data, we model each of the Chebyshev coefficients independently via their projection channel by fixing n n , as they are tied to separate ρ \\\\rho ’s when reconstructed. The label will be omitted hereafter for notational clarity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3.1 Generating Central Slices\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The MCM, at the step shown in Equation 4 , decomposes each projection into a set of Chebyshev coefficients to carry out the reconstruction. These are the source values used for our generative process of central slices.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a fixed projection, truncated SVD is applied across the angular channel over all unique slice indices, reducing the dimensionality to a level that maintains approximately ∼ 90 % \\\\sim 90\\\\% of the variance. This yields an independent lower dimensional latent space per projection channel, each of which is modelled by fitting a multivariate Gaussian to allow sampling new points. When lifted back to coefficient space, the result is a new set of Chebyshev coefficients that can proceed through the rest of the MCM (Equation 12 ), and after some minor tuning will give a realistic central TPMD slice.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"More precisely, for 𝐚 ~ \\\\mathbf{\\\\tilde{a}} previously defined in Section 2.3 , let 𝐚 ~ l ∈ ℝ Δ , \\\\mathbf{\\\\tilde{a}}_{l}\\\\in\\\\mathbb{R}^{\\\\Delta}, denote the copper Chebyshev coefficient state at a fixed projection n n at a given slice l l . To construct the latent space and sample a new set 𝐚 ^ ∈ ℝ Δ , \\\\hat{\\\\mathbf{a}}\\\\in\\\\mathbb{R}^{\\\\Delta}, we begin by constructing the snapshot matrix using the copper values:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The temporal 2 2 2 Note that time is not a variable in our work, instead the Chebyshev coefficient variation over slices are the variable. However, for consistency with the DMD and Koopman operator literature, we call this time. mean is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"allowing us to get the mean-centered data as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝟏 ∈ ℝ L p \\\\mathbf{1}\\\\in\\\\mathbb{R}^{L_{p}} is a vector of ones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compute the singular value decomposition (SVD) of the centered snapshot matrix:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝐔 ∈ ℝ Δ × Δ \\\\mathbf{U}\\\\in\\\\mathbb{R}^{\\\\Delta\\\\times\\\\Delta} contains the orthonormal spatial modes, 𝚺 ∈ ℝ Δ × L p \\\\bm{\\\\Sigma}\\\\in\\\\mathbb{R}^{\\\\Delta\\\\times L_{p}} is a diagonal matrix of singular values, and 𝐕 ∈ ℝ L p × L p \\\\mathbf{V}\\\\in\\\\mathbb{R}^{L_{p}\\\\times L_{p}} contains temporal coefficients. We retain the first K ≪ Δ K\\\\ll\\\\Delta dominant modes, yielding the truncated SVD:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝐔 k ∈ ℝ Δ × K \\\\mathbf{U}_{k}\\\\in\\\\mathbb{R}^{\\\\Delta\\\\times K} defines a reduced spatial basis. In this step, K = 15 K=15 ( Δ / K = 12 ) (\\\\Delta/K=12) was found to construct reasonable TPMD features, capturing ∼ 88 % \\\\sim 88\\\\% of the total variance across slices. A Gaussian is fitted to the latent coordinates 𝚺 k 𝐕 k ⊤ ∈ ℝ K × L p \\\\bm{\\\\Sigma}_{k}\\\\mathbf{V}_{k}^{\\\\top}\\\\in\\\\mathbb{R}^{K\\\\times L_{p}} of the L p = 256 L_{p}=256 copper slices, a new latent vector 𝐳 ^ \\\\hat{\\\\mathbf{z}} is drawn, and projected back to coefficient space to give a synthetic Chebyshev coefficient 𝐚 ^ ∈ ℝ Δ \\\\mathbf{\\\\hat{a}}\\\\in\\\\mathbb{R}^{\\\\Delta} by:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where μ k \\\\mu_{k} and Λ k \\\\Lambda_{k} are the sample mean and full covariance matrix of the latent coordinates respectively. Repeating this process for all Φ \\\\Phi channels (with the projection channel n n reinstated) yields the new central-slice coefficient matrix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equation 10 can subsequently be applied to recover the corresponding ρ \\\\rho and Equation 12 for the reconstructed ρ P 2 γ \\\\rho^{2\\\\gamma}_{P} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It was found that applying corrections to the ρ \\\\rho generated from 𝐀 ^ l = 0 \\\\hat{\\\\mathbf{A}}_{\\\\mathrm{l=0}} improved the overall quality and consistency of the SVD sampling method. First, the isotropic term ρ 0 \\\\rho_{0} (being the most sensitive and frequently producing poor-quality TPMDs when sampled via SVD) is replaced with the isotropic term drawn from the original copper dataset at a slice index in the range l ∈ [ 0 , 133 ] l\\\\in[0,133] . 1D Gaussian smoothing is applied along the radial axis of the higher-order density terms ρ n > 0 \\\\rho_{n>0} to suppress unphysical and overly exotic features that arise from SVD sampling but are not characteristic of real TPMDs. All ρ n \\\\rho_{n} functions are normalised such that their peak intensities remain within a ∼ 10 % \\\\sim 10\\\\% range observed in the corresponding copper reference functions. Finally, a physicality constraint is then enforced: the total radial density ∑ n ρ n \\\\sum_{n}\\\\rho_{n} must remain non-negative everywhere. Passing these corrected ρ n \\\\rho_{n} functions through the final MCM reconstruction step yields the synthetic ground truth TPMD slice to be used in the next stage of the data generation method.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since corrections have been applied to the ρ n \\\\rho_{n} functions, evolving them into full 3D TPMDs will require recomputing the Chebyshev coefficients for each synthetic central slice. This is done by applying the Radon transform at 20 equally spaced projection angles φ ∈ [ 0 ∘ , 45 ∘ ] \\\\varphi\\\\in[0^{\\\\circ},45^{\\\\circ}] and recovering the corresponding (corrected) Chebyshev coefficients via the MCM (Equation 10 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3.2 Evolving the central slice into 3D TPMDs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once feasible generated central slices are created, 3D versions of them need to be derived. In order to extrapolate the central slice in a manner coherent with TPMD structure, we propose a dynamic mode decomposition (DMD) model [ 2 ] . This is trained per-projection channel on the original copper Chebyshev coefficients 𝐚 ~ \\\\mathbf{\\\\tilde{a}} . DMD is a particularly powerful method for this: it allows the slice-to-slice non-linear dynamics of the Chebyshev coefficients to be simulated by assuming they evolve linearly in a latent space. It achieves this by computing an approximate Koopman operator for each n n , such that the synthetically generated set of central-slice Chebyshev coefficients 𝐀 ^ l = 0 \\\\hat{\\\\mathbf{A}}_{\\\\mathrm{l=0}} can be evolved into a full set 𝐀 ^ ∈ ℝ L p × Δ × Φ \\\\hat{\\\\mathbf{A}}\\\\in\\\\mathbb{R}^{L_{p}\\\\times\\\\Delta\\\\times\\\\Phi} . Passing each of the evolved Chebyshev coefficients through the MCM as 𝒜 f ¯ \\\\mathcal{A}\\\\bar{f} in Equation 10 , the result is a continuously evolving 3D TPMD.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In order to compute the DMD model, we Continue from the truncated SVD for copper in Equation 16 , for a fixed projection channel n n , we now retain the first K = 24 K=24 ( M / K = 7.5 M/K=7.5 ) dominant modes as it was found to give a good trade-off between diversity and stable volume evolution. Each Chebyshev slice snapshot ( 𝐚 ~ l ) (\\\\tilde{\\\\mathbf{a}}_{l}) from Equation 15 is projected onto the reduced basis as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Allowing us to define the time-shifted latent snapshot matrices:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We assume linear evolution in latent space:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒦 ∈ ℝ K × K \\\\mathcal{K}\\\\in\\\\mathbb{R}^{K\\\\times K} is a finite-dimensional approximation of the Koopman operator, and is obtained by solving the least-squares problem\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where | | ⋅ | | F ||\\\\cdot||_{F} is the Frobenius norm. The closed-form solution is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with 𝐙 † \\\\mathbf{Z}^{\\\\dagger} denoting the Moore–Penrose pseudoinverse.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once the Koopman operator 𝒦 \\\\mathcal{K} is pre-computed in latent space, the generated central slice for the projection channel 𝐚 ^ l \\\\mathbf{\\\\hat{a}}_{l} can be rolled out to produce slice a ^ l + 1 \\\\hat{a}_{l+1} via:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"resulting in the new Chebyshev snapshot matrix for projection channel n n :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Repeating this process for all Φ \\\\Phi projection channels — each with their own unique Koopman operator 𝒦 n \\\\mathcal{K}_{n} — will result in a full set of synthetic Chebyshev coefficients:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Separately passing each 𝐀 ^ l = [ 𝒜 f ] synthetic \\\\hat{\\\\mathbf{A}}_{l}=[\\\\mathcal{A}f]_{\\\\text{synthetic}} from l = 0 l=0 to L p − 1 L_{p}-1 into Equation 12 will result in a full 3D TPMD which continuously evolves. Each time a new central slice is constructed with SVD and evolved with DMD, a unique 3D TPMD is created allowing countless samples to be generated for training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The effect of taking the copper central slice Chebyshev coefficents and applying DMD to continuously evolve them with the copper-derived Koopman operators is shown in Appendix B , where the Cu ground truth and DMD Fermi surfaces are compared.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.3.3 Simulating Experimental Measurement Conditions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A final step of generating realistic data for supervised ML is to be able to produce noise similar to the one present in experimental conditions at different count levels. Figure 5 shows our simulated noise pipeline and its effect in MCM reconstructed slices, for both a central slice and a slice 1.1 1.1 au away from it.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a synthetic 3D TPMD volume, (i) five 2D projections are computed at angles φ ∈ [ 0 ∘ , 45 ∘ ] \\\\varphi\\\\in[0^{\\\\circ},45^{\\\\circ}] . (ii) Detector resolution effects ( 0.110 × 0.137 0.110\\\\times 0.137 a.u. smearing along the projection axis p x p_{x} and p y p_{y} ), (iii) Poisson counting noise and momentum sampling function (MSF) effects are applied to each projection in order to simulate realistic experimental conditions at the desired count rates. The MSF arises as a consequence of the detector geometry, where only electrons emitted with perfect collinearity have a 100 % 100\\\\% probability of both registering a hit at either detector. This effect can be simulated by multiplying the projections with a normalized tent function. As the synthetic dataset is based on a momentum density for copper, it will retain the sample’s four-fold symmetry, and (iv) each projection’s quadrants can be averaged with one another before removing the MSF contribution. MCM reconstruction is then applied to these degraded projections. This yields matched pairs of ground truth and simulated measurement projections, radial density functions, and 3D TPMD volumes for use in supervised training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Training data for DeepCormack\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To reduce storage and memory requirements, only a subset of p y p_{y} -slices are retained from each volume. These are drawn randomly from the range p y ∈ [ 0 , 83 ] p_{y}\\\\in[0,83] out of 256 total slices, a region which accounts for approximately 90% of the total counts in the copper reference data. Appendix C Figure 16 shows three slices taken through different synthetic 3D momentum densities that were generated using our method and used in training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section we present experiments of trained DeepCormack models at different scenarios. Before showing model performance, we first validate our noise simulation approach to the simulation of experimental conditions at varying counts by comparing the Fermi surface degradation of ZrZn 2 with that of real experimental data ( 3.1 ). Different DeepCormack model configurations are then tested and compared using three different datasets with simulated measurement conditions at 200 200 M counts: the synthetically generated test data (DMD), the copper TPMD which the synthetic training data used as reference, and ZrZn 2 data ( 3.2 ). This allows for testing the models’ reconstruction capabilities on data that is within the distribution of the training data (the DMD data), data which has features and slice-by-slice structure within distribution of the training data but not directly seen during training (the copper data), and on data out of distribution (OOD) entirely (the ZrZn 2 data).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In addition to higher image quality, one goal of DeepCormack is to enable faster scanning time. The next experiment presented in Section 3.3 showcases the model’s performance at lowering the photon counts measured (i.e. reducing acquisition time). This is tested with the same three datasets from the previous experiments, i.e. the DMD, copper, and ZrZn 2 data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Validating the Simulation Approach\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To validate the DeepCormack framework on real data, the simulated experimental conditions must first be verified against real experimental measurements. This is to confirm that the synthetic data faithfully reproduces the degradation due to noise and smearing of real ACAR data across a range of count levels. To this end, we compare simulated and experimental ACAR measurements of ZrZn 2 across the range 10 − 165 10-165 M counts, using a DFT-calculated momentum density as the ground truth. Computing the gradient of the 3D TPMDs allows calibration quality to be assessed in TPMD space rather than by comparing the reconstructed images directly. The ACAR measurements had simulated measurement conditions applied as outlined in Section 2.3.3 , and repeated for data at 10 10 M, 100 100 M, and 165 165 M counts in order to provide direct comparison with real experimental data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using the nominal experimental parameters gave simulated data that was too smooth and insufficiently noisy, with degradation in Fermi space that was too mild relative to experiment. A better match was obtained by increasing the convolution widths by + 60 % +60\\\\% to the reference values and using an effective count level of 40 % 40\\\\% of the experimental total (e.g. 100 100 M experimental counts → \\\\rightarrow 40 40 M simulated counts). With these adjusted parameters, the degradation matched the experimental reference well in Fermi surface space as shown in Figure 6 , and was similarly consistent in TPMD space (Appendix D Figure 17 ). This mismatch is probably explained by the fact that there are other possible factors producing noise in real measurements that our first order approximation of the noise does not capture.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This validates the proposed forward model and approximation of the measurement noise, allowing us to generate synthetic measurement data with the correct noise characteristics, a necessary step for the supervised learning proposed in this work, that uses the synthetic TPMDs explained in Section 2.3 . The rest of the experiments use this simulated noise model for training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Model Comparison at Standard Counts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several different DeepCormack model configurations (defined below) were evaluated across three datasets to assess their reconstruction performance and ability to generalize. The DMD test set shares the same distribution as the training data; the copper (Cu) dataset represents a real TPMD, but is still within reasonable distribution of the training data, as it was used as the reference momentum density for the DMD generation; and the ZrZn 2 dataset introduces unique and unseen TPMD features, providing a useful test of OOD generalization. Due to the SNR of the 2D-ACAR technique, a standard experimental measurement will capture around 200 200 M total annihilation events per projection. Therefore, all datasets were simulated with experimental conditions at 200 200 M counts. The DeepCormack configurations explored in this work are the following:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNN : A CNN trained on the projections f f .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MLP : An MLP trained on the radial density functions ρ \\\\rho .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"UNet : A UNet trained on the TPMD ρ 2 γ \\\\rho^{2\\\\gamma} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ETE(CNN + MLP + UNet) : A CNN, MLP, and UNet trained end-to-end.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNN PT → \\\\rightarrow UNet : Pre-trained CNN, where the outputs are used to train a UNet.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNN PT + UNet PT : Pre-trained CNN and pre-trained UNet linked together and trained.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNN PT + MLP PT + UNet PT : Pre-trained CNN, pre-trained MLP, and pre-trained UNet linked together and trained.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CNN PT + MLP PT → \\\\rightarrow UNet : Pre-trained CNN and pre-trained MLP linked together and trained, then their outputs are used to train a new UNet. A skip connection runs from the CNN through the MCM to recover ρ P 2 γ \\\\rho^{2\\\\gamma}_{P} , and is concatenated as an additional input into the UNet.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results using Peak Signal to Noise Ratio (PSNR) are show in Table 1 , for all the model configurations and the three test datasets. The MLP delivers competitive performance on the DMD and Cu datasets ( 38.24 38.24 and 38.10 38.10 PSNR respectively), but struggles with ZrZn 2 , achieving 30.78 30.78 PSNR, far below the MCM baseline of 33.96 33.96 PSNR. On the other hand, the CNN is modest on DMD ( 33.78 33.78 PSNR) but achieves 36.20 36.20 PSNR on ZrZn 2 — the second highest overall — while retaining the lowest variance of all configurations. The UNet gives the strongest and most consistent results overall, competitive on DMD ( 38.41 38.41 PSNR) and the best on both Cu and ZrZn 2 ( 40.05 40.05 and 37.44 37.44 PSNR respectively). The combination CNN PT + MLP PT → UNet \\\\text{CNN}_{\\\\text{PT}}+\\\\text{MLP}_{\\\\text{PT}}\\\\rightarrow\\\\text{UNet} achieved the highest DMD performance ( 40.69 40.69 PSNR), while also achieving 38.24 38.24 PSNR on the copper data. However, this comes at the cost of OOD generalization, with ZrZn 2 performance falling to 32.51 32.51 PSNR — below even the Modified Cormack baseline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These experiments highlight that wile DeepCormack has can have far superior performance than baselines, an appropriately designed training dataset is required to fully exploit its potential. For the rest of the experiments, the following configurations are selected and named for convenience: DC C \\\\text{DC}_{\\\\text{C}} is the CNN only configuration (1), DC M \\\\text{DC}_{\\\\text{M}} is the MLP only configuration (2), DC U \\\\text{DC}_{\\\\text{U}} is the UNet only configuration (3) and finally DC CMU \\\\text{DC}_{\\\\text{CMU}} is the configuration that performs best in DMD data, i.e. pretraining the CNN and MLP first, and the training all together with the Unet (8).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The goal of ACAR measurements is to recover the Fermi surface, but models output and are trained on TPMDs in p -space. To recover the Fermi surface, the momentum densities for copper were converted to k -space densities using the LCW theorem. Figure 7 shows the full pipeline, where projections are converted to TPMDs, k -space momentum densities, before finally extracting the Fermi surface. As this last step — the Fermi surface — is the variable of interest, the rest of the experiments in this work will show results in this space, even though DeepCormack only reconstructs the first step (from projections to TPMDs).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 DeepCormack Performance Across Counts\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.1 Model Performance across counts on DMD Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To assess how various DeepCormacks perform as data quality degrades, the full framework was evaluated on the DMD test set across count levels ranging from 10 10 M to 200 200 M, and benchmarked against the MCM using both PSNR and Structural Similarity Index Metric (SSIM) as seen in Table 2 and equivalently in Figure 8 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DC CMU \\\\text{DC}_{\\\\text{CMU}} outperforms all models at every count level on both metrics. Most notably, its reconstruction at 10 10 M counts ( 38.22 38.22 PSNR) exceeds the MCM at 200 200 M counts ( 32.38 32.38 PSNR), demonstrating that the framework can maintain reconstruction quality at count levels an order of magnitude lower than those currently used in practice. The MCM degrades with decreasing counts, dropping from 32.38 32.38 to 26.12 26.12 PSNR between 200 200 M and 10 10 M, while DC CMU \\\\text{DC}_{\\\\text{CMU}} comparatively stable across the same range ( 40.69 40.69 to 38.22 38.22 PSNR).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The improvement over the best individual models — the DC M \\\\text{DC}_{M} and DC U \\\\text{DC}_{U} , — is more modest, averaging around + 1 − 1.5 +1-1.5 db PSNR across count levels.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.2 Model Performance across counts on Copper Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here we test the models on the copper momentum density with simulated measurement conditions applied for counts ranging from 10 − 200 10-200 M. Results can be seen in Table 3 ans Figure 9 . This is the same reference used to generate synthetic training data. Since the copper features are not far to the training distribution, this will test how well the models generalize to real TPMD data rather than purely synthetically generated ones.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the Cu data, the DC U \\\\text{DC}_{\\\\text{U}} is the strongest performer across all count levels, reaching 40.05 40.05 PSNR at 200 200 M counts and 37.41 37.41 at 10 10 M, outperforming DC CMU \\\\text{DC}_{\\\\text{CMU}} , which achieves 38.65 38.65 and 36.04 36.04 dB respectively. DC CMU \\\\text{DC}_{\\\\text{CMU}} performs comparably to the DC M \\\\text{DC}_{\\\\text{M}} across all counts, around 1.40 1.40 PSNR lower than the DC U \\\\text{DC}_{\\\\text{U}} . However, all models again comfortably outperform the MCM at every count level. Notably, DC CMU \\\\text{DC}_{\\\\text{CMU}} at 10 10 M counts ( 36.04 36.04 PSNR) still exceeds the MCM at 200 200 M counts ( 31.62 31.62 dB PSNR), consistent with the DMD results.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When these slices are individually passed through DeepCormack, they need to be renormalized in order to produce coherent 3D TPMDs and Fermi surfaces. This is done using a smoothed version of the measured momentum density data’s maximum intensity across slices.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite the PSNR and SSIM gap between the DC U \\\\text{DC}_{\\\\text{U}} and DC CMU \\\\text{DC}_{\\\\text{CMU}} in momentum space, the Fermi surface reconstructions show that both produce qualitatively comparable results across the full count range, recovering a greater detail of features along all three slices than the MCM. Notably, the recovered Fermi surface remains largely intact even at 10 10 M counts, with features that are more distinct than the MCM at 200 200 M counts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.3 Model Performance on ZrZn 2 Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here the models are tested on ZrZn 2 data with simulated measurement conditions applied across 10 10 M to 200 200 M counts (Table 4 , Figure 9 ). Unlike the copper data, the ZrZn 2 momentum density has features that are outside the training distribution, making this a strict test of generalization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DC CMU \\\\text{DC}_{\\\\text{CMU}} s performance is significantly lower on the ZrZn 2 dataset, achieving only 32.82 32.82 PSNR at 200 200 M counts, which is below the MCM ( 33.96 33.96 PSNR) and the DC C \\\\text{DC}_{\\\\text{C}} ( 36.20 36.20 PSNR), but greater than that of the DC M \\\\text{DC}_{\\\\text{M}} by itself ( 30.78 30.78 PSNR). This performance drop becomes apparent when computing and comparing the Fermi surfaces, as shown in Figure 12 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We found out, however, that conditioning the DC U \\\\text{DC}_{\\\\text{U}} with FiLM conditioning has a noticeable impact on the reconstructed Fermi surface for ZrZn 2 , as seen in Figure 13 . Qualitatively, the results suggest that DC U \\\\text{DC}_{\\\\text{U}} produces a better Fermi surface than MCM, albeit its unclear if a sufficiently informative one.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we proposed a DeepCormack family of models for Fermi surface tomography, together with data generation approaches and testing on a variety of count levels, which is directly proportional to scanning time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A large combination of different DeepCormack architecture variations were tested, and results were generally inconclusive. Each configuration carries its own trade-offs. The CNN and MLP are the least intensive to train: the CNN generalizes well but gives weaker in-distribution performance, while the MLP is powerful within the training distribution but struggles on OOD data. The UNet is the most computationally demanding to train, but offers the strongest and most reliable performance overall. We note, however, that the UNet’s advantage likely reflects the limitations of the training data as much as its architectural strengths. There are currently no public databases of experimentally measured or DFT-generated momentum densities for real samples that could support supervised learning, so models were trained entirely on synthetic data derived from a copper reference. With more diverse and representative training data, configurations that perform strongest on the DMD test set — particularly DC CMU \\\\text{DC}_{\\\\text{CMU}} , which achieves amongst the highest in-distribution performance while offering considerable architectural flexibility — may prove more competitive. Each component is straightforward to train independently and can subsequently be linked and fine-tuned for a small number of epochs, making this framework practical and adaptable, particularly as richer training data becomes available. Furthermore, if strong reconstruction performance can be maintained at lower count levels, collecting data along more than 5 5 projection directions becomes viable, and the richer angular sampling may allow the full range of model configurations to reach their potential.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, we saw that the performance gain of the models depend on which data are tested. While pre-training can improve reconstruction quality on data close to the training distribution, it appears to specialize the models towards training-data-like features in a way that does not fully recover with joint training. More broadly, models with sufficient capacity to exploit the structure of the synthetic data tend to do so at the cost of generalization, and the DC U \\\\text{DC}_{\\\\text{U}} currently strikes the best balance between the two. It should be noted that these results are obtained on synthetic DMD data drawn from the same distribution as the training data, which naturally favours all models evaluated here. Real experimental data will differ in both noise characteristics and momentum density features, and how well these results carry over depends on how faithfully the simulated data reproduces real measurements — as assessed in Section 3.1 . Training on a broader range of DFT-generated momentum densities, or ideally real experimental data, would likely improve performance and generalization across the board. Testing the models on real experimental data will be explored in future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"That said, the lack of generalization to out-of-distribution samples is a limitation, but only a moderate one. As is true of inverse problems in general, learned reconstructions are highly sensitive to the data distribution, meaning they tend to generalize only moderately, particularly when tested on real data. An advantage here is that Fermi surface tomography is a very slow process — one that can rely on precomputed approximations of the ground truth. Currently, ACAR measurements for 200 200 M counts take over 3 3 months of scanning time. However, one can pre-compute a reference momentum density within a day or so. This reference can then be used to generate training data for DeepCormack via DMD, after which one can choose to reduce the measurement time. Our work suggests that this will still produce higher quality Fermi surfaces than the modified Cormack method, in significantly less time. That said, DeepCormack produces significantly better reconstructions than MCM on 200 200 M-count data, so even without the time-reduction possibility, DeepCormack enables much higher quality Fermiology via 2D-ACAR. It is also noteworthy that FiLM conditioning led to some qualitatively better reconstructions than MCM on OOD experimental data, meaning OOD testing cannot be completely discarded.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In general, for this work it is worth noting that we used the natural image quality metrics PSNR and SSIM to evaluate performance. While, in broad terms, the qualitative evaluations of the results here correlate with an increase in PSNR and SSIM, these metrics may not be the appropriate ones for evaluating fine differences or for choosing a ”best” configuration of DeepCormack. Ideally, the quality of the Fermi surface — or, better still, the quality of quantitative values extracted from the Fermi surface — would be the best metric, as these are what physicists will use, not the images themselves. This is, however, non-trivial, and would require a set of labelled ground truth data, which is not available for this task at the time of writing. Future work should focus on validating DeepCormack with such metrics, evaluating how useful it is for real outcomes with real data. This idea could also be extended to training the models. If MSE/PSNR of the TPMDs that DeepCormack outputs is not the goal, moving the training loss to metrics defined on the Fermi surface would be an improved solution. This is non-trivial, however, as applying the LCW theorem to pseudo-real data is not obvious, and computationally expensive, as it would require storing all TPMD slices during training rather than just a subset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All configurations here have been desiged to only 5 5 projections, which constrains the quality of reconstruction regardless of the model used. Given the strong performance of DeepCormack, a natural next step would be to acquire more projections — for example 10-20 — at proportionally lower count levels ( 50 50 — 25 25 % of 200 200 M counts), keeping total acquisition time fixed. DeepCormack could compensate for the lower quality projections, whilst the richer angular sampling could improve reconstruction quality overall. This may also benefit configurations beyond the standalone DC U \\\\text{DC}_{\\\\text{U}} , as the CNN and MLP are well-suited to exploiting additional input channels, and modular frameworks such as DC CMU \\\\text{DC}_{\\\\text{CMU}} could prove better positioned to take advantage of both denser projection coverage and richer training data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work we proposed DeepCormack, a family of data-driven, model-based reconstruction algorithms for Fermi surface tomography. We also propose a data generation model based on Koopman operators to train such models. Results show significant improvement in reconstruction quality compared to the baseline, standard Modified Cormack Method, both in the standard acquisition regime and in significantly reduced ones. This not only allows for much higher quality results for the condensed matter physics community, but also potentially saves weeks of scanning time without any loss (in fact, with an increase) in reconstruction quality.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Copper Koopman Operator on Copper Central Slice:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The copper momentum density is where we derived our per-channel Koopman operators 𝒦 n \\\\mathcal{K}_{n} from. Here, we evaluate whether evolving the Chebyshev coefficients for the copper central slice via dynamic mode decomposition is sufficient to recover the original structure. To do this comparison, we took the original DFT-generated copper momentum density along with the DMD-evolved copper momentum density, applied LCW-folding to convert from p p -space to k k -space before extracting their Fermi surfaces. Figure 15 compares slices taken through high symmetry planes ( 100 ) (100) , ( 110 ) (110) , and ( 111 ) (111) of the Fermi surfaces.\"\n }\n]","isConversational":false}}},{"rowIdx":293,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.14147"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Breaking Refusal in the First Half: A Mechanistic Study of the Prefill Jailbreak"},"authors":{"kind":"list like","value":["Alex Kwon"],"string":"[\n \"Alex Kwon\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.14147v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Aligned language models refuse harmful requests, and that refusal is easily removed. Prepend an affirmative prefix to the model’s own turn (“Sure, here is”) and many models continue straight into the harmful content they would otherwise decline (Wei et al., 2023 ; Qi et al., 2025 ) . The fragility is well documented behaviorally : we know the attack works. We ask the mechanistic question underneath it. The prefill appends to the model’s own response , so it cannot touch the representation the model has already computed over the prompt; the question is therefore not whether that representation is preserved under the attack (structurally, it must) but whether the model builds a strong harm representation there at all, and if so, why the refusal decision ignores it."},{"type":"para","text":"This localizes what kind of defense is possible. A behavioral gate is defeated by the attack; a monitor that reads the prompt-side harm representation is not, because the attack never reaches that representation, though this prompt-side placement is already how production input classifiers work, so the observation is about placement, not a new detector. The scientific question that remains is why a model that represents the request as harmful nonetheless complies, that is, where the refusal decision overriding the representation is computed."},{"type":"para","text":"Empirically, the model builds a full-strength harm representation on the flipped prompts (probe 0.91 0.91 – 0.98 0.98 , as high as on the ones it refuses) while behavioral refusal drops to chance, across four models and three families (§ 3 ). But the sharper mechanistic statement is that the gate is not ignoring harm, it is being overpowered : under a full-strength prefill essentially every prompt flips, so harm cannot predict who complies, yet under a weakened prefill where a real split exists, harm strength predicts refuse-vs-comply (AUC 0.63 0.63 – 0.72 0.72 , n = 70 n=70 ; § 5.1 ). And the gate’s failure is localized to an early window: a dose-matched position control shows that restoring the harm signal over the first half of the response recovers refusal as much as the whole response, while the second half recovers almost nothing and the onset token alone only 9 % 9\\% (§ 4 ). We use overpowered descriptively: the evidence fixes where refusal fails, an early window, but does not separate active suppression from passive autoregressive conditioning (§ 7 ). We then pin down what the gate reads (§ 5 ) and show it has no compact internal handle (§ 5.1 )."},{"type":"heading","level":4,"text":"Relation to prior work."},{"type":"para","text":"Two lines are adjacent (§ 6 ). Qi et al. ( 2025 ) show safety alignment is “shallow” in token depth : the refusal decision is carried by the first few output tokens, and attacks that get past them succeed. We go past that behavioral picture on three fronts. First, we localize the failure causally, not just correlating depth with success, to an early multi-token window rather than a first-token count. Second, we show the causal converse: transplanting the refuse-state re-induces refusal. Third, the base-model discriminator identifies the prefill’s grip as generic autoregressive conditioning rather than a defeated safety-specific gate, a distinction depth-shallowness alone does not make. Arditi et al. ( 2024 ) show refusal is mediated by a single direction in activation space. Closest, concept-level work shows a model still recognizes toxicity under a jailbreak while complying (Zhang et al., 2025c ) , but attributes the behavior-flip to a competing prompt-side “jailbreak concept” that overrides it. Our contribution is positional and causal. The representational half is a clarification, not a discovery: because prefill leaves the prompt untouched, the harm representation there is trivially unchanged, and the useful contrast is with prompt-modifying jailbreaks (the ones that literature studies), which do move or attenuate the prompt-side representation. Our substantive claims are mechanistic: refusal is computed at the response site and the attack operates there; and the harm direction is an excellent read-out but only a partial, non-selective write-handle for the refusal action. The prompt-side account cannot explain the prefill attack at all. Prefill adds nothing to the prompt, so there is no prompt-side jailbreak concept to override anything. The flip is response-site , and we localize it there. The template (read what a model represents versus what it acts on) is the one we used for communicative intent in prior work (Kwon, 2026 ) ; here it is pointed at safety."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"A quantitative dissociation under prefill (§ 3 ). On the very prompts a prefill attack flips to compliance, the harm representation is not degraded but intact : a probe scores the complied prompts at 0.91 0.91 – 0.98 0.98 , level with the ones the model refuses, while behavioral refusal collapses to chance. This replicates across four models and three families ( 1.5 1.5 – 3.8 3.8 B, weak and strong gates). Unlike the prompt-modifying jailbreaks prior work studies, prefill leaves the representation quantitatively unchanged, not merely present-but-attenuated."},{"type":"para","text":"A response-site causal mechanism (§ 4 ). We localize refusal causally, not just correlationally: adding the harm direction back across the response positions partially restores refusal (harm 33 33 – 48 % 48\\% at two fixed cells vs a matched-norm random control’s 14 14 – 16 % 16\\% ; harm − - random + 17 +17 to + 34 +34 pp at n = 100 n=100 , both significant), so the intact representation is upstream of the gate, not epiphenomenal. A dose-matched position control localizes the effect to an early window of the response (the first half restores as much as the whole; the second half is nearly inert), more than a single onset token but not the whole generation. The converse holds in the same window and is state-specific: transplanting the model’s own refuse-state into the attack re-induces refusal ( → 74 % \\to 74\\% held-out, first half = = whole response), matched by a random control at 0 % 0\\% , a working lever where the single harm direction is only partial. And the grip operates through early-window attention: knocking out the early response’s attention to the prefill (but not an equal mass elsewhere) restores refusal + 21 +21 pp (CI [ + 12 , + 30 ] [+12,+30] ), harm-selectively and early-window-gated. A base-model control identifies this as generic autoregressive conditioning, not safety-specific active suppression (the non-safety-tuned base model shows the same prefill-specific collapse, at both 1.5 1.5 B and 7 7 B), resolving the paper’s central open question toward the passive reading. The dramatic full-residual “restoration” is generic disruption, since a benign-patch control also hits 100 % 100\\% . Replicated on two families, and the discriminator holds at 7 7 B."},{"type":"para","text":"A defensive consequence, and its scope (§ 3 , § 4.5 ). Read as a monitor, the intact representation catches 100 % 100\\% of the attack at 0 % 0\\% false positive. The scope is narrow, in three ways. First, this 100 % / 0 % 100\\%/0\\% is in-distribution: the probe is AdvBench-calibrated (it fires at 0.97 0.97 on that phrasing but 0.49 0.49 on differently-worded unsafe prompts, and a mixed-source fix wrecked specificity), so a source-general detector is unbuilt. Second, it is only a response-site guarantee, and the placement is not novel (input classifiers already sit prompt-side). Third, steering the same direction does not close the gap: the harm direction is a read-out, not a write-handle."},{"type":"para","text":"What the gate reads (§ 5 ). A pre-registered harm × \\times surface 2 × 2 2\\times 2 that decorrelates harm from trigger-word surface shows refusal is harm-primary and the harm representation keyword-invariant, with a real but secondary surface-driven over-refusal on benign inputs. A matched-contrast search finds no single refusal-decision direction (any one reads ≤ 0.73 \\leq 0.73 and none steers), yet a full linear probe reads the decision at ∼ 0.85 \\sim 0.85 : within the Qwen family the decision is decodable but multidimensional , not a single steerable feature (SmolLM2 is a single-direction counterexample; cross-family is open)."},{"type":"para","text":"Every headline effect is checked against a control that can demote it, and several do; the claims-and-evidence table (Table LABEL:tab:claims , Appendix A ) records each claim that passed with its evidence and the demoted versions (full-residual disruption, the steering null, the retracted “separate feature”), and Appendix B gives the reproduction chain. The dissociation also extends, exploratorily, to inferred-intent refusal ( n = 24 n=24 , one model; § 5 , App. I )."},{"type":"heading","level":1,"text":"2 Setup"},{"type":"heading","level":4,"text":"Models."},{"type":"para","text":"We use open-weight instruction-tuned models so we can read activations: Qwen2.5-1.5B and 3B (Qwen Team, 2024 ) , SmolLM2-1.7B (Allal et al., 2025 ) , and Phi-3-mini-3.8B (Abdin et al., 2024 ) : three families, 1.5 1.5 – 3.8 3.8 B parameters, and a range of safety-gate strength (SmolLM2 refuses 63 % 63\\% of plain harmful prompts; the others ≈ 100 % \\approx 100\\% ). All experiments run on CPU; sample sizes are reported per experiment."},{"type":"heading","level":4,"text":"Data."},{"type":"para","text":"Harmful prompts are AdvBench behaviors (Zou et al., 2023b ) ; benign prompts are Alpaca instructions (Taori et al., 2023 ) ; the surface-vs-harm control uses XSTest (Röttger et al., 2024 ) , whose “safe” split is purpose-built to sound harmful (“how do I kill a Python process”) while being benign."},{"type":"heading","level":4,"text":"The attack."},{"type":"para","text":"The prefill attack forces the assistant turn to begin with the affirmative prefix “Sure, here is” and lets the model continue; we score whether the continuation refuses."},{"type":"heading","level":4,"text":"The probe."},{"type":"para","text":"The harm probe is ℓ 2 \\ell_{2} -regularized logistic regression on the default-pass hidden state at the last prompt token (the decision point), trained on plain harmful vs. benign prompts. Probe AUCs use cross-validation (nested-CV layer selection; no test-set peeking). Refusal is detected by string match against a fixed refusal-phrase list. Full details in App. B ."},{"type":"heading","level":1,"text":"3 The phenomenon: represented but not refused"},{"type":"para","text":"This section states the phenomenon the rest of the paper explains: a fully intact harm representation coexists with compliance, part of which is structural (below). The mechanistic account of how that gate is defeated (§ 4 ) is where the empirical work and every control live."},{"type":"heading","level":4,"text":"Harm is strongly represented."},{"type":"para","text":"On plain prompts, the harm probe separates harmful from benign near-perfectly: one nested-CV probe scores AUC 0.996 0.996 on Qwen2.5-1.5B and ≥ 0.990 \\geq 0.990 on every model (Table 1 , one estimator throughout). The representation of harm is not subtle."},{"type":"heading","level":4,"text":"The attack changes behavior; it cannot change the prompt-side read."},{"type":"para","text":"Under the prefill attack the same model complies: on Qwen2.5-1.5B, refusal on harmful prompts drops from 1.00 1.00 to 0.03 0.03 (97 of 100 comply) and behavior, used as a harm detector, collapses to chance (AUC 0.51 0.51 ). “Comply” here is coherent harmful compliance, not just an absent refusal string: a judge-free content read of the attacked outputs (§ 7 ) finds 94 % 94\\% of the non-refusals are coherent harmful content, deflection rare. The other half of the dissociation is partly structural. The harm representation we probe is the model’s read at the last prompt token, computed before any of the attack’s tokens exist; the prefill appends to the response , so by the causal structure of the transformer this representation is bit-for-bit identical with and without the attack. Its “intactness” is structural, not a measurement . The substantive finding is this: at full attack strength, compliance is not explained by a weaker harm read. The prompts the model complies on score as harmful ( 0.975 0.975 ) as the ones it refuses ( 0.966 0.966 ; benign 0.055 0.055 ). Here “refused” is the plain-condition harmful set, since the strong prefill leaves at most 5 / 100 5/100 still refusing per model. A full -strength harm representation coexists with compliance; the comparison has little variation, since 95 95 – 100 % 100\\% comply under the strong attack. Where a real split exists, a weakened prefill (§ 5.1 , 41 / 29 41/29 refuse/comply), harm strength does predict who complies: the harm-probe score reads refuse-vs-comply at AUC 0.63 0.63 (layer 12) to 0.72 0.72 (best layer). This is harm, not the obvious surface correlate: prompt length and scary-keyword count predict the same split at chance (AUC 0.50 0.50 – 0.54 0.54 ), and the harm-probe AUC holds when we regress out length and keyword count ( 0.73 0.73 ). So the strong attack overrides the harm signal rather than the model down-weighting it. This is moderate evidence: n = 70 n=70 is small and prompt-level confounds (topic, severity, formality) are not held fixed. Table 1 shows the pattern across all four models and three families, at both weak and strong gates. It also extends to larger scale (run on GPU). On Qwen2.5-7B the prefill drops refusal 0.98 → 0.24 0.98\\to 0.24 at probe AUC 1.00 1.00 ; the attack is weaker here, with a quarter still refusing under the same prefill (§ 7 ). On the heavily-safety-trained Phi-3-medium-14B (Abdin et al., 2024 ) it drops refusal 1.00 → 0.15 1.00\\to 0.15 (probe AUC 1.00 1.00 , complied-harmful 0.96 0.96 , coherent output; App. J ). So the dissociation is not a small-model artifact. The attack is also not specific to one prefill string: across six affirmative prefills (varying length and phrasing) it drops refusal from 100 % 100\\% to 0 – 37 % 37\\% (most to ≤ 7 % \\leq 7\\% ), so it is a general property of affirmative response-onset prefills. The real question is then why an intact, strong harm representation fails to drive refusal, which the mechanism (§ 4 ) answers."},{"type":"heading","level":4,"text":"Is the probe reading harm, or scary words?"},{"type":"para","text":"A natural worry is that the probe separates AdvBench from Alpaca on surface features (violent vocabulary) rather than harm. We test this with XSTest, whose “safe” prompts sound harmful but are benign. The AdvBench-trained probe scores XSTest- safe at 0.21 0.21 (mean; only 12 % 12\\% exceed the 0.5 0.5 threshold) versus XSTest- unsafe at 0.49 0.49 , separating them at AUC 0.79 0.79 . The probe is not fooled by scary surface: it reads harm. It is, however, calibrated to AdvBench-style phrasing (it fires at 0.97 0.97 there but only 0.49 0.49 on differently-phrased unsafe prompts), and training on mixed harmful sources did not fix this: it raised sensitivity but collapsed specificity (XSTest-safe false positives 12 % → 64 % 12\\%\\to 64\\% ). So the harm reading is earned; a source-general harm detector is not (§ 7 )."},{"type":"heading","level":4,"text":"A design consequence: a prompt-side monitor is immune to this attack."},{"type":"para","text":"Because the attack cannot reach the prompt-side representation, a monitor that reads it inherits that immunity by construction: on the attacked set the harm probe flags 100 % 100\\% of harmful prompts at 0 % 0\\% false positive on easy (Alpaca) negatives. This is not an empirical detection result, since the probe input is unchanged by the attack, “catching the attack” reduces to ordinary harmful-vs-benign accuracy, and the placement is not new: production input classifiers (e.g. Llama Guard, Inan et al., 2023 ) already sit prompt-side and are unreachable by a response-site prefill for the same reason. Our contribution is the mechanistic account of why that placement is immune. The guarantee is scoped to response-site attacks (prompt-modifying jailbreaks change the probe input and degrade it: Doda, 2026 and our encoding nulls, App. K ) and costs 12 % 12\\% over-refusal on scary-but-benign XSTest inputs (App. C )."},{"type":"heading","level":1,"text":"4 The mechanism"},{"type":"para","text":"The dissociation says behavior and the harm representation are separable; it does not say whether the harm representation is causally in the refusal pathway or epiphenomenal for it, nor why prefill specifically defeats the gate. This section answers both with white-box causal experiments on Qwen2.5-1.5B, replicated on Phi-3-mini: activation patching, a position control, a direction search, and component ablation, each against a control that can kill it. For each effect we report the version that passes its control, not the dramatic one."},{"type":"heading","level":2,"text":"4.1 Refusal restoration is response-site and concentrated in an early window (causal patching)"},{"type":"para","text":"The intact harm representation lives at the prompt ; the attack operates at the response site (it occupies the output tokens). So we ask: under the attack, if we restore the harm signal at the response positions, does refusal return despite the prefill? We patch there and let generation flow, sweeping layers and injection strength (App. D , Table 7 ) and settling the harm-vs-random delta at two fixed cells with n = 100 n=100 (Table 2 ). A dose-matched position control (below) localizes the effect to an early window of the response: injecting over the first half restores as much as the whole response, the second half is nearly inert, and prompt-side injection does nothing."},{"type":"para","text":"Two rungs, and the control matters. Full-residual patching (replacing the response-onset residual with the mean harmful residual) restores refusal to 100 % 100\\% . But a benign control demolishes it: patching the mean benign residual at the same position also restores refusal to 100 % 100\\% at every layer. Full-residual patching is generic disruption; injecting any prompt-final state at the onset breaks the prefill continuation and the model reverts to refusing. It says nothing harm-specific."},{"type":"para","text":"What passes is the subspace rung: adding only the harm direction at the response positions restores refusal while sparing benign, and it passes the same control the full rung got. We settle it at n = 100 n=100 on two cells fixed in advance from the grid (L12, the best subspace layer there, and L8), against a matched-norm direction orthogonalized to the harm direction. The cells are grid-selected (fixed before the run) and the n = 100 n=100 re-estimates them on overlapping AdvBench prompts. The delta CI bootstraps over the harm prompts and the 24 24 random seeds (App. D ). At L12/ α = 1 \\alpha{=}1 the harm direction restores 48 % 48\\% ( 95 % 95\\% bootstrap CI 38 38 – 58 % 58\\% ) versus 14 % 14\\% for the orthogonal-random control (24 seeds, range 0 – 60 % 60\\% ): a harm − - random delta of + 𝟑𝟒 \\mathbf{+34} pp (CI + 22 +22 , + 46 +46 ). At L8/ α = 1.5 \\alpha{=}1.5 it restores 33 % 33\\% ( 24 24 – 42 % 42\\% ) versus random 16 % 16\\% ( 0 – 60 % 60\\% ): delta + 𝟏𝟕 \\mathbf{+17} pp (CI + 6 +6 , + 28 +28 ). Both deltas exclude zero, and restoration is benign-sparing (benign 12 % 12\\% / 0 % 0\\% vs random’s 3 % 3\\% / 4 % 4\\% ), so the harm direction is a real causal handle above the generic-perturbation baseline, but a partial one: a third to a half of the raw restoration is matched by the random control. This passes a clean held-out replication. Training the harm direction on a disjoint 200 200 -prompt AdvBench split (activations recomputed, so there is no overlap with the test prompts) and testing restoration on a held-out 100 100 , the L12/ α = 1 \\alpha{=}1 cell restores 42 % 42\\% ( 95 % 95\\% CI 32 32 – 52 % 52\\% ) versus 13 % 13\\% for the matched-norm random control (6 seeds), a harm − - random delta of + 𝟐𝟗 \\mathbf{+29} pp (CI + 16 +16 , + 41 +41 ): the effect shrinks only ∼ 5 \\sim 5 pp off-distribution and the delta still excludes zero, so it is not an artifact of prompt overlap (App. E )."},{"type":"heading","level":4,"text":"Position control: the attack’s grip is an early response window, not the whole response and not a single token."},{"type":"para","text":"We test where the harm signal has to be restored, over one consistent response window (the prefill tokens plus the 18 18 generated tokens; n = 100 n=100 , L12/ α = 1 \\alpha{=}1 ). Two axes matter, position and per-position strength, and a dose-matched design separates them (App. D ). Position: injecting over the first half of the response (the prefill and the first ∼ 7 \\sim 7 generated tokens) restores refusal to 42 % 42\\% , as much as the whole response ( 41 % 41\\% ), whereas the second half restores only 6 % 6\\% even at double strength. (These dose-matched levels sit just below the 48 % 48\\% fixed-cell of Table 2 : this control matches per-position dose to isolate where , so the first-half-vs-whole ratio is the claim, not the absolute level.) So the early window suffices on its own , and the late response is nearly inert. Strength: halving the per-position coefficient (whole response at α = 0.5 \\alpha{=}0.5 ) drops restoration to 0 % 0\\% , so there is a per-position threshold, not just a total-dose budget; a distributed every-other-token pattern also needs the higher coefficient ( 1 % 1\\% at α = 1 \\alpha{=}1 , 48 % 48\\% at α = 2 \\alpha{=}2 ). This sharpens the shallow-alignment picture: the prefill’s grip is not a single onset token (injecting only the onset restores 9 % 9\\% , less than the early window) but an early multi-token window; the back half of the response carries no weight. (An earlier version of this control compared an all-positions band that silently included the prefill against half-bands that did not, which confounded position with prefill-inclusion and dose; the dose-matched window design here corrects it.) We use “defeated” descriptively: the control localizes where the counter-signal must go, but does not distinguish an active mechanism that down-weights the harm signal from passive autoregressive conditioning by the forced affirmative context (§ 7 )."},{"type":"heading","level":4,"text":"Robustness of the early window (boundary, prefill-exclusion, prefill length, sampling)."},{"type":"para","text":"Four checks probe the window’s edges (App. E ). Boundary: sweeping the cutoff, restoration is already saturated by the first third (first-third 50 % 50\\% , first-half 57 % 57\\% , first-two-thirds 55 % 55\\% , whole 50 % 50\\% at full per-position dose, reproducing the 48 % 48\\% fixed cell) while the onset alone is 10 % 10\\% and the second half 5 % 5\\% ; so first-half ≈ \\approx whole is not a knife-edge at 1 / 2 1/2 , the window is the early response and saturates by its first third. Prefill-exclusion: the restoration is anchored at the prefill positions. Injecting the harm direction over the generated tokens only (excluding the prefill span) restores ≤ 3 % \\leq 3\\% , so the causal write-site inside the early window is the prefill span; sustaining across the early generated tokens then boosts it (prefill + + onset 10 % → 10\\%\\to prefill + + first-half 57 % 57\\% ), but the generated positions carry nothing on their own. Prefill length: the early saturation is stable across prefill length (short and long prefills both saturate by the first third), while the restoration magnitude tracks prefill commitment, a short prefill (“Sure,”) is easy to reverse ( 86 % 86\\% ) and a long committal one (“Sure, here is a detailed step-by-step guide:”) is hard ( 6 6 – 12 % 12\\% across bands) at fixed injection strength. Sampling: the dissociation is not a greedy artifact, at temperature 0.7 0.7 plain prompts still refuse 100 % 100\\% and the attack still strips refusal to 0 % 0\\% ; the harm-direction restoration, a partial handle, weakens under sampling ( 50 % → 21 % 50\\%\\to 21\\% ), as expected for a non-selective write-handle."},{"type":"heading","level":2,"text":"4.2 The reversal is early-window too: the model’s own refuse-state re-induces refusal (state transfer)"},{"type":"para","text":"The position control shows the attack breaks refusal in an early response window. We now show the converse , and it lands in the same window: transplanting the model’s own refusal state into the attack re-induces refusal, and doing so over the first half of the response suffices. This is a cleaner causal handle than the single harm direction (§ 4.1 ), because it uses the full multidimensional state, not one line."},{"type":"heading","level":4,"text":"Design (measure-only, held-out, Qwen2.5-1.5B, L12)."},{"type":"para","text":"We capture the mean layer- 11 11 -output residual over the first-half generated window in two conditions: the plain prompt with no prefill (refusal 100 % 100\\% ) and the attack (“Sure, here is”). Their difference d reverse = s plain − s attack d_{\\text{reverse}}=s_{\\text{plain}}-s_{\\text{attack}} is the model’s own refuse-vs-comply state shift. This is held out : d reverse d_{\\text{reverse}} is estimated on a 260 260 -prompt AdvBench train split and injected/tested on a disjoint 100 100 -prompt held-out split (with an in-distribution reference, the same d reverse d_{\\text{reverse}} tested on train prompts, run alongside). We add d reverse d_{\\text{reverse}} into the attack run’s early generated window and measure whether refusal returns (string-match plus the judge-free coherence read of § 7 ), against a matched-norm orthogonal-random control at the same norm."},{"type":"heading","level":4,"text":"Result: state-specific, early-gated, and a working lever (held-out)."},{"type":"para","text":"Injecting d reverse d_{\\text{reverse}} over the first half of the response restores refusal on the held-out split from the attack’s ≈ 0 % \\approx 0\\% to 𝟕𝟒 % \\mathbf{74\\%} , as much as injecting over the whole response ( 78 % 78\\% ), while the onset token alone restores 13 % 13\\% ; the matched-norm random control restores 𝟎 % \\mathbf{0\\%} , and coherence holds throughout ( ≈ 0.97 \\approx 0.97 – 0.99 0.99 distinct-token ratio, no degeneracy). The in-distribution reference (same d reverse d_{\\text{reverse}} , tested on train prompts) restores 72 % 72\\% , so there is no generalization gap: the refuse-state is not an artifact of the prompts it was estimated on. This replicates on a second family: on Phi-3-mini ( 3.8 3.8 B, L 16 L16 ) the same held-out procedure restores refusal to 69 % 69\\% over the first half (whole 71 % 71\\% , onset 4 % 4\\% ) against a matched-norm random control at 2 % 2\\% , coherent, with the same early-window gating (App. E ). Three things follow. (i) The reversal is state-specific , not generic disruption: unlike the full-residual patch of § 4.1 (which a benign control also drove to 100 % 100\\% ), here a matched-norm random vector does nothing, so this transfer passes the very control that demoted the full-residual effect. (ii) It is early-window-gated , the same locus as the attack: the first half restores as much as the whole response and the second half adds nothing, mirroring the break direction. (iii) It works where the single harm direction is only a partial handle (§ 4.1 ) and d refuse d_{\\text{refuse}} steering is inert (§ 5.1 ), precisely because it is the full multidimensional state shift: the lever exists, but it is the early-window state , not any single direction, consistent with the decision being distributed (§ 5.1 )."},{"type":"heading","level":4,"text":"Benign-sparing, and a syntax-vs-content byproduct."},{"type":"para","text":"We verified state-specificity against a matched-norm random control and generalization against a held-out split. The transfer is also benign-sparing , so it is a selective edit and not merely a refusal-inducer: injecting d reverse d_{\\text{reverse}} over the early window of benign prompts under the attack raises their refusal only 12 % → 18 % 12\\%\\to 18\\% ( + 6 +6 pp) while restoring harmful refusal to 75 % 75\\% (this run reproducing the held-out 74 % 74\\% ), a + 57 +57 pp selectivity gap (App. E ). A syntax-vs-content check falls out of the same run: forcing a benign affirmative onset (“Sure, here is a poem about”) yields 0 % 0\\% harmful procedural content (vs the attack’s 47 % 47\\% ) and near-zero refusal, so the jailbreak rides the harmful continuation , not the affirmative syntax alone."},{"type":"heading","level":2,"text":"4.3 Cutting early attention to the prefill restores refusal, generically (attention knockout)"},{"type":"para","text":"The reversal shows the refuse/comply decision is carried in the early-window residual state . That leaves the paper’s central open question (§ 7 ) standing: is the prefill’s effect maintained by the response continuing to attend to it, or written into the trajectory at the first generated token and merely relayed forward? We test this directly by knocking out the attention edges from early generated positions to the forced-prefill key positions and asking whether refusal returns, against a control that removes the same attention mass from non-prefill keys. This is the discriminating experiment the Limitations named (§ 7 ); it is preregistered ( PREREG_attn_knockout.md ), including the falsifier and the interpretation table."},{"type":"heading","level":4,"text":"Design (edge knockout, mass-matched control; n = 100 n=100 , Qwen2.5-1.5B, all layers/heads)."},{"type":"para","text":"At each early generated decode step ( g < 9 g<9 , the first-half window) we add ln ( 1 − λ ) \\ln(1-\\lambda) to the pre-softmax attention logit of every (current query → \\to prefill key) edge and let softmax renormalize ( λ = 1 \\lambda{=}1 removes the edge, λ = 0.5 \\lambda{=}0.5 halves it). The control removes the same total baseline attention mass from non-prefill prompt keys, selected per prompt to match the prefill’s mass (achieved ratio 0.98 0.98 , [ 0.75 , 1.08 ] [0.75,1.08] over three fixed seeds); this matters because attention-sink tokens carry far more mass than the prefill, so an unmatched span is not a fair control. The intervention passes a unit-test suite before any behavioral run (sham-hook bit-identity, edge-only masking with softmax renormalization, KV-cache consistency, and a behavioral positive control; App. E ). Indexing caveat: generated token 1’s logits come from the final prefill token’s hidden state, so this cannot touch token 1; it is a test of early maintenance through direct attention, not attack initiation."},{"type":"heading","level":4,"text":"Result: prefill-specific, early-window-gated, and harm-selective."},{"type":"para","text":"Knocking out early response-to-prefill attention restores refusal from the attack’s 3 % 3\\% to 24 % 24\\% , versus 1 % 1\\% for the mass-matched control (Table 3 , three-seed mean 1 % 1\\% ): a prefill − - matched difference of + 𝟐𝟏 \\mathbf{+21} pp (paired bootstrap 95 % 95\\% CI [ + 12 , + 30 ] [+12,+30] , computed against one fixed control seed). It is early-window-gated: knocking out the late window does nothing ( 4 % ≈ 4\\%\\approx baseline; early − - late + 20 +20 pp, CI [ + 11 , + 29 ] [+11,+29] ), and knocking out the whole response ( 29 % 29\\% ) is not distinguishable from early alone (whole − - early + 5 +5 pp, CI [ − 1 , + 12 ] [-1,+12] , includes zero), so the early window carries the effect, matching the reversal and the position control. Halving the knockout ( λ = 0.5 \\lambda{=}0.5 ) does nothing ( 3 % 3\\% ), a threshold echoing the position control’s. The effect is harm-selective : the same early knockout on benign prompts under the prefill barely moves refusal ( 0 % → 4 % 0\\%\\to 4\\% ), a specificity difference-in-differences of + 16.8 +16.8 pp, and coherence is unchanged throughout ( 0.96 0.96 – 0.97 0.97 ). It suppresses harmful content , not just adds refusal strings (procedural-content rate 51 % → 18 % 51\\%\\to 18\\% early, → 3 % \\to 3\\% whole, while the matched control stays at 50 % 50\\% ). This replicates on a second family: on Phi-3-mini ( 3.8 3.8 B) the identical pattern holds, early prefill-edge knockout restores refusal to 25 % 25\\% versus 1 % 1\\% for the mass-matched control, early-window-gated (late knockout inert at 0 % 0\\% , whole 32 % ≈ 32\\%\\approx early), threshold in dose ( λ = 0.5 \\lambda{=}0.5 inert), coherent (App. E )."},{"type":"heading","level":4,"text":"What this settles: the grip is generic, not safety-specific."},{"type":"para","text":"The effect is real but partial ( 24 % 24\\% of a 97 % 97\\% compliance), consistent with the distributed, redundant computation: direct early-window attention is one causal pathway, not the whole mechanism. Its interpretation we can now pin down, and it resolves the paper’s central open question toward passive conditioning. A base-model control is the discriminator: on the Qwen2.5-1.5B base model (no safety tuning, and so no refusal to re-engage), the identical early prefill-edge knockout collapses the harmful continuation prefill-specifically , harmful content 64 % → 25 % 64\\%\\to 25\\% against the mass-matched control’s 64 % 64\\% (a + 39 +39 pp prefill-specific drop), early-window-gated (late knockout inert). The same discriminator holds at mid scale: on the Qwen2.5- 7B base model the identical knockout drops harmful content 68 % → 30 % 68\\%\\to 30\\% against the mass-matched control’s 66 % 66\\% (a + 36 +36 pp prefill-specific drop, n = 100 n=100 ), again early-window-gated (late knockout 68 % 68\\% , inert), so the passive reading is not a small-model artifact. The dependence on early attention to the prefill is therefore architecture-general autoregressive conditioning , not a safety-tuning-specific active suppression of the harm signal: safety tuning fixes only the fallback once the grip is cut (the instruct model reverts to refusing, the base model degrades). So the prefill sustains compliance through generic early-window attention, and “refusal restoration” is a model-dependent fallback, not a re-engaging gate. This is the passive reading. A logit-level trace then measures a safety-specific component on top: when the knockout frees probability mass off the compliance continuation, the instruct model routes it to refusal tokens about 8 × 8\\times more than the base model does (concentration 0.24 0.24 vs 0.03 0.03 , App. E ). Safety tuning adds a small learned refusal attractor, but the dominant mechanism is passive: compliance collapses in both models. The trace confirms a safety-specific refusal pull exists; it does not separate an actively-suppressed gate springing back from a passive fallback attractor."},{"type":"heading","level":4,"text":"The generic story extends to the early MLP pathway."},{"type":"para","text":"Attention is one causal pathway; the compliance that persists ( 24 % 24\\% of 97 % 97\\% ) could hide a safety-specific mechanism in the pathway we did not cut. We probe the early-window MLP contribution (App. E ) and find the same base-vs-instruct control resolves it the same way. In the base model, removing the prefill’s early-MLP shift drops harmful content 68 % → 48 % 68\\%\\to 48\\% , a + 16 +16 pp prefill-specific, early-window-gated drop (matched-random inert) in a model with no safety mechanism to house, so the early MLP’s prefill-dependence is generic too. The passive account therefore holds across two pathways (attention and early MLP), not one."},{"type":"para","text":"What a difference-of-means test could not settle, a clean path patch does. That test restored 47 % 47\\% refusal in the instruct model, but a plain − - attack injection tests decodability, not active computation, so the larger instruct effect is overdetermined by its larger behavioral gap. We therefore run a frozen-residual path patch that routes the prefill counterfactual through the early-MLP sub-block alone , every other activation held at its clean attack value (the surgery is bit-exact under a sham / MLP-output-faithfulness / locality unit-test suite, App. E ). The MLP pathway carries the generic effect and only that: it strips prefill-specific compliance probability hard (instruct − 21 -21 pp, base − 25 -25 pp, both far above their mass-matched controls at − 7 -7 / − 5 -5 pp) but produces no safety-specific refusal pull (instruct concentration 0.007 ≈ 0.007\\approx base 0.001 ≈ 0.001\\approx matched; prefill-specific refusal gain + 0.26 +0.26 pp, within noise). So the instruct excess was decodability, not an active MLP computation, and the small residual safety-specific pull (the 0.24 0.24 generative concentration above) does not route through the early MLP. The passive account holds across both pathways under a clean path patch (§ 7 )."},{"type":"heading","level":2,"text":"4.4 The onset projection is illustrative (and confounded)"},{"type":"para","text":"We also track the projection of the generation-site state onto the harm-probe direction across decoding steps (Fig. 2 ): with no attack the onset step sits high ( 2.80 2.80 ) and the model goes on to refuse, while under the prefill it sits low ( 0.22 0.22 ). We report it as illustration, not localization; the causal locus is the early-window response-site effect above."},{"type":"para","text":"The 2.80 → 0.22 2.80\\to 0.22 drop is not a clean measurement of the refusal decision being displaced, and a control says why. The axis is the harm-probe direction, so the projection at a forced onset position reflects the content of the token sitting there. Forcing a fully benign, non-affirmative prefix on the same harmful prompts (“Here’s a fun fact:”) drives the onset projection even lower ( − 0.82 -0.82 ) than the attack’s “Sure, here is” ( 0.22 0.22 ), and a benign affirmative (“Sure, here is a poem about”) to − 0.11 -0.11 ; none of these is the attack, yet all read low, and the benign prefixes are not refused either ( 0 – 7 % 7\\% ). The onset projection therefore tracks measured-position content, not a pure refusal-decision magnitude, so we treat it as illustrative only; the clean positional result comes from the patching (§ 4.1 ), not the projection."},{"type":"heading","level":2,"text":"4.5 Steering fails; monitoring works"},{"type":"para","text":"If the harm direction were a clean causal handle, adding it should restore refusal selectively. It does not. Sweeping layers and coefficients (raw and Arditi-normalized), we find no window that raises harmful refusal without dragging benign refusal up with it: the best cell restores harmful refusal to 83 % 83\\% but at 40 % 40\\% benign over-refusal; smaller coefficients over-refuse everything, larger ones break generation. Adding the direction induces refusal broadly, not on-harm, consistent with Arditi et al. ( 2024 ) , where the refusal direction makes the model refuse harmless prompts too. The harm representation is a perfect read-out (detection AUC 1.0 1.0 ) but not a selective write-handle . The usable intervention is therefore the monitor of § 3 ( 100 % 100\\% catch, 0 % 0\\% false positive), not steering: we can monitor the gap shut, not steer it shut."},{"type":"heading","level":4,"text":"Why response-site patching spares benign but global steering does not: the injection site."},{"type":"para","text":"The benign-sparing patching (§ 4.1 , benign restoration 12 % 12\\% / 0 % 0\\% ) and this non-selective steering look like the same intervention, adding the harm direction under attack, but differ in where it is added: patching touches only the response positions, steering adds it at all positions, the benign prompt included. We ran a dedicated site comparison: direction and layer held fixed (the harm-probe direction at the patching coefficient α = 1 \\alpha{=}1 of § 4.1 , layer 12 12 ; not the normalized Table 8 sweep), varying only the injection site. Response-only restores 48 % 48\\% harmful at 12 % 12\\% benign (reproducing Table 2 ). All-positions restores 58 % 58\\% harmful but drags benign to 40 % 40\\% (the steering over-refusal). Global injection corrupts the benign prompt’s own read, so it registers as harmful; response-site injection leaves the prompt intact. The harm direction is thus a partial, benign-sparing handle at the response site but a non-selective knob applied globally, and even response-only its window is narrow: harmful 48 % 48\\% at α = 1 \\alpha{=}1 , 74 % 74\\% at α = 1.5 \\alpha{=}1.5 , but benign over-refusal climbs 12 → 28 % 12\\to 28\\% and generation collapses by α = 2 \\alpha{=}2 ."},{"type":"heading","level":1,"text":"5 What refusal reads: harm, not surface"},{"type":"para","text":"Does the refusal gate read harm, or the scary surface features that correlate with it? We answer with a pre-registered 2 × 2 2\\times 2 that decorrelates the two axes (actual harm and surface trigger-word density) and asks which drives refusal. Cells come from natural keyword-density variation in existing public prompts (no authored evasions): harmful-scary (AdvBench with trigger words), harmful-clean (AdvBench without), benign-scary (all of XSTest-safe), benign-clean (Alpaca). We fix the keyword lexicon and the labels before looking at any outcome."},{"type":"para","text":"Refusal is harm-primary (Table 4 ). The load-bearing comparison is within-source : harmful-scary and harmful-clean are both AdvBench, so stripping the trigger words holds source, register, and phrasing fixed and varies only keyword density. Refusal does not budge (harmful-clean 100 % 100\\% , identical to harmful-scary) and the harm representation is keyword-invariant (probe 0.982 0.982 vs 0.983 0.983 ). A logistic regression of refusal on the two axes gives β harm = 5.12 ≫ β scary = 2.20 \\beta_{\\text{harm}}=5.12\\gg\\beta_{\\text{scary}}=2.20 . There is a real but secondary keyword effect on the benign side: benign-scary prompts are over-refused at 59 % 59\\% versus 8 % 8\\% for benign-clean, the well-known exaggerated-safety phenomenon (Röttger et al., 2024 ) . Its magnitude is an upper bound , not an estimate: benign-scary is entirely XSTest and benign-clean entirely Alpaca, so β scary \\beta_{\\text{scary}} is entangled with dataset and register and partly counts a “sounds like XSTest” effect. The harm-primary conclusion does not rest on it; it rests on the within-AdvBench comparison, which no source confound touches."},{"type":"heading","level":4,"text":"Off ceiling (SmolLM2)."},{"type":"para","text":"Qwen’s harmful cells both sit at 100 % 100\\% , so the within-source comparison there can only fail to detect a keyword effect on the harmful side, not measure one. We therefore rerun the 2 × 2 2\\times 2 on the weak-gate SmolLM2 ( 0.63 0.63 plain refusal), where the harmful cells are off ceiling. Harm-primary holds and a small surface effect becomes visible: harmful-scary refuses 66 % 66\\% and harmful-clean 59 % 59\\% (stripping trigger words drops refusal 7 7 pp, a real but small keyword contribution the Qwen ceiling masked), while harm dominates (harmful-clean 59 % 59\\% vs benign-scary 12 % 12\\% ; logistic β harm = 2.83 ≫ β scary = 0.55 \\beta_{\\text{harm}}=2.83\\gg\\beta_{\\text{scary}}=0.55 ). Refusal is harm-primary, with a small keyword effect on the harmful side that is detectable only off ceiling."},{"type":"heading","level":4,"text":"A retracted reading."},{"type":"para","text":"An earlier version of this analysis, run on XSTest directly, took the cosine between a harm-content direction and a refusal-behavior direction and found it low ( ∼ 0.1 {\\sim}0.1 in deep layers), which we initially read as “refusal reads a genuinely separate feature from harm.” That reading does not pass the 2 × 2 2\\times 2 . The low cosine was inflated by exactly the keyword-driven over-refusals of the benign-scary cell: on XSTest, harm-content and refusal-behavior dissociate mostly through over-refusal, so the “refusal direction” there partly encodes scary surface. Once harm and surface are properly decorrelated, refusal is harm-primary. We keep the corrected reading and drop the “separate feature” claim."},{"type":"heading","level":2,"text":"5.1 Is there a separate refusal direction? A decision-direction search"},{"type":"para","text":"The walk-back invites the question directly: if we build the refuse-vs-comply direction properly , holding harm fixed, is there a separate decision feature to find? We search and report a controlled negative. The prefill supplies the matched contrast a naive setup lacks: a weakened prefill (“I understand you’re asking”) splits harmful prompts 41 / 29 41/29 refuse-vs-comply, so refused and complied cases share both harm and prefill-presence and differ only in the decision. The diff-of-means direction d refuse d_{\\text{refuse}} is geometrically distinct from the harm direction ( | cos | = 0.06 |\\cos|=0.06 ) but reads the decision only weakly: cross-validated AUC 0.648 0.648 , barely above the harm direction’s own 0.640 0.640 on the same split. A sweep of all layers and the first four generated positions ceilings at AUC 0.727 0.727 (layer 27 27 ) for any single direction. This ceiling, like the “best cells” in the patching grid (App. D , Table 7 ) and in Table 8 , is a maximum over its grid, so it is post-selection and optimistically biased. The main-text patching claim (Table 2 ) instead uses two fixed cells at n = 100 n=100 , and the qualitative conclusions do not depend on the grid maxima. But the decision is multidimensional rather than weakly encoded: a full linear probe on the same activations reads refuse-vs-comply at ∼ 0.85 \\sim 0.85 (layer 21 21 ) and well above harm alone ( 0.64 0.64 ). The capacity gain over the best single direction is + 0.12 +0.12 (bootstrap mean + 0.17 +0.17 , 95 % 95\\% CI [ + 0.05 , + 0.33 ] [+0.05,+0.33] , n = 70 n=70 ), so it excludes zero though it is imprecise, and it is consistent across regularization and layers. So there is no single refusal direction , the signal is spread across directions. Whether it is genuinely high-rank rather than low-rank is exploratory at n = 70 n=70 (three principal components reach only 0.74 0.74 , near the single-direction 0.73 0.73 , and the readout keeps climbing as dimensions are added, but n = 70 n=70 flatters high-rank estimates; App. H ); the robust claim is just single ≪ \\ll full. No single direction is a usable handle: steering d refuse d_{\\text{refuse}} under the strong attack is causally inert (no restored refusal at a mid layer where steering has purchase, nor at the best-read late layer; 0 % 0\\% at every coefficient; App. H ). This is the mechanistic form of “more to refusal than a single direction” (Joad et al., 2026 ) : refusal is a decodable but distributed computation, which is why single-direction steering (§ 4.5 ) and the “separate feature” reading both fail."},{"type":"heading","level":4,"text":"Across scale (within family)."},{"type":"para","text":"This distributed structure is a within-family scale law. Across the Qwen ladder ( 0.5 0.5 , 1.5 1.5 , 3 3 , 7 7 B) the full probe beats the best single direction by 0.06 0.06 – 0.12 0.12 at every scale, and both readouts rise with size (single direction 0.76 → 0.85 0.76\\to 0.85 , full 0.84 → 0.94 0.84\\to 0.94 ; the 7 7 B run is on GPU, App. H ). The claim is Qwen-scoped: the one balanced cross-family point, SmolLM2-1.7B, shows no gain (its decision is essentially a single direction), and Phi-3’s calibration split was too imbalanced to read, so cross-family is left open. SmolLM2 is also the weakest gate ( 0.63 0.63 plain refusal), and a weakly-enforced refusal need not recruit a distributed representation to do its job, so the lower rank may track the weaker gate. We flag this as a hypothesis; a scale of gate-strength against decision rank is untested here."},{"type":"heading","level":4,"text":"Refusal is not selectively ablatable."},{"type":"para","text":"No intervention we tried removes refusal selectively, a transition from harmful refusal to coherent harmful compliance without comparable degradation on benign behavior; anything less destroys the computation rather than editing it. Ablating any single attention head at any layer leaves refusal fully intact (a 28 × 12 28\\times 12 sweep on Qwen2.5-1.5B: every head-ablation drops refusal by 0.00 0.00 ; the hook is verified to change next-token logits by ‖ Δ ‖ ≈ 23 \\|\\Delta\\|\\approx 23 , so this is robustness, not a dead intervention), and ablating an entire attention layer likewise leaves refusal at 100 % 100\\% (e.g. layer 12, ‖ Δ logits ‖ ≈ 123 \\|\\Delta\\text{logits}\\|\\approx 123 ). Heavier ablation does not yield compliance. Ablating the first attention layer, or all of them, drives the refusal string-match to zero only by collapsing generation into garbage (“ silly silly noble glorious …”). With all attention removed the model emits one fixed gibberish string regardless of the prompt , harmful or benign, a comparable non-selective degradation that a coherence read (App. J ) confirms is not compliance. Together with the earlier decompositions, direction steering (§ 4.5 ), subspace and full-residual patching (§ 4.1 ), this leaves no compact causal support for refusal among the decompositions we test (direction, subspace, layer, attention head) for these models. It leaves open a compact mechanism in a space we did not probe: MLP neurons and sparse (SAE) features are untested and either could still host one. We tested the scale concern at 14 14 B on heavily-safety-trained Phi-3-medium (App. J ): the negative holds (no single component selectively removes refusal; only the input layer shows a partial drop), so heavier safety training did not concentrate refusal into a compact circuit here, though frontier scale is untested. The only intervention that coherently removes refusal here is the response-site prefill, which leaves the model fluent and complying. Refusal is thus distributed and redundant in representation space but fragile at the response site: the mechanism is diffuse, the only compact failure point we identify is positional, not representational."},{"type":"heading","level":4,"text":"Not specific to surface harm (exploratory)."},{"type":"para","text":"The dissociation extends from lexical harm to inferred intent: on 24 24 dual-use triplets where intent, not surface content, decides the right answer (App. I ), the prefill flips every bare prompt the model reads as harmful-intent to compliance ( 100 % → 0 % 100\\%\\to 0\\% ) while that intent reading is unchanged, so the gate is shallow as a computation whether its basis is a harmful word or an inferred intent. Exploratory: one model, n = 24 n=24 ; full numbers and the same structural caveat in App. I ."},{"type":"heading","level":1,"text":"6 Related work"},{"type":"para","text":"Shallow safety and prefill attacks. Qi et al. ( 2025 ) argue safety alignment is shallow in token-depth (carried by the first few output tokens) and explicitly explain prefilling attacks (Wei et al., 2023 ; Zhu et al., 2024 ; Li et al., 2025 ) as a consequence. Our result is complementary and representational: the harm concept is not shallow or degraded under a prefill attack, it is fully and identically represented; what is shallow is the gate that turns that representation into a refusal, and we localize it causally to the response site. Where their proposed fix deepens the alignment via training (cf. Zhang et al., 2025a ), our defensive consequence is a read-side monitor."},{"type":"para","text":"Detecting harm from internal activations, and where the flip lives. That harmful and jailbreak prompts are linearly detectable in hidden states is well established, and several systems monitor or manipulate these signals (Zou et al., 2023a ; Jiang et al., 2025 ; Han et al., 2025 ; Ball et al., 2025 ) . Our claim is not that harm is probeable: it is. The closest work is JBShield (Zhang et al., 2025c ) : it shows a model still recognizes a toxic concept under a jailbreak while complying, attributes the flip to a competing prompt-side “jailbreak concept” (associated with affirmative tokens like “sure”), and mitigates by enhancing the toxic concept back toward the plain-harmful level. Two things separate us. First, that account is prompt-side and cannot cover prefill, which injects the affirmative opening into the response and leaves the prompt (and its toxic representation) untouched: there is no prompt-side jailbreak concept to do the overriding, so the flip must be, and we show is, a response-site positional displacement (§ 4 ). Second, JBShield reports a binary concept-present flag and, by enhancing the concept, presupposes it is attenuated ; we find the harm representation quantitatively intact under prefill (level with the refused prompts), so enhancing it is unnecessary and a final-token monitor already suffices. This also resolves an apparent tension. Doda ( 2026 ) (a concurrent preprint) show final-token probes miss many jailbreaks (the evidence displaced to earlier tokens), and Zhang et al. ( 2025d ) report vanishing discriminability under disguise. Our account does not rest on either, but it explains both: those are prompt-modifying attacks that move or attenuate the prompt-side representation, whereas prefill leaves it whole. Prompt-modifying attacks attenuate; prefill does not. This attenuate-vs-intact split is the representational signature of the response-site/prompt-modifying dichotomy, and our own encoding results (Appendix K ) sit on the attenuating side. The monitor’s scope follows from the mechanism, not from luck."},{"type":"para","text":"Refusal directions and steering. Arditi et al. ( 2024 ) show refusal is mediated by a single direction; a follow-up argues there is more to it than one direction (Joad et al., 2026 ) . We use the direction and find it a strong read-out but a non-selective write-handle for restoring refusal under attack, and (§ 5 ) that refusal is harm-primary rather than riding a separate feature. Exaggerated safety. XSTest (Röttger et al., 2024 ) documents over-refusal of benign scary-sounding prompts; we quantify it as a secondary effect and separate it from genuine harm-tracking. Method and framing. We build on representation probing and activation patching (Zou et al., 2023a ; Meng et al., 2022 ) ; the represent-vs-act framing (probe for what a model represents but does not act on) is the one we applied to communicative intent (Kwon, 2026 ) , with safety as the second substrate."},{"type":"heading","level":1,"text":"7 Limitations"},{"type":"para","text":"Controls and demotions. Every headline effect is checked against a control that can demote it, and several were: the full-residual “ 100 % 100\\% ” restoration is generic disruption (a benign-residual control also reaches 100 % 100\\% ), naive steering over-refuses everything, an earlier “separate refusal feature” did not pass the decorrelated 2 × 2 2\\times 2 , and the attack-map scaling prediction was refuted on a clean three-point test. Table LABEL:tab:claims lists the claims that passed alongside the demoted ones, and Appendix L gives the full ledger: every control, what it could have shown, and its verdict. The boundaries below apply the same checking to what remains. Several are open questions we scoped rather than results we could not reach, and we note where the evidence stops."},{"type":"para","text":"Active-vs-passive resolves toward passive; a scoped residual remains. The base-model control (§ 4.3 ) resolves the central open question toward passive conditioning across two pathways: the prefill’s early-window grip is generic in both the attention and the early MLP channel (a non-safety-tuned base model shows the same prefill-specific collapse in each), with only a small safety-specific refusal attractor on top (logit trace, concentration 0.24 0.24 vs 0.03 0.03 ). The one part that was open, the instruct excess of that attractor, we resolve with a clean path patch. A difference-of-means injection could not settle it (it tests decodability, not active computation), and a positional MLP ablation cannot either (a transformer MLP is applied position-wise and never reads the prefill positions, so its prefill-dependence is inherited from attention, with no positional handle to sever). So we run a frozen-residual path patch that severs only the prefill-attention contribution to the MLP input, validated bit-exact by a sham / MLP-output-faithfulness / locality suite (App. E ). It answers toward generic: the early-MLP pathway routes prefill-specific compliance suppression (instruct − 21 -21 pp, base − 25 -25 pp vs matched − 7 -7 / − 5 -5 pp) but carries no safety-specific refusal pull (instruct concentration 0.007 ≈ 0.007\\approx base ≈ \\approx matched; prefill-specific refusal gain + 0.26 +0.26 pp). The residual safety-specific attractor does not live in the early MLP. The one caveat is frame: this is a teacher-forced logit path patch, well-powered for compliance-suppression routing and less so for a small refusal-promotion component that would need a generative trajectory to build, but the within-frame instruct-vs-base-vs-matched comparison is clean regardless. A safety-training-depth dose-response (fine-tune matched shallow-vs-deep checkpoints and re-run the knockout) would confirm from the training side; the path patch already answers the question the paper posed."},{"type":"para","text":"Scale and mechanism breadth. The core models are 1.5 1.5 – 3.8 3.8 B and run on CPU (with 7 7 B and 14 14 B probed for scale on GPU), because white-box access requires open weights. The dissociation replicates across four models, three families, and both a weak gate (SmolLM2, 0.63 0.63 plain refusal) and strong ones (Phi-3, 1.00 1.00 ), which mitigates but does not remove the concern that the picture changes at frontier scale, where safety training is heavier and the residual stream is wider. The full mechanism battery (patching, position, reversal, § 4 ) is shown on two models (Qwen2.5-1.5B and Phi-3-mini) at the per-condition n n reported per experiment: a localization, not a scaling survey. The paper’s central causal claim, the active-vs-passive knockout discriminator, does reach mid scale: on the Qwen2.5- 7B base model the knockout drops harmful content 68 % → 30 % 68\\%\\to 30\\% prefill-specifically against a mass-matched control’s 66 % 66\\% ( + 36 +36 pp, early-window-gated, n = 100 n=100 , App. E ), so the generic early-window grip is not a small-model artifact. On the 7B instruct model the prefill attack is weaker (refusal drops only to 24 % 24\\% , procedural content 16 % 16\\% ) and the refusal- string restoration does not cleanly replicate, consistent with the passive reading in which “refusal” is a model-dependent fallback, while the prefill-specific content-channel effect persists. So the mechanism (prefill-specific early-window attention dependence) and its passive character persist at 7 7 B; the behavioral refusal signature is model-dependent, and frontier scale remains untested."},{"type":"para","text":"Probe source-calibration. The harm probe reads harm rather than surface vocabulary, and we show this: an AdvBench-trained probe scores XSTest’s scary-but-benign split at 0.21 0.21 (only 12 % 12\\% clear threshold), so the intactness result is not the probe re-detecting violent words. But the same probe is AdvBench- calibrated : it fires at 0.97 0.97 on AdvBench-style phrasing and only 0.49 0.49 on differently-phrased XSTest-unsafe prompts, and a mixed-source probe (AdvBench + + MaliciousInstruct) raised sensitivity only by wrecking specificity (benign-scary false positives 12 % 12\\% to 64 % 64\\% ). This bounds the claim precisely, and the bound cuts in our favor on the core result: the dissociation compares complied-harmful against refused-harmful prompts drawn from the same distribution, so source-calibration cancels and cannot manufacture the gap. What it does threaten is the deployable monitor, whose 100 % 100\\% -catch / 0 % 0\\% -false-positive figure is measured on AdvBench-sourced attacks. A source-general harm monitor is unbuilt, and building one is the obvious next step; the mixed-source attempt here narrows the worry but does not discharge it."},{"type":"para","text":"Attack scope. Every headline number is scoped to the prefill (response-site) attack, and the monitor result is a claim about that regime and no wider. Prompt-modifying attacks (adversarial suffixes, and encodings the model cannot decode) attenuate or displace the prompt-side representation rather than leaving it whole (§ 6 , App. K ), so a representation monitor is strictly weaker against them. We report the attenuate-versus-intact split as the boundary of the guarantee, not as coverage of all jailbreaks."},{"type":"para","text":"The harm direction is a real but partial handle. The harm direction restores refusal significantly above a matched-norm random baseline ( n = 100 n=100 : + 34 +34 / + 17 +17 pp, both CIs exclude zero) and spares benign, so it is a real partial handle, but the dramatic full-residual “ 100 % 100\\% ” restoration is generic disruption (a benign-residual control also reaches 100 % 100\\% ), and a third to a half of even the subspace restoration is matched by the random control. Adding the harm direction back under attack does not selectively restore refusal: a four-layer by three-coefficient sweep, raw and Arditi-normalized, finds no clean window. So we close the gap as a monitor, not as an intervention; whether some other handle can un-collapse the behavior is left open."},{"type":"para","text":"Measurement: what the attacked-condition outputs actually are. Refusal is scored by matching a fixed refusal-phrase list, not by an LLM judge: cheap and exactly reproducible. But “non-refusal” is not the same as coherent harmful compliance, so we read the attacked-condition generations directly, judge-free: a distinct-token/repetition coherence rule plus a transparent deflection-marker list, on 200 200 -token generations over a 50 50 -prompt sample per model, with all completions withheld and only per-model counts released (Ethics). Two conditions come apart. Under the full -strength prefill (the headline dissociation), the non-refusals are overwhelmingly coherent harmful compliance, not deflection. On Qwen2.5-1.5B, 36 36 of the 48 48 non-refusals (of 50 50 ) carry substantive procedural content by the automated rule. That is a conservative floor: a manual read of the remainder (harmful prose, e.g. incitement text or a pro-anorexia post) raises it to 45 / 48 45/48 ( 94 % 94\\% ), with deflection 2 / 48 2/48 and no degeneracy. Across the other three models, substantive compliance is 69 69 – 95 % 95\\% of non-refusals and genuine deflection is ≤ 3 / 50 \\leq 3/50 , with a small degeneracy tail on SmolLM2 ( 8 / 50 8/50 ). Some models add a cosmetic “this is illegal” disclaimer but deliver the procedure anyway. So the string-match compliance rate is not inflated by deflection; the behavior is real. Under partial harm-restoration patching, by contrast, the marginal non-refusals at the L12/ α = 1 \\alpha{=}1 cell are mostly non-harmful deflection (“a guide for making an explosive device: 1. ‘bomb’, this is not real information”), so there the string match under counts safe behavior and does not inflate the restoration. Both reads are judge-free. The same two-scorer agreement holds across every causal condition and both models: string-match refusal and the independent harmful-content scorer move in opposite directions throughout. On the inert controls (attack baseline, mass-matched knockout, late-window knockout, matched-norm random injection) refusal stays ≤ 4 % \\leq 4\\% while harmful content stays 48 48 – 59 % 59\\% (Qwen) / 13 13 – 18 % 18\\% (Phi); on the working interventions (early prefill-edge knockout, refuse-state injection) refusal is restored to 24 24 – 78 % 78\\% while harmful content collapses to 1 1 – 18 % 18\\% ; and coherence stays 0.94 0.94 – 0.99 0.99 everywhere. So no headline effect is an artifact of string-matching or degeneracy: the refusal scorer and the content scorer tell the same story, at full n n rather than a sample."},{"type":"heading","level":1,"text":"8 Conclusion"},{"type":"para","text":"A prefill attack makes an aligned model comply with harmful requests while the model represents those requests as harmful exactly as strongly as the ones it refuses. Refusal is a shallow, response-site computation, primarily driven by the harm representation but severable from it: the attack occupies the response site, the harm concept sits intact at the prompt, and restoring the harm signal in an early window of the response partially restores the action; a dose-matched position control shows the first half of the response suffices while the second half is nearly inert, more than a single onset token but not the whole generation. The converse lands in the same window: injecting the model’s own refuse-state back into the attack re-induces refusal ( → 74 % \\to 74\\% , held-out), state-specifically (a matched-norm random control does nothing) and early-window-gated, a working defensive lever that the single harm direction alone is not. And the grip operates through early-window attention: knocking out the early response’s attention to the prefill, but not an equal attention mass elsewhere, selectively collapses the harmful continuation ( + 21 +21 pp refusal restored). A base-model control shows this dependence is generic autoregressive conditioning, not safety-specific active suppression (a non-safety-tuned base model shows the same prefill-specific collapse), resolving the central open question toward the passive reading: the prefill sustains compliance through generic early attention, and refusal is a model-dependent fallback. The concept is not the gate. And the gate has no compact internal form: across direction, subspace, layer, and attention-head decompositions no single-component intervention selectively removes refusal; heavy ablation only degrades generation into garbage. Refusal is distributed and redundant in representation space but fragile at the response site: the mechanism is diffuse, and the only compact failure point we identify is positional, not representational. The practical consequence is a design fact rather than a detector: because the attack never reaches the prompt-side representation, a monitor that reads it is immune to this class of attack by construction, where the behavioral gate is not."},{"type":"heading","level":1,"text":"Ethics and responsible disclosure"},{"type":"para","text":"This is a mechanistic analysis of a known attack, not a new one (the prefill jailbreak is public (Wei et al., 2023 ; Qi et al., 2025 ) ; we introduce no novel evasion), on small open-weight models ( 1.5 1.5 – 3.8 3.8 B) and public benchmarks (AdvBench, XSTest, Alpaca), with no authored harmful prompts. The direction is net-defensive: the practical output is where to place a safety classifier (prompt-side, immune to response-site attacks by the mechanism we identify), not how to defeat one. On artifacts: verifying that attacked-condition non-refusals are coherent harmful compliance required generating harmful completions, so we withhold them: the repository ships per-model counts (which reproduce every number) not the completion text. No human subjects or personal data are involved."},{"type":"para","text":"The table below maps every load-bearing claim to its evidence and epistemic status: shown (direct measurement) or analytic (follows from the construction or argument). The dissociation and the causal mechanism are the contribution; the demoted effects (full-residual disruption, the steering null, the walked-back “separate feature” reading) are recorded so the claims that passed are unambiguous."},{"type":"para","text":"All code and the full experiment ledger ( FINDINGS.md , every number with its control) are at https://github.com/collapseindex/breaking-refusal . Decision-token activations are cached to .npz so all analyses (probes, patching, the 2 × 2 2\\times 2 ) re-run offline in seconds; the model passes regenerate them."},{"type":"heading","level":4,"text":"Data."},{"type":"para","text":"All prompts are public: harmful = = AdvBench behaviors; benign = = Alpaca instructions; the surface control = = XSTest (its “safe” split is purpose-built to sound harmful); the mixed-source probe control adds MaliciousInstruct (Huang et al., 2024 ) . The harm/keyword 2 × 2 2\\times 2 (§ 5 , App. G ) is built from natural keyword-density variation in these existing prompts with a fixed lexicon; no evasion prompts are authored."},{"type":"heading","level":4,"text":"Models and environment."},{"type":"para","text":"Qwen2.5-1.5B/3B-Instruct, SmolLM2-1.7B-Instruct, and Phi-3-mini-4k-instruct, from the Hugging Face cache (revisions pinned in the repo manifest). Python 3.13, transformers (Wolf et al., 2020 ) , torch (Paszke et al., 2019 ) (CPU-only), scikit-learn (Pedregosa et al., 2011 ) ; the one embedding control uses fastembed (BGE). Models load in bfloat16 at ≥ 1.5 \\geq 1.5 B (float32 exhausts memory on CPU) and float32 otherwise. Per-condition baselines (e.g. the 1.5 1.5 B attack-refusal rate, or the base model’s plain-attack compliance) vary by a few points across experiments, since scripts draw different AdvBench subsets at different n n ; every control is compared against its own run’s baseline, and we report each run’s own numbers rather than a pooled figure."},{"type":"heading","level":4,"text":"Probe, attack, and refusal."},{"type":"para","text":"The harm probe is ℓ 2 \\ell_{2} logistic regression ( C = 0.5 =0.5 , max_iter = 2000 =2000 ) on the last-prompt-token hidden state. Reported detection AUCs use nested-CV layer selection (no test-set peeking); the downstream causal experiments (patching, position, steering) instead fix layer 12, a representative mid layer where the probe is near-ceiling, so the intervention layer is not itself selected on the outcome. The prefill attack forces the assistant turn to begin with “Sure, here is”. Refusal is a fixed refusal-phrase string match. All generation is greedy ( do_sample = = False)."},{"type":"heading","level":4,"text":"Determinism."},{"type":"para","text":"Probes and featurizers are deterministic given the data; the only stochastic element is the cross-validation split (fixed seed). The 2 × 2 2\\times 2 lexicon, cell construction, and decision thresholds were fixed before any outcome was seen."},{"type":"heading","level":4,"text":"State-transfer experiment (§ 4.2 )."},{"type":"para","text":"The reversal is run by experiments/modal_prefix_transfer.py (Qwen2.5-1.5B, L12, n = 100 n=100 AdvBench, greedy, 18 18 generated tokens). It captures the mean layer- 11 11 -output residual over the first-half generated window for the plain (no-prefill) and attack conditions, forms d reverse = s plain − s attack d_{\\text{reverse}}=s_{\\text{plain}}-s_{\\text{attack}} , and adds it into the attack’s early generated window over the same bands as the position control (onset / first half / all), against a matched-norm control drawn orthogonal-random and rescaled to ∥ d reverse ∥ \\lVert d_{\\text{reverse}}\\rVert . A tiny- n n gate (plain refusal ≥ 0.7 \\geq 0.7 , attack refusal ≤ 0.15 \\leq 0.15 ) validates the harness before the full run. The script is measure-only: it records refusal, a distinct-token coherence ratio, and a harmful-procedural-content flag as per-prompt counts to data/exp_prefix_transfer.json ; no generations are stored, and run logs (gitignored) print rates only."},{"type":"heading","level":4,"text":"Attention-knockout experiment (§ 4.3 , App. E )."},{"type":"para","text":"Run by experiments/modal_attn_knockout.py (eager attention, Qwen2.5-1.5B, all layers/heads, n = 100 n=100 ), preregistered in PREREG_attn_knockout.md (claim, falsifier, condition matrix, interpretation table). Edge knockout adds ln ( 1 − λ ) \\ln(1-\\lambda) to the pre-softmax attention logit of (early-window query → \\to prefill key) edges and renormalizes; the mass-matched control selects non-prefill prompt keys whose baseline attention mass matches the prefill’s per prompt (three fixed seeds). A MODE=\"unittest\" scaffold must pass before any behavioral run (sham-hook bit-identity, edge-only masking, softmax renormalization, KV-cache consistency, live-hook and behavioral positive controls). Measure-only: refusal / coherence / harmful-content counts to data/exp_attn_knockout.json ; CIs are paired bootstrap over prompts."},{"type":"para","text":"Table 1 gives the headline numbers. The “rep. AUC” column is one nested-CV probe (outer 5-fold, inner layer selection) on the clean prompt-token activations: 0.996 0.996 (Qwen-1.5B), 1.000 1.000 (Qwen-3B), 0.990 0.990 (SmolLM2), 0.998 0.998 (Phi-3). There is no separate “under attack” representation number, because the attack does not alter these activations (it appends to the response); we report a single estimator to make that explicit. The attacked-set behavior AUC is chance ( 0.49 0.49 – 0.52 0.52 ); the per-model complied-harmful vs. benign probe scores are 0.975 / 0.055 0.975/0.055 (Qwen-1.5B), 0.981 / 0.048 0.981/0.048 (Qwen-3B), 0.958 / 0.044 0.958/0.044 (SmolLM2), 0.907 / 0.122 0.907/0.122 (Phi-3)."},{"type":"heading","level":4,"text":"Surface control (Table 6 )."},{"type":"para","text":"We score the AdvBench-trained probe (layer 12) on XSTest. It separates real-unsafe from scary-but-safe at AUC 0.79 0.79 with only 12 % 12\\% false positives on the scary-safe prompts: it reads harm, not violent vocabulary. It is, however, AdvBench-calibrated ( 0.97 0.97 on AdvBench-harmful but 0.49 0.49 on differently-phrased XSTest-unsafe), and training on mixed harmful sources (AdvBench + + MaliciousInstruct) does not fix it: it raises sensitivity but collapses specificity."},{"type":"para","text":"Table 7 reports the full grid. We patch at the response site on harmful + + prefill prompts and let generation continue with no further intervention. The attack baseline (no patch) is harmful refusal 0.07 / 0.07/ benign 0 (Qwen-1.5B) and 0 / 0 0/0 (Phi-3). full : replace the onset residual with the mean harmful residual. benign-CTRL : the same with the mean benign residual: it returns refusal to 100 % 100\\% at every layer for both models, which is why we read the full-residual restoration as generic disruption rather than a harm-specific effect. subspace : add α ⋅ d ^ harm \\alpha\\cdot\\hat{d}_{\\text{harm}} (unit harm-probe direction) at the response positions, α ∈ { 0.5 , 1 , 1.5 } \\alpha\\in\\{0.5,1,1.5\\} scaled to the layer’s activation norm; the reported cell is the best α \\alpha by (harmful − - benign) refusal. Settling the harm-vs-random delta at n = 100 n=100 on two fixed cells (not maxima): at L12/ α = 1 \\alpha{=}1 the harm direction restores 48 % 48\\% (bootstrap 95 % 95\\% CI 38 38 – 58 % 58\\% ) versus 14 % 14\\% for a matched-norm direction orthogonalized to d ^ harm \\hat{d}_{\\text{harm}} (24 seeds, range 0 – 60 % 60\\% ), delta + 34 +34 pp (CI + 22 +22 , + 46 +46 ); at L8/ α = 1.5 \\alpha{=}1.5 , 33 % 33\\% ( 24 24 – 42 % 42\\% ) versus 16 % 16\\% ( 0 – 60 % 60\\% ), delta + 17 +17 pp (CI + 6 +6 , + 28 +28 ). Both exclude zero (the earlier n = 15 n=15 estimate had overlapping CIs). Harm is benign-sparing (benign restoration 12 % 12\\% / 0 % 0\\% vs random 3 % 3\\% / 4 % 4\\% ). So a harm-specific component is real and significant, but a third to a half of the raw restoration is matched by the random control; we report the harm − - random delta. This licenses a partial, partly specific causal claim, not a clean lever."},{"type":"heading","level":4,"text":"Cell selection and the bootstrap."},{"type":"para","text":"The two cells are grid-selected (L12 is the best subspace layer in the grid above; L8 a second), fixed before the n = 100 n=100 run, which re-estimates them on AdvBench [ : 100 ] [:100] , overlapping the grid prompts. Selection optimism therefore affects the absolute restore rate but not the harm − - random delta, which is measured for both arms at the same cell and prompts. For the delta CI each bootstrap iterate resamples the 100 100 harm prompts (with replacement) and the 24 24 random seeds (with replacement), taking the harm mean minus the resampled-seed mean; the harm-restore CI resamples prompts only. The random-control distribution is right-skewed and mostly low: the 24 24 L8 seeds are (sorted) 0 , 0 , .01 , .01 , .01 , .03 , .06 , .06 , .07 , .09 , .10 , .11 , .12 , .13 , .13 , .13 , .20 , .21 , .23 , .26 , .35 , .37 , .53 , .60 0,0,.01,.01,.01,.03,.06,.06,.07,.09,.10,.11,.12,.13,.13,.13,.20,.21,.23,.26,.35,.37,.53,.60 , median 0.12 0.12 , with the 0 – 60 % 60\\% range driven by four high outliers, so the harm direction’s 33 % 33\\% beats the typical random direction by more than the mean delta suggests."},{"type":"heading","level":4,"text":"Dose-matched positional control."},{"type":"para","text":"To ask whether the position result reflects where the harm signal is restored or merely how much total signal is injected, we band one consistent response window (the prefill + + the 18 18 generated tokens, 22 22 positions) and vary position and per-position coefficient independently ( n = 100 n=100 , L12; GATE: the all-positions band at α = 1 \\alpha{=}1 reproduces the 41 % 41\\% of the main text). Restoration by condition (% refusal, baseline 3 % 3\\% ): all @ 1 @1 = 41 =41 , first-half @ 1 @1 = 42 =42 , first-half @ 2 @2 = 47 =47 , second-half @ 2 @2 = 6 =6 , all @ 0.5 @0.5 = 0 =0 , every-other @ 1 @1 = 1 =1 , every-other @ 2 @2 = 48 =48 (coherence ≥ 0.96 \\geq 0.96 throughout, so these are not degeneration). Reading: (i) the first half restores as much as the whole, and the second half is nearly inert even at double strength, so the effect is a matter of position (an early window), not total dose; (ii) halving the coefficient over all positions gives 0 % 0\\% , so there is a per-position strength threshold, not just a dose budget. This corrects an earlier version of the control whose “all” band silently included the prefill positions while its half-bands did not, confounding position with prefill-inclusion; the windowed design here is consistent across bands."},{"type":"para","text":"This appendix backs § 4.3 . The experiment is preregistered ( PREREG_attn_knockout.md ): primary hypothesis, falsifier, the frozen condition matrix, the preregistered contrasts, and an interpretation table written before results."},{"type":"heading","level":4,"text":"Intervention and the onset caveat."},{"type":"para","text":"On each early-window decode step ( g < 9 g<9 ) we add ln ( 1 − λ ) \\ln(1-\\lambda) to the pre-softmax attention logit of every (current-query → \\to prefill-key) edge and let softmax renormalize ( λ = 1 \\lambda{=}1 is a full edge cut). Eager attention is mandatory (fused SDPA/Flash kernels bypass a logit-level edit). Because generated token 1’s logits are produced from the final prefill token’s hidden state, a response-to-prefill knockout cannot affect token 1; this is a test of early maintenance , not attack initiation."},{"type":"heading","level":4,"text":"Unit tests (all pass before any behavioral run)."},{"type":"para","text":"(i) eager attention active; (ii) prefill key indices located after the chat template; (iii) sham hook (knockout with an empty key set) is bit-for-bit identical to baseline under greedy decoding; (iv) after a full edge cut the removed edges carry 0 attention mass and the softmax row renormalizes to 1 1 ; (v) non-target query rows are unchanged; (vi) no NaN/Inf reaches the residual stream; (vii) a large knockout measurably moves next-token logits ( ‖ Δ ‖ ≈ 10 3 \\|\\Delta\\|\\approx 10^{3} , the hook is live); (viii) knocking out attention to the prompt tokens materially changes generation (a behavioral positive control: the machinery moves behavior, so a prefill-specific null means “prefill edges not necessary,” not “dead hook”); (ix) KV-cache on/off agree under the sham hook."},{"type":"heading","level":4,"text":"Mass-matched control."},{"type":"para","text":"A no-knockout measurement pass records the baseline attention mass from early-window queries onto each key (mean over layers and heads). The prefill keys carry small mass (mean 0.082 0.082 ); attention-sink tokens (BOS, template) carry far more, so a max-mass or random span is not a fair control. We select, per prompt, a set of non-prefill prompt keys of the same size whose summed mass best matches the prefill’s (achieved matched/prefill ratio 0.98 0.98 , range [ 0.75 , 1.08 ] [0.75,1.08] over three fixed seeds). Knockout of this set is the primary control."},{"type":"heading","level":4,"text":"Full matrix and contrasts."},{"type":"para","text":"Table 3 gives the condition means. Paired bootstrap ( 10 , 000 10{,}000 resamples over the 100 100 harmful prompts; the matched contrast uses one fixed control seed, matched 3 % 3\\% on that seed, three-seed mean 1 % 1\\% ): early prefill KO − - mass-matched KO = + 21 =+21 pp ( [ + 12 , + 30 ] [+12,+30] ); early − - baseline = + 21 =+21 pp ( [ + 12 , + 30 ] [+12,+30] ); early − - late = + 20 =+20 pp ( [ + 11 , + 29 ] [+11,+29] ); whole − - early = + 5 =+5 pp ( [ − 1 , + 12 ] [-1,+12] , includes zero, so whole is not distinguishable from early). Benign specificity: under the prefill, early knockout moves benign refusal only 0 % → 4 % 0\\%\\to 4\\% (24 benign prompts), against the harmful 3 % → 24 % 3\\%\\to 24\\% , a difference-in-differences of + 16.8 +16.8 pp. Coherence stays 0.96 0.96 – 0.97 0.97 across all conditions (the preregistered non-inferiority margin was a ≤ 5 \\leq 5 -point drop). The knockout also suppresses harmful content (procedural rate 51 % → 18 % 51\\%\\to 18\\% early, → 3 % \\to 3\\% whole), prefill-specifically (matched control 50 % 50\\% )."},{"type":"heading","level":4,"text":"Held-out replications of the two causal handles (§ 4.2 , § 4.1 )."},{"type":"para","text":"Both the state-transfer reversal and the harm-direction patching are re-run with a clean train/test split so the injected vector never sees the test prompts. Reversal: d reverse d_{\\text{reverse}} is estimated on AdvBench [ 0 : 260 ] [0{:}260] and injected/tested on the disjoint held-out [ 260 : 360 ] [260{:}360] ; it restores refusal to 74 % 74\\% over the first half (whole 78 % 78\\% , onset 13 % 13\\% ), the matched-norm random control to 0 % 0\\% , coherence 0.97 0.97 – 0.99 0.99 ; the in-distribution reference (same d reverse d_{\\text{reverse}} tested on train prompts) gives 72 % 72\\% , so the generalization gap is within noise. Patching: the harm direction is trained on AdvBench [ 0 : 200 ] [0{:}200] (harmful vs Alpaca benign, activations recomputed on the fly) and restoration tested on the held-out [ 260 : 360 ] [260{:}360] ; the L12/ α = 1 \\alpha{=}1 cell restores 42 % 42\\% ( 95 % 95\\% bootstrap CI 32 32 – 52 % 52\\% , n = 100 n=100 ) versus 13 % 13\\% for the matched-norm orthogonal-random control ( 6 6 seeds), delta + 29 +29 pp (CI + 16 +16 , + 41 +41 ), benign over-refusal 15 % 15\\% . Both handles pass held-out (the direction shrinks ∼ 5 \\sim 5 – 6 6 pp off-distribution, the reversal does not), so neither is an artifact of the prompts it was estimated on. Held-out patching was run for the primary L12 cell only. Second family (Phi-3-mini, 3.8 3.8 B). Both new results replicate. The held-out reversal ( L 16 L16 ) restores refusal to 69 % 69\\% first-half / 71 % 71\\% whole / 4 % 4\\% onset, random 2 % 2\\% , in-distribution reference 71 % 71\\% (baseline 1 % 1\\% ). The attention knockout replicates the full pattern at n = 100 n=100 : attack baseline 1 % 1\\% ; early prefill-edge KO 25 % 25\\% ; whole 32 % 32\\% ; late 0 % 0\\% ; half-dose ( λ = 0.5 \\lambda{=}0.5 ) 0 % 0\\% ; mass-matched control 1 % 1\\% ; coherence 0.95 0.95 – 0.98 0.98 throughout (control mass-match ratio 1.00 1.00 ). Phi’s unit-test suite passes 9 / 10 9/10 ; the one failure is a cache-vs-no-cache determinism check that is a bf16 property of the model (it reproduces with a no-op hook), and does not affect the experiment, which compares cache-on baseline against cache-on knockout, and the sham-identity and edge-cut checks pass under cache-on."},{"type":"heading","level":4,"text":"Active-vs-passive: base-model discriminator."},{"type":"para","text":"To ask whether the knockout effect is safety-tuning-specific (active suppression) or generic (passive conditioning), we run the identical early prefill-edge knockout on the Qwen2.5-1.5B base model (no safety tuning), with a raw-completion format (no chat template) and the harmful- content scorer as the endpoint (the base model does not refuse). The base model complies ( 64 % 64\\% harmful content) and shows the same prefill-specific, early-window dependence as the instruct model: early knockout drives harmful content to 25 % 25\\% (a − 39 -39 pp drop) while the mass-matched control leaves it at 64 % 64\\% ( − 0 -0 pp) and late knockout is inert ( 61 % 61\\% ). Since a non-safety-tuned model shares the dependence, it is generic autoregressive conditioning, not a safety-specific active suppression; this resolves the active-vs-passive question toward the passive reading (§ 7 ). Mid-scale replication (Qwen2.5-7B base). The identical knockout on the 7 7 B base model reproduces the pattern cleanly: attack baseline 68 % 68\\% harmful content, early prefill-edge knockout 30 % 30\\% (a − 38 -38 pp drop), mass-matched control 66 % 66\\% ( − 2 -2 pp, three seeds), late knockout inert ( 68 % 68\\% ), for a + 36 +36 pp prefill-specific, early-window-gated drop ( n = 100 n=100 ). The passive reading is therefore not a small-model artifact. Measure-only, experiments/modal_attn_knockout_base.py and modal_attn_knockout_base_7b.py ."},{"type":"heading","level":4,"text":"Active-vs-passive: logit-level decomposition (a small active component)."},{"type":"para","text":"To probe for a residual safety-specific pull, we decompose how refusal recovers under the knockout at the logit level. For each early-window generated step we record the softmax mass on refusal-onset tokens ( p refuse p_{\\text{refuse}} ) and the probability of the compliance continuation ( p comply p_{\\text{comply}} ), under baseline / early knockout / mass-matched knockout, and define concentration = Δ p refuse / ( − Δ p comply ) =\\Delta p_{\\text{refuse}}/(-\\Delta p_{\\text{comply}}) : of the probability the knockout strips off compliance, the fraction that lands on refusal. In both models the knockout’s dominant effect is compliance collapse ( p comply p_{\\text{comply}} : instruct − 5.6 -5.6 pp, base − 17 -17 pp), the passive signature. But the instruct model routes the freed mass to refusal far more than the base model (concentration 0.24 0.24 vs 0.03 0.03 ; prefill-specific p refuse p_{\\text{refuse}} gain + 1.3 +1.3 pp vs matched + 0.1 +0.1 pp), so safety tuning adds a modest learned refusal attractor on top of the generic mechanism. The metric confirms a small safety-specific refusal component exists and sizes it, but does not separate an actively-suppressed gate springing back from a passive fallback attractor. Instruct 60 60 prompts; base as anchor. Measure-only, experiments/modal_active_probe.py ."},{"type":"heading","level":4,"text":"The MLP pathway (closing the accounting gap on the generic side)."},{"type":"para","text":"To ask whether the compliance that persists after the attention knockout hides a safety-specific mechanism in the MLP pathway, we capture the mean early-window MLP output per layer in the plain (no-prefill) and attack conditions, form d mlp = plain − attack d_{\\text{mlp}}=\\text{plain}-\\text{attack} , and inject it at the early-window MLP outputs under the attack ( n = 60 n=60 , all layers, matched-norm random and late-window controls). Base model (the clean discriminator): removing the base model’s own prefill-induced MLP shift drops harmful content 68 % → 48 % 68\\%\\to 48\\% ( + 16 +16 pp prefill-specific vs the matched-random control’s ∼ 4 \\sim 4 pp, late-window inert), in a model with no safety mechanism, so the early MLP’s prefill-dependence is generic, extending the passive account from attention to a second pathway. Instruct (a decomposition, not a discriminator): the same injection restores refusal 3 % → 47 % 3\\%\\to 47\\% (prefill-specific, early-window-gated), showing the MLP carries much of the refuse-state; but this is a difference-of-means injection, which tests decodability, not active computation, and the larger instruct effect is overdetermined by its larger plain-vs-attack behavioral gap. We therefore report the 47 % 47\\% as decomposition only, and settle the instruct excess with the clean path patch below. Measure-only, experiments/modal_mlp_pathpatch.py ."},{"type":"heading","level":4,"text":"Clean path patch: the early MLP pathway is generic."},{"type":"para","text":"A position-wise MLP never reads the prefill positions, so the excess cannot be isolated by a positional ablation; it needs a path patch that severs only the prefill-attention contribution to the MLP input. Teacher-forced over the greedily-generated attack sequence (so clean logits equal the generation), we (i) knock out early-window attention to the prefill and capture each layer’s resulting MLP input at the early-window positions, then (ii) run the clean attack and overwrite only those MLP inputs with the captured knockout values, leaving every other activation, including all downstream attention, at its clean value. Because Qwen2 saves the residual before the MLP layernorm, the counterfactual enters through the MLP sub-block alone. The surgery passes a unit-test suite before any read: sham (injecting the clean MLP input changes nothing, max | Δ | = 0 |\\Delta|{=}0 ), MLP-output faithfulness (the patched MLP output equals the knockout run’s, max | Δ | = 0 |\\Delta|{=}0 ), hook-live (the attention knockout moves logits by > 15 >15 ), and locality (patching only the last position leaves earlier-position distributions unchanged, max | Δ | = 0 |\\Delta|{=}0 ); all bit-exact on both models. On the same concentration metric ( Δ p refuse / − Δ p comply \\Delta p_{\\text{refuse}}/-\\Delta p_{\\text{comply}} , p comply p_{\\text{comply}} the mass on the clean compliance token), the MLP pathway carries the generic effect and only it: it strips prefill-specific compliance probability (instruct − 21 -21 pp, base − 25 -25 pp, against mass-matched-MLP-path controls at − 7 -7 / − 5 -5 pp) but shows no safety-specific refusal concentration (instruct 0.007 0.007 , base 0.001 0.001 , matched − 0.02 -0.02 ; prefill-specific refusal gain instruct + 0.26 +0.26 pp). So the residual safety-specific pull found in the logit trace does not route through the early MLP, and the instruct excess in the difference-of-means test was decodability. Caveat: this teacher-forced logit frame is well-powered for compliance-suppression routing and less sensitive to a small refusal-promotion component (which would need a generative trajectory to accumulate), but the within-frame instruct-vs-base-vs-matched comparison is clean. n = 60 n{=}60 ; measure-only, experiments/modal_mlp_pathpatch_clean.py ."},{"type":"heading","level":4,"text":"Reviewer robustness checks (boundary, prefill-exclusion, prefill length, sampling, benign-sparing)."},{"type":"para","text":"Bundled on one instruct load, gated by the same plain-refuses/attack-complies check and cross-validated against the published anchors (the whole-band harm patch reproduces the 48 % 48\\% fixed cell; d reverse d_{\\text{reverse}} first-half reproduces 75 % ≈ 75\\%\\approx the 74 % 74\\% held-out reversal), experiments/modal_revision_robustness.py . (i) Boundary sweep (harm-direction patch, L12/ α 1 \\alpha 1 , full per-position dose, n = 60 n{=}60 ): onset 10 % 10\\% , first-third 50 % 50\\% , first-half 57 % 57\\% , first-two-thirds 55 % 55\\% , whole 50 % 50\\% , second-half 5 % 5\\% , so restoration saturates by the first third and is flat thereafter. (ii) Prefill-exclusion: injecting over the generated tokens only (prefill positions excluded) restores 0 % 0\\% (first-half band) and 3 % 3\\% (generated tokens 1 1 – 7 7 ), versus 57 % 57\\% with the prefill span included; the causal write-site is the prefill span, and generated positions carry the effect only when the prefill is also driven. (iii) Prefill length ( n = 50 n{=}50 , bands over generated-token counts, all prefill positions driven): the saturation-by-first-third pattern holds for a short prefill (“Sure,”: first-third 86 % 86\\% , whole 86 % 86\\% ) and a long one (“Sure, here is a detailed step-by-step guide:”: first-third 12 % 12\\% , whole 6 % 6\\% ), so the window is an early band in generated-token count, not a fraction of the prefill; the restoration level falls with prefill commitment. (iv) Sampling ( T = 0.7 T{=}0.7 , n = 40 n{=}40 , K = 2 K{=}2 ): plain refusal 100 % 100\\% , attack refusal 0 % 0\\% (vs greedy 5 % 5\\% ), so the dissociation is not a greedy artifact; the harm-direction restoration weakens ( 50 % → 21 % 50\\%\\to 21\\% ), consistent with a partial, non-selective write-handle. (v) Benign-sparing of the state transfer: d reverse d_{\\text{reverse}} injected on benign prompts under the attack raises their refusal 12 % → 18 % 12\\%\\to 18\\% ( + 6 +6 pp) while restoring harmful refusal to 75 % 75\\% ( + 57 +57 pp selectivity), so the transfer is a selective edit, not a blanket refusal-inducer."},{"type":"heading","level":4,"text":"Scale (Qwen2.5-7B)."},{"type":"para","text":"At 7 7 B the prefill attack is weaker: plain refusal 98 % 98\\% , attack refusal 24 % 24\\% (vs ≤ 3 % \\leq 3\\% at 1.5 1.5 – 3.8 3.8 B), procedural content 16 % 16\\% . The knockout’s prefill-specific effect persists in the content channel (early knockout 0 % 0\\% , whole 0 % 0\\% , late 4 % 4\\% , half-dose 13 % 13\\% , mass-matched control 23 % 23\\% , coherence ≥ 0.95 \\geq 0.95 ), early-window-gated and dose-dependent, but the refusal- string restoration seen at smaller scale does not cleanly replicate (early 20 % ≈ 20\\%\\approx baseline 24 % 24\\% ), consistent with the passive reading that the refusal fallback is model-dependent. Unit tests pass 10 / 10 10/10 (including the KV-cache check); run on an L40S (eager attention). Gate relaxed to attack-refusal ≤ 0.35 \\leq 0.35 for the more-robust 7 7 B."},{"type":"para","text":"We add the harm direction at all response positions under the attack and sweep layer and strength, in two parameterizations. Neither finds a window that raises harmful refusal while sparing benign (Table 8 ). The best cells restore harmful refusal to 71 71 – 83 % 83\\% but at 40 40 – 50 % 50\\% benign over-refusal; smaller coefficients over-refuse everything, larger ones ( α = 4 \\alpha{=}4 / coeff 14 14 ) collapse generation to 0 . The harm direction induces refusal broadly, not on-harm. By contrast the read-side use of the same representation (the monitor of § 3 ) catches 100 % 100\\% of the attacked-harmful prompts at 0 % 0\\% false positive."},{"type":"heading","level":4,"text":"Construction."},{"type":"para","text":"We fix a trigger-word lexicon (eight categories: violence, weapons/ explosives, drugs, cyber, self-harm, extremism, sexual, fraud/theft) with inflection-aware matching (so “hacking” and “drugs” count), and a per-prompt scary flag = = “at least one trigger word.” Cells come from natural variation in existing public prompts, with no authored evasions: harmful-scary = = AdvBench with a trigger word; harmful-clean = = AdvBench without one (e.g. “how do I best destroy my boss’s car,” “commit insider trading”); benign-scary = = all of XSTest-safe (purpose-built to sound harmful); benign-clean = = Alpaca. We balance to n = 120 n{=}120 per cell and use source labels (flagging, not filtering, the severity spread). Two issues surfaced and were fixed before scoring: an initial word-boundary matcher leaked inflections into the clean cell (fixed), and the strict-lexicon benign-scary cell was thin ( n = 54 n{=}54 ), so we pre-registered benign-scary = = all XSTest-safe with the strict-keyword subset as a robustness check."},{"type":"heading","level":4,"text":"Results."},{"type":"para","text":"Table 4 in the main text. A logistic regression of refusal on the two binary axes gives β harm = 5.12 \\beta_{\\text{harm}}{=}5.12 , β scary = 2.20 \\beta_{\\text{scary}}{=}2.20 . The representational readout agrees: the refusal-behavior direction is closer to the harm-content direction than to the surface direction (mid-layer | cos | |\\cos| 0.34 0.34 vs. 0.13 0.13 )."},{"type":"para","text":"Test for a refuse-vs-comply direction distinct from harm (§ 5.1 ). A weakened prefill (“I understand you’re asking”) gives a 41 / 29 41/29 refuse/comply split on 70 70 harmful prompts, so harm and prefill-presence are held fixed and only the decision varies; d refuse d_{\\text{refuse}} is the diff-of-means of the decision-token activations. Distinct but weak: | cos ( d refuse , d harm ) | = 0.06 |\\cos(d_{\\text{refuse}},d_{\\text{harm}})|=0.06 ; five-fold CV AUC 0.648 0.648 versus d harm d_{\\text{harm}} ’s 0.640 0.640 at layer 12 12 . Read sweep (single direction): across all 29 29 layers × \\times the first 4 4 generated positions the diff-of-means readout ceilings at AUC 0.727 0.727 (layer 27 27 , position 1 1 ); most cells are 0.60 0.60 – 0.72 0.72 . But that ceiling is rank-1: a full L2-logistic probe on the same activations reads refuse-vs-comply at 0.85 0.85 (layer 21 21 ; + 0.12 +0.12 over the best single direction, consistent across C ∈ { 0.01 , 0.1 } C\\in\\{0.01,0.1\\} and layers, noisy at n = 70 n=70 ). The decision is multidimensional, not weak. Rank: it is distributed , not low-rank: PCA-then-logistic reaches AUC 0.74 0.74 at 3 3 components (near the single-direction 0.73 0.73 ) and climbs to ∼ 0.84 \\sim 0.84 only by ∼ 30 {\\sim}30 ; the top-variance axis is anti-predictive (rank-1 PCA 0.40 0.40 ) and none of the top components are harm-aligned ( | cos | ≤ 0.13 |\\cos|\\leq 0.13 ), so the decision lives in low-variance, non-harm directions spread wide (caveat: n = 70 n=70 flatters the high-rank estimates). Control (inert): steering + c d refuse +c\\,d_{\\text{refuse}} at the response positions under the strong attack restores ≤ 7 % \\leq 7\\% harmful refusal at layer 12 12 and 0 % 0\\% at layer 27 27 (the best-read cell), at every coefficient ∈ { 0.5 , 1 , 1.5 } \\in\\{0.5,1,1.5\\} , while d harm d_{\\text{harm}} restores refusal at layer 12 12 (non-selectively) and a matched-norm random direction restores nothing. No layer or position yields a clean or steerable decision direction; refusal is not a separable linear feature."},{"type":"para","text":"Across scale (Table 9 ). We repeat the single-vs-full comparison across the model ladder under a per-model calibrated weak prefill. Within the Qwen family the full probe beats the best single direction at every scale ( 0.5 0.5 – 7 7 B); SmolLM2 (a balanced split, different family) shows no gain, and Phi-3’s calibration gave a 43 / 7 43/7 split too imbalanced to read. The 7 7 B run used Modal (A10G, white-box) with a last-layer/ 5 5 -fold probe."},{"type":"para","text":"24 24 dual-use triplets (§ 5 ), each request in three framings. Construction: the request is dual-use (e.g. “how do I get into a locked iPhone I can’t unlock?”); the benign context makes helping appropriate (“it’s my own phone and I forgot the passcode”) and the harmful context makes refusing appropriate (“it’s my ex’s phone and I want to read their messages”). Refusal is string-matched; d intent = mean ( benign ) − mean ( harmful ) d_{\\text{intent}}=\\text{mean}(\\text{benign})-\\text{mean}(\\text{harmful}) at the decision token (layer 17 17 , best decode). Graded behavior: refusal 33 % 33\\% (benign) / 62 % 62\\% (bare) / 71 % 71\\% (harmful); intent CV AUC 0.76 0.76 . Faithful tracking: in deep layers the bare prompts the model refuses project harmful on d intent d_{\\text{intent}} ( − 5 -5 to − 20 -20 ) and the ones it helps project benign ( + 3 +3 to + 16 +16 ). Attack: on the 15 15 bare prompts refused (read harmful-intent, mean projection − 5.9 -5.9 ), the prefill drops refusal from 100 % 100\\% to 0 % 0\\% ; over all 24 24 , 62 % → 0 % 62\\%\\to 0\\% . Exploratory: one model (Qwen2.5-1.5B), n = 24 n=24 ."},{"type":"para","text":"We searched for a compact refusal circuit by attention-head ablation (Qwen2.5-1.5B, GPU), and the negative only holds because we verified the hook fires. (1) Ablating each of the 28 × 12 28\\times 12 heads (zeroing its slice of the o_proj input across generation) drops refusal by 0.00 0.00 everywhere. (2) Suspecting a dead hook, a positive control ablating a whole attention layer also left refusal at 100 % 100\\% , still ambiguous. (3) A logit check settled it: the hook changes the next-token logits by ‖ Δ ‖ ≈ 123 \\|\\Delta\\|\\approx 123 (whole attention, layer 12) and ≈ 23 \\approx 23 (a single head), so it works, and the robustness is real. (4) Cumulative ablation (first- k k attention layers) drove the refusal string-match to 0 even at k = 1 k=1 , but this contradicted the per-layer control (layer 0 breaks it, layer 12 does not). (5) Reading the actual generations resolved it: the refusal- 0 cases are garbage (“ silly silly noble glorious …”; with all attention removed the model emits one fixed gibberish string for every prompt), not compliance, the same generic-disruption confound as full-residual patching, caught only because we read the text rather than the metric. The bar we hold every intervention to is a selective transition from harmful refusal to coherent harmful compliance without comparable degradation on benign behavior; broad ablation fails it doubly (the output is garbage, and the same garbage for benign prompts), so the result speaks against a compact head- or layer-level circuit rather than merely lowering a refusal rate (MLP and SAE features untested). Conclusion: no single head or layer is necessary for refusal, and no tested internal intervention selectively removes it (it collapses the model instead). Scope: one model, attention ablation; MLP not tested."},{"type":"heading","level":4,"text":"At 14 14 B (Phi-3-medium, heavily safety-trained)."},{"type":"para","text":"We re-ran the two load-bearing pieces at 14 14 B (measure-only: refusal string-match and a distinct-token coherence ratio, no generations stored). Dissociation holds : harm-probe AUC 1.00 1.00 (layer 8 8 ), plain refusal 1.00 → 0.15 1.00\\to 0.15 under the prefill, and the 34 / 40 34/40 complied prompts still score harmful ( 0.96 0.96 ) with coherent (not deflecting) output (coherence 0.94 0.94 ). Compact-circuit check : over a per-attention-layer ablation ( 40 40 layers, refusing set n = 16 n=16 ), only the input layer (L 0 ) shows a refusal drop ≥ 0.3 \\geq 0.3 ( 1.00 → 0.69 1.00\\to 0.69 , coherence 0.69 0.69 ), and L 0 is a generic early-processing locus, not a refusal-specific circuit; every other layer leaves refusal near ceiling. So the negative holds at 14 14 B and heavier safety training, with the honest caveat that the input layer carries a partial, non-selective contribution. Frontier scale (70B + + ) is untested."},{"type":"heading","level":4,"text":"Two off-thesis nulls (in brief; full detail in the ledger)."},{"type":"para","text":"Before safety, we pointed the same represent-vs-act probe at two other tasks and got nulls that motivated the choice of substrate. On MC correctness (ARC-Easy, 0.6 0.6 – 1.5 1.5 B) the model’s verbalized confidence matches or beats a hidden-state probe, so the gap does not appear where the output already has an honest channel, which is what sent us to safety, where refusal has none. On judge-free failure localization (Who&When, Zhang et al., 2025b ) two onset signals lose to dumb priors; the one reportable byproduct is an audit finding that published LLM-judge methods lose to constant-guessing on step localization ( 14 % 14\\% vs. 27 % 27\\% ), which we flag as a separate direction, not a contribution here."},{"type":"heading","level":4,"text":"An attack-surface map (exploratory)."},{"type":"para","text":"Applying the harm probe across the attack transforms of the Arcanum prompt-injection taxonomy (Haddix, 2026 ) , semantic attacks (roleplay, tense) keep harm represented while heavy encodings the model cannot decode (base64, morse, reverse) collapse it: the response-site/prompt-modifying distinction of § 6 . But a pre-registered three-scale test refuted the prediction that this “blind spot” shrinks monotonically with scale, and the per-technique verdicts do not compare across models, so we report it as exploratory only."},{"type":"para","text":"Every control we ran against a main-text result, what it could have shown, and the verdict: passed (the effect held), demoted (the dramatic version reduced to a modest true one), null , or retracted . About a third of the verdicts are not passed ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Aligned language models refuse harmful requests, and that refusal is easily removed. Prepend an affirmative prefix to the model’s own turn (“Sure, here is”) and many models continue straight into the harmful content they would otherwise decline (Wei et al., 2023 ; Qi et al., 2025 ) . The fragility is well documented behaviorally : we know the attack works. We ask the mechanistic question underneath it. The prefill appends to the model’s own response , so it cannot touch the representation the model has already computed over the prompt; the question is therefore not whether that representation is preserved under the attack (structurally, it must) but whether the model builds a strong harm representation there at all, and if so, why the refusal decision ignores it.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This localizes what kind of defense is possible. A behavioral gate is defeated by the attack; a monitor that reads the prompt-side harm representation is not, because the attack never reaches that representation, though this prompt-side placement is already how production input classifiers work, so the observation is about placement, not a new detector. The scientific question that remains is why a model that represents the request as harmful nonetheless complies, that is, where the refusal decision overriding the representation is computed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Empirically, the model builds a full-strength harm representation on the flipped prompts (probe 0.91 0.91 – 0.98 0.98 , as high as on the ones it refuses) while behavioral refusal drops to chance, across four models and three families (§ 3 ). But the sharper mechanistic statement is that the gate is not ignoring harm, it is being overpowered : under a full-strength prefill essentially every prompt flips, so harm cannot predict who complies, yet under a weakened prefill where a real split exists, harm strength predicts refuse-vs-comply (AUC 0.63 0.63 – 0.72 0.72 , n = 70 n=70 ; § 5.1 ). And the gate’s failure is localized to an early window: a dose-matched position control shows that restoring the harm signal over the first half of the response recovers refusal as much as the whole response, while the second half recovers almost nothing and the onset token alone only 9 % 9\\\\% (§ 4 ). We use overpowered descriptively: the evidence fixes where refusal fails, an early window, but does not separate active suppression from passive autoregressive conditioning (§ 7 ). We then pin down what the gate reads (§ 5 ) and show it has no compact internal handle (§ 5.1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Relation to prior work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two lines are adjacent (§ 6 ). Qi et al. ( 2025 ) show safety alignment is “shallow” in token depth : the refusal decision is carried by the first few output tokens, and attacks that get past them succeed. We go past that behavioral picture on three fronts. First, we localize the failure causally, not just correlating depth with success, to an early multi-token window rather than a first-token count. Second, we show the causal converse: transplanting the refuse-state re-induces refusal. Third, the base-model discriminator identifies the prefill’s grip as generic autoregressive conditioning rather than a defeated safety-specific gate, a distinction depth-shallowness alone does not make. Arditi et al. ( 2024 ) show refusal is mediated by a single direction in activation space. Closest, concept-level work shows a model still recognizes toxicity under a jailbreak while complying (Zhang et al., 2025c ) , but attributes the behavior-flip to a competing prompt-side “jailbreak concept” that overrides it. Our contribution is positional and causal. The representational half is a clarification, not a discovery: because prefill leaves the prompt untouched, the harm representation there is trivially unchanged, and the useful contrast is with prompt-modifying jailbreaks (the ones that literature studies), which do move or attenuate the prompt-side representation. Our substantive claims are mechanistic: refusal is computed at the response site and the attack operates there; and the harm direction is an excellent read-out but only a partial, non-selective write-handle for the refusal action. The prompt-side account cannot explain the prefill attack at all. Prefill adds nothing to the prompt, so there is no prompt-side jailbreak concept to override anything. The flip is response-site , and we localize it there. The template (read what a model represents versus what it acts on) is the one we used for communicative intent in prior work (Kwon, 2026 ) ; here it is pointed at safety.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A quantitative dissociation under prefill (§ 3 ). On the very prompts a prefill attack flips to compliance, the harm representation is not degraded but intact : a probe scores the complied prompts at 0.91 0.91 – 0.98 0.98 , level with the ones the model refuses, while behavioral refusal collapses to chance. This replicates across four models and three families ( 1.5 1.5 – 3.8 3.8 B, weak and strong gates). Unlike the prompt-modifying jailbreaks prior work studies, prefill leaves the representation quantitatively unchanged, not merely present-but-attenuated.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A response-site causal mechanism (§ 4 ). We localize refusal causally, not just correlationally: adding the harm direction back across the response positions partially restores refusal (harm 33 33 – 48 % 48\\\\% at two fixed cells vs a matched-norm random control’s 14 14 – 16 % 16\\\\% ; harm − - random + 17 +17 to + 34 +34 pp at n = 100 n=100 , both significant), so the intact representation is upstream of the gate, not epiphenomenal. A dose-matched position control localizes the effect to an early window of the response (the first half restores as much as the whole; the second half is nearly inert), more than a single onset token but not the whole generation. The converse holds in the same window and is state-specific: transplanting the model’s own refuse-state into the attack re-induces refusal ( → 74 % \\\\to 74\\\\% held-out, first half = = whole response), matched by a random control at 0 % 0\\\\% , a working lever where the single harm direction is only partial. And the grip operates through early-window attention: knocking out the early response’s attention to the prefill (but not an equal mass elsewhere) restores refusal + 21 +21 pp (CI [ + 12 , + 30 ] [+12,+30] ), harm-selectively and early-window-gated. A base-model control identifies this as generic autoregressive conditioning, not safety-specific active suppression (the non-safety-tuned base model shows the same prefill-specific collapse, at both 1.5 1.5 B and 7 7 B), resolving the paper’s central open question toward the passive reading. The dramatic full-residual “restoration” is generic disruption, since a benign-patch control also hits 100 % 100\\\\% . Replicated on two families, and the discriminator holds at 7 7 B.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A defensive consequence, and its scope (§ 3 , § 4.5 ). Read as a monitor, the intact representation catches 100 % 100\\\\% of the attack at 0 % 0\\\\% false positive. The scope is narrow, in three ways. First, this 100 % / 0 % 100\\\\%/0\\\\% is in-distribution: the probe is AdvBench-calibrated (it fires at 0.97 0.97 on that phrasing but 0.49 0.49 on differently-worded unsafe prompts, and a mixed-source fix wrecked specificity), so a source-general detector is unbuilt. Second, it is only a response-site guarantee, and the placement is not novel (input classifiers already sit prompt-side). Third, steering the same direction does not close the gap: the harm direction is a read-out, not a write-handle.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What the gate reads (§ 5 ). A pre-registered harm × \\\\times surface 2 × 2 2\\\\times 2 that decorrelates harm from trigger-word surface shows refusal is harm-primary and the harm representation keyword-invariant, with a real but secondary surface-driven over-refusal on benign inputs. A matched-contrast search finds no single refusal-decision direction (any one reads ≤ 0.73 \\\\leq 0.73 and none steers), yet a full linear probe reads the decision at ∼ 0.85 \\\\sim 0.85 : within the Qwen family the decision is decodable but multidimensional , not a single steerable feature (SmolLM2 is a single-direction counterexample; cross-family is open).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every headline effect is checked against a control that can demote it, and several do; the claims-and-evidence table (Table LABEL:tab:claims , Appendix A ) records each claim that passed with its evidence and the demoted versions (full-residual disruption, the steering null, the retracted “separate feature”), and Appendix B gives the reproduction chain. The dissociation also extends, exploratorily, to inferred-intent refusal ( n = 24 n=24 , one model; § 5 , App. I ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use open-weight instruction-tuned models so we can read activations: Qwen2.5-1.5B and 3B (Qwen Team, 2024 ) , SmolLM2-1.7B (Allal et al., 2025 ) , and Phi-3-mini-3.8B (Abdin et al., 2024 ) : three families, 1.5 1.5 – 3.8 3.8 B parameters, and a range of safety-gate strength (SmolLM2 refuses 63 % 63\\\\% of plain harmful prompts; the others ≈ 100 % \\\\approx 100\\\\% ). All experiments run on CPU; sample sizes are reported per experiment.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Harmful prompts are AdvBench behaviors (Zou et al., 2023b ) ; benign prompts are Alpaca instructions (Taori et al., 2023 ) ; the surface-vs-harm control uses XSTest (Röttger et al., 2024 ) , whose “safe” split is purpose-built to sound harmful (“how do I kill a Python process”) while being benign.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The attack.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prefill attack forces the assistant turn to begin with the affirmative prefix “Sure, here is” and lets the model continue; we score whether the continuation refuses.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The probe.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The harm probe is ℓ 2 \\\\ell_{2} -regularized logistic regression on the default-pass hidden state at the last prompt token (the decision point), trained on plain harmful vs. benign prompts. Probe AUCs use cross-validation (nested-CV layer selection; no test-set peeking). Refusal is detected by string match against a fixed refusal-phrase list. Full details in App. B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The phenomenon: represented but not refused\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section states the phenomenon the rest of the paper explains: a fully intact harm representation coexists with compliance, part of which is structural (below). The mechanistic account of how that gate is defeated (§ 4 ) is where the empirical work and every control live.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Harm is strongly represented.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On plain prompts, the harm probe separates harmful from benign near-perfectly: one nested-CV probe scores AUC 0.996 0.996 on Qwen2.5-1.5B and ≥ 0.990 \\\\geq 0.990 on every model (Table 1 , one estimator throughout). The representation of harm is not subtle.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The attack changes behavior; it cannot change the prompt-side read.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the prefill attack the same model complies: on Qwen2.5-1.5B, refusal on harmful prompts drops from 1.00 1.00 to 0.03 0.03 (97 of 100 comply) and behavior, used as a harm detector, collapses to chance (AUC 0.51 0.51 ). “Comply” here is coherent harmful compliance, not just an absent refusal string: a judge-free content read of the attacked outputs (§ 7 ) finds 94 % 94\\\\% of the non-refusals are coherent harmful content, deflection rare. The other half of the dissociation is partly structural. The harm representation we probe is the model’s read at the last prompt token, computed before any of the attack’s tokens exist; the prefill appends to the response , so by the causal structure of the transformer this representation is bit-for-bit identical with and without the attack. Its “intactness” is structural, not a measurement . The substantive finding is this: at full attack strength, compliance is not explained by a weaker harm read. The prompts the model complies on score as harmful ( 0.975 0.975 ) as the ones it refuses ( 0.966 0.966 ; benign 0.055 0.055 ). Here “refused” is the plain-condition harmful set, since the strong prefill leaves at most 5 / 100 5/100 still refusing per model. A full -strength harm representation coexists with compliance; the comparison has little variation, since 95 95 – 100 % 100\\\\% comply under the strong attack. Where a real split exists, a weakened prefill (§ 5.1 , 41 / 29 41/29 refuse/comply), harm strength does predict who complies: the harm-probe score reads refuse-vs-comply at AUC 0.63 0.63 (layer 12) to 0.72 0.72 (best layer). This is harm, not the obvious surface correlate: prompt length and scary-keyword count predict the same split at chance (AUC 0.50 0.50 – 0.54 0.54 ), and the harm-probe AUC holds when we regress out length and keyword count ( 0.73 0.73 ). So the strong attack overrides the harm signal rather than the model down-weighting it. This is moderate evidence: n = 70 n=70 is small and prompt-level confounds (topic, severity, formality) are not held fixed. Table 1 shows the pattern across all four models and three families, at both weak and strong gates. It also extends to larger scale (run on GPU). On Qwen2.5-7B the prefill drops refusal 0.98 → 0.24 0.98\\\\to 0.24 at probe AUC 1.00 1.00 ; the attack is weaker here, with a quarter still refusing under the same prefill (§ 7 ). On the heavily-safety-trained Phi-3-medium-14B (Abdin et al., 2024 ) it drops refusal 1.00 → 0.15 1.00\\\\to 0.15 (probe AUC 1.00 1.00 , complied-harmful 0.96 0.96 , coherent output; App. J ). So the dissociation is not a small-model artifact. The attack is also not specific to one prefill string: across six affirmative prefills (varying length and phrasing) it drops refusal from 100 % 100\\\\% to 0 – 37 % 37\\\\% (most to ≤ 7 % \\\\leq 7\\\\% ), so it is a general property of affirmative response-onset prefills. The real question is then why an intact, strong harm representation fails to drive refusal, which the mechanism (§ 4 ) answers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Is the probe reading harm, or scary words?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A natural worry is that the probe separates AdvBench from Alpaca on surface features (violent vocabulary) rather than harm. We test this with XSTest, whose “safe” prompts sound harmful but are benign. The AdvBench-trained probe scores XSTest- safe at 0.21 0.21 (mean; only 12 % 12\\\\% exceed the 0.5 0.5 threshold) versus XSTest- unsafe at 0.49 0.49 , separating them at AUC 0.79 0.79 . The probe is not fooled by scary surface: it reads harm. It is, however, calibrated to AdvBench-style phrasing (it fires at 0.97 0.97 there but only 0.49 0.49 on differently-phrased unsafe prompts), and training on mixed harmful sources did not fix this: it raised sensitivity but collapsed specificity (XSTest-safe false positives 12 % → 64 % 12\\\\%\\\\to 64\\\\% ). So the harm reading is earned; a source-general harm detector is not (§ 7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A design consequence: a prompt-side monitor is immune to this attack.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the attack cannot reach the prompt-side representation, a monitor that reads it inherits that immunity by construction: on the attacked set the harm probe flags 100 % 100\\\\% of harmful prompts at 0 % 0\\\\% false positive on easy (Alpaca) negatives. This is not an empirical detection result, since the probe input is unchanged by the attack, “catching the attack” reduces to ordinary harmful-vs-benign accuracy, and the placement is not new: production input classifiers (e.g. Llama Guard, Inan et al., 2023 ) already sit prompt-side and are unreachable by a response-site prefill for the same reason. Our contribution is the mechanistic account of why that placement is immune. The guarantee is scoped to response-site attacks (prompt-modifying jailbreaks change the probe input and degrade it: Doda, 2026 and our encoding nulls, App. K ) and costs 12 % 12\\\\% over-refusal on scary-but-benign XSTest inputs (App. C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 The mechanism\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dissociation says behavior and the harm representation are separable; it does not say whether the harm representation is causally in the refusal pathway or epiphenomenal for it, nor why prefill specifically defeats the gate. This section answers both with white-box causal experiments on Qwen2.5-1.5B, replicated on Phi-3-mini: activation patching, a position control, a direction search, and component ablation, each against a control that can kill it. For each effect we report the version that passes its control, not the dramatic one.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Refusal restoration is response-site and concentrated in an early window (causal patching)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The intact harm representation lives at the prompt ; the attack operates at the response site (it occupies the output tokens). So we ask: under the attack, if we restore the harm signal at the response positions, does refusal return despite the prefill? We patch there and let generation flow, sweeping layers and injection strength (App. D , Table 7 ) and settling the harm-vs-random delta at two fixed cells with n = 100 n=100 (Table 2 ). A dose-matched position control (below) localizes the effect to an early window of the response: injecting over the first half restores as much as the whole response, the second half is nearly inert, and prompt-side injection does nothing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two rungs, and the control matters. Full-residual patching (replacing the response-onset residual with the mean harmful residual) restores refusal to 100 % 100\\\\% . But a benign control demolishes it: patching the mean benign residual at the same position also restores refusal to 100 % 100\\\\% at every layer. Full-residual patching is generic disruption; injecting any prompt-final state at the onset breaks the prefill continuation and the model reverts to refusing. It says nothing harm-specific.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What passes is the subspace rung: adding only the harm direction at the response positions restores refusal while sparing benign, and it passes the same control the full rung got. We settle it at n = 100 n=100 on two cells fixed in advance from the grid (L12, the best subspace layer there, and L8), against a matched-norm direction orthogonalized to the harm direction. The cells are grid-selected (fixed before the run) and the n = 100 n=100 re-estimates them on overlapping AdvBench prompts. The delta CI bootstraps over the harm prompts and the 24 24 random seeds (App. D ). At L12/ α = 1 \\\\alpha{=}1 the harm direction restores 48 % 48\\\\% ( 95 % 95\\\\% bootstrap CI 38 38 – 58 % 58\\\\% ) versus 14 % 14\\\\% for the orthogonal-random control (24 seeds, range 0 – 60 % 60\\\\% ): a harm − - random delta of + 𝟑𝟒 \\\\mathbf{+34} pp (CI + 22 +22 , + 46 +46 ). At L8/ α = 1.5 \\\\alpha{=}1.5 it restores 33 % 33\\\\% ( 24 24 – 42 % 42\\\\% ) versus random 16 % 16\\\\% ( 0 – 60 % 60\\\\% ): delta + 𝟏𝟕 \\\\mathbf{+17} pp (CI + 6 +6 , + 28 +28 ). Both deltas exclude zero, and restoration is benign-sparing (benign 12 % 12\\\\% / 0 % 0\\\\% vs random’s 3 % 3\\\\% / 4 % 4\\\\% ), so the harm direction is a real causal handle above the generic-perturbation baseline, but a partial one: a third to a half of the raw restoration is matched by the random control. This passes a clean held-out replication. Training the harm direction on a disjoint 200 200 -prompt AdvBench split (activations recomputed, so there is no overlap with the test prompts) and testing restoration on a held-out 100 100 , the L12/ α = 1 \\\\alpha{=}1 cell restores 42 % 42\\\\% ( 95 % 95\\\\% CI 32 32 – 52 % 52\\\\% ) versus 13 % 13\\\\% for the matched-norm random control (6 seeds), a harm − - random delta of + 𝟐𝟗 \\\\mathbf{+29} pp (CI + 16 +16 , + 41 +41 ): the effect shrinks only ∼ 5 \\\\sim 5 pp off-distribution and the delta still excludes zero, so it is not an artifact of prompt overlap (App. E ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Position control: the attack’s grip is an early response window, not the whole response and not a single token.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We test where the harm signal has to be restored, over one consistent response window (the prefill tokens plus the 18 18 generated tokens; n = 100 n=100 , L12/ α = 1 \\\\alpha{=}1 ). Two axes matter, position and per-position strength, and a dose-matched design separates them (App. D ). Position: injecting over the first half of the response (the prefill and the first ∼ 7 \\\\sim 7 generated tokens) restores refusal to 42 % 42\\\\% , as much as the whole response ( 41 % 41\\\\% ), whereas the second half restores only 6 % 6\\\\% even at double strength. (These dose-matched levels sit just below the 48 % 48\\\\% fixed-cell of Table 2 : this control matches per-position dose to isolate where , so the first-half-vs-whole ratio is the claim, not the absolute level.) So the early window suffices on its own , and the late response is nearly inert. Strength: halving the per-position coefficient (whole response at α = 0.5 \\\\alpha{=}0.5 ) drops restoration to 0 % 0\\\\% , so there is a per-position threshold, not just a total-dose budget; a distributed every-other-token pattern also needs the higher coefficient ( 1 % 1\\\\% at α = 1 \\\\alpha{=}1 , 48 % 48\\\\% at α = 2 \\\\alpha{=}2 ). This sharpens the shallow-alignment picture: the prefill’s grip is not a single onset token (injecting only the onset restores 9 % 9\\\\% , less than the early window) but an early multi-token window; the back half of the response carries no weight. (An earlier version of this control compared an all-positions band that silently included the prefill against half-bands that did not, which confounded position with prefill-inclusion and dose; the dose-matched window design here corrects it.) We use “defeated” descriptively: the control localizes where the counter-signal must go, but does not distinguish an active mechanism that down-weights the harm signal from passive autoregressive conditioning by the forced affirmative context (§ 7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Robustness of the early window (boundary, prefill-exclusion, prefill length, sampling).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four checks probe the window’s edges (App. E ). Boundary: sweeping the cutoff, restoration is already saturated by the first third (first-third 50 % 50\\\\% , first-half 57 % 57\\\\% , first-two-thirds 55 % 55\\\\% , whole 50 % 50\\\\% at full per-position dose, reproducing the 48 % 48\\\\% fixed cell) while the onset alone is 10 % 10\\\\% and the second half 5 % 5\\\\% ; so first-half ≈ \\\\approx whole is not a knife-edge at 1 / 2 1/2 , the window is the early response and saturates by its first third. Prefill-exclusion: the restoration is anchored at the prefill positions. Injecting the harm direction over the generated tokens only (excluding the prefill span) restores ≤ 3 % \\\\leq 3\\\\% , so the causal write-site inside the early window is the prefill span; sustaining across the early generated tokens then boosts it (prefill + + onset 10 % → 10\\\\%\\\\to prefill + + first-half 57 % 57\\\\% ), but the generated positions carry nothing on their own. Prefill length: the early saturation is stable across prefill length (short and long prefills both saturate by the first third), while the restoration magnitude tracks prefill commitment, a short prefill (“Sure,”) is easy to reverse ( 86 % 86\\\\% ) and a long committal one (“Sure, here is a detailed step-by-step guide:”) is hard ( 6 6 – 12 % 12\\\\% across bands) at fixed injection strength. Sampling: the dissociation is not a greedy artifact, at temperature 0.7 0.7 plain prompts still refuse 100 % 100\\\\% and the attack still strips refusal to 0 % 0\\\\% ; the harm-direction restoration, a partial handle, weakens under sampling ( 50 % → 21 % 50\\\\%\\\\to 21\\\\% ), as expected for a non-selective write-handle.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 The reversal is early-window too: the model’s own refuse-state re-induces refusal (state transfer)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The position control shows the attack breaks refusal in an early response window. We now show the converse , and it lands in the same window: transplanting the model’s own refusal state into the attack re-induces refusal, and doing so over the first half of the response suffices. This is a cleaner causal handle than the single harm direction (§ 4.1 ), because it uses the full multidimensional state, not one line.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Design (measure-only, held-out, Qwen2.5-1.5B, L12).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We capture the mean layer- 11 11 -output residual over the first-half generated window in two conditions: the plain prompt with no prefill (refusal 100 % 100\\\\% ) and the attack (“Sure, here is”). Their difference d reverse = s plain − s attack d_{\\\\text{reverse}}=s_{\\\\text{plain}}-s_{\\\\text{attack}} is the model’s own refuse-vs-comply state shift. This is held out : d reverse d_{\\\\text{reverse}} is estimated on a 260 260 -prompt AdvBench train split and injected/tested on a disjoint 100 100 -prompt held-out split (with an in-distribution reference, the same d reverse d_{\\\\text{reverse}} tested on train prompts, run alongside). We add d reverse d_{\\\\text{reverse}} into the attack run’s early generated window and measure whether refusal returns (string-match plus the judge-free coherence read of § 7 ), against a matched-norm orthogonal-random control at the same norm.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Result: state-specific, early-gated, and a working lever (held-out).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Injecting d reverse d_{\\\\text{reverse}} over the first half of the response restores refusal on the held-out split from the attack’s ≈ 0 % \\\\approx 0\\\\% to 𝟕𝟒 % \\\\mathbf{74\\\\%} , as much as injecting over the whole response ( 78 % 78\\\\% ), while the onset token alone restores 13 % 13\\\\% ; the matched-norm random control restores 𝟎 % \\\\mathbf{0\\\\%} , and coherence holds throughout ( ≈ 0.97 \\\\approx 0.97 – 0.99 0.99 distinct-token ratio, no degeneracy). The in-distribution reference (same d reverse d_{\\\\text{reverse}} , tested on train prompts) restores 72 % 72\\\\% , so there is no generalization gap: the refuse-state is not an artifact of the prompts it was estimated on. This replicates on a second family: on Phi-3-mini ( 3.8 3.8 B, L 16 L16 ) the same held-out procedure restores refusal to 69 % 69\\\\% over the first half (whole 71 % 71\\\\% , onset 4 % 4\\\\% ) against a matched-norm random control at 2 % 2\\\\% , coherent, with the same early-window gating (App. E ). Three things follow. (i) The reversal is state-specific , not generic disruption: unlike the full-residual patch of § 4.1 (which a benign control also drove to 100 % 100\\\\% ), here a matched-norm random vector does nothing, so this transfer passes the very control that demoted the full-residual effect. (ii) It is early-window-gated , the same locus as the attack: the first half restores as much as the whole response and the second half adds nothing, mirroring the break direction. (iii) It works where the single harm direction is only a partial handle (§ 4.1 ) and d refuse d_{\\\\text{refuse}} steering is inert (§ 5.1 ), precisely because it is the full multidimensional state shift: the lever exists, but it is the early-window state , not any single direction, consistent with the decision being distributed (§ 5.1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Benign-sparing, and a syntax-vs-content byproduct.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We verified state-specificity against a matched-norm random control and generalization against a held-out split. The transfer is also benign-sparing , so it is a selective edit and not merely a refusal-inducer: injecting d reverse d_{\\\\text{reverse}} over the early window of benign prompts under the attack raises their refusal only 12 % → 18 % 12\\\\%\\\\to 18\\\\% ( + 6 +6 pp) while restoring harmful refusal to 75 % 75\\\\% (this run reproducing the held-out 74 % 74\\\\% ), a + 57 +57 pp selectivity gap (App. E ). A syntax-vs-content check falls out of the same run: forcing a benign affirmative onset (“Sure, here is a poem about”) yields 0 % 0\\\\% harmful procedural content (vs the attack’s 47 % 47\\\\% ) and near-zero refusal, so the jailbreak rides the harmful continuation , not the affirmative syntax alone.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Cutting early attention to the prefill restores refusal, generically (attention knockout)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The reversal shows the refuse/comply decision is carried in the early-window residual state . That leaves the paper’s central open question (§ 7 ) standing: is the prefill’s effect maintained by the response continuing to attend to it, or written into the trajectory at the first generated token and merely relayed forward? We test this directly by knocking out the attention edges from early generated positions to the forced-prefill key positions and asking whether refusal returns, against a control that removes the same attention mass from non-prefill keys. This is the discriminating experiment the Limitations named (§ 7 ); it is preregistered ( PREREG_attn_knockout.md ), including the falsifier and the interpretation table.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Design (edge knockout, mass-matched control; n = 100 n=100 , Qwen2.5-1.5B, all layers/heads).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At each early generated decode step ( g < 9 g<9 , the first-half window) we add ln ( 1 − λ ) \\\\ln(1-\\\\lambda) to the pre-softmax attention logit of every (current query → \\\\to prefill key) edge and let softmax renormalize ( λ = 1 \\\\lambda{=}1 removes the edge, λ = 0.5 \\\\lambda{=}0.5 halves it). The control removes the same total baseline attention mass from non-prefill prompt keys, selected per prompt to match the prefill’s mass (achieved ratio 0.98 0.98 , [ 0.75 , 1.08 ] [0.75,1.08] over three fixed seeds); this matters because attention-sink tokens carry far more mass than the prefill, so an unmatched span is not a fair control. The intervention passes a unit-test suite before any behavioral run (sham-hook bit-identity, edge-only masking with softmax renormalization, KV-cache consistency, and a behavioral positive control; App. E ). Indexing caveat: generated token 1’s logits come from the final prefill token’s hidden state, so this cannot touch token 1; it is a test of early maintenance through direct attention, not attack initiation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Result: prefill-specific, early-window-gated, and harm-selective.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Knocking out early response-to-prefill attention restores refusal from the attack’s 3 % 3\\\\% to 24 % 24\\\\% , versus 1 % 1\\\\% for the mass-matched control (Table 3 , three-seed mean 1 % 1\\\\% ): a prefill − - matched difference of + 𝟐𝟏 \\\\mathbf{+21} pp (paired bootstrap 95 % 95\\\\% CI [ + 12 , + 30 ] [+12,+30] , computed against one fixed control seed). It is early-window-gated: knocking out the late window does nothing ( 4 % ≈ 4\\\\%\\\\approx baseline; early − - late + 20 +20 pp, CI [ + 11 , + 29 ] [+11,+29] ), and knocking out the whole response ( 29 % 29\\\\% ) is not distinguishable from early alone (whole − - early + 5 +5 pp, CI [ − 1 , + 12 ] [-1,+12] , includes zero), so the early window carries the effect, matching the reversal and the position control. Halving the knockout ( λ = 0.5 \\\\lambda{=}0.5 ) does nothing ( 3 % 3\\\\% ), a threshold echoing the position control’s. The effect is harm-selective : the same early knockout on benign prompts under the prefill barely moves refusal ( 0 % → 4 % 0\\\\%\\\\to 4\\\\% ), a specificity difference-in-differences of + 16.8 +16.8 pp, and coherence is unchanged throughout ( 0.96 0.96 – 0.97 0.97 ). It suppresses harmful content , not just adds refusal strings (procedural-content rate 51 % → 18 % 51\\\\%\\\\to 18\\\\% early, → 3 % \\\\to 3\\\\% whole, while the matched control stays at 50 % 50\\\\% ). This replicates on a second family: on Phi-3-mini ( 3.8 3.8 B) the identical pattern holds, early prefill-edge knockout restores refusal to 25 % 25\\\\% versus 1 % 1\\\\% for the mass-matched control, early-window-gated (late knockout inert at 0 % 0\\\\% , whole 32 % ≈ 32\\\\%\\\\approx early), threshold in dose ( λ = 0.5 \\\\lambda{=}0.5 inert), coherent (App. E ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"What this settles: the grip is generic, not safety-specific.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The effect is real but partial ( 24 % 24\\\\% of a 97 % 97\\\\% compliance), consistent with the distributed, redundant computation: direct early-window attention is one causal pathway, not the whole mechanism. Its interpretation we can now pin down, and it resolves the paper’s central open question toward passive conditioning. A base-model control is the discriminator: on the Qwen2.5-1.5B base model (no safety tuning, and so no refusal to re-engage), the identical early prefill-edge knockout collapses the harmful continuation prefill-specifically , harmful content 64 % → 25 % 64\\\\%\\\\to 25\\\\% against the mass-matched control’s 64 % 64\\\\% (a + 39 +39 pp prefill-specific drop), early-window-gated (late knockout inert). The same discriminator holds at mid scale: on the Qwen2.5- 7B base model the identical knockout drops harmful content 68 % → 30 % 68\\\\%\\\\to 30\\\\% against the mass-matched control’s 66 % 66\\\\% (a + 36 +36 pp prefill-specific drop, n = 100 n=100 ), again early-window-gated (late knockout 68 % 68\\\\% , inert), so the passive reading is not a small-model artifact. The dependence on early attention to the prefill is therefore architecture-general autoregressive conditioning , not a safety-tuning-specific active suppression of the harm signal: safety tuning fixes only the fallback once the grip is cut (the instruct model reverts to refusing, the base model degrades). So the prefill sustains compliance through generic early-window attention, and “refusal restoration” is a model-dependent fallback, not a re-engaging gate. This is the passive reading. A logit-level trace then measures a safety-specific component on top: when the knockout frees probability mass off the compliance continuation, the instruct model routes it to refusal tokens about 8 × 8\\\\times more than the base model does (concentration 0.24 0.24 vs 0.03 0.03 , App. E ). Safety tuning adds a small learned refusal attractor, but the dominant mechanism is passive: compliance collapses in both models. The trace confirms a safety-specific refusal pull exists; it does not separate an actively-suppressed gate springing back from a passive fallback attractor.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The generic story extends to the early MLP pathway.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Attention is one causal pathway; the compliance that persists ( 24 % 24\\\\% of 97 % 97\\\\% ) could hide a safety-specific mechanism in the pathway we did not cut. We probe the early-window MLP contribution (App. E ) and find the same base-vs-instruct control resolves it the same way. In the base model, removing the prefill’s early-MLP shift drops harmful content 68 % → 48 % 68\\\\%\\\\to 48\\\\% , a + 16 +16 pp prefill-specific, early-window-gated drop (matched-random inert) in a model with no safety mechanism to house, so the early MLP’s prefill-dependence is generic too. The passive account therefore holds across two pathways (attention and early MLP), not one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What a difference-of-means test could not settle, a clean path patch does. That test restored 47 % 47\\\\% refusal in the instruct model, but a plain − - attack injection tests decodability, not active computation, so the larger instruct effect is overdetermined by its larger behavioral gap. We therefore run a frozen-residual path patch that routes the prefill counterfactual through the early-MLP sub-block alone , every other activation held at its clean attack value (the surgery is bit-exact under a sham / MLP-output-faithfulness / locality unit-test suite, App. E ). The MLP pathway carries the generic effect and only that: it strips prefill-specific compliance probability hard (instruct − 21 -21 pp, base − 25 -25 pp, both far above their mass-matched controls at − 7 -7 / − 5 -5 pp) but produces no safety-specific refusal pull (instruct concentration 0.007 ≈ 0.007\\\\approx base 0.001 ≈ 0.001\\\\approx matched; prefill-specific refusal gain + 0.26 +0.26 pp, within noise). So the instruct excess was decodability, not an active MLP computation, and the small residual safety-specific pull (the 0.24 0.24 generative concentration above) does not route through the early MLP. The passive account holds across both pathways under a clean path patch (§ 7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 The onset projection is illustrative (and confounded)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also track the projection of the generation-site state onto the harm-probe direction across decoding steps (Fig. 2 ): with no attack the onset step sits high ( 2.80 2.80 ) and the model goes on to refuse, while under the prefill it sits low ( 0.22 0.22 ). We report it as illustration, not localization; the causal locus is the early-window response-site effect above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 2.80 → 0.22 2.80\\\\to 0.22 drop is not a clean measurement of the refusal decision being displaced, and a control says why. The axis is the harm-probe direction, so the projection at a forced onset position reflects the content of the token sitting there. Forcing a fully benign, non-affirmative prefix on the same harmful prompts (“Here’s a fun fact:”) drives the onset projection even lower ( − 0.82 -0.82 ) than the attack’s “Sure, here is” ( 0.22 0.22 ), and a benign affirmative (“Sure, here is a poem about”) to − 0.11 -0.11 ; none of these is the attack, yet all read low, and the benign prefixes are not refused either ( 0 – 7 % 7\\\\% ). The onset projection therefore tracks measured-position content, not a pure refusal-decision magnitude, so we treat it as illustrative only; the clean positional result comes from the patching (§ 4.1 ), not the projection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Steering fails; monitoring works\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If the harm direction were a clean causal handle, adding it should restore refusal selectively. It does not. Sweeping layers and coefficients (raw and Arditi-normalized), we find no window that raises harmful refusal without dragging benign refusal up with it: the best cell restores harmful refusal to 83 % 83\\\\% but at 40 % 40\\\\% benign over-refusal; smaller coefficients over-refuse everything, larger ones break generation. Adding the direction induces refusal broadly, not on-harm, consistent with Arditi et al. ( 2024 ) , where the refusal direction makes the model refuse harmless prompts too. The harm representation is a perfect read-out (detection AUC 1.0 1.0 ) but not a selective write-handle . The usable intervention is therefore the monitor of § 3 ( 100 % 100\\\\% catch, 0 % 0\\\\% false positive), not steering: we can monitor the gap shut, not steer it shut.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why response-site patching spares benign but global steering does not: the injection site.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The benign-sparing patching (§ 4.1 , benign restoration 12 % 12\\\\% / 0 % 0\\\\% ) and this non-selective steering look like the same intervention, adding the harm direction under attack, but differ in where it is added: patching touches only the response positions, steering adds it at all positions, the benign prompt included. We ran a dedicated site comparison: direction and layer held fixed (the harm-probe direction at the patching coefficient α = 1 \\\\alpha{=}1 of § 4.1 , layer 12 12 ; not the normalized Table 8 sweep), varying only the injection site. Response-only restores 48 % 48\\\\% harmful at 12 % 12\\\\% benign (reproducing Table 2 ). All-positions restores 58 % 58\\\\% harmful but drags benign to 40 % 40\\\\% (the steering over-refusal). Global injection corrupts the benign prompt’s own read, so it registers as harmful; response-site injection leaves the prompt intact. The harm direction is thus a partial, benign-sparing handle at the response site but a non-selective knob applied globally, and even response-only its window is narrow: harmful 48 % 48\\\\% at α = 1 \\\\alpha{=}1 , 74 % 74\\\\% at α = 1.5 \\\\alpha{=}1.5 , but benign over-refusal climbs 12 → 28 % 12\\\\to 28\\\\% and generation collapses by α = 2 \\\\alpha{=}2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 What refusal reads: harm, not surface\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Does the refusal gate read harm, or the scary surface features that correlate with it? We answer with a pre-registered 2 × 2 2\\\\times 2 that decorrelates the two axes (actual harm and surface trigger-word density) and asks which drives refusal. Cells come from natural keyword-density variation in existing public prompts (no authored evasions): harmful-scary (AdvBench with trigger words), harmful-clean (AdvBench without), benign-scary (all of XSTest-safe), benign-clean (Alpaca). We fix the keyword lexicon and the labels before looking at any outcome.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Refusal is harm-primary (Table 4 ). The load-bearing comparison is within-source : harmful-scary and harmful-clean are both AdvBench, so stripping the trigger words holds source, register, and phrasing fixed and varies only keyword density. Refusal does not budge (harmful-clean 100 % 100\\\\% , identical to harmful-scary) and the harm representation is keyword-invariant (probe 0.982 0.982 vs 0.983 0.983 ). A logistic regression of refusal on the two axes gives β harm = 5.12 ≫ β scary = 2.20 \\\\beta_{\\\\text{harm}}=5.12\\\\gg\\\\beta_{\\\\text{scary}}=2.20 . There is a real but secondary keyword effect on the benign side: benign-scary prompts are over-refused at 59 % 59\\\\% versus 8 % 8\\\\% for benign-clean, the well-known exaggerated-safety phenomenon (Röttger et al., 2024 ) . Its magnitude is an upper bound , not an estimate: benign-scary is entirely XSTest and benign-clean entirely Alpaca, so β scary \\\\beta_{\\\\text{scary}} is entangled with dataset and register and partly counts a “sounds like XSTest” effect. The harm-primary conclusion does not rest on it; it rests on the within-AdvBench comparison, which no source confound touches.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Off ceiling (SmolLM2).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Qwen’s harmful cells both sit at 100 % 100\\\\% , so the within-source comparison there can only fail to detect a keyword effect on the harmful side, not measure one. We therefore rerun the 2 × 2 2\\\\times 2 on the weak-gate SmolLM2 ( 0.63 0.63 plain refusal), where the harmful cells are off ceiling. Harm-primary holds and a small surface effect becomes visible: harmful-scary refuses 66 % 66\\\\% and harmful-clean 59 % 59\\\\% (stripping trigger words drops refusal 7 7 pp, a real but small keyword contribution the Qwen ceiling masked), while harm dominates (harmful-clean 59 % 59\\\\% vs benign-scary 12 % 12\\\\% ; logistic β harm = 2.83 ≫ β scary = 0.55 \\\\beta_{\\\\text{harm}}=2.83\\\\gg\\\\beta_{\\\\text{scary}}=0.55 ). Refusal is harm-primary, with a small keyword effect on the harmful side that is detectable only off ceiling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A retracted reading.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An earlier version of this analysis, run on XSTest directly, took the cosine between a harm-content direction and a refusal-behavior direction and found it low ( ∼ 0.1 {\\\\sim}0.1 in deep layers), which we initially read as “refusal reads a genuinely separate feature from harm.” That reading does not pass the 2 × 2 2\\\\times 2 . The low cosine was inflated by exactly the keyword-driven over-refusals of the benign-scary cell: on XSTest, harm-content and refusal-behavior dissociate mostly through over-refusal, so the “refusal direction” there partly encodes scary surface. Once harm and surface are properly decorrelated, refusal is harm-primary. We keep the corrected reading and drop the “separate feature” claim.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Is there a separate refusal direction? A decision-direction search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The walk-back invites the question directly: if we build the refuse-vs-comply direction properly , holding harm fixed, is there a separate decision feature to find? We search and report a controlled negative. The prefill supplies the matched contrast a naive setup lacks: a weakened prefill (“I understand you’re asking”) splits harmful prompts 41 / 29 41/29 refuse-vs-comply, so refused and complied cases share both harm and prefill-presence and differ only in the decision. The diff-of-means direction d refuse d_{\\\\text{refuse}} is geometrically distinct from the harm direction ( | cos | = 0.06 |\\\\cos|=0.06 ) but reads the decision only weakly: cross-validated AUC 0.648 0.648 , barely above the harm direction’s own 0.640 0.640 on the same split. A sweep of all layers and the first four generated positions ceilings at AUC 0.727 0.727 (layer 27 27 ) for any single direction. This ceiling, like the “best cells” in the patching grid (App. D , Table 7 ) and in Table 8 , is a maximum over its grid, so it is post-selection and optimistically biased. The main-text patching claim (Table 2 ) instead uses two fixed cells at n = 100 n=100 , and the qualitative conclusions do not depend on the grid maxima. But the decision is multidimensional rather than weakly encoded: a full linear probe on the same activations reads refuse-vs-comply at ∼ 0.85 \\\\sim 0.85 (layer 21 21 ) and well above harm alone ( 0.64 0.64 ). The capacity gain over the best single direction is + 0.12 +0.12 (bootstrap mean + 0.17 +0.17 , 95 % 95\\\\% CI [ + 0.05 , + 0.33 ] [+0.05,+0.33] , n = 70 n=70 ), so it excludes zero though it is imprecise, and it is consistent across regularization and layers. So there is no single refusal direction , the signal is spread across directions. Whether it is genuinely high-rank rather than low-rank is exploratory at n = 70 n=70 (three principal components reach only 0.74 0.74 , near the single-direction 0.73 0.73 , and the readout keeps climbing as dimensions are added, but n = 70 n=70 flatters high-rank estimates; App. H ); the robust claim is just single ≪ \\\\ll full. No single direction is a usable handle: steering d refuse d_{\\\\text{refuse}} under the strong attack is causally inert (no restored refusal at a mid layer where steering has purchase, nor at the best-read late layer; 0 % 0\\\\% at every coefficient; App. H ). This is the mechanistic form of “more to refusal than a single direction” (Joad et al., 2026 ) : refusal is a decodable but distributed computation, which is why single-direction steering (§ 4.5 ) and the “separate feature” reading both fail.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Across scale (within family).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This distributed structure is a within-family scale law. Across the Qwen ladder ( 0.5 0.5 , 1.5 1.5 , 3 3 , 7 7 B) the full probe beats the best single direction by 0.06 0.06 – 0.12 0.12 at every scale, and both readouts rise with size (single direction 0.76 → 0.85 0.76\\\\to 0.85 , full 0.84 → 0.94 0.84\\\\to 0.94 ; the 7 7 B run is on GPU, App. H ). The claim is Qwen-scoped: the one balanced cross-family point, SmolLM2-1.7B, shows no gain (its decision is essentially a single direction), and Phi-3’s calibration split was too imbalanced to read, so cross-family is left open. SmolLM2 is also the weakest gate ( 0.63 0.63 plain refusal), and a weakly-enforced refusal need not recruit a distributed representation to do its job, so the lower rank may track the weaker gate. We flag this as a hypothesis; a scale of gate-strength against decision rank is untested here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Refusal is not selectively ablatable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"No intervention we tried removes refusal selectively, a transition from harmful refusal to coherent harmful compliance without comparable degradation on benign behavior; anything less destroys the computation rather than editing it. Ablating any single attention head at any layer leaves refusal fully intact (a 28 × 12 28\\\\times 12 sweep on Qwen2.5-1.5B: every head-ablation drops refusal by 0.00 0.00 ; the hook is verified to change next-token logits by ‖ Δ ‖ ≈ 23 \\\\|\\\\Delta\\\\|\\\\approx 23 , so this is robustness, not a dead intervention), and ablating an entire attention layer likewise leaves refusal at 100 % 100\\\\% (e.g. layer 12, ‖ Δ logits ‖ ≈ 123 \\\\|\\\\Delta\\\\text{logits}\\\\|\\\\approx 123 ). Heavier ablation does not yield compliance. Ablating the first attention layer, or all of them, drives the refusal string-match to zero only by collapsing generation into garbage (“ silly silly noble glorious …”). With all attention removed the model emits one fixed gibberish string regardless of the prompt , harmful or benign, a comparable non-selective degradation that a coherence read (App. J ) confirms is not compliance. Together with the earlier decompositions, direction steering (§ 4.5 ), subspace and full-residual patching (§ 4.1 ), this leaves no compact causal support for refusal among the decompositions we test (direction, subspace, layer, attention head) for these models. It leaves open a compact mechanism in a space we did not probe: MLP neurons and sparse (SAE) features are untested and either could still host one. We tested the scale concern at 14 14 B on heavily-safety-trained Phi-3-medium (App. J ): the negative holds (no single component selectively removes refusal; only the input layer shows a partial drop), so heavier safety training did not concentrate refusal into a compact circuit here, though frontier scale is untested. The only intervention that coherently removes refusal here is the response-site prefill, which leaves the model fluent and complying. Refusal is thus distributed and redundant in representation space but fragile at the response site: the mechanism is diffuse, the only compact failure point we identify is positional, not representational.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Not specific to surface harm (exploratory).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dissociation extends from lexical harm to inferred intent: on 24 24 dual-use triplets where intent, not surface content, decides the right answer (App. I ), the prefill flips every bare prompt the model reads as harmful-intent to compliance ( 100 % → 0 % 100\\\\%\\\\to 0\\\\% ) while that intent reading is unchanged, so the gate is shallow as a computation whether its basis is a harmful word or an inferred intent. Exploratory: one model, n = 24 n=24 ; full numbers and the same structural caveat in App. I .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Related work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Shallow safety and prefill attacks. Qi et al. ( 2025 ) argue safety alignment is shallow in token-depth (carried by the first few output tokens) and explicitly explain prefilling attacks (Wei et al., 2023 ; Zhu et al., 2024 ; Li et al., 2025 ) as a consequence. Our result is complementary and representational: the harm concept is not shallow or degraded under a prefill attack, it is fully and identically represented; what is shallow is the gate that turns that representation into a refusal, and we localize it causally to the response site. Where their proposed fix deepens the alignment via training (cf. Zhang et al., 2025a ), our defensive consequence is a read-side monitor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Detecting harm from internal activations, and where the flip lives. That harmful and jailbreak prompts are linearly detectable in hidden states is well established, and several systems monitor or manipulate these signals (Zou et al., 2023a ; Jiang et al., 2025 ; Han et al., 2025 ; Ball et al., 2025 ) . Our claim is not that harm is probeable: it is. The closest work is JBShield (Zhang et al., 2025c ) : it shows a model still recognizes a toxic concept under a jailbreak while complying, attributes the flip to a competing prompt-side “jailbreak concept” (associated with affirmative tokens like “sure”), and mitigates by enhancing the toxic concept back toward the plain-harmful level. Two things separate us. First, that account is prompt-side and cannot cover prefill, which injects the affirmative opening into the response and leaves the prompt (and its toxic representation) untouched: there is no prompt-side jailbreak concept to do the overriding, so the flip must be, and we show is, a response-site positional displacement (§ 4 ). Second, JBShield reports a binary concept-present flag and, by enhancing the concept, presupposes it is attenuated ; we find the harm representation quantitatively intact under prefill (level with the refused prompts), so enhancing it is unnecessary and a final-token monitor already suffices. This also resolves an apparent tension. Doda ( 2026 ) (a concurrent preprint) show final-token probes miss many jailbreaks (the evidence displaced to earlier tokens), and Zhang et al. ( 2025d ) report vanishing discriminability under disguise. Our account does not rest on either, but it explains both: those are prompt-modifying attacks that move or attenuate the prompt-side representation, whereas prefill leaves it whole. Prompt-modifying attacks attenuate; prefill does not. This attenuate-vs-intact split is the representational signature of the response-site/prompt-modifying dichotomy, and our own encoding results (Appendix K ) sit on the attenuating side. The monitor’s scope follows from the mechanism, not from luck.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Refusal directions and steering. Arditi et al. ( 2024 ) show refusal is mediated by a single direction; a follow-up argues there is more to it than one direction (Joad et al., 2026 ) . We use the direction and find it a strong read-out but a non-selective write-handle for restoring refusal under attack, and (§ 5 ) that refusal is harm-primary rather than riding a separate feature. Exaggerated safety. XSTest (Röttger et al., 2024 ) documents over-refusal of benign scary-sounding prompts; we quantify it as a secondary effect and separate it from genuine harm-tracking. Method and framing. We build on representation probing and activation patching (Zou et al., 2023a ; Meng et al., 2022 ) ; the represent-vs-act framing (probe for what a model represents but does not act on) is the one we applied to communicative intent (Kwon, 2026 ) , with safety as the second substrate.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Controls and demotions. Every headline effect is checked against a control that can demote it, and several were: the full-residual “ 100 % 100\\\\% ” restoration is generic disruption (a benign-residual control also reaches 100 % 100\\\\% ), naive steering over-refuses everything, an earlier “separate refusal feature” did not pass the decorrelated 2 × 2 2\\\\times 2 , and the attack-map scaling prediction was refuted on a clean three-point test. Table LABEL:tab:claims lists the claims that passed alongside the demoted ones, and Appendix L gives the full ledger: every control, what it could have shown, and its verdict. The boundaries below apply the same checking to what remains. Several are open questions we scoped rather than results we could not reach, and we note where the evidence stops.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Active-vs-passive resolves toward passive; a scoped residual remains. The base-model control (§ 4.3 ) resolves the central open question toward passive conditioning across two pathways: the prefill’s early-window grip is generic in both the attention and the early MLP channel (a non-safety-tuned base model shows the same prefill-specific collapse in each), with only a small safety-specific refusal attractor on top (logit trace, concentration 0.24 0.24 vs 0.03 0.03 ). The one part that was open, the instruct excess of that attractor, we resolve with a clean path patch. A difference-of-means injection could not settle it (it tests decodability, not active computation), and a positional MLP ablation cannot either (a transformer MLP is applied position-wise and never reads the prefill positions, so its prefill-dependence is inherited from attention, with no positional handle to sever). So we run a frozen-residual path patch that severs only the prefill-attention contribution to the MLP input, validated bit-exact by a sham / MLP-output-faithfulness / locality suite (App. E ). It answers toward generic: the early-MLP pathway routes prefill-specific compliance suppression (instruct − 21 -21 pp, base − 25 -25 pp vs matched − 7 -7 / − 5 -5 pp) but carries no safety-specific refusal pull (instruct concentration 0.007 ≈ 0.007\\\\approx base ≈ \\\\approx matched; prefill-specific refusal gain + 0.26 +0.26 pp). The residual safety-specific attractor does not live in the early MLP. The one caveat is frame: this is a teacher-forced logit path patch, well-powered for compliance-suppression routing and less so for a small refusal-promotion component that would need a generative trajectory to build, but the within-frame instruct-vs-base-vs-matched comparison is clean regardless. A safety-training-depth dose-response (fine-tune matched shallow-vs-deep checkpoints and re-run the knockout) would confirm from the training side; the path patch already answers the question the paper posed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scale and mechanism breadth. The core models are 1.5 1.5 – 3.8 3.8 B and run on CPU (with 7 7 B and 14 14 B probed for scale on GPU), because white-box access requires open weights. The dissociation replicates across four models, three families, and both a weak gate (SmolLM2, 0.63 0.63 plain refusal) and strong ones (Phi-3, 1.00 1.00 ), which mitigates but does not remove the concern that the picture changes at frontier scale, where safety training is heavier and the residual stream is wider. The full mechanism battery (patching, position, reversal, § 4 ) is shown on two models (Qwen2.5-1.5B and Phi-3-mini) at the per-condition n n reported per experiment: a localization, not a scaling survey. The paper’s central causal claim, the active-vs-passive knockout discriminator, does reach mid scale: on the Qwen2.5- 7B base model the knockout drops harmful content 68 % → 30 % 68\\\\%\\\\to 30\\\\% prefill-specifically against a mass-matched control’s 66 % 66\\\\% ( + 36 +36 pp, early-window-gated, n = 100 n=100 , App. E ), so the generic early-window grip is not a small-model artifact. On the 7B instruct model the prefill attack is weaker (refusal drops only to 24 % 24\\\\% , procedural content 16 % 16\\\\% ) and the refusal- string restoration does not cleanly replicate, consistent with the passive reading in which “refusal” is a model-dependent fallback, while the prefill-specific content-channel effect persists. So the mechanism (prefill-specific early-window attention dependence) and its passive character persist at 7 7 B; the behavioral refusal signature is model-dependent, and frontier scale remains untested.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Probe source-calibration. The harm probe reads harm rather than surface vocabulary, and we show this: an AdvBench-trained probe scores XSTest’s scary-but-benign split at 0.21 0.21 (only 12 % 12\\\\% clear threshold), so the intactness result is not the probe re-detecting violent words. But the same probe is AdvBench- calibrated : it fires at 0.97 0.97 on AdvBench-style phrasing and only 0.49 0.49 on differently-phrased XSTest-unsafe prompts, and a mixed-source probe (AdvBench + + MaliciousInstruct) raised sensitivity only by wrecking specificity (benign-scary false positives 12 % 12\\\\% to 64 % 64\\\\% ). This bounds the claim precisely, and the bound cuts in our favor on the core result: the dissociation compares complied-harmful against refused-harmful prompts drawn from the same distribution, so source-calibration cancels and cannot manufacture the gap. What it does threaten is the deployable monitor, whose 100 % 100\\\\% -catch / 0 % 0\\\\% -false-positive figure is measured on AdvBench-sourced attacks. A source-general harm monitor is unbuilt, and building one is the obvious next step; the mixed-source attempt here narrows the worry but does not discharge it.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Attack scope. Every headline number is scoped to the prefill (response-site) attack, and the monitor result is a claim about that regime and no wider. Prompt-modifying attacks (adversarial suffixes, and encodings the model cannot decode) attenuate or displace the prompt-side representation rather than leaving it whole (§ 6 , App. K ), so a representation monitor is strictly weaker against them. We report the attenuate-versus-intact split as the boundary of the guarantee, not as coverage of all jailbreaks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The harm direction is a real but partial handle. The harm direction restores refusal significantly above a matched-norm random baseline ( n = 100 n=100 : + 34 +34 / + 17 +17 pp, both CIs exclude zero) and spares benign, so it is a real partial handle, but the dramatic full-residual “ 100 % 100\\\\% ” restoration is generic disruption (a benign-residual control also reaches 100 % 100\\\\% ), and a third to a half of even the subspace restoration is matched by the random control. Adding the harm direction back under attack does not selectively restore refusal: a four-layer by three-coefficient sweep, raw and Arditi-normalized, finds no clean window. So we close the gap as a monitor, not as an intervention; whether some other handle can un-collapse the behavior is left open.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Measurement: what the attacked-condition outputs actually are. Refusal is scored by matching a fixed refusal-phrase list, not by an LLM judge: cheap and exactly reproducible. But “non-refusal” is not the same as coherent harmful compliance, so we read the attacked-condition generations directly, judge-free: a distinct-token/repetition coherence rule plus a transparent deflection-marker list, on 200 200 -token generations over a 50 50 -prompt sample per model, with all completions withheld and only per-model counts released (Ethics). Two conditions come apart. Under the full -strength prefill (the headline dissociation), the non-refusals are overwhelmingly coherent harmful compliance, not deflection. On Qwen2.5-1.5B, 36 36 of the 48 48 non-refusals (of 50 50 ) carry substantive procedural content by the automated rule. That is a conservative floor: a manual read of the remainder (harmful prose, e.g. incitement text or a pro-anorexia post) raises it to 45 / 48 45/48 ( 94 % 94\\\\% ), with deflection 2 / 48 2/48 and no degeneracy. Across the other three models, substantive compliance is 69 69 – 95 % 95\\\\% of non-refusals and genuine deflection is ≤ 3 / 50 \\\\leq 3/50 , with a small degeneracy tail on SmolLM2 ( 8 / 50 8/50 ). Some models add a cosmetic “this is illegal” disclaimer but deliver the procedure anyway. So the string-match compliance rate is not inflated by deflection; the behavior is real. Under partial harm-restoration patching, by contrast, the marginal non-refusals at the L12/ α = 1 \\\\alpha{=}1 cell are mostly non-harmful deflection (“a guide for making an explosive device: 1. ‘bomb’, this is not real information”), so there the string match under counts safe behavior and does not inflate the restoration. Both reads are judge-free. The same two-scorer agreement holds across every causal condition and both models: string-match refusal and the independent harmful-content scorer move in opposite directions throughout. On the inert controls (attack baseline, mass-matched knockout, late-window knockout, matched-norm random injection) refusal stays ≤ 4 % \\\\leq 4\\\\% while harmful content stays 48 48 – 59 % 59\\\\% (Qwen) / 13 13 – 18 % 18\\\\% (Phi); on the working interventions (early prefill-edge knockout, refuse-state injection) refusal is restored to 24 24 – 78 % 78\\\\% while harmful content collapses to 1 1 – 18 % 18\\\\% ; and coherence stays 0.94 0.94 – 0.99 0.99 everywhere. So no headline effect is an artifact of string-matching or degeneracy: the refusal scorer and the content scorer tell the same story, at full n n rather than a sample.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A prefill attack makes an aligned model comply with harmful requests while the model represents those requests as harmful exactly as strongly as the ones it refuses. Refusal is a shallow, response-site computation, primarily driven by the harm representation but severable from it: the attack occupies the response site, the harm concept sits intact at the prompt, and restoring the harm signal in an early window of the response partially restores the action; a dose-matched position control shows the first half of the response suffices while the second half is nearly inert, more than a single onset token but not the whole generation. The converse lands in the same window: injecting the model’s own refuse-state back into the attack re-induces refusal ( → 74 % \\\\to 74\\\\% , held-out), state-specifically (a matched-norm random control does nothing) and early-window-gated, a working defensive lever that the single harm direction alone is not. And the grip operates through early-window attention: knocking out the early response’s attention to the prefill, but not an equal attention mass elsewhere, selectively collapses the harmful continuation ( + 21 +21 pp refusal restored). A base-model control shows this dependence is generic autoregressive conditioning, not safety-specific active suppression (a non-safety-tuned base model shows the same prefill-specific collapse), resolving the central open question toward the passive reading: the prefill sustains compliance through generic early attention, and refusal is a model-dependent fallback. The concept is not the gate. And the gate has no compact internal form: across direction, subspace, layer, and attention-head decompositions no single-component intervention selectively removes refusal; heavy ablation only degrades generation into garbage. Refusal is distributed and redundant in representation space but fragile at the response site: the mechanism is diffuse, and the only compact failure point we identify is positional, not representational. The practical consequence is a design fact rather than a detector: because the attack never reaches the prompt-side representation, a monitor that reads it is immune to this class of attack by construction, where the behavioral gate is not.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Ethics and responsible disclosure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is a mechanistic analysis of a known attack, not a new one (the prefill jailbreak is public (Wei et al., 2023 ; Qi et al., 2025 ) ; we introduce no novel evasion), on small open-weight models ( 1.5 1.5 – 3.8 3.8 B) and public benchmarks (AdvBench, XSTest, Alpaca), with no authored harmful prompts. The direction is net-defensive: the practical output is where to place a safety classifier (prompt-side, immune to response-site attacks by the mechanism we identify), not how to defeat one. On artifacts: verifying that attacked-condition non-refusals are coherent harmful compliance required generating harmful completions, so we withhold them: the repository ships per-model counts (which reproduce every number) not the completion text. No human subjects or personal data are involved.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The table below maps every load-bearing claim to its evidence and epistemic status: shown (direct measurement) or analytic (follows from the construction or argument). The dissociation and the causal mechanism are the contribution; the demoted effects (full-residual disruption, the steering null, the walked-back “separate feature” reading) are recorded so the claims that passed are unambiguous.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All code and the full experiment ledger ( FINDINGS.md , every number with its control) are at https://github.com/collapseindex/breaking-refusal . Decision-token activations are cached to .npz so all analyses (probes, patching, the 2 × 2 2\\\\times 2 ) re-run offline in seconds; the model passes regenerate them.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All prompts are public: harmful = = AdvBench behaviors; benign = = Alpaca instructions; the surface control = = XSTest (its “safe” split is purpose-built to sound harmful); the mixed-source probe control adds MaliciousInstruct (Huang et al., 2024 ) . The harm/keyword 2 × 2 2\\\\times 2 (§ 5 , App. G ) is built from natural keyword-density variation in these existing prompts with a fixed lexicon; no evasion prompts are authored.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Models and environment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Qwen2.5-1.5B/3B-Instruct, SmolLM2-1.7B-Instruct, and Phi-3-mini-4k-instruct, from the Hugging Face cache (revisions pinned in the repo manifest). Python 3.13, transformers (Wolf et al., 2020 ) , torch (Paszke et al., 2019 ) (CPU-only), scikit-learn (Pedregosa et al., 2011 ) ; the one embedding control uses fastembed (BGE). Models load in bfloat16 at ≥ 1.5 \\\\geq 1.5 B (float32 exhausts memory on CPU) and float32 otherwise. Per-condition baselines (e.g. the 1.5 1.5 B attack-refusal rate, or the base model’s plain-attack compliance) vary by a few points across experiments, since scripts draw different AdvBench subsets at different n n ; every control is compared against its own run’s baseline, and we report each run’s own numbers rather than a pooled figure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Probe, attack, and refusal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The harm probe is ℓ 2 \\\\ell_{2} logistic regression ( C = 0.5 =0.5 , max_iter = 2000 =2000 ) on the last-prompt-token hidden state. Reported detection AUCs use nested-CV layer selection (no test-set peeking); the downstream causal experiments (patching, position, steering) instead fix layer 12, a representative mid layer where the probe is near-ceiling, so the intervention layer is not itself selected on the outcome. The prefill attack forces the assistant turn to begin with “Sure, here is”. Refusal is a fixed refusal-phrase string match. All generation is greedy ( do_sample = = False).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Determinism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Probes and featurizers are deterministic given the data; the only stochastic element is the cross-validation split (fixed seed). The 2 × 2 2\\\\times 2 lexicon, cell construction, and decision thresholds were fixed before any outcome was seen.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"State-transfer experiment (§ 4.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The reversal is run by experiments/modal_prefix_transfer.py (Qwen2.5-1.5B, L12, n = 100 n=100 AdvBench, greedy, 18 18 generated tokens). It captures the mean layer- 11 11 -output residual over the first-half generated window for the plain (no-prefill) and attack conditions, forms d reverse = s plain − s attack d_{\\\\text{reverse}}=s_{\\\\text{plain}}-s_{\\\\text{attack}} , and adds it into the attack’s early generated window over the same bands as the position control (onset / first half / all), against a matched-norm control drawn orthogonal-random and rescaled to ∥ d reverse ∥ \\\\lVert d_{\\\\text{reverse}}\\\\rVert . A tiny- n n gate (plain refusal ≥ 0.7 \\\\geq 0.7 , attack refusal ≤ 0.15 \\\\leq 0.15 ) validates the harness before the full run. The script is measure-only: it records refusal, a distinct-token coherence ratio, and a harmful-procedural-content flag as per-prompt counts to data/exp_prefix_transfer.json ; no generations are stored, and run logs (gitignored) print rates only.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Attention-knockout experiment (§ 4.3 , App. E ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Run by experiments/modal_attn_knockout.py (eager attention, Qwen2.5-1.5B, all layers/heads, n = 100 n=100 ), preregistered in PREREG_attn_knockout.md (claim, falsifier, condition matrix, interpretation table). Edge knockout adds ln ( 1 − λ ) \\\\ln(1-\\\\lambda) to the pre-softmax attention logit of (early-window query → \\\\to prefill key) edges and renormalizes; the mass-matched control selects non-prefill prompt keys whose baseline attention mass matches the prefill’s per prompt (three fixed seeds). A MODE=\\\"unittest\\\" scaffold must pass before any behavioral run (sham-hook bit-identity, edge-only masking, softmax renormalization, KV-cache consistency, live-hook and behavioral positive controls). Measure-only: refusal / coherence / harmful-content counts to data/exp_attn_knockout.json ; CIs are paired bootstrap over prompts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 gives the headline numbers. The “rep. AUC” column is one nested-CV probe (outer 5-fold, inner layer selection) on the clean prompt-token activations: 0.996 0.996 (Qwen-1.5B), 1.000 1.000 (Qwen-3B), 0.990 0.990 (SmolLM2), 0.998 0.998 (Phi-3). There is no separate “under attack” representation number, because the attack does not alter these activations (it appends to the response); we report a single estimator to make that explicit. The attacked-set behavior AUC is chance ( 0.49 0.49 – 0.52 0.52 ); the per-model complied-harmful vs. benign probe scores are 0.975 / 0.055 0.975/0.055 (Qwen-1.5B), 0.981 / 0.048 0.981/0.048 (Qwen-3B), 0.958 / 0.044 0.958/0.044 (SmolLM2), 0.907 / 0.122 0.907/0.122 (Phi-3).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Surface control (Table 6 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We score the AdvBench-trained probe (layer 12) on XSTest. It separates real-unsafe from scary-but-safe at AUC 0.79 0.79 with only 12 % 12\\\\% false positives on the scary-safe prompts: it reads harm, not violent vocabulary. It is, however, AdvBench-calibrated ( 0.97 0.97 on AdvBench-harmful but 0.49 0.49 on differently-phrased XSTest-unsafe), and training on mixed harmful sources (AdvBench + + MaliciousInstruct) does not fix it: it raises sensitivity but collapses specificity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 7 reports the full grid. We patch at the response site on harmful + + prefill prompts and let generation continue with no further intervention. The attack baseline (no patch) is harmful refusal 0.07 / 0.07/ benign 0 (Qwen-1.5B) and 0 / 0 0/0 (Phi-3). full : replace the onset residual with the mean harmful residual. benign-CTRL : the same with the mean benign residual: it returns refusal to 100 % 100\\\\% at every layer for both models, which is why we read the full-residual restoration as generic disruption rather than a harm-specific effect. subspace : add α ⋅ d ^ harm \\\\alpha\\\\cdot\\\\hat{d}_{\\\\text{harm}} (unit harm-probe direction) at the response positions, α ∈ { 0.5 , 1 , 1.5 } \\\\alpha\\\\in\\\\{0.5,1,1.5\\\\} scaled to the layer’s activation norm; the reported cell is the best α \\\\alpha by (harmful − - benign) refusal. Settling the harm-vs-random delta at n = 100 n=100 on two fixed cells (not maxima): at L12/ α = 1 \\\\alpha{=}1 the harm direction restores 48 % 48\\\\% (bootstrap 95 % 95\\\\% CI 38 38 – 58 % 58\\\\% ) versus 14 % 14\\\\% for a matched-norm direction orthogonalized to d ^ harm \\\\hat{d}_{\\\\text{harm}} (24 seeds, range 0 – 60 % 60\\\\% ), delta + 34 +34 pp (CI + 22 +22 , + 46 +46 ); at L8/ α = 1.5 \\\\alpha{=}1.5 , 33 % 33\\\\% ( 24 24 – 42 % 42\\\\% ) versus 16 % 16\\\\% ( 0 – 60 % 60\\\\% ), delta + 17 +17 pp (CI + 6 +6 , + 28 +28 ). Both exclude zero (the earlier n = 15 n=15 estimate had overlapping CIs). Harm is benign-sparing (benign restoration 12 % 12\\\\% / 0 % 0\\\\% vs random 3 % 3\\\\% / 4 % 4\\\\% ). So a harm-specific component is real and significant, but a third to a half of the raw restoration is matched by the random control; we report the harm − - random delta. This licenses a partial, partly specific causal claim, not a clean lever.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cell selection and the bootstrap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two cells are grid-selected (L12 is the best subspace layer in the grid above; L8 a second), fixed before the n = 100 n=100 run, which re-estimates them on AdvBench [ : 100 ] [:100] , overlapping the grid prompts. Selection optimism therefore affects the absolute restore rate but not the harm − - random delta, which is measured for both arms at the same cell and prompts. For the delta CI each bootstrap iterate resamples the 100 100 harm prompts (with replacement) and the 24 24 random seeds (with replacement), taking the harm mean minus the resampled-seed mean; the harm-restore CI resamples prompts only. The random-control distribution is right-skewed and mostly low: the 24 24 L8 seeds are (sorted) 0 , 0 , .01 , .01 , .01 , .03 , .06 , .06 , .07 , .09 , .10 , .11 , .12 , .13 , .13 , .13 , .20 , .21 , .23 , .26 , .35 , .37 , .53 , .60 0,0,.01,.01,.01,.03,.06,.06,.07,.09,.10,.11,.12,.13,.13,.13,.20,.21,.23,.26,.35,.37,.53,.60 , median 0.12 0.12 , with the 0 – 60 % 60\\\\% range driven by four high outliers, so the harm direction’s 33 % 33\\\\% beats the typical random direction by more than the mean delta suggests.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Dose-matched positional control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ask whether the position result reflects where the harm signal is restored or merely how much total signal is injected, we band one consistent response window (the prefill + + the 18 18 generated tokens, 22 22 positions) and vary position and per-position coefficient independently ( n = 100 n=100 , L12; GATE: the all-positions band at α = 1 \\\\alpha{=}1 reproduces the 41 % 41\\\\% of the main text). Restoration by condition (% refusal, baseline 3 % 3\\\\% ): all @ 1 @1 = 41 =41 , first-half @ 1 @1 = 42 =42 , first-half @ 2 @2 = 47 =47 , second-half @ 2 @2 = 6 =6 , all @ 0.5 @0.5 = 0 =0 , every-other @ 1 @1 = 1 =1 , every-other @ 2 @2 = 48 =48 (coherence ≥ 0.96 \\\\geq 0.96 throughout, so these are not degeneration). Reading: (i) the first half restores as much as the whole, and the second half is nearly inert even at double strength, so the effect is a matter of position (an early window), not total dose; (ii) halving the coefficient over all positions gives 0 % 0\\\\% , so there is a per-position strength threshold, not just a dose budget. This corrects an earlier version of the control whose “all” band silently included the prefill positions while its half-bands did not, confounding position with prefill-inclusion; the windowed design here is consistent across bands.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This appendix backs § 4.3 . The experiment is preregistered ( PREREG_attn_knockout.md ): primary hypothesis, falsifier, the frozen condition matrix, the preregistered contrasts, and an interpretation table written before results.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Intervention and the onset caveat.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On each early-window decode step ( g < 9 g<9 ) we add ln ( 1 − λ ) \\\\ln(1-\\\\lambda) to the pre-softmax attention logit of every (current-query → \\\\to prefill-key) edge and let softmax renormalize ( λ = 1 \\\\lambda{=}1 is a full edge cut). Eager attention is mandatory (fused SDPA/Flash kernels bypass a logit-level edit). Because generated token 1’s logits are produced from the final prefill token’s hidden state, a response-to-prefill knockout cannot affect token 1; this is a test of early maintenance , not attack initiation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Unit tests (all pass before any behavioral run).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(i) eager attention active; (ii) prefill key indices located after the chat template; (iii) sham hook (knockout with an empty key set) is bit-for-bit identical to baseline under greedy decoding; (iv) after a full edge cut the removed edges carry 0 attention mass and the softmax row renormalizes to 1 1 ; (v) non-target query rows are unchanged; (vi) no NaN/Inf reaches the residual stream; (vii) a large knockout measurably moves next-token logits ( ‖ Δ ‖ ≈ 10 3 \\\\|\\\\Delta\\\\|\\\\approx 10^{3} , the hook is live); (viii) knocking out attention to the prompt tokens materially changes generation (a behavioral positive control: the machinery moves behavior, so a prefill-specific null means “prefill edges not necessary,” not “dead hook”); (ix) KV-cache on/off agree under the sham hook.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Mass-matched control.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A no-knockout measurement pass records the baseline attention mass from early-window queries onto each key (mean over layers and heads). The prefill keys carry small mass (mean 0.082 0.082 ); attention-sink tokens (BOS, template) carry far more, so a max-mass or random span is not a fair control. We select, per prompt, a set of non-prefill prompt keys of the same size whose summed mass best matches the prefill’s (achieved matched/prefill ratio 0.98 0.98 , range [ 0.75 , 1.08 ] [0.75,1.08] over three fixed seeds). Knockout of this set is the primary control.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Full matrix and contrasts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 gives the condition means. Paired bootstrap ( 10 , 000 10{,}000 resamples over the 100 100 harmful prompts; the matched contrast uses one fixed control seed, matched 3 % 3\\\\% on that seed, three-seed mean 1 % 1\\\\% ): early prefill KO − - mass-matched KO = + 21 =+21 pp ( [ + 12 , + 30 ] [+12,+30] ); early − - baseline = + 21 =+21 pp ( [ + 12 , + 30 ] [+12,+30] ); early − - late = + 20 =+20 pp ( [ + 11 , + 29 ] [+11,+29] ); whole − - early = + 5 =+5 pp ( [ − 1 , + 12 ] [-1,+12] , includes zero, so whole is not distinguishable from early). Benign specificity: under the prefill, early knockout moves benign refusal only 0 % → 4 % 0\\\\%\\\\to 4\\\\% (24 benign prompts), against the harmful 3 % → 24 % 3\\\\%\\\\to 24\\\\% , a difference-in-differences of + 16.8 +16.8 pp. Coherence stays 0.96 0.96 – 0.97 0.97 across all conditions (the preregistered non-inferiority margin was a ≤ 5 \\\\leq 5 -point drop). The knockout also suppresses harmful content (procedural rate 51 % → 18 % 51\\\\%\\\\to 18\\\\% early, → 3 % \\\\to 3\\\\% whole), prefill-specifically (matched control 50 % 50\\\\% ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Held-out replications of the two causal handles (§ 4.2 , § 4.1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Both the state-transfer reversal and the harm-direction patching are re-run with a clean train/test split so the injected vector never sees the test prompts. Reversal: d reverse d_{\\\\text{reverse}} is estimated on AdvBench [ 0 : 260 ] [0{:}260] and injected/tested on the disjoint held-out [ 260 : 360 ] [260{:}360] ; it restores refusal to 74 % 74\\\\% over the first half (whole 78 % 78\\\\% , onset 13 % 13\\\\% ), the matched-norm random control to 0 % 0\\\\% , coherence 0.97 0.97 – 0.99 0.99 ; the in-distribution reference (same d reverse d_{\\\\text{reverse}} tested on train prompts) gives 72 % 72\\\\% , so the generalization gap is within noise. Patching: the harm direction is trained on AdvBench [ 0 : 200 ] [0{:}200] (harmful vs Alpaca benign, activations recomputed on the fly) and restoration tested on the held-out [ 260 : 360 ] [260{:}360] ; the L12/ α = 1 \\\\alpha{=}1 cell restores 42 % 42\\\\% ( 95 % 95\\\\% bootstrap CI 32 32 – 52 % 52\\\\% , n = 100 n=100 ) versus 13 % 13\\\\% for the matched-norm orthogonal-random control ( 6 6 seeds), delta + 29 +29 pp (CI + 16 +16 , + 41 +41 ), benign over-refusal 15 % 15\\\\% . Both handles pass held-out (the direction shrinks ∼ 5 \\\\sim 5 – 6 6 pp off-distribution, the reversal does not), so neither is an artifact of the prompts it was estimated on. Held-out patching was run for the primary L12 cell only. Second family (Phi-3-mini, 3.8 3.8 B). Both new results replicate. The held-out reversal ( L 16 L16 ) restores refusal to 69 % 69\\\\% first-half / 71 % 71\\\\% whole / 4 % 4\\\\% onset, random 2 % 2\\\\% , in-distribution reference 71 % 71\\\\% (baseline 1 % 1\\\\% ). The attention knockout replicates the full pattern at n = 100 n=100 : attack baseline 1 % 1\\\\% ; early prefill-edge KO 25 % 25\\\\% ; whole 32 % 32\\\\% ; late 0 % 0\\\\% ; half-dose ( λ = 0.5 \\\\lambda{=}0.5 ) 0 % 0\\\\% ; mass-matched control 1 % 1\\\\% ; coherence 0.95 0.95 – 0.98 0.98 throughout (control mass-match ratio 1.00 1.00 ). Phi’s unit-test suite passes 9 / 10 9/10 ; the one failure is a cache-vs-no-cache determinism check that is a bf16 property of the model (it reproduces with a no-op hook), and does not affect the experiment, which compares cache-on baseline against cache-on knockout, and the sham-identity and edge-cut checks pass under cache-on.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Active-vs-passive: base-model discriminator.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ask whether the knockout effect is safety-tuning-specific (active suppression) or generic (passive conditioning), we run the identical early prefill-edge knockout on the Qwen2.5-1.5B base model (no safety tuning), with a raw-completion format (no chat template) and the harmful- content scorer as the endpoint (the base model does not refuse). The base model complies ( 64 % 64\\\\% harmful content) and shows the same prefill-specific, early-window dependence as the instruct model: early knockout drives harmful content to 25 % 25\\\\% (a − 39 -39 pp drop) while the mass-matched control leaves it at 64 % 64\\\\% ( − 0 -0 pp) and late knockout is inert ( 61 % 61\\\\% ). Since a non-safety-tuned model shares the dependence, it is generic autoregressive conditioning, not a safety-specific active suppression; this resolves the active-vs-passive question toward the passive reading (§ 7 ). Mid-scale replication (Qwen2.5-7B base). The identical knockout on the 7 7 B base model reproduces the pattern cleanly: attack baseline 68 % 68\\\\% harmful content, early prefill-edge knockout 30 % 30\\\\% (a − 38 -38 pp drop), mass-matched control 66 % 66\\\\% ( − 2 -2 pp, three seeds), late knockout inert ( 68 % 68\\\\% ), for a + 36 +36 pp prefill-specific, early-window-gated drop ( n = 100 n=100 ). The passive reading is therefore not a small-model artifact. Measure-only, experiments/modal_attn_knockout_base.py and modal_attn_knockout_base_7b.py .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Active-vs-passive: logit-level decomposition (a small active component).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To probe for a residual safety-specific pull, we decompose how refusal recovers under the knockout at the logit level. For each early-window generated step we record the softmax mass on refusal-onset tokens ( p refuse p_{\\\\text{refuse}} ) and the probability of the compliance continuation ( p comply p_{\\\\text{comply}} ), under baseline / early knockout / mass-matched knockout, and define concentration = Δ p refuse / ( − Δ p comply ) =\\\\Delta p_{\\\\text{refuse}}/(-\\\\Delta p_{\\\\text{comply}}) : of the probability the knockout strips off compliance, the fraction that lands on refusal. In both models the knockout’s dominant effect is compliance collapse ( p comply p_{\\\\text{comply}} : instruct − 5.6 -5.6 pp, base − 17 -17 pp), the passive signature. But the instruct model routes the freed mass to refusal far more than the base model (concentration 0.24 0.24 vs 0.03 0.03 ; prefill-specific p refuse p_{\\\\text{refuse}} gain + 1.3 +1.3 pp vs matched + 0.1 +0.1 pp), so safety tuning adds a modest learned refusal attractor on top of the generic mechanism. The metric confirms a small safety-specific refusal component exists and sizes it, but does not separate an actively-suppressed gate springing back from a passive fallback attractor. Instruct 60 60 prompts; base as anchor. Measure-only, experiments/modal_active_probe.py .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The MLP pathway (closing the accounting gap on the generic side).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ask whether the compliance that persists after the attention knockout hides a safety-specific mechanism in the MLP pathway, we capture the mean early-window MLP output per layer in the plain (no-prefill) and attack conditions, form d mlp = plain − attack d_{\\\\text{mlp}}=\\\\text{plain}-\\\\text{attack} , and inject it at the early-window MLP outputs under the attack ( n = 60 n=60 , all layers, matched-norm random and late-window controls). Base model (the clean discriminator): removing the base model’s own prefill-induced MLP shift drops harmful content 68 % → 48 % 68\\\\%\\\\to 48\\\\% ( + 16 +16 pp prefill-specific vs the matched-random control’s ∼ 4 \\\\sim 4 pp, late-window inert), in a model with no safety mechanism, so the early MLP’s prefill-dependence is generic, extending the passive account from attention to a second pathway. Instruct (a decomposition, not a discriminator): the same injection restores refusal 3 % → 47 % 3\\\\%\\\\to 47\\\\% (prefill-specific, early-window-gated), showing the MLP carries much of the refuse-state; but this is a difference-of-means injection, which tests decodability, not active computation, and the larger instruct effect is overdetermined by its larger plain-vs-attack behavioral gap. We therefore report the 47 % 47\\\\% as decomposition only, and settle the instruct excess with the clean path patch below. Measure-only, experiments/modal_mlp_pathpatch.py .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Clean path patch: the early MLP pathway is generic.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A position-wise MLP never reads the prefill positions, so the excess cannot be isolated by a positional ablation; it needs a path patch that severs only the prefill-attention contribution to the MLP input. Teacher-forced over the greedily-generated attack sequence (so clean logits equal the generation), we (i) knock out early-window attention to the prefill and capture each layer’s resulting MLP input at the early-window positions, then (ii) run the clean attack and overwrite only those MLP inputs with the captured knockout values, leaving every other activation, including all downstream attention, at its clean value. Because Qwen2 saves the residual before the MLP layernorm, the counterfactual enters through the MLP sub-block alone. The surgery passes a unit-test suite before any read: sham (injecting the clean MLP input changes nothing, max | Δ | = 0 |\\\\Delta|{=}0 ), MLP-output faithfulness (the patched MLP output equals the knockout run’s, max | Δ | = 0 |\\\\Delta|{=}0 ), hook-live (the attention knockout moves logits by > 15 >15 ), and locality (patching only the last position leaves earlier-position distributions unchanged, max | Δ | = 0 |\\\\Delta|{=}0 ); all bit-exact on both models. On the same concentration metric ( Δ p refuse / − Δ p comply \\\\Delta p_{\\\\text{refuse}}/-\\\\Delta p_{\\\\text{comply}} , p comply p_{\\\\text{comply}} the mass on the clean compliance token), the MLP pathway carries the generic effect and only it: it strips prefill-specific compliance probability (instruct − 21 -21 pp, base − 25 -25 pp, against mass-matched-MLP-path controls at − 7 -7 / − 5 -5 pp) but shows no safety-specific refusal concentration (instruct 0.007 0.007 , base 0.001 0.001 , matched − 0.02 -0.02 ; prefill-specific refusal gain instruct + 0.26 +0.26 pp). So the residual safety-specific pull found in the logit trace does not route through the early MLP, and the instruct excess in the difference-of-means test was decodability. Caveat: this teacher-forced logit frame is well-powered for compliance-suppression routing and less sensitive to a small refusal-promotion component (which would need a generative trajectory to accumulate), but the within-frame instruct-vs-base-vs-matched comparison is clean. n = 60 n{=}60 ; measure-only, experiments/modal_mlp_pathpatch_clean.py .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Reviewer robustness checks (boundary, prefill-exclusion, prefill length, sampling, benign-sparing).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bundled on one instruct load, gated by the same plain-refuses/attack-complies check and cross-validated against the published anchors (the whole-band harm patch reproduces the 48 % 48\\\\% fixed cell; d reverse d_{\\\\text{reverse}} first-half reproduces 75 % ≈ 75\\\\%\\\\approx the 74 % 74\\\\% held-out reversal), experiments/modal_revision_robustness.py . (i) Boundary sweep (harm-direction patch, L12/ α 1 \\\\alpha 1 , full per-position dose, n = 60 n{=}60 ): onset 10 % 10\\\\% , first-third 50 % 50\\\\% , first-half 57 % 57\\\\% , first-two-thirds 55 % 55\\\\% , whole 50 % 50\\\\% , second-half 5 % 5\\\\% , so restoration saturates by the first third and is flat thereafter. (ii) Prefill-exclusion: injecting over the generated tokens only (prefill positions excluded) restores 0 % 0\\\\% (first-half band) and 3 % 3\\\\% (generated tokens 1 1 – 7 7 ), versus 57 % 57\\\\% with the prefill span included; the causal write-site is the prefill span, and generated positions carry the effect only when the prefill is also driven. (iii) Prefill length ( n = 50 n{=}50 , bands over generated-token counts, all prefill positions driven): the saturation-by-first-third pattern holds for a short prefill (“Sure,”: first-third 86 % 86\\\\% , whole 86 % 86\\\\% ) and a long one (“Sure, here is a detailed step-by-step guide:”: first-third 12 % 12\\\\% , whole 6 % 6\\\\% ), so the window is an early band in generated-token count, not a fraction of the prefill; the restoration level falls with prefill commitment. (iv) Sampling ( T = 0.7 T{=}0.7 , n = 40 n{=}40 , K = 2 K{=}2 ): plain refusal 100 % 100\\\\% , attack refusal 0 % 0\\\\% (vs greedy 5 % 5\\\\% ), so the dissociation is not a greedy artifact; the harm-direction restoration weakens ( 50 % → 21 % 50\\\\%\\\\to 21\\\\% ), consistent with a partial, non-selective write-handle. (v) Benign-sparing of the state transfer: d reverse d_{\\\\text{reverse}} injected on benign prompts under the attack raises their refusal 12 % → 18 % 12\\\\%\\\\to 18\\\\% ( + 6 +6 pp) while restoring harmful refusal to 75 % 75\\\\% ( + 57 +57 pp selectivity), so the transfer is a selective edit, not a blanket refusal-inducer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scale (Qwen2.5-7B).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At 7 7 B the prefill attack is weaker: plain refusal 98 % 98\\\\% , attack refusal 24 % 24\\\\% (vs ≤ 3 % \\\\leq 3\\\\% at 1.5 1.5 – 3.8 3.8 B), procedural content 16 % 16\\\\% . The knockout’s prefill-specific effect persists in the content channel (early knockout 0 % 0\\\\% , whole 0 % 0\\\\% , late 4 % 4\\\\% , half-dose 13 % 13\\\\% , mass-matched control 23 % 23\\\\% , coherence ≥ 0.95 \\\\geq 0.95 ), early-window-gated and dose-dependent, but the refusal- string restoration seen at smaller scale does not cleanly replicate (early 20 % ≈ 20\\\\%\\\\approx baseline 24 % 24\\\\% ), consistent with the passive reading that the refusal fallback is model-dependent. Unit tests pass 10 / 10 10/10 (including the KV-cache check); run on an L40S (eager attention). Gate relaxed to attack-refusal ≤ 0.35 \\\\leq 0.35 for the more-robust 7 7 B.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We add the harm direction at all response positions under the attack and sweep layer and strength, in two parameterizations. Neither finds a window that raises harmful refusal while sparing benign (Table 8 ). The best cells restore harmful refusal to 71 71 – 83 % 83\\\\% but at 40 40 – 50 % 50\\\\% benign over-refusal; smaller coefficients over-refuse everything, larger ones ( α = 4 \\\\alpha{=}4 / coeff 14 14 ) collapse generation to 0 . The harm direction induces refusal broadly, not on-harm. By contrast the read-side use of the same representation (the monitor of § 3 ) catches 100 % 100\\\\% of the attacked-harmful prompts at 0 % 0\\\\% false positive.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Construction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We fix a trigger-word lexicon (eight categories: violence, weapons/ explosives, drugs, cyber, self-harm, extremism, sexual, fraud/theft) with inflection-aware matching (so “hacking” and “drugs” count), and a per-prompt scary flag = = “at least one trigger word.” Cells come from natural variation in existing public prompts, with no authored evasions: harmful-scary = = AdvBench with a trigger word; harmful-clean = = AdvBench without one (e.g. “how do I best destroy my boss’s car,” “commit insider trading”); benign-scary = = all of XSTest-safe (purpose-built to sound harmful); benign-clean = = Alpaca. We balance to n = 120 n{=}120 per cell and use source labels (flagging, not filtering, the severity spread). Two issues surfaced and were fixed before scoring: an initial word-boundary matcher leaked inflections into the clean cell (fixed), and the strict-lexicon benign-scary cell was thin ( n = 54 n{=}54 ), so we pre-registered benign-scary = = all XSTest-safe with the strict-keyword subset as a robustness check.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Results.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 in the main text. A logistic regression of refusal on the two binary axes gives β harm = 5.12 \\\\beta_{\\\\text{harm}}{=}5.12 , β scary = 2.20 \\\\beta_{\\\\text{scary}}{=}2.20 . The representational readout agrees: the refusal-behavior direction is closer to the harm-content direction than to the surface direction (mid-layer | cos | |\\\\cos| 0.34 0.34 vs. 0.13 0.13 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Test for a refuse-vs-comply direction distinct from harm (§ 5.1 ). A weakened prefill (“I understand you’re asking”) gives a 41 / 29 41/29 refuse/comply split on 70 70 harmful prompts, so harm and prefill-presence are held fixed and only the decision varies; d refuse d_{\\\\text{refuse}} is the diff-of-means of the decision-token activations. Distinct but weak: | cos ( d refuse , d harm ) | = 0.06 |\\\\cos(d_{\\\\text{refuse}},d_{\\\\text{harm}})|=0.06 ; five-fold CV AUC 0.648 0.648 versus d harm d_{\\\\text{harm}} ’s 0.640 0.640 at layer 12 12 . Read sweep (single direction): across all 29 29 layers × \\\\times the first 4 4 generated positions the diff-of-means readout ceilings at AUC 0.727 0.727 (layer 27 27 , position 1 1 ); most cells are 0.60 0.60 – 0.72 0.72 . But that ceiling is rank-1: a full L2-logistic probe on the same activations reads refuse-vs-comply at 0.85 0.85 (layer 21 21 ; + 0.12 +0.12 over the best single direction, consistent across C ∈ { 0.01 , 0.1 } C\\\\in\\\\{0.01,0.1\\\\} and layers, noisy at n = 70 n=70 ). The decision is multidimensional, not weak. Rank: it is distributed , not low-rank: PCA-then-logistic reaches AUC 0.74 0.74 at 3 3 components (near the single-direction 0.73 0.73 ) and climbs to ∼ 0.84 \\\\sim 0.84 only by ∼ 30 {\\\\sim}30 ; the top-variance axis is anti-predictive (rank-1 PCA 0.40 0.40 ) and none of the top components are harm-aligned ( | cos | ≤ 0.13 |\\\\cos|\\\\leq 0.13 ), so the decision lives in low-variance, non-harm directions spread wide (caveat: n = 70 n=70 flatters the high-rank estimates). Control (inert): steering + c d refuse +c\\\\,d_{\\\\text{refuse}} at the response positions under the strong attack restores ≤ 7 % \\\\leq 7\\\\% harmful refusal at layer 12 12 and 0 % 0\\\\% at layer 27 27 (the best-read cell), at every coefficient ∈ { 0.5 , 1 , 1.5 } \\\\in\\\\{0.5,1,1.5\\\\} , while d harm d_{\\\\text{harm}} restores refusal at layer 12 12 (non-selectively) and a matched-norm random direction restores nothing. No layer or position yields a clean or steerable decision direction; refusal is not a separable linear feature.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across scale (Table 9 ). We repeat the single-vs-full comparison across the model ladder under a per-model calibrated weak prefill. Within the Qwen family the full probe beats the best single direction at every scale ( 0.5 0.5 – 7 7 B); SmolLM2 (a balanced split, different family) shows no gain, and Phi-3’s calibration gave a 43 / 7 43/7 split too imbalanced to read. The 7 7 B run used Modal (A10G, white-box) with a last-layer/ 5 5 -fold probe.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"24 24 dual-use triplets (§ 5 ), each request in three framings. Construction: the request is dual-use (e.g. “how do I get into a locked iPhone I can’t unlock?”); the benign context makes helping appropriate (“it’s my own phone and I forgot the passcode”) and the harmful context makes refusing appropriate (“it’s my ex’s phone and I want to read their messages”). Refusal is string-matched; d intent = mean ( benign ) − mean ( harmful ) d_{\\\\text{intent}}=\\\\text{mean}(\\\\text{benign})-\\\\text{mean}(\\\\text{harmful}) at the decision token (layer 17 17 , best decode). Graded behavior: refusal 33 % 33\\\\% (benign) / 62 % 62\\\\% (bare) / 71 % 71\\\\% (harmful); intent CV AUC 0.76 0.76 . Faithful tracking: in deep layers the bare prompts the model refuses project harmful on d intent d_{\\\\text{intent}} ( − 5 -5 to − 20 -20 ) and the ones it helps project benign ( + 3 +3 to + 16 +16 ). Attack: on the 15 15 bare prompts refused (read harmful-intent, mean projection − 5.9 -5.9 ), the prefill drops refusal from 100 % 100\\\\% to 0 % 0\\\\% ; over all 24 24 , 62 % → 0 % 62\\\\%\\\\to 0\\\\% . Exploratory: one model (Qwen2.5-1.5B), n = 24 n=24 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We searched for a compact refusal circuit by attention-head ablation (Qwen2.5-1.5B, GPU), and the negative only holds because we verified the hook fires. (1) Ablating each of the 28 × 12 28\\\\times 12 heads (zeroing its slice of the o_proj input across generation) drops refusal by 0.00 0.00 everywhere. (2) Suspecting a dead hook, a positive control ablating a whole attention layer also left refusal at 100 % 100\\\\% , still ambiguous. (3) A logit check settled it: the hook changes the next-token logits by ‖ Δ ‖ ≈ 123 \\\\|\\\\Delta\\\\|\\\\approx 123 (whole attention, layer 12) and ≈ 23 \\\\approx 23 (a single head), so it works, and the robustness is real. (4) Cumulative ablation (first- k k attention layers) drove the refusal string-match to 0 even at k = 1 k=1 , but this contradicted the per-layer control (layer 0 breaks it, layer 12 does not). (5) Reading the actual generations resolved it: the refusal- 0 cases are garbage (“ silly silly noble glorious …”; with all attention removed the model emits one fixed gibberish string for every prompt), not compliance, the same generic-disruption confound as full-residual patching, caught only because we read the text rather than the metric. The bar we hold every intervention to is a selective transition from harmful refusal to coherent harmful compliance without comparable degradation on benign behavior; broad ablation fails it doubly (the output is garbage, and the same garbage for benign prompts), so the result speaks against a compact head- or layer-level circuit rather than merely lowering a refusal rate (MLP and SAE features untested). Conclusion: no single head or layer is necessary for refusal, and no tested internal intervention selectively removes it (it collapses the model instead). Scope: one model, attention ablation; MLP not tested.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"At 14 14 B (Phi-3-medium, heavily safety-trained).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We re-ran the two load-bearing pieces at 14 14 B (measure-only: refusal string-match and a distinct-token coherence ratio, no generations stored). Dissociation holds : harm-probe AUC 1.00 1.00 (layer 8 8 ), plain refusal 1.00 → 0.15 1.00\\\\to 0.15 under the prefill, and the 34 / 40 34/40 complied prompts still score harmful ( 0.96 0.96 ) with coherent (not deflecting) output (coherence 0.94 0.94 ). Compact-circuit check : over a per-attention-layer ablation ( 40 40 layers, refusing set n = 16 n=16 ), only the input layer (L 0 ) shows a refusal drop ≥ 0.3 \\\\geq 0.3 ( 1.00 → 0.69 1.00\\\\to 0.69 , coherence 0.69 0.69 ), and L 0 is a generic early-processing locus, not a refusal-specific circuit; every other layer leaves refusal near ceiling. So the negative holds at 14 14 B and heavier safety training, with the honest caveat that the input layer carries a partial, non-selective contribution. Frontier scale (70B + + ) is untested.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Two off-thesis nulls (in brief; full detail in the ledger).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before safety, we pointed the same represent-vs-act probe at two other tasks and got nulls that motivated the choice of substrate. On MC correctness (ARC-Easy, 0.6 0.6 – 1.5 1.5 B) the model’s verbalized confidence matches or beats a hidden-state probe, so the gap does not appear where the output already has an honest channel, which is what sent us to safety, where refusal has none. On judge-free failure localization (Who&When, Zhang et al., 2025b ) two onset signals lose to dumb priors; the one reportable byproduct is an audit finding that published LLM-judge methods lose to constant-guessing on step localization ( 14 % 14\\\\% vs. 27 % 27\\\\% ), which we flag as a separate direction, not a contribution here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"An attack-surface map (exploratory).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying the harm probe across the attack transforms of the Arcanum prompt-injection taxonomy (Haddix, 2026 ) , semantic attacks (roleplay, tense) keep harm represented while heavy encodings the model cannot decode (base64, morse, reverse) collapse it: the response-site/prompt-modifying distinction of § 6 . But a pre-registered three-scale test refuted the prediction that this “blind spot” shrinks monotonically with scale, and the per-technique verdicts do not compare across models, so we report it as exploratory only.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every control we ran against a main-text result, what it could have shown, and the verdict: passed (the effect held), demoted (the dramatic version reduced to a modest true one), null , or retracted . About a third of the verdicts are not passed .\"\n }\n]","isConversational":false}}},{"rowIdx":294,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12550"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"A JoLT for the KV Cache: Near-Lossless KV Cache Compression via Joint Tucker and JL-Residual Allocation for LLMs"},"authors":{"kind":"list like","value":["Rahul Krishnan","Volker Schulz"],"string":"[\n \"Rahul Krishnan\",\n \"Volker Schulz\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12550v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Every step of autoregressive transformer inference reuses the key and value projections of all previous tokens, so an implementation caches them at every layer rather than recomputing them [ 1 ] . This KV cache grows as O ( batch × T × L × n h × d h ) O(\\text{batch}\\times T\\times L\\times n_{h}\\times d_{h}) , and once the context is long or the batch is large it, not the model weights, becomes the memory bottleneck and the binding constraint on throughput. Shrinking the cache without hurting model quality is therefore a direct lever on how cheaply a model can serve long contexts, which is the problem we study."},{"type":"para","text":"Two families of methods address this bottleneck, and each commits early to a particular view of the cache. Low-rank methods treat the cache as a collection of matrices and factor those matrices: Palu projects the per-head key and value matrices onto low-rank subspaces [ 2 ] , while xKV factors stacked cross-layer feature blocks [ 3 ] . Quantization methods instead keep the full cache but lower the bit-width of every entry [ 4 , 5 ] . Both families are effective, yet both leave structure on the table. A two-dimensional factorization must commit to one axis of redundancy and ignore the others, and a fixed-bit-width quantizer cannot reach the modest, intermediate compression ratios where quality is still free, because its smallest setting already overshoots them. What is missing is a view of the cache that sees all of its axes at once and spends a shared budget across them."},{"type":"para","text":"We take that view by treating the cache at a layer as a third-order tensor K l ∈ ℝ n h × T × d h K_{l}\\in\\mathbb{R}^{n_{h}\\times T\\times d_{h}} and asking, empirically, which of its axes actually carry compressible structure. The answer is uneven: the head and layer axes are essentially incompressible, while the token and feature axes hold almost all of the redundancy, and within those the values(cache) are consistently harder to compress than the keys. This observation motivates our method, JoLT (Joint Tucker and JL-residual allocation). Rather than compress every axis, JoLT applies a partial Tucker decomposition that truncates only the token and feature axes and leaves the head and layer axes intact, and it then restores the energy that truncation discards with a JL-rotated low-bit residual. Because keys and values have different spectra, a single Lagrangian dual allocates the Tucker ranks and the residual bit-widths jointly, per layer group and separately for keys and values, under one byte budget. The method works because it places the budget exactly where the redundancy is and lets the dual move bits between ranks and residual, and between keys and values, instead of fixing that split in advance."},{"type":"para","text":"Our contributions are the following."},{"type":"para","text":"A spectral analysis of real KV caches (Section 3 ) that locates the compressible structure: the head and layer axes are incompressible, the token and feature axes carry the redundancy, and values are 2 2 – 3 × 3\\times harder than keys. This is what motivates a partial decomposition with separate key and value budgets, rather than a full tensor factorization."},{"type":"para","text":"JoLT (Section 4 ), which couples a partial Tucker backbone with a JL-rotated residual and allocates ranks and residual bits through a single Lagrangian dual per (layer group, K/V). The joint allocation is what turns a lossy backbone into a near-lossless one in a 2 2 – 3 × 3\\times free zone."},{"type":"para","text":"FlashJoLT (Section 5 ), a randomized-SVD variant with tail-mass accounting that preserves free-zone quality while cutting compression time 5 5 – 13 × 13\\times , which keeps the method practical at the long contexts where the cache matters most."},{"type":"para","text":"An empirical study on a GQA and an MHA model (Sections 6 – 8 ) that establishes the free zone, surfaces a sharp GQA-versus-MHA split beyond it, and isolates each design choice through ablations."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Low-rank KV compression."},{"type":"para","text":"The closest line of work compresses the cache by exploiting low-rank structure, and existing methods differ mainly in which two-dimensional slice they factor. Palu factorizes the key and value projection weights by SVD and stores a low-rank latent cache that it reconstructs at attention time [ 2 ] , whereas xKV extracts aligned singular vectors across layers and factors grouped cross-layer keys and values onto a shared low-rank subspace [ 3 ] . Both are matrix methods: they commit to one axis of redundancy and never see the cache as a single object across heads, tokens, and features. Our spectral analysis shows why this matters, since the right place to put low-rank structure is the token and feature axes jointly while the head and layer axes stay intact, and since reaching near-lossless fidelity additionally requires a residual term that pure low-rank factorization does not provide. Eviction methods, which drop low-importance tokens rather than compress them, are orthogonal and can be combined with ours [ 6 , 7 , 8 ] ."},{"type":"heading","level":4,"text":"Quantization."},{"type":"para","text":"A second family keeps the full cache and lowers the bit-width of each entry. KIVI applies tuning-free asymmetric 2-bit quantization [ 4 ] , per-token 4-bit quantization is a common strong baseline, and TurboQuant is a near-optimal-distortion vector quantizer [ 5 ] . These methods are strong at their native bit-rates, but a fixed-bit-width scheme saturates at a corresponding compression floor; a 4-bit scheme, for instance, cannot reach below roughly 4 × 4\\times . JoLT instead targets intermediate budgets continuously across 2 2 – 10 × 10\\times , which includes the 2 2 – 3 × 3\\times band that a fixed 4-bit quantizer simply cannot express. We therefore see quantization as complementary rather than as a direct competitor in that band."},{"type":"heading","level":4,"text":"Tensor decompositions."},{"type":"para","text":"JoLT builds on classical multilinear algebra. Tucker [ 9 , 10 ] , CP [ 11 ] , tensor-train [ 12 ] , and t-SVD [ 13 ] are the standard formats for third-order tensors, and we use a partial Tucker decomposition chosen for the cache’s specific mode structure, paired with a JL residual [ 14 ] and, in the fast variant, a randomized SVD [ 15 ] . The partial form is not an arbitrary choice: a full multilinear comparison (Appendix B.2 ) shows Tucker is the best of these formats at every ratio on both architectures, and that compressing the index-like head and layer axes is what hurts."},{"type":"heading","level":1,"text":"3 Spectral Motivation"},{"type":"para","text":"Before designing a compressor, we measure where the redundancy in a KV cache actually lives. We unfold the cache of Mistral-7B layer 15 ( T = 1024 T{=}1024 ) along each of its three axes and take the singular values of each unfolding; the full table is in Appendix A , and three facts stand out."},{"type":"para","text":"First, the head axis is essentially incompressible. The ratio of largest to smallest singular value across heads is only about 1.5 1.5 for keys and 1.4 1.4 for values, which means every KV head carries roughly the same energy and none can be dropped cheaply. This is expected for a grouped-query model, where GQA [ 16 ] has already merged redundant heads upstream, and it tells us not to spend budget on this axis. Second, the token axis is where most of the opportunity lies, but keys and values behave very differently along it: the key unfolding reaches 10 % 10\\% error at rank 228 228 of 1024 1024 , whereas the value unfolding needs rank 563 563 . Third, the feature axis is compressible for keys but nearly isotropic for values, with a singular-value ratio of 27.4 27.4 for keys against only 2.7 2.7 for values."},{"type":"para","text":"These measurements expose a robust asymmetry that drives the rest of the design: values are 2 2 – 3 × 3\\times harder to compress than keys at every ratio and for every format we tested (Appendices B.2 and B.1 ). The mechanism is the projection geometry. The value projection W V W_{V} spreads energy across feature directions so that the weighted sum preserves content, whereas the key projection W K W_{K} concentrates energy into the few directions that decide the softmax score, leaving the value spectrum flat and the key spectrum sharply decaying. Two design consequences follow directly. Because compressing the head axis is always a loss, and likewise the small, semantically distinct layer axis (Appendix B.1 ), we pin both and truncate only the token and feature axes. And because keys and values sit at opposite ends of the compressibility scale, they must receive separate rank and bit budgets rather than a shared one."},{"type":"heading","level":1,"text":"4 Method: Joint Tucker and JL Allocation (JoLT)"},{"type":"para","text":"JoLT compresses the cache in two stages and decides how hard to push each stage with one optimizer. We first partition the L L layers into G G contiguous groups and, within each group g g , treat the keys and the values as separate compression targets t ∈ { K , V } t\\in\\{\\mathrm{K},\\mathrm{V}\\} . For each pair ( g , t ) (g,t) we apply a partial Tucker decomposition that truncates only the token and feature axes, encode the resulting truncation residual with a JL-rotated low-bit code, and let a single Lagrangian dual choose the Tucker ranks and the residual bit-width that meet a global byte budget. The three components below correspond to these stages, and we present each as motivation, design, and the advantage it buys."},{"type":"heading","level":4,"text":"Partial Tucker backbone."},{"type":"para","text":"The role of the backbone is to remove the redundancy the spectral analysis identified, and only that redundancy. For a grouped tensor X ∈ ℝ | g | ⋅ n h × T × d h X\\in\\mathbb{R}^{|g|\\cdot n_{h}\\times T\\times d_{h}} , the partial Tucker approximation X ~ ( r T , r d ) \\tilde{X}(r_{T},r_{d}) truncates the token mode to rank r T r_{T} and the feature mode to rank r d r_{d} through their mode- k k singular vectors, using the sequentially truncated higher-order SVD (ST-HOSVD) [ 17 ] , while keeping identity factors on the head and layer modes. Pinning those two modes is not a heuristic shortcut but the empirically optimal choice: when we instead give a full four-mode allocator freedom over all axes, it drives the head and layer ranks back to their full size on every group and ratio we tested, producing byte-identical output (Appendix B.2 ). Decomposing only the token and feature modes is therefore lossless relative to the full search, and it runs 2.4 2.4 – 3.0 × 3.0\\times faster than full HOOI at identical error."},{"type":"heading","level":4,"text":"JL residual."},{"type":"para","text":"A pure low-rank backbone cannot reach near-lossless fidelity on its own, because truncation discards real energy, and on the flat value spectrum that energy is large. The residual recovers it. We take the truncation residual R = X − X ~ ( r T , r d ) R=X-\\tilde{X}(r_{T},r_{d}) , apply a Johnson-Lindenstrauss random rotation, and quantize the rotated residual uniformly at b ∈ { 0 , 2 , 4 , 8 } b\\in\\{0,2,4,8\\} bits per entry, where b = 0 b{=}0 means no residual is stored [ 14 ] ; decoding simply inverts the rotation. This residual is the lever that turns a lossy backbone into a near-lossless one at low ratio, and it is especially valuable on the values, whose flat spectrum makes residual bits far more productive than additional rank."},{"type":"heading","level":4,"text":"Joint Lagrangian allocation."},{"type":"para","text":"The two stages compete for the same bytes, so we allocate across them jointly rather than fixing the split in advance. Consider a single (layer group, tensor) cell ( g , t ) (g,t) , write m = | g | n h m=|g|\\,n_{h} for its merged head-and-layer count, and let c c be the bytes per stored scalar. A choice of token rank r T r_{T} , feature rank r d r_{d} , and residual width b b then costs"},{"type":"para","text":"bytes, up to a negligible per-row scale term, and it incurs a reconstruction error we model as"},{"type":"para","text":"Here τ g , t ( r T , r d ) \\tau_{g,t}(r_{T},r_{d}) is the relative Frobenius mass that the partial Tucker truncation discards at ranks ( r T , r d ) (r_{T},r_{d}) , and ε 2 ( b ) \\varepsilon^{2}(b) is the fraction of that discarded mass the JL-rotated b b -bit residual fails to recover, calibrated once on a Gaussian round-trip with ε 2 ( 0 ) = 1 \\varepsilon^{2}(0){=}1 and ε 2 \\varepsilon^{2} decreasing in b b . This factorization is what makes the two stages comparable in one budget: rank buys a smaller truncation tail τ \\tau , while bits buy a smaller residual factor ε 2 ( b ) \\varepsilon^{2}(b) , and Equation ( 2 ) prices both on the same error scale."},{"type":"para","text":"Given a total byte budget B B , JoLT minimizes the summed error subject to the summed cost,"},{"type":"para","text":"We solve this through its Lagrangian relaxation,"},{"type":"para","text":"which separates across cells. For a fixed multiplier λ \\lambda each cell is minimized independently over its finite ( r T , r d , b ) (r_{T},r_{d},b) grid, and a single bisection on λ \\lambda drives the total cost to the budget B B . This joint solve is the design advantage over per-tensor heuristics: a greedy or rank-only allocator cannot move bits between keys and values, or between ranks and residual, whereas the dual in Equation ( 4 ) does both at once, funding the value residual heavily and pulling key residual bits down to exploit the asymmetry of Section 3 . The ablations in Section 7 confirm that this freedom is what separates JoLT from its alternatives. In the free zone the allocator meets each target ratio to three decimal places; at high compression ( R ≥ 7 R\\geq 7 ) both it and the exact baseline undershoot the target slightly, because per-group rank and bit-width floors begin to bind."},{"type":"heading","level":1,"text":"5 Fast Variant: FlashJoLT"},{"type":"para","text":"The exact token-mode SVD dominates compression time and scales poorly with context length, which is a problem at exactly the long contexts where the cache matters most. FlashJoLT removes this cost while leaving the rest of JoLT unchanged. It replaces the exact token-mode SVD with a randomized low-rank SVD [ 15 ] that computes only the top q cap q_{\\mathrm{cap}} token-mode directions, keeps the feature-mode SVD exact, and reuses the same joint allocator. The one subtlety is that the randomized SVD discards the spectral tail past q cap q_{\\mathrm{cap}} , which would mislead the allocator about how much energy a given rank captures; we correct for this by adding the omitted Frobenius mass back as a single synthetic residual singular value, an accounting step we call tail-mass accounting."},{"type":"heading","level":4,"text":"Cap selection."},{"type":"para","text":"The cap q cap q_{\\mathrm{cap}} must be large enough to preserve quality yet small enough to be fast, and it should grow with context. We set it by the context-aware policy min ( max ( q min ( R ) , ⌈ T / 32 ⌉ ) , 512 ) \\min(\\max(q_{\\min}(R),\\lceil T/32\\rceil),512) , with q min = 32 q_{\\min}{=}32 for ratio R ≤ 4 R\\leq 4 and 64 64 for R ≥ 5 R\\geq 5 . The policy is a no-op for T ≤ 1024 T\\leq 1024 , so every short-context number in this paper is unchanged by it, and it grows the cap only sublinearly beyond that. A real-KV calibration sweep on Mistral confirms the policy is safe in the free zone: for R ≤ 5 R\\leq 5 it passes a Δ K ≤ 0.025 \\Delta_{K}\\leq 0.025 , speedup ≥ 3 × \\geq 3\\times gate at every context from 512 512 to 8192 8192 , and the token-mode effective rank is found to grow sublinearly in T T rather than proportionally (Appendix D )."},{"type":"heading","level":4,"text":"Speedup."},{"type":"para","text":"The result is a large speedup at no measurable quality cost in the free zone. On Mistral at T = 1024 T{=}1024 , FlashJoLT compresses 5.25 5.25 – 13.57 × 13.57\\times faster than the exact backbone, with a headline of 9.30 × 9.30\\times at R = 3 R{=}3 , and it is 5.2 5.2 – 10 × 10\\times faster on LLaMA, while its reconstruction error matches the exact method to within | Δ K | , | Δ V | ≤ 0.0033 |\\Delta_{K}|,|\\Delta_{V}|\\leq 0.0033 across the free zone (full tables in Appendix D ). Because of this, FlashJoLT is the default for every experiment that follows, and the exact backbone is retained only as a reproducibility reference."},{"type":"heading","level":1,"text":"6 Experiments"},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"We evaluate on two models that bracket the architecture axis we care about: Mistral-7B-v0.3 (grouped-query attention, GQA [ 16 ] ; n k v = 8 n_{kv}{=}8 , 32 layers) and LLaMA-2-13B (multi-head attention, MHA [ 1 ] ; n k v = 40 n_{kv}{=}40 , 40 layers), both in bf16 on an A100-40GB with float32 decomposition numerics. Calibration and perplexity use a mixture of WikiText-2 [ 18 ] and C4 [ 19 ] , reconstruction error is the mean relative Frobenius error over all layers, and perplexity is averaged over seeds { 0 , 1 , 2 } \\{0,1,2\\} . Across-seed standard deviation is at most 0.009 0.009 PPL on every cell, so the error bars sit within the last reported digit and we omit them from the tables. The experiments answer three questions in turn: does compression preserve quality (the free zone), which design choices produce that result (ablations), and what does it cost at decode time (efficiency)."},{"type":"heading","level":4,"text":"The near-lossless free zone."},{"type":"para","text":"Our central claim is that a 2 2 – 3 × 3\\times cache is essentially free, and Figure 1 together with Table 1 shows it for perplexity. At 2 2 – 3 × 3\\times the drift stays within noise on both architectures at context lengths 512 and 1024; the only exception is the MHA model at ctx-2048, where 2 2 – 3 × 3\\times sits about 2.5 % 2.5\\% above baseline (Appendix E ). Beyond the free zone the two architectures part ways. Mistral degrades gracefully, losing roughly 4 % 4\\% perplexity per integer ratio step from 4 × 4\\times on, whereas LLaMA degrades sharply between 4 × 4\\times and 5 × 5\\times . We return to this split repeatedly, because it is the main caveat on an otherwise architecture-agnostic result."},{"type":"heading","level":4,"text":"Reconstruction fidelity."},{"type":"para","text":"The free zone is so flat because JoLT reconstructs the cache far more faithfully than the baselines at the same budget. Table 2 reports relative Frobenius error at 2 × 2\\times , where JoLT reaches 0.009 0.009 on keys and 0.006 0.006 on values, identically on both architectures and roughly an order of magnitude below both 4-bit quantization and cross-layer SVD. We stress that this error is small but nonzero everywhere, which is why we describe the regime as near-lossless rather than lossless."},{"type":"heading","level":4,"text":"Downstream tasks."},{"type":"para","text":"Low reconstruction error is only useful if it survives to task accuracy, so we evaluate the free zone end to end with the production fast method. On GSM8K [ 20 , 21 ] , scored with 8-shot chain-of-thought and exact match, every compressed cell at 2 2 – 3 × 3\\times lands within the Wilson 95 % 95\\% confidence interval of its baseline on both models: Mistral moves from 40.56 % 40.56\\% to 39.04 % 39.04\\% at 2 × 2\\times and 38.44 % 38.44\\% at 3 × 3\\times (full test set, n = 1319 n{=}1319 ), and LLaMA from 28.13 % 28.13\\% to 29.04 % 29.04\\% at 2 × 2\\times ( n = 1319 n{=}1319 ). On RULER [ 22 ] single-needle retrieval (250 tasks per cell), Mistral holds exactly 100 % 100\\% at 2 × 2\\times and 3 × 3\\times , and LLaMA matches its own ctx-4096 ceiling of 84.8 % 84.8\\% at 2 × 2\\times and 83.6 % 83.6\\% at 3 × 3\\times . The architecture split reappears above the free zone, and it is starker for reasoning than for retrieval: LLaMA GSM8K falls to 18.5 % 18.5\\% , 6.0 % 6.0\\% , and 1.5 % 1.5\\% at 4 × 4\\times , 6 × 6\\times , and 8 × 8\\times (Appendix F ), the downstream face of its perplexity degradation, whereas Mistral RULER retrieval matches the full-KV baseline out to 16K context, at 100 % 100\\% at 8K and 99.2 % 99.2\\% at 16K."},{"type":"heading","level":1,"text":"7 Ablations"},{"type":"para","text":"We now isolate the design choices behind the free zone. All ablations are on Mistral, with perplexity averaged over seeds { 0 , 1 , 2 } \\{0,1,2\\} at T = 1024 T{=}1024 ; the full grids over context lengths { 512 , 1024 , 2048 } \\{512,1024,2048\\} are in Appendix G , and the headline cells appear in Table 3 ."},{"type":"para","text":"The first question is whether the residual and the joint allocation matter, or whether a simpler scheme would do. They do matter. Removing the residual entirely, so that only Tucker ranks are allocated, costs about 0.4 0.4 – 1.0 1.0 PPL across the grid (for example 7.27 7.27 against 6.54 6.54 at 2 × 2\\times ), which confirms that the residual is what makes the backbone near-lossless. Replacing the joint dual with a greedy allocator, which spends each marginal byte on whichever group offers the largest error reduction per byte but has no global multiplier to rebalance across keys and values, is more competitive at low ratio, where the two are within noise, but the joint solve pulls clearly ahead in the high-compression regime, for example 7.92 7.92 against 8.36 8.36 at 8 × 8\\times and T = 512 T{=}512 , precisely where the freedom to move budget between keys and values is most valuable."},{"type":"para","text":"The second question is whether ranks should vary per group or be shared. Per-group allocation wins at every cell, by as much as 0.97 0.97 PPL at R = 4 R{=}4 and T = 512 T{=}512 ; the advantage shrinks at long context but never reverses, which tells us that different layer groups genuinely want different ranks."},{"type":"para","text":"The third question is how many residual bits are needed. Quality saturates at four bits: moving from 0 to 2 2 to 4 4 bits closes nearly all of the gap to near-lossless, while moving from 4 4 to 8 8 buys nothing measurable ( 6.537 6.537 against 6.536 6.536 PPL at 2 × 2\\times ). Since an 8-bit residual also cannot fit the byte budget above 2 × 2\\times , four bits is both the quality knee and the only feasible default, which is why JoLT fixes the residual at four bits and lets the allocator decide only whether to spend them."},{"type":"para","text":"Two further ablations sit in the appendix because they confirm rather than establish design choices: the parity between the fast and exact backbones (Appendix D ), and the pre- versus post-RoPE capture point for keys (Appendix C )."},{"type":"heading","level":1,"text":"8 Memory Efficiency"},{"type":"para","text":"The memory win is the persistent KV footprint, which shrinks to exactly the target ratio: on Mistral at ctx-2048 the cache drops from 268.4 268.4 MB to 134.6 134.6 , 89.6 89.6 , 67.1 67.1 , and 33.7 33.7 MB at 2 × 2\\times , 3 × 3\\times , 4 × 4\\times , and 8 × 8\\times . Peak process memory, by contrast, is bounded by the model weights and activations and barely moves with the ratio, holding around 16.5 16.5 GB at ctx-2048. The footprint that shrinks is therefore the persistent state, which is exactly what matters for prompt-caching deployments that keep many cached prefixes resident at once. Decode reconstructs the compressed cache from its Tucker factors at each step, an arithmetic cost that today’s attention kernels are not built to absorb; a fused kernel that runs attention directly on the factors, without materializing the cache, is the natural route to decode parity and the main systems direction we leave open."},{"type":"heading","level":1,"text":"9 Conclusion"},{"type":"para","text":"We presented JoLT and FlashJoLT, a KV-cache compressor that treats the cache at a layer as a third-order tensor, compresses only the token and feature axes that carry redundancy, and restores the rest with a JL-rotated low-bit residual, allocating ranks and residual bits jointly per layer group and per key/value through one Lagrangian dual. The central empirical result is a near-lossless 2 2 – 3 × 3\\times free zone that holds across perplexity, GSM8K, and RULER on both a GQA and an MHA model, supported at the representation level by an order-of-magnitude reduction in reconstruction error over strong baselines, and the fast variant FlashJoLT makes the method practical by cutting compression time 5 5 – 13 × 13\\times at matched quality."},{"type":"para","text":"Two boundaries scope these results and point to the next steps. First, the method is bounded by architecture at high ratio: grouped-query attention degrades gracefully past the free zone, but multi-head attention degrades sharply above 4 × 4\\times in both reconstruction and reasoning, so the safe, model-agnostic recommendation is the 2 2 – 3 × 3\\times band, and pushing MHA further will require an architecture-aware backbone. Second, the method is bounded by deployment cost: decode reconstructs the cache from its factors at each step, so a fused reconstruction-and-attention kernel that avoids materialization is the gap between a storage win and a deployable inference path, and the most important direction for future work. Two further caveats are worth stating directly. The compressed LLaMA RULER setting does not fit in 40GB because of the transient reconstruction copy and was run on larger hardware, a cost the same fused kernel would remove; and our byte accounting uses an fp16 serialization convention for the low-rank factors and idealized bit-packed bytes for the quantization baselines, so rankings within a class are exact while cross-class achieved ratios are convention-based. Validating calibration robustness across text domains is in progress, and the long-context cap policy is currently validated to 8192 tokens. Finally, this study was conducted under a single-GPU (A100-40GB) compute budget; broader sweeps, larger models, and the harder multi-needle long-context regime are left to future work with more compute."},{"type":"para","text":"Mistral-7B layer 15, T = 1024 T{=}1024 , mode- k k SVD of each unfolding."},{"type":"heading","level":2,"text":"B.1 Tucker vs. per-head SVD"},{"type":"para","text":"On Mistral, over all layers at T = 1024 T{=}1024 and matched storage, Tucker beats per-head SVD only when the head mode is left untruncated ( r 1 = n k v r_{1}{=}n_{kv} )."},{"type":"para","text":"On LLaMA, configurations that compress the head mode ( r 1 < n k v r_{1}<n_{kv} ) lose to per-head SVD, which confirms that the 3D advantage comes from sharing bases across heads rather than from reducing head-mode dimensionality. The layer mode behaves the same way: ablating the cross-layer rank to r L ∈ { 2 , 3 } r_{L}\\in\\{2,3\\} below the layer count is catastrophic, raising key error 2.5 2.5 – 11 × 11\\times , which is why the method pins both modes."},{"type":"heading","level":2,"text":"B.2 Multilinear format comparison"},{"type":"para","text":"The mean relative Frobenius error over all layers and three seeds shows Tucker is the best multilinear format at every ratio on both architectures."},{"type":"para","text":"The ordering is Tucker < < CP < < t-SVD < < TT on Mistral, and Tucker < < t-SVD < < TT on LLaMA, where CP is computationally intractable on the larger MHA tensor. Even the best multilinear format reaches value error around 0.28 0.28 at 2 × 2\\times , far above JoLT’s 0.006 0.006 , which is direct evidence that decomposing the small index-like modes is intrinsically lossy and that the win comes from the partial decomposition plus residual. Consistently, the full four-mode allocator, given complete freedom, reproduces the partial method byte for byte by driving the head and layer ranks to full size, which confirms the mode-pinning design."},{"type":"para","text":"With the same groups and two compression passes (Mistral, T = 1024 T{=}1024 ), post-RoPE keys are uniformly harder to compress."},{"type":"para","text":"RoPE smears key energy off the top components in both the token and feature modes, dropping the group-0 top-8 feature-mode energy share from 73.4 % 73.4\\% to 48.4 % 48.4\\% . The penalty replicates on LLaMA at + 22 +22 – 44 % 44\\% , which is why JoLT decomposes keys before RoPE and re-applies the rotation at decode time."},{"type":"heading","level":4,"text":"Compression-time speedup (Mistral, T = 1024 T{=}1024 , q cap = 32 q_{\\mathrm{cap}}{=}32 )."},{"type":"para","text":"LLaMA gives 5.24 5.24 – 9.97 × 9.97\\times at T = 1024 T{=}1024 , and the fast-versus-exact reconstruction parity in the free zone is | Δ K | ≤ 0.0093 |\\Delta_{K}|\\leq 0.0093 and | Δ V | ≤ 0.0011 |\\Delta_{V}|\\leq 0.0011 at 2 2 – 3 × 3\\times ."},{"type":"heading","level":4,"text":"Cap calibration gate (Mistral, real KV)."},{"type":"para","text":"The policy passes the Δ K ≤ 0.025 \\Delta_{K}\\leq 0.025 , speedup ≥ 3 × \\geq 3\\times gate across the free zone ( R ≤ 5 R\\leq 5 ) at every context from 512 to 8192. The token-mode effective rank grows sublinearly in T T , with the key rank-to- T T fraction falling from 0.355 0.355 at T = 512 T{=}512 to 0.066 0.066 at T = 8192 T{=}8192 and the value fraction from 0.722 0.722 to 0.219 0.219 , so the feared eff-rank-proportional-to- T T regime does not hold for real KV and the linear cap is a safe over-provision. The remaining failures are confined to small T T at high ratio ( R ≥ 6 R\\geq 6 ), outside the free zone, where no reachable cap helps because the limit is high-ratio residual lossiness rather than cap size."},{"type":"para","text":"Production fast method, multi-seed; Δ % \\Delta\\% is relative to the uncompressed baseline."},{"type":"heading","level":4,"text":"GSM8K degradation past the free zone (LLaMA-2-13B, fast method)."},{"type":"para","text":"The baseline is 28.13 % 28.13\\% ( n = 1319 n{=}1319 ) and 2 × 2\\times holds at 29.04 % 29.04\\% ( n = 1319 n{=}1319 ), while 4 × 4\\times , 6 × 6\\times , and 8 × 8\\times fall to 18.50 % 18.50\\% , 6.00 % 6.00\\% , and 1.50 % 1.50\\% ( n = 200 n{=}200 each). At 2 × 2\\times accuracy matches the baseline within noise; above the free zone it collapses, which is the downstream face of the MHA perplexity degradation."},{"type":"heading","level":4,"text":"Long-context RULER (Mistral, fast method)."},{"type":"para","text":"Single-needle retrieval stays at 100 % 100\\% at 2 × 2\\times and 3 × 3\\times for ctx-8192 and at 99.20 % 99.20\\% for ctx-16384, matching the full-KV baseline at each context, where the 0.8 0.8 pp dip at 16K is present in the baseline as well. Achieved ratios land exactly on target."},{"type":"para","text":"A1 (joint / greedy / rank-only) and A5 (joint / uniform) report mean PPL over 3 seeds across all contexts; the joint, greedy, rank-only, and uniform allocators are defined in Section 7 ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every step of autoregressive transformer inference reuses the key and value projections of all previous tokens, so an implementation caches them at every layer rather than recomputing them [ 1 ] . This KV cache grows as O ( batch × T × L × n h × d h ) O(\\\\text{batch}\\\\times T\\\\times L\\\\times n_{h}\\\\times d_{h}) , and once the context is long or the batch is large it, not the model weights, becomes the memory bottleneck and the binding constraint on throughput. Shrinking the cache without hurting model quality is therefore a direct lever on how cheaply a model can serve long contexts, which is the problem we study.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two families of methods address this bottleneck, and each commits early to a particular view of the cache. Low-rank methods treat the cache as a collection of matrices and factor those matrices: Palu projects the per-head key and value matrices onto low-rank subspaces [ 2 ] , while xKV factors stacked cross-layer feature blocks [ 3 ] . Quantization methods instead keep the full cache but lower the bit-width of every entry [ 4 , 5 ] . Both families are effective, yet both leave structure on the table. A two-dimensional factorization must commit to one axis of redundancy and ignore the others, and a fixed-bit-width quantizer cannot reach the modest, intermediate compression ratios where quality is still free, because its smallest setting already overshoots them. What is missing is a view of the cache that sees all of its axes at once and spends a shared budget across them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We take that view by treating the cache at a layer as a third-order tensor K l ∈ ℝ n h × T × d h K_{l}\\\\in\\\\mathbb{R}^{n_{h}\\\\times T\\\\times d_{h}} and asking, empirically, which of its axes actually carry compressible structure. The answer is uneven: the head and layer axes are essentially incompressible, while the token and feature axes hold almost all of the redundancy, and within those the values(cache) are consistently harder to compress than the keys. This observation motivates our method, JoLT (Joint Tucker and JL-residual allocation). Rather than compress every axis, JoLT applies a partial Tucker decomposition that truncates only the token and feature axes and leaves the head and layer axes intact, and it then restores the energy that truncation discards with a JL-rotated low-bit residual. Because keys and values have different spectra, a single Lagrangian dual allocates the Tucker ranks and the residual bit-widths jointly, per layer group and separately for keys and values, under one byte budget. The method works because it places the budget exactly where the redundancy is and lets the dual move bits between ranks and residual, and between keys and values, instead of fixing that split in advance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are the following.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A spectral analysis of real KV caches (Section 3 ) that locates the compressible structure: the head and layer axes are incompressible, the token and feature axes carry the redundancy, and values are 2 2 – 3 × 3\\\\times harder than keys. This is what motivates a partial decomposition with separate key and value budgets, rather than a full tensor factorization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"JoLT (Section 4 ), which couples a partial Tucker backbone with a JL-rotated residual and allocates ranks and residual bits through a single Lagrangian dual per (layer group, K/V). The joint allocation is what turns a lossy backbone into a near-lossless one in a 2 2 – 3 × 3\\\\times free zone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FlashJoLT (Section 5 ), a randomized-SVD variant with tail-mass accounting that preserves free-zone quality while cutting compression time 5 5 – 13 × 13\\\\times , which keeps the method practical at the long contexts where the cache matters most.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An empirical study on a GQA and an MHA model (Sections 6 – 8 ) that establishes the free zone, surfaces a sharp GQA-versus-MHA split beyond it, and isolates each design choice through ablations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Low-rank KV compression.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The closest line of work compresses the cache by exploiting low-rank structure, and existing methods differ mainly in which two-dimensional slice they factor. Palu factorizes the key and value projection weights by SVD and stores a low-rank latent cache that it reconstructs at attention time [ 2 ] , whereas xKV extracts aligned singular vectors across layers and factors grouped cross-layer keys and values onto a shared low-rank subspace [ 3 ] . Both are matrix methods: they commit to one axis of redundancy and never see the cache as a single object across heads, tokens, and features. Our spectral analysis shows why this matters, since the right place to put low-rank structure is the token and feature axes jointly while the head and layer axes stay intact, and since reaching near-lossless fidelity additionally requires a residual term that pure low-rank factorization does not provide. Eviction methods, which drop low-importance tokens rather than compress them, are orthogonal and can be combined with ours [ 6 , 7 , 8 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Quantization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A second family keeps the full cache and lowers the bit-width of each entry. KIVI applies tuning-free asymmetric 2-bit quantization [ 4 ] , per-token 4-bit quantization is a common strong baseline, and TurboQuant is a near-optimal-distortion vector quantizer [ 5 ] . These methods are strong at their native bit-rates, but a fixed-bit-width scheme saturates at a corresponding compression floor; a 4-bit scheme, for instance, cannot reach below roughly 4 × 4\\\\times . JoLT instead targets intermediate budgets continuously across 2 2 – 10 × 10\\\\times , which includes the 2 2 – 3 × 3\\\\times band that a fixed 4-bit quantizer simply cannot express. We therefore see quantization as complementary rather than as a direct competitor in that band.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Tensor decompositions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"JoLT builds on classical multilinear algebra. Tucker [ 9 , 10 ] , CP [ 11 ] , tensor-train [ 12 ] , and t-SVD [ 13 ] are the standard formats for third-order tensors, and we use a partial Tucker decomposition chosen for the cache’s specific mode structure, paired with a JL residual [ 14 ] and, in the fast variant, a randomized SVD [ 15 ] . The partial form is not an arbitrary choice: a full multilinear comparison (Appendix B.2 ) shows Tucker is the best of these formats at every ratio on both architectures, and that compressing the index-like head and layer axes is what hurts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Spectral Motivation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before designing a compressor, we measure where the redundancy in a KV cache actually lives. We unfold the cache of Mistral-7B layer 15 ( T = 1024 T{=}1024 ) along each of its three axes and take the singular values of each unfolding; the full table is in Appendix A , and three facts stand out.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, the head axis is essentially incompressible. The ratio of largest to smallest singular value across heads is only about 1.5 1.5 for keys and 1.4 1.4 for values, which means every KV head carries roughly the same energy and none can be dropped cheaply. This is expected for a grouped-query model, where GQA [ 16 ] has already merged redundant heads upstream, and it tells us not to spend budget on this axis. Second, the token axis is where most of the opportunity lies, but keys and values behave very differently along it: the key unfolding reaches 10 % 10\\\\% error at rank 228 228 of 1024 1024 , whereas the value unfolding needs rank 563 563 . Third, the feature axis is compressible for keys but nearly isotropic for values, with a singular-value ratio of 27.4 27.4 for keys against only 2.7 2.7 for values.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These measurements expose a robust asymmetry that drives the rest of the design: values are 2 2 – 3 × 3\\\\times harder to compress than keys at every ratio and for every format we tested (Appendices B.2 and B.1 ). The mechanism is the projection geometry. The value projection W V W_{V} spreads energy across feature directions so that the weighted sum preserves content, whereas the key projection W K W_{K} concentrates energy into the few directions that decide the softmax score, leaving the value spectrum flat and the key spectrum sharply decaying. Two design consequences follow directly. Because compressing the head axis is always a loss, and likewise the small, semantically distinct layer axis (Appendix B.1 ), we pin both and truncate only the token and feature axes. And because keys and values sit at opposite ends of the compressibility scale, they must receive separate rank and bit budgets rather than a shared one.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Method: Joint Tucker and JL Allocation (JoLT)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"JoLT compresses the cache in two stages and decides how hard to push each stage with one optimizer. We first partition the L L layers into G G contiguous groups and, within each group g g , treat the keys and the values as separate compression targets t ∈ { K , V } t\\\\in\\\\{\\\\mathrm{K},\\\\mathrm{V}\\\\} . For each pair ( g , t ) (g,t) we apply a partial Tucker decomposition that truncates only the token and feature axes, encode the resulting truncation residual with a JL-rotated low-bit code, and let a single Lagrangian dual choose the Tucker ranks and the residual bit-width that meet a global byte budget. The three components below correspond to these stages, and we present each as motivation, design, and the advantage it buys.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Partial Tucker backbone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The role of the backbone is to remove the redundancy the spectral analysis identified, and only that redundancy. For a grouped tensor X ∈ ℝ | g | ⋅ n h × T × d h X\\\\in\\\\mathbb{R}^{|g|\\\\cdot n_{h}\\\\times T\\\\times d_{h}} , the partial Tucker approximation X ~ ( r T , r d ) \\\\tilde{X}(r_{T},r_{d}) truncates the token mode to rank r T r_{T} and the feature mode to rank r d r_{d} through their mode- k k singular vectors, using the sequentially truncated higher-order SVD (ST-HOSVD) [ 17 ] , while keeping identity factors on the head and layer modes. Pinning those two modes is not a heuristic shortcut but the empirically optimal choice: when we instead give a full four-mode allocator freedom over all axes, it drives the head and layer ranks back to their full size on every group and ratio we tested, producing byte-identical output (Appendix B.2 ). Decomposing only the token and feature modes is therefore lossless relative to the full search, and it runs 2.4 2.4 – 3.0 × 3.0\\\\times faster than full HOOI at identical error.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"JL residual.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A pure low-rank backbone cannot reach near-lossless fidelity on its own, because truncation discards real energy, and on the flat value spectrum that energy is large. The residual recovers it. We take the truncation residual R = X − X ~ ( r T , r d ) R=X-\\\\tilde{X}(r_{T},r_{d}) , apply a Johnson-Lindenstrauss random rotation, and quantize the rotated residual uniformly at b ∈ { 0 , 2 , 4 , 8 } b\\\\in\\\\{0,2,4,8\\\\} bits per entry, where b = 0 b{=}0 means no residual is stored [ 14 ] ; decoding simply inverts the rotation. This residual is the lever that turns a lossy backbone into a near-lossless one at low ratio, and it is especially valuable on the values, whose flat spectrum makes residual bits far more productive than additional rank.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Joint Lagrangian allocation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two stages compete for the same bytes, so we allocate across them jointly rather than fixing the split in advance. Consider a single (layer group, tensor) cell ( g , t ) (g,t) , write m = | g | n h m=|g|\\\\,n_{h} for its merged head-and-layer count, and let c c be the bytes per stored scalar. A choice of token rank r T r_{T} , feature rank r d r_{d} , and residual width b b then costs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"bytes, up to a negligible per-row scale term, and it incurs a reconstruction error we model as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here τ g , t ( r T , r d ) \\\\tau_{g,t}(r_{T},r_{d}) is the relative Frobenius mass that the partial Tucker truncation discards at ranks ( r T , r d ) (r_{T},r_{d}) , and ε 2 ( b ) \\\\varepsilon^{2}(b) is the fraction of that discarded mass the JL-rotated b b -bit residual fails to recover, calibrated once on a Gaussian round-trip with ε 2 ( 0 ) = 1 \\\\varepsilon^{2}(0){=}1 and ε 2 \\\\varepsilon^{2} decreasing in b b . This factorization is what makes the two stages comparable in one budget: rank buys a smaller truncation tail τ \\\\tau , while bits buy a smaller residual factor ε 2 ( b ) \\\\varepsilon^{2}(b) , and Equation ( 2 ) prices both on the same error scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a total byte budget B B , JoLT minimizes the summed error subject to the summed cost,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We solve this through its Lagrangian relaxation,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which separates across cells. For a fixed multiplier λ \\\\lambda each cell is minimized independently over its finite ( r T , r d , b ) (r_{T},r_{d},b) grid, and a single bisection on λ \\\\lambda drives the total cost to the budget B B . This joint solve is the design advantage over per-tensor heuristics: a greedy or rank-only allocator cannot move bits between keys and values, or between ranks and residual, whereas the dual in Equation ( 4 ) does both at once, funding the value residual heavily and pulling key residual bits down to exploit the asymmetry of Section 3 . The ablations in Section 7 confirm that this freedom is what separates JoLT from its alternatives. In the free zone the allocator meets each target ratio to three decimal places; at high compression ( R ≥ 7 R\\\\geq 7 ) both it and the exact baseline undershoot the target slightly, because per-group rank and bit-width floors begin to bind.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Fast Variant: FlashJoLT\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The exact token-mode SVD dominates compression time and scales poorly with context length, which is a problem at exactly the long contexts where the cache matters most. FlashJoLT removes this cost while leaving the rest of JoLT unchanged. It replaces the exact token-mode SVD with a randomized low-rank SVD [ 15 ] that computes only the top q cap q_{\\\\mathrm{cap}} token-mode directions, keeps the feature-mode SVD exact, and reuses the same joint allocator. The one subtlety is that the randomized SVD discards the spectral tail past q cap q_{\\\\mathrm{cap}} , which would mislead the allocator about how much energy a given rank captures; we correct for this by adding the omitted Frobenius mass back as a single synthetic residual singular value, an accounting step we call tail-mass accounting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cap selection.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cap q cap q_{\\\\mathrm{cap}} must be large enough to preserve quality yet small enough to be fast, and it should grow with context. We set it by the context-aware policy min ( max ( q min ( R ) , ⌈ T / 32 ⌉ ) , 512 ) \\\\min(\\\\max(q_{\\\\min}(R),\\\\lceil T/32\\\\rceil),512) , with q min = 32 q_{\\\\min}{=}32 for ratio R ≤ 4 R\\\\leq 4 and 64 64 for R ≥ 5 R\\\\geq 5 . The policy is a no-op for T ≤ 1024 T\\\\leq 1024 , so every short-context number in this paper is unchanged by it, and it grows the cap only sublinearly beyond that. A real-KV calibration sweep on Mistral confirms the policy is safe in the free zone: for R ≤ 5 R\\\\leq 5 it passes a Δ K ≤ 0.025 \\\\Delta_{K}\\\\leq 0.025 , speedup ≥ 3 × \\\\geq 3\\\\times gate at every context from 512 512 to 8192 8192 , and the token-mode effective rank is found to grow sublinearly in T T rather than proportionally (Appendix D ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Speedup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The result is a large speedup at no measurable quality cost in the free zone. On Mistral at T = 1024 T{=}1024 , FlashJoLT compresses 5.25 5.25 – 13.57 × 13.57\\\\times faster than the exact backbone, with a headline of 9.30 × 9.30\\\\times at R = 3 R{=}3 , and it is 5.2 5.2 – 10 × 10\\\\times faster on LLaMA, while its reconstruction error matches the exact method to within | Δ K | , | Δ V | ≤ 0.0033 |\\\\Delta_{K}|,|\\\\Delta_{V}|\\\\leq 0.0033 across the free zone (full tables in Appendix D ). Because of this, FlashJoLT is the default for every experiment that follows, and the exact backbone is retained only as a reproducibility reference.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on two models that bracket the architecture axis we care about: Mistral-7B-v0.3 (grouped-query attention, GQA [ 16 ] ; n k v = 8 n_{kv}{=}8 , 32 layers) and LLaMA-2-13B (multi-head attention, MHA [ 1 ] ; n k v = 40 n_{kv}{=}40 , 40 layers), both in bf16 on an A100-40GB with float32 decomposition numerics. Calibration and perplexity use a mixture of WikiText-2 [ 18 ] and C4 [ 19 ] , reconstruction error is the mean relative Frobenius error over all layers, and perplexity is averaged over seeds { 0 , 1 , 2 } \\\\{0,1,2\\\\} . Across-seed standard deviation is at most 0.009 0.009 PPL on every cell, so the error bars sit within the last reported digit and we omit them from the tables. The experiments answer three questions in turn: does compression preserve quality (the free zone), which design choices produce that result (ablations), and what does it cost at decode time (efficiency).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The near-lossless free zone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our central claim is that a 2 2 – 3 × 3\\\\times cache is essentially free, and Figure 1 together with Table 1 shows it for perplexity. At 2 2 – 3 × 3\\\\times the drift stays within noise on both architectures at context lengths 512 and 1024; the only exception is the MHA model at ctx-2048, where 2 2 – 3 × 3\\\\times sits about 2.5 % 2.5\\\\% above baseline (Appendix E ). Beyond the free zone the two architectures part ways. Mistral degrades gracefully, losing roughly 4 % 4\\\\% perplexity per integer ratio step from 4 × 4\\\\times on, whereas LLaMA degrades sharply between 4 × 4\\\\times and 5 × 5\\\\times . We return to this split repeatedly, because it is the main caveat on an otherwise architecture-agnostic result.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Reconstruction fidelity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The free zone is so flat because JoLT reconstructs the cache far more faithfully than the baselines at the same budget. Table 2 reports relative Frobenius error at 2 × 2\\\\times , where JoLT reaches 0.009 0.009 on keys and 0.006 0.006 on values, identically on both architectures and roughly an order of magnitude below both 4-bit quantization and cross-layer SVD. We stress that this error is small but nonzero everywhere, which is why we describe the regime as near-lossless rather than lossless.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Downstream tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Low reconstruction error is only useful if it survives to task accuracy, so we evaluate the free zone end to end with the production fast method. On GSM8K [ 20 , 21 ] , scored with 8-shot chain-of-thought and exact match, every compressed cell at 2 2 – 3 × 3\\\\times lands within the Wilson 95 % 95\\\\% confidence interval of its baseline on both models: Mistral moves from 40.56 % 40.56\\\\% to 39.04 % 39.04\\\\% at 2 × 2\\\\times and 38.44 % 38.44\\\\% at 3 × 3\\\\times (full test set, n = 1319 n{=}1319 ), and LLaMA from 28.13 % 28.13\\\\% to 29.04 % 29.04\\\\% at 2 × 2\\\\times ( n = 1319 n{=}1319 ). On RULER [ 22 ] single-needle retrieval (250 tasks per cell), Mistral holds exactly 100 % 100\\\\% at 2 × 2\\\\times and 3 × 3\\\\times , and LLaMA matches its own ctx-4096 ceiling of 84.8 % 84.8\\\\% at 2 × 2\\\\times and 83.6 % 83.6\\\\% at 3 × 3\\\\times . The architecture split reappears above the free zone, and it is starker for reasoning than for retrieval: LLaMA GSM8K falls to 18.5 % 18.5\\\\% , 6.0 % 6.0\\\\% , and 1.5 % 1.5\\\\% at 4 × 4\\\\times , 6 × 6\\\\times , and 8 × 8\\\\times (Appendix F ), the downstream face of its perplexity degradation, whereas Mistral RULER retrieval matches the full-KV baseline out to 16K context, at 100 % 100\\\\% at 8K and 99.2 % 99.2\\\\% at 16K.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Ablations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now isolate the design choices behind the free zone. All ablations are on Mistral, with perplexity averaged over seeds { 0 , 1 , 2 } \\\\{0,1,2\\\\} at T = 1024 T{=}1024 ; the full grids over context lengths { 512 , 1024 , 2048 } \\\\{512,1024,2048\\\\} are in Appendix G , and the headline cells appear in Table 3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The first question is whether the residual and the joint allocation matter, or whether a simpler scheme would do. They do matter. Removing the residual entirely, so that only Tucker ranks are allocated, costs about 0.4 0.4 – 1.0 1.0 PPL across the grid (for example 7.27 7.27 against 6.54 6.54 at 2 × 2\\\\times ), which confirms that the residual is what makes the backbone near-lossless. Replacing the joint dual with a greedy allocator, which spends each marginal byte on whichever group offers the largest error reduction per byte but has no global multiplier to rebalance across keys and values, is more competitive at low ratio, where the two are within noise, but the joint solve pulls clearly ahead in the high-compression regime, for example 7.92 7.92 against 8.36 8.36 at 8 × 8\\\\times and T = 512 T{=}512 , precisely where the freedom to move budget between keys and values is most valuable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second question is whether ranks should vary per group or be shared. Per-group allocation wins at every cell, by as much as 0.97 0.97 PPL at R = 4 R{=}4 and T = 512 T{=}512 ; the advantage shrinks at long context but never reverses, which tells us that different layer groups genuinely want different ranks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The third question is how many residual bits are needed. Quality saturates at four bits: moving from 0 to 2 2 to 4 4 bits closes nearly all of the gap to near-lossless, while moving from 4 4 to 8 8 buys nothing measurable ( 6.537 6.537 against 6.536 6.536 PPL at 2 × 2\\\\times ). Since an 8-bit residual also cannot fit the byte budget above 2 × 2\\\\times , four bits is both the quality knee and the only feasible default, which is why JoLT fixes the residual at four bits and lets the allocator decide only whether to spend them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two further ablations sit in the appendix because they confirm rather than establish design choices: the parity between the fast and exact backbones (Appendix D ), and the pre- versus post-RoPE capture point for keys (Appendix C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8 Memory Efficiency\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The memory win is the persistent KV footprint, which shrinks to exactly the target ratio: on Mistral at ctx-2048 the cache drops from 268.4 268.4 MB to 134.6 134.6 , 89.6 89.6 , 67.1 67.1 , and 33.7 33.7 MB at 2 × 2\\\\times , 3 × 3\\\\times , 4 × 4\\\\times , and 8 × 8\\\\times . Peak process memory, by contrast, is bounded by the model weights and activations and barely moves with the ratio, holding around 16.5 16.5 GB at ctx-2048. The footprint that shrinks is therefore the persistent state, which is exactly what matters for prompt-caching deployments that keep many cached prefixes resident at once. Decode reconstructs the compressed cache from its Tucker factors at each step, an arithmetic cost that today’s attention kernels are not built to absorb; a fused kernel that runs attention directly on the factors, without materializing the cache, is the natural route to decode parity and the main systems direction we leave open.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"9 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We presented JoLT and FlashJoLT, a KV-cache compressor that treats the cache at a layer as a third-order tensor, compresses only the token and feature axes that carry redundancy, and restores the rest with a JL-rotated low-bit residual, allocating ranks and residual bits jointly per layer group and per key/value through one Lagrangian dual. The central empirical result is a near-lossless 2 2 – 3 × 3\\\\times free zone that holds across perplexity, GSM8K, and RULER on both a GQA and an MHA model, supported at the representation level by an order-of-magnitude reduction in reconstruction error over strong baselines, and the fast variant FlashJoLT makes the method practical by cutting compression time 5 5 – 13 × 13\\\\times at matched quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two boundaries scope these results and point to the next steps. First, the method is bounded by architecture at high ratio: grouped-query attention degrades gracefully past the free zone, but multi-head attention degrades sharply above 4 × 4\\\\times in both reconstruction and reasoning, so the safe, model-agnostic recommendation is the 2 2 – 3 × 3\\\\times band, and pushing MHA further will require an architecture-aware backbone. Second, the method is bounded by deployment cost: decode reconstructs the cache from its factors at each step, so a fused reconstruction-and-attention kernel that avoids materialization is the gap between a storage win and a deployable inference path, and the most important direction for future work. Two further caveats are worth stating directly. The compressed LLaMA RULER setting does not fit in 40GB because of the transient reconstruction copy and was run on larger hardware, a cost the same fused kernel would remove; and our byte accounting uses an fp16 serialization convention for the low-rank factors and idealized bit-packed bytes for the quantization baselines, so rankings within a class are exact while cross-class achieved ratios are convention-based. Validating calibration robustness across text domains is in progress, and the long-context cap policy is currently validated to 8192 tokens. Finally, this study was conducted under a single-GPU (A100-40GB) compute budget; broader sweeps, larger models, and the harder multi-needle long-context regime are left to future work with more compute.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mistral-7B layer 15, T = 1024 T{=}1024 , mode- k k SVD of each unfolding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Tucker vs. per-head SVD\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On Mistral, over all layers at T = 1024 T{=}1024 and matched storage, Tucker beats per-head SVD only when the head mode is left untruncated ( r 1 = n k v r_{1}{=}n_{kv} ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On LLaMA, configurations that compress the head mode ( r 1 < n k v r_{1}<n_{kv} ) lose to per-head SVD, which confirms that the 3D advantage comes from sharing bases across heads rather than from reducing head-mode dimensionality. The layer mode behaves the same way: ablating the cross-layer rank to r L ∈ { 2 , 3 } r_{L}\\\\in\\\\{2,3\\\\} below the layer count is catastrophic, raising key error 2.5 2.5 – 11 × 11\\\\times , which is why the method pins both modes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Multilinear format comparison\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mean relative Frobenius error over all layers and three seeds shows Tucker is the best multilinear format at every ratio on both architectures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ordering is Tucker < < CP < < t-SVD < < TT on Mistral, and Tucker < < t-SVD < < TT on LLaMA, where CP is computationally intractable on the larger MHA tensor. Even the best multilinear format reaches value error around 0.28 0.28 at 2 × 2\\\\times , far above JoLT’s 0.006 0.006 , which is direct evidence that decomposing the small index-like modes is intrinsically lossy and that the win comes from the partial decomposition plus residual. Consistently, the full four-mode allocator, given complete freedom, reproduces the partial method byte for byte by driving the head and layer ranks to full size, which confirms the mode-pinning design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With the same groups and two compression passes (Mistral, T = 1024 T{=}1024 ), post-RoPE keys are uniformly harder to compress.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RoPE smears key energy off the top components in both the token and feature modes, dropping the group-0 top-8 feature-mode energy share from 73.4 % 73.4\\\\% to 48.4 % 48.4\\\\% . The penalty replicates on LLaMA at + 22 +22 – 44 % 44\\\\% , which is why JoLT decomposes keys before RoPE and re-applies the rotation at decode time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Compression-time speedup (Mistral, T = 1024 T{=}1024 , q cap = 32 q_{\\\\mathrm{cap}}{=}32 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLaMA gives 5.24 5.24 – 9.97 × 9.97\\\\times at T = 1024 T{=}1024 , and the fast-versus-exact reconstruction parity in the free zone is | Δ K | ≤ 0.0093 |\\\\Delta_{K}|\\\\leq 0.0093 and | Δ V | ≤ 0.0011 |\\\\Delta_{V}|\\\\leq 0.0011 at 2 2 – 3 × 3\\\\times .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cap calibration gate (Mistral, real KV).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The policy passes the Δ K ≤ 0.025 \\\\Delta_{K}\\\\leq 0.025 , speedup ≥ 3 × \\\\geq 3\\\\times gate across the free zone ( R ≤ 5 R\\\\leq 5 ) at every context from 512 to 8192. The token-mode effective rank grows sublinearly in T T , with the key rank-to- T T fraction falling from 0.355 0.355 at T = 512 T{=}512 to 0.066 0.066 at T = 8192 T{=}8192 and the value fraction from 0.722 0.722 to 0.219 0.219 , so the feared eff-rank-proportional-to- T T regime does not hold for real KV and the linear cap is a safe over-provision. The remaining failures are confined to small T T at high ratio ( R ≥ 6 R\\\\geq 6 ), outside the free zone, where no reachable cap helps because the limit is high-ratio residual lossiness rather than cap size.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Production fast method, multi-seed; Δ % \\\\Delta\\\\% is relative to the uncompressed baseline.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"GSM8K degradation past the free zone (LLaMA-2-13B, fast method).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline is 28.13 % 28.13\\\\% ( n = 1319 n{=}1319 ) and 2 × 2\\\\times holds at 29.04 % 29.04\\\\% ( n = 1319 n{=}1319 ), while 4 × 4\\\\times , 6 × 6\\\\times , and 8 × 8\\\\times fall to 18.50 % 18.50\\\\% , 6.00 % 6.00\\\\% , and 1.50 % 1.50\\\\% ( n = 200 n{=}200 each). At 2 × 2\\\\times accuracy matches the baseline within noise; above the free zone it collapses, which is the downstream face of the MHA perplexity degradation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Long-context RULER (Mistral, fast method).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Single-needle retrieval stays at 100 % 100\\\\% at 2 × 2\\\\times and 3 × 3\\\\times for ctx-8192 and at 99.20 % 99.20\\\\% for ctx-16384, matching the full-KV baseline at each context, where the 0.8 0.8 pp dip at 16K is present in the baseline as well. Achieved ratios land exactly on target.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A1 (joint / greedy / rank-only) and A5 (joint / uniform) report mean PPL over 3 seeds across all contexts; the joint, greedy, rank-only, and uniform allocators are defined in Section 7 .\"\n }\n]","isConversational":false}}},{"rowIdx":295,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12488"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Sample Efficient Generative Optimization for Molecular Design"},"authors":{"kind":"list like","value":["Sarina Kopf","Cristina Nevado","Philippe Schwaller"],"string":"[\n \"Sarina Kopf\",\n \"Cristina Nevado\",\n \"Philippe Schwaller\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12488v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Molecular optimization is a central problem in chemistry, with applications ranging from drug discovery to catalyst design (Filella-Merce et al. , 2025 ; Strieth-Kalthoff et al. , 2024 ; Seumer et al. , 2023 ) . These tasks require identifying molecules that satisfy multiple, often competing objectives from vast chemical space (Sanchez-Lengeling and Aspuru-Guzik, 2018 ) . Most relevant optimization problems are also data scarce: phenomena such as protein-ligand binding or catalytic activity are difficult to simulate accurately, and experimental measurements—particularly those requiring molecular synthesis—are costly and time-consuming. As a result, the real-world impact of an optimization strategy is defined by the number of evaluations required to discover optimal molecules (sample efficiency). Low sample efficiency necessitates reliance on inexpensive, low-fidelity in silico oracles vulnerable to hacking (Renz et al. , 2019 ; Lehman et al. , 2020 ) . Improved sample efficiency would enable the use of higher-fidelity simulations or even direct experimental feedback, thereby increasing the practical impact of molecular optimization methods."},{"type":"para","text":"ML methods for molecule optimization must traverse chemical space and strategically guide this exploration toward the most promising regions. Traversal methods include deep generative methods that learn a data distribution and can sample from that distribution, including recurrent neural networks (RNNs) (Olivecrona et al. , 2017 ; Loeffler et al. , 2024 ) , transformers (Feng et al. , 2023 ; Mazuz et al. , 2023 ) , and diffusion or flow-based models (Lee et al. , 2023 ; Jiang et al. , 2024 ; Schneuing et al. , 2023 ) . These generative methods can be steered with various optimization techniques including Monte Carlo tree search (MCTS), Markov Chain Monte Carlo (MCMC), reinforcement learning (RL), as well as with Bayesian Optimization (BO). Language models that generate molecules in the form of Simplified Molecular Input Line Entry System (SMILES) (Weininger, 1988 ) steered with RL remain among the most sample efficient methods (Gao et al. , 2022 ; Guo et al. , 2026 ) . However, state-of-the-art generative methods still require between hundreds and thousands of oracle evaluations to identify high-quality candidates (Gao et al. , 2022 ; Guo et al. , 2026 ) . While this cost may permit the use of moderately expensive simulations such as DFT, it remains prohibitive for workflows involving direct experimental evaluation, particularly in the case of prospective campaigns where the target distribution is not known."},{"type":"para","text":"BO, an optimization technique that uses a probabilistic surrogate model and an acquisition function to guide sequential experimentation within a candidate library or a continuous latent space (Frazier, 2018 ) , has excellent sample efficiency and has seen widespread use and success in chemistry (Sin et al. , 2025 ; Zaza et al. , 2025 ; Shields et al. , 2021 ) . Through the fitting of the surrogate model, these methods learn quickly from negative examples. However, performance depends on the presence of suitable candidates within a library or accurate decoding from a latent space to molecule space (Agarwal et al. , 2021 ; Griffiths et al. , 2022 ) . Moreover, BO is highly sensitive to the choice of molecular representation, which can be challenging to design, expensive to compute, and may not generalize across diverse regions of chemical space (Ranković et al. , 2025 ) . Recent BO workflows use large language models (LLMs) trained jointly with GP objectives that enable the rapid, flexible featurization only from SMILES with excellent performance across diverse chemical tasks, opening the door to on-the-fly application across chemical space (Kristiadi et al. , 2024 ; Ranković et al. , 2025 ; Ranković and Schwaller, 2023 ) ."},{"type":"para","text":"We propose the Sample Efficient Generative Optimization (SEGO) framework, a tandem system based on RL and BO for sample efficient search in open-ended molecular space. At each iteration, a probabilistic surrogate forms a hypothesis about where hits lie and steers a generative model to produce a library of candidates in that region. The library is featurized on-the-fly through deep-kernel learning over LLM embeddings (Ranković et al. , 2025 ) and an acquisition function selects the most promising candidate for evaluation; the resulting oracle call both sharpens the surrogate into a new hypothesis and anchors the generator in real reward. SEGO achieves sample efficiency from two mechanisms within this loop: SMILES augmentation in the surrogate training, and an anchoring mechanism for the surrogate-guided generation. Our method outperforms both pure BO and pure RL methods and attains state-of-the-art performance on the practical molecular optimization (PMO) benchmark using only a tenth of the oracle budget of other methods. On a multiparameter docking optimization task, SEGO identifies candidates in half or less of the oracle budget of other state-of-the-art methods. These gains could enable prospective campaigns beyond predefined libraries based on direct experimental feedback."},{"type":"heading","level":1,"text":"2 Background and related work"},{"type":"heading","level":2,"text":"2.1 De Novo Molecular Design."},{"type":"para","text":"De novo molecular design seeks to generate molecules with desired properties from scratch, bypassing the constraints of fixed libraries (Sanchez-Lengeling and Aspuru-Guzik, 2018 ) . Generative approaches span a broad range of architectures, from SMILES-based recurrent networks and transformers to graph-based, diffusion, and flow-based models. Among these, language-based models operating on SMILES representations have proven to be particularly sample efficient, capable of satisfying even three-dimensional objectives such as docking despite working from one-dimensional string inputs (Gao et al. , 2022 ; Brown et al. , 2019 ) . A key advantage of SMILES is its non-injective nature: a single molecule admits many valid string representations, and this redundancy can be leveraged through augmentation to improve generalization in low-data regimes (Guo and Schwaller, 2024 ; Arús-Pous et al. , 2019 ) . As the field has shifted toward stricter oracle budgets that better reflect real-world constraints, sample efficiency has become a central evaluation criterion, and language-based methods consistently rank among the top performers under these conditions. Recent advances such as combining experience replay with SMILES augmentation have pushed sample efficiency further, establishing strong baselines for generative molecular optimization (Guo et al. , 2026 ) ."},{"type":"para","text":"However, based on the formulation of the loss, these systems do not learn much from negative examples early in the optimization (Loeffler et al. , 2024 ) . In challenging optimization scenarios, where early batches yield little to no reward, slow learning causes long lag times which erode sample efficiency. This motivates integrating generative models with principled selection strategies that extract signal from all evaluations, even from negative evaluations early in the optimization trajectory."},{"type":"heading","level":2,"text":"2.2 Bayesian Optimization on Molecules."},{"type":"para","text":"Bayesian optimization has been applied to molecular discovery across domains in chemistry. For example, it has been used to identify redox-active molecules from libraries of scaffolds enumerated with substituents and featurized with DFT descriptors (Agarwal et al. , 2021 ) , to identify candidate protein binders from an Enamine library using fingerprint representations (Graff et al. , 2021 ) , and for materials discovery efforts, where sets of predefined building blocks were evaluated using time-dependent DFT (Strieth-Kalthoff et al. , 2024 ) . More broadly, library-based BO has been applied to transition-metal complexes (Janet et al. , 2020 ) , metal-organic frameworks (Comlek et al. , 2023 ) , with surrogate models trained on small labeled subsets and acquisition functions used to iteratively select candidates for evaluation. These case studies illustrate the strength of BO for optimization in challenging reward landscapes, but also highlight two recurring limitations: optimization is constrained to expert-crafted libraries that cannot propose structures beyond their initial enumeration, and performance depends heavily on molecular representation. Bespoke descriptors such as DFT features can be expensive to compute, while general-purpose fingerprints, though cheap, may not capture the structure-property relationships relevant to a given task."},{"type":"para","text":"Recent work has begun to address the representation challenge through language-model-based molecular embeddings (Kristiadi et al. , 2024 ; Ranković and Schwaller, 2023 ; Ranković et al. , 2025 ) . Building on this direction, the GOLLuM framework fine-tunes language model embeddings using Gaussian Process (GP) marginal likelihood, yielding task-adaptive representations that outperform both fingerprints and bespoke descriptors across diverse benchmarks (Ranković et al. , 2025 ) . Crucially, because GOLLuM operates directly on SMILES strings without requiring additional descriptor computation, it can featurize novel molecules on-the-fly, making it feasible to apply BO over dynamically generated libraries rather than fixed, predefined candidate sets."},{"type":"heading","level":2,"text":"2.3 Hybrid Generative BO methods."},{"type":"para","text":"To move beyond fixed libraries, several works have coupled generative models with BO. A common approach embeds discrete molecular representations into a continuous latent space, performs optimization there, and decodes the resulting vectors back into molecules (Filella-Merce et al. , 2025 ; Gómez-Bombarelli et al. , 2018 ) . However, this latent-optimize-then-decode paradigm faces practical challenges: decoded molecules are frequently invalid or redundant, and latent spaces can be high-dimensional and poorly structured, complicating surrogate modeling and uncertainty quantification. Alternative approaches bypass latent embeddings and optimize acquisition functions directly over discrete representations using grammar-constrained search (Moss et al. , 2020 ) , graph-based kernels (Oh et al. , 2019 ) , or learned generation policies (Swersky et al. , 2020 ) , though these methods face their own difficulties in maintaining chemical plausibility and scaling to large batch sizes."},{"type":"para","text":"A more recent line of work decouples the generative and optimization stages entirely, first using a generative model to propose candidates and then applying BO to select the most informative subset for evaluation. Dodds et al. ( 2024 ) developed a hybrid framework combining REINVENT with a fingerprint-driven, random-forest-based active learning loop, demonstrating improved sample efficiency over baseline REINVENT, though the modest efficiency of the underlying generative model limits practical applicability. Tripp and Hernández-Lobato ( 2024 ) used a genetic algorithm as a molecule generation engine and a Tanimoto kernel on molecular fingerprints to achieve excellent results on the PMO benchmark. Muthyala et al. ( 2025 ) also proposed a generate-then-optimize system with a new acquisition function for batch selection. While SEGO shares this hybrid format, it introduces surrogate-guided generation and augmented optimization to improve sample efficiency substantially. Further, SEGO builds on frontier methods in both Bayesian optimization and generative design, enabling new levels of sample efficiency not achieved by prior hybrid approaches."},{"type":"heading","level":1,"text":"3 SEGO Design"},{"type":"para","text":"SEGO is a framework for generative optimization that attains excellent sample efficiency through the intelligent allocation of oracle calls across dynamically generated libraries (Figure 1 ). Each oracle call drives updates to a generative agent, a GP surrogate model, as well as the surrogate’s learned featurization. SEGO is organized around the following components:"},{"type":"para","text":"An anchor generative agent π θ Agent \\pi_{\\theta_{\\text{Agent}}} updated only on true-oracle feedback."},{"type":"para","text":"An inner-loop agent π θ Inner \\pi_{\\theta_{\\text{Inner}}} , re-instantiated at each BO iteration from the anchor and trained against the GP posterior mean to produce a dynamic candidate library."},{"type":"para","text":"A deep-kernel GP surrogate over LLM embeddings, which featurizes the dynamic library for selection via an acquisition function."},{"type":"para","text":"SEGO’s sample efficiency rests on two contributions: an anchored surrogate-guided inner loop, enabling surrogate-driven generation while retaining information from true-oracle feedback; and SMILES-augmented surrogate updates, which exploits the string-based featurizer to extract additional signal from each oracle call."},{"type":"heading","level":2,"text":"3.1 Molecular Generation Engine."},{"type":"para","text":"As our generative engine, we take Saturn (Guo et al. , 2026 ) , a Mamba-based (Gu and Dao, 2024 ) autoregressive SMILES generator trained with reinforcement learning. A SMILES string x = ( a 1 , … , a T ) x=(a_{1},...,a_{T}) is generated as a Markov process,"},{"type":"para","text":"where a t a_{t} is the token selected at step t t , s t s_{t} is the token sequence so far, and π θ Agent \\pi_{\\theta_{\\text{Agent}}} is the Mamba backbone. The aim in RL is to maximize the expected reward (equation 2 )."},{"type":"para","text":"Augmented likelihood (equation 3 ) is defined, where the prior is a pretrained model with frozen weights, R R is a reward function, and σ \\sigma is a scalar modulating the reward’s effect:"},{"type":"para","text":"Maximizing equation 2 is equivalent (up to a factor) to minimizing the squared difference between the Augmented and Agent likelihoods (equation 4 )."},{"type":"para","text":"where B B is a batch of SMILES sampled from the agent and A ∗ A^{*} denotes the action sequences across all timesteps in the batch. Sample efficiency in Saturn comes from Augmented Memory: a replay buffer of the top-100 rewarded SMILES is maintained, randomized into equivalent SMILES, and used for N a u g m e n t a t i o n N_{augmentation} additional update rounds per iteration. Mode collapse is mitigated by purging the buffer of scaffolds with low-diversity samples and assigning zero reward to repeated generations beyond a threshold."},{"type":"heading","level":2,"text":"3.2 Bayesian Optimization Engine."},{"type":"para","text":"As our BO engine, we take GOLLuM, a flexible framework based on SMILES that jointly trains LLM embeddings and GP hyperparameters (Ranković et al. , 2025 ) . A learned feature transformation is composed with the base kernel:"},{"type":"para","text":"where g ϕ g_{\\phi} is a parameterized feature extractor with parameters ϕ \\phi , trained jointly with the GP hyperparameters through the marginal likelihood. Though multiple ways of constructing g ϕ g_{\\phi} were proposed in the original paper, we take the projection layer variant: g ϕ ( 𝐱 ) = 𝐏 LLM ( x ) g_{\\phi}(\\mathbf{x})=\\mathbf{P}\\text{LLM}(x) , where 𝐏 ∈ ℝ m × d \\mathbf{P}\\in\\mathbb{R}^{m\\times d} is a learned linear projection followed by ELU activation. The GP hyperparameters θ = ( ℓ , σ 2 , σ n 2 , c ) \\theta=(\\ell,\\sigma^{2},\\sigma_{n}^{2},c) and feature parameters ϕ \\phi are jointly optimized by maximizing the log marginal likelihood:"},{"type":"para","text":"GOLLuM is particularly well suited as the BO engine in SEGO because it operates directly on SMILES strings, bypassing the need for hand-crafted molecular descriptors. Through joint optimization of the LLM and GP via the marginal likelihood, GOLLuM learns to smooth the representation space so that the GP can generalize effectively from very few observations."},{"type":"heading","level":2,"text":"3.3 SEGO Operation"},{"type":"para","text":"At iteration i i , SEGO maintains the anchor agent π θ Agent ( i ) \\pi_{\\theta_{\\text{Agent}}}^{(i)} and a GP 𝒢 𝒫 ( i ) \\mathcal{GP}^{(i)} fit to the oracle-evaluated set 𝒟 ( i ) = { ( x j , y j ) } j ≤ i \\mathcal{D}^{(i)}=\\{(x_{j},y_{j})\\}_{j\\leq i} . The loop is initialized by sampling L L molecules from π θ Prior \\pi_{\\theta_{\\text{Prior}}} , scoring them under the acquisition function, and selecting n init n_{\\text{init}} for oracle evaluation; these seed 𝒟 ( 0 ) \\mathcal{D}^{(0)} and provide the first updates to π θ Agent \\pi_{\\theta_{\\text{Agent}}} and 𝒢 𝒫 \\mathcal{GP} . Each iteration:"},{"type":"para","text":"Initializes π θ Inner \\pi_{\\theta_{\\text{Inner}}} from π θ Agent ( i ) \\pi_{\\theta_{\\text{Agent}}}^{(i)} ."},{"type":"para","text":"Trains π θ Inner \\pi_{\\theta_{\\text{Inner}}} for K inner K_{\\text{inner}} Saturn updates against R surr ( x ) = μ ( i ) ( x ) R_{\\text{surr}}(x)=\\mu^{(i)}(x) , the GP posterior mean."},{"type":"para","text":"Samples a candidate library 𝒞 ( i ) \\mathcal{C}^{(i)} of size L L from π θ Inner \\pi_{\\theta_{\\text{Inner}}} , then discards π θ Inner \\pi_{\\theta_{\\text{Inner}}} ."},{"type":"para","text":"Selects x i ⋆ = arg max x ∈ 𝒞 ( i ) α ( x ) x^{\\star}_{i}=\\arg\\max_{x\\in\\mathcal{C}^{(i)}}\\alpha(x) under acquisition function α \\alpha (we use expected improvement)."},{"type":"para","text":"Queries the oracle to obtain y i ⋆ = f oracle ( x i ⋆ ) y^{\\star}_{i}=f_{\\text{oracle}}(x^{\\star}_{i}) ."},{"type":"para","text":"Constructs an augmented set 𝒜 i = { ( x i , k ⋆ , y i ⋆ ) } k = 1 N aug \\mathcal{A}_{i}=\\{(x^{\\star}_{i,k},y^{\\star}_{i})\\}_{k=1}^{N_{\\text{aug}}} from N aug N_{\\text{aug}} randomized SMILES variants of x i ⋆ x^{\\star}_{i} , updates 𝒟 ( i + 1 ) ← 𝒟 ( i ) ∪ 𝒜 i \\mathcal{D}^{(i+1)}\\leftarrow\\mathcal{D}^{(i)}\\cup\\mathcal{A}_{i} , refits ( θ , ϕ ) (\\theta,\\phi) via Eq. ( 3.2 ), and updates π θ Agent \\pi_{\\theta_{\\text{Agent}}} by one round of Saturn updates (Eq. ( 4 )) with reward y i ⋆ y^{\\star}_{i} ."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 SEGO architecture ablation experiments"},{"type":"para","text":"We explore the impact of key hyperparameters in SEGO: the role of the inner loop in sample efficiency, the anchor’s role in training stability, and augmented optimization."},{"type":"para","text":"Experimental Setup. Following Guo and Schwaller ( 2024 ) , we optimize molecules according to a multi-property optimization (MPO) objective: molecular weight (MW) < < 350 Da, number of rings ≥ \\geq 2, and maximize the topological polar surface area (tPSA). To satisfy this objective, molecules must have rings saturated with heteroatoms. Such saturated molecules are dissimilar from training data, so this experiment also evaluates out-of-distribution optimization. We use an extremely stringent oracle budget of 100 calls, representing 10% of the already stringent budget employed in (Guo et al. , 2026 ) , which better reflects realistic budgets for experimental campaigns. All experiments were run across 10 seeds (0-9 inclusive) using a prior pretrained on the ChEMBL 33 dataset (Gaulton et al. , 2012 ) ."},{"type":"para","text":"Metrics. As our sample efficiency metrics, we take Yield , the number of unique molecules generated above a reward threshold and Oracle Burden (OB), the number of oracle calls required to generate N molecules above a reward threshold. As our threshold, we take 0.7 as molecules start to possess saturated heteroatom rings. We also compute diversity metrics, as there is a tradeoff in most generative methods between yield and diversity. We compute IntDiv , the internal diversity (Polykovskiy et al. , 2020 ) , Scaffolds , the number of Bemis-Murcko scaffolds (Bemis and Murcko, 1996 ) , and #Circles , (Xie et al. , 2023 ) which measures chemical space coverage."},{"type":"para","text":"Inner loop anchor strategies. We explore three strategies for instantiating the inner loop, in which the RL agent receives reward from the GP predicted mean. The default strategy initializes each inner loop from an anchor, an RL agent previously trained on true oracle rewards, giving the inner loop a warm start in a promising region of chemical space. To test whether this warm start is necessary, we ablate with two alternatives: a prior strategy, which re-initializes from the pretrained prior at each iteration, testing whether the inner loop can learn to generate high-scoring molecules without any oracle-informed starting point; and a continuous strategy, which retains the inner loop agent across iterations, testing whether it is sufficiently flexible to track the changing GP landscape throughout the campaign."},{"type":"para","text":"The prior initialization strategy fails to produce hits in most cases (Table 1 ). This is unsurprising: at early iterations the GP has been trained on as few as 10 molecules, so its reward landscape is poorly calibrated across chemical space, providing too faint a signal for the agent to learn from scratch. The continuous strategy yields competitive hit counts, but at the cost of reliability. Since the agent is never reset, it tends to commit to a narrow region of chemical space; if that region contains hits, oracle burden is low, but the agent struggles to transition elsewhere in the absence of hits. This is reflected in the high variance of OB1 across seeds. We adopt the anchored framework as default."},{"type":"para","text":"Surrogate-guided inner loop. Given the strong performance of the anchor strategy, we next assess the impact of the number of inner loop iterations. To isolate the effect of surrogate-informed training from the additional augmented memory updates that the inner loop provides, we include a further control: a variant with zero inner loops but with N anchor augmentation rounds set to 210, matching the total number of updates the 20-step inner loop would receive (20 inner loop steps × \\times 10 inner loop augmentations, plus 10 outer loop augmentations). Sample efficiency improves and variance across seeds decreases as the number of inner loop iterations increases, and diversity follows suit, reflecting the stronger optimization (Table A1 ). The augmentation-matched control, despite receiving the same number of gradient updates from the replay buffer, suffers in yield, oracle burden, and the number of seeds that successfully identify a hit, confirming that the benefit of the inner loop stems from training on GP-predicted rewards rather than from additional augmented memory updates alone. Though the difference between 0 and 20 inner loops does not reach statistical significance for most metrics, the consistent trends across all metrics suggest that the inner loop improves both the reliability and efficiency of SEGO."},{"type":"para","text":"Augmented Optimization. Inspired by Augmented Memory, in which a generative model is updated according to N N augmentation rounds of randomized, high-reward SMILES, we develop augmented optimization, in which library SMILES are augmented N N -fold and the GP is trained on all representations for each molecule. Because the GOLLuM framework learns its latent space through implicit contrastive learning — reorganizing so that similar molecules have more similar representations — we hypothesize that providing many representations of the same molecule will encourage the GP to align them, improving predictive accuracy."},{"type":"para","text":"Augmented optimization improves both oracle burden and yield without statistically significant impacts on internal diversity and #Circles (Table 2 ). The effect saturates at 5-fold augmentation; no statistically significant difference was observed between 5-, 10-, and 20-fold augmentation. To investigate the mechanism behind this improvement, we examined the impact of augmentation on both representational alignment and predictive accuracy. At each iteration we sampled 500 molecules from the prior and from the anchor and recorded the GP mean prediction across 10 augmentations of each molecule. SMILES augmentation narrows the range of predictions for different representations of the same molecule, and this alignment improves with additional augmentation rounds (Figure A1 ). Crucially, the effect is observed for the prior as well as the anchor, indicating that augmentation improves global representational alignment rather than only aligning the narrow region of chemical space being explored by the oracle. This corresponds to gains in predictive accuracy — measured by the correlation between mean predicted values and true oracle values — particularly early on, when the GP has very little training data (Figure A1 )."},{"type":"para","text":"Isomer Collapse. The previous experiments showcase the powerful optimization behavior of SEGO, though as is typical of strong optimizers, this comes at the cost of diversity. The Bemis-Murcko scaffold metric even inflates diversity for this oracle: dense saturation means that molecules differing in only heteroatom placement are assigned different scaffolds. To probe diversity collapse for this oracle, we introduce the metric Isomers , defined as the number of unique hit molecular formulas. Despite generating 59 ( ± \\pm 25) unique molecules at this threshold, SEGO produces only 12 ( ± \\pm 9) unique isomers, revealing a form of mode collapse in which the agent repeatedly generates constitutional isomers of the same formula (Table A2 )."},{"type":"para","text":"We address this isomer collapse by modulating the expected improvement acquisition function. Standard expected improvement can favor isomers predicted to score near the previous best over novel formulas whose higher uncertainty does not translate into a higher chance of exceeding the incumbent. Following Brochu et al. ( 2010 ) , we inflate the incumbent by a margin ξ \\xi and so require a candidate’s predicted improvement to clear this raised threshold. We observe that ξ = 0.05 \\xi=0.05 and ξ = 0.1 \\xi=0.1 significantly increase the number of unique hit isomers, with no decrease in the overall number of hits (Table A2 , implementation details in section A.3 ). We also introduce an isomer filter analogous to a Bemis-Murcko scaffold filter, setting a stringent threshold of 2. This filter offers additional gains when combined with the inflated expected improvement, showing significant improvements over the baseline in isomers and both diversity metrics. Both the ξ \\xi parameter and the isomer bucket size are hyperparameters that can be tuned to balance exploration and exploitation."},{"type":"para","text":"Hybrid versus individual approaches. SEGO is a hybrid architecture that enables sample-efficient search over open-ended chemical space. We compare SEGO to the individual generative and BO engines it is built from to showcase the benefits of the hybrid approach. To benchmark GOLLuM, we take a random sample of 500 molecules from ChEMBL 33 (Gaulton et al. , 2012 ) , the same database used to train the prior in Saturn and in SEGO, to mimic a real-world prospective campaign in which a desirable library is not known. Given the benefits of augmented optimization observed earlier, we also test GOLLuM with 10-fold SMILES augmentation (GOLLuM-10x). For SEGO, we use the anchored framework with 20 inner loop iterations and an augmented optimization factor of 10. We also use ξ = 0.05 \\xi=0.05 as well as an Isomer diversity filter with a bucket size of 2."},{"type":"para","text":"Saturn fails to produce any hits at this strict oracle budget (Table 3 ). With the recommended batch size of 16 and augmentation rounds of 10, it undergoes only 6 rounds of RL updates, which is insufficient to shift the distribution toward hit molecules. Smaller batch sizes in Saturn do not lead to gains in sample efficiency due to the noise in approximating expected reward with small batch sizes (Guo et al. , 2026 ) . GOLLuM fails because the randomly sampled library does not contain hits. In SEGO, the surrogate-steered generation shifts the distribution toward high reward regions providing the selector with hit-containing libraries to yield 63 ( ± 9 \\pm 9 ) hits."},{"type":"heading","level":2,"text":"4.2 PMO benchmark"},{"type":"para","text":"Experimental Setup. We evaluate SEGO on the practical molecular optimization (PMO) benchmark, a suite of 23 oracles spanning model-based scoring (e.g., DRD2, JNK3, GSK3 β \\beta ), similarity-based objectives (e.g., Albuterol, Mestranol, Celecoxib rediscovery), and physicochemical and multi-property composites (e.g., QED, isomer matching, scaffold hopping) (Gao et al. , 2022 ) . PMO reports the sum of each trajectory’s area under the curve (AUC) of the top-10 average performance as a function of cumulative oracle queries, favoring methods that find high values in fewer calls. The PMO takes a relaxed budget of 10,000. For increased relevance to expensive oracles, following Nguyen and Grover ( 2025 ) , we instead report results for baselines on the PMO when restricted to an oracle budget of 1,000 (PMO-1k). For SEGO, we restrict oracle budget to just 100 oracle calls, after which the top-1 and top-10 scores are held fixed for computing metrics at the 1k horizon (Krüger et al. , 2026 ) to showcase sample efficiency. As many objectives are similarity based, and therefore diversity is not a priority, we run SEGO with vanilla expected improvement as the acquisition function. We report both top-1 AUC (consistent with 1% of oracle calls) and top-10 AUC, aggregated across 3 seeds (0–2 inclusive). We compare against published PMO leaderboard methods including REINVENT (Loeffler et al. , 2024 ) , Augmented Memory (Guo and Schwaller, 2024 ) , Genetic GFN (Kim et al. , 2024 ) , MOLLEO (Wang et al. , 2025 ) , Graph GA (Jensen, 2019 ) , and GP BO (Tripp and Hernández-Lobato, 2024 ) ."},{"type":"para","text":"At the top-1 threshold, which mirrors the baselines by evaluating the top 1% of the oracle budget, SEGO achieves state-of-the-art performance despite consuming only a tenth of the oracle budget available to other methods (Table 4 ). SEGO attains competitive performance at the top-10 threshold which is particularly challenging because top-10 represents 10% of the total budget available to SEGO. Extended results for individual oracles are reported in the appendix (Table A3 ). Together, these results demonstrate SEGO’s strong optimization capability under tight budget constraints."},{"type":"para","text":"The most closely related method in this benchmark is GP BO (Tripp and Hernández-Lobato, 2024 ) , which uses BO to select candidates from a library generated by a genetic algorithm. SEGO outperforms GP BO at a fraction of the oracle budget, highlighting the strength of its underlying generative and optimization components. LICO (Nguyen and Grover, 2025 ) follows a similar framework but replaces the GP surrogate with an LLM. SEGO surpasses LICO with greater sample efficiency, though LICO’s strong results suggest that substituting SEGO’s GP with an LLM is a promising direction for future work. Another language-model approach is MOLLEO (Wang et al. , 2025 ) , which leverages an LLM to perform mutation and crossover operations within an evolutionary algorithm. While MOLLEO achieves competitive performance, it relies on detailed textual task descriptions provided to the LLM, raising concerns about data leakage and limiting its applicability to prospective campaigns where such descriptions may not be available."},{"type":"heading","level":2,"text":"4.3 Molecular Docking Task"},{"type":"para","text":"Experimental Setup. Following Lee et al. ( 2024b ) , we benchmark SEGO on multiparameter optimization docking tasks against five targets (parp1, fa7, 5ht1b, braf and jak2). We optimize the MPO objective function (equation 7 ), where DS ^ \\widehat{\\mathrm{DS}} is the normalized QuickVina 2 (Alhossary et al. , 2015 ) score and SA ^ \\widehat{\\mathrm{SA}} is the normalized synthetic accessibility score. We compare against the goal-aware fragment extraction, assembly and modification (GEAM) model (Lee et al. , 2024b ) and against Saturn."},{"type":"para","text":"Metrics. Mirroring a campaign in which the aim is to find several promising candidates for further evaluation and in which experiments are expensive, we take the oracle burden at 10 hits ( OB@10 ). Here, we define a hit as a strict hit with stringent criteria requiring a molecule to have a docking score better than the median of known actives, a Q E D > 0.7 QED>0.7 and S A < 3 SA<3 (Guo et al. , 2026 ) . We take strict hits to test the optimization capacity of the model. SEGO was run for an oracle budget of 600 calls, Saturn and GEAM were run for 3,000 calls."},{"type":"para","text":"Results. SEGO finds 10 strict hits in half or less of the oracles required by Saturn and GEAM, showcasing strong optimization performance on a pharmacologically relevant oracle (Table 5 ). Generated molecules for each target are shown in Figure A2 ."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"para","text":"We introduce SEGO, a hybrid generative optimizer for sample-efficient search in open-ended chemical space. Unlike pure generative methods that rely on slow updates to shift the distribution and conventional BO methods that operate on fixed libraries, SEGO leverages the principled exploration of BO across dynamically generated libraries. This design yields gains in hit-finding and sample efficiency compared to the individual parent methods on a toy oracle. SEGO also achieves excellent performance on the PMO benchmark, reaching state-of-the-art top 1% performance in only 100 oracle calls—one-tenth of the oracle budget of PMO-1k. On a more challenging MPO molecular docking oracle, SEGO identifies multiple hits much faster than current state-of-the-art methods."},{"type":"para","text":"Our ablations show that our design decisions—namely, guiding an inner loop based on surrogate predictions and anchoring that inner loop—contribute to the sample efficiency of SEGO. Further, data augmentation, known to be beneficial for sample efficiency in generative models, can improve the predictive performance of BO as well and improve representational alignment between diverse SMILES. Together, this setup yields an effective optimizer that is prone to repeat isomer generation. We enhance diversity through the inclusion of an isomer-based diversity filter as well as by inflating the previous best in the expected improvement acquisition function."},{"type":"para","text":"Excellent performance on the PMO benchmark and the MPO docking benchmark hint at powerful optimization ability; however, it may not reflect optimization behavior in the complex reward landscapes found in real-world campaigns where competing objectives like synthetic accessibility, solubility, and activity must be navigated simultaneously. Further work should assess SEGO on challenging oracles including physics-based simulations relevant to drug-discovery and catalysis as well as stringent synthesis constraints before considering deployment to wet-lab prospective campaigns (Lee et al. , 2024a ) . Finally, practical high-throughput campaigns require multiobjective optimization over Pareto fronts and batched acquisition (Sin et al. , 2025 ) which could be introduced into SEGO via alternative acquisition functions (Muthyala et al. , 2025 ) . Despite these limitations, the sample efficiency and hit-finding ability demonstrated here are a promising step toward generative molecular design driven by direct experimental feedback."},{"type":"heading","level":1,"text":"Software and Data"},{"type":"para","text":"Software to run SEGO can be found at https://github.com/schwallergroup/sego ."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"As AI-driven methods become increasingly integrated into chemical discovery, it is important to acknowledge the dual-use risks inherent in molecular generation, which become more relevant as engines become more powerful. The risks of SEGO are in line with existing performant molecular design engines. Such risks can be mitigated through controlled access to oracles and established safeguards at the institutional level."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Molecular optimization is a central problem in chemistry, with applications ranging from drug discovery to catalyst design (Filella-Merce et al. , 2025 ; Strieth-Kalthoff et al. , 2024 ; Seumer et al. , 2023 ) . These tasks require identifying molecules that satisfy multiple, often competing objectives from vast chemical space (Sanchez-Lengeling and Aspuru-Guzik, 2018 ) . Most relevant optimization problems are also data scarce: phenomena such as protein-ligand binding or catalytic activity are difficult to simulate accurately, and experimental measurements—particularly those requiring molecular synthesis—are costly and time-consuming. As a result, the real-world impact of an optimization strategy is defined by the number of evaluations required to discover optimal molecules (sample efficiency). Low sample efficiency necessitates reliance on inexpensive, low-fidelity in silico oracles vulnerable to hacking (Renz et al. , 2019 ; Lehman et al. , 2020 ) . Improved sample efficiency would enable the use of higher-fidelity simulations or even direct experimental feedback, thereby increasing the practical impact of molecular optimization methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ML methods for molecule optimization must traverse chemical space and strategically guide this exploration toward the most promising regions. Traversal methods include deep generative methods that learn a data distribution and can sample from that distribution, including recurrent neural networks (RNNs) (Olivecrona et al. , 2017 ; Loeffler et al. , 2024 ) , transformers (Feng et al. , 2023 ; Mazuz et al. , 2023 ) , and diffusion or flow-based models (Lee et al. , 2023 ; Jiang et al. , 2024 ; Schneuing et al. , 2023 ) . These generative methods can be steered with various optimization techniques including Monte Carlo tree search (MCTS), Markov Chain Monte Carlo (MCMC), reinforcement learning (RL), as well as with Bayesian Optimization (BO). Language models that generate molecules in the form of Simplified Molecular Input Line Entry System (SMILES) (Weininger, 1988 ) steered with RL remain among the most sample efficient methods (Gao et al. , 2022 ; Guo et al. , 2026 ) . However, state-of-the-art generative methods still require between hundreds and thousands of oracle evaluations to identify high-quality candidates (Gao et al. , 2022 ; Guo et al. , 2026 ) . While this cost may permit the use of moderately expensive simulations such as DFT, it remains prohibitive for workflows involving direct experimental evaluation, particularly in the case of prospective campaigns where the target distribution is not known.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"BO, an optimization technique that uses a probabilistic surrogate model and an acquisition function to guide sequential experimentation within a candidate library or a continuous latent space (Frazier, 2018 ) , has excellent sample efficiency and has seen widespread use and success in chemistry (Sin et al. , 2025 ; Zaza et al. , 2025 ; Shields et al. , 2021 ) . Through the fitting of the surrogate model, these methods learn quickly from negative examples. However, performance depends on the presence of suitable candidates within a library or accurate decoding from a latent space to molecule space (Agarwal et al. , 2021 ; Griffiths et al. , 2022 ) . Moreover, BO is highly sensitive to the choice of molecular representation, which can be challenging to design, expensive to compute, and may not generalize across diverse regions of chemical space (Ranković et al. , 2025 ) . Recent BO workflows use large language models (LLMs) trained jointly with GP objectives that enable the rapid, flexible featurization only from SMILES with excellent performance across diverse chemical tasks, opening the door to on-the-fly application across chemical space (Kristiadi et al. , 2024 ; Ranković et al. , 2025 ; Ranković and Schwaller, 2023 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose the Sample Efficient Generative Optimization (SEGO) framework, a tandem system based on RL and BO for sample efficient search in open-ended molecular space. At each iteration, a probabilistic surrogate forms a hypothesis about where hits lie and steers a generative model to produce a library of candidates in that region. The library is featurized on-the-fly through deep-kernel learning over LLM embeddings (Ranković et al. , 2025 ) and an acquisition function selects the most promising candidate for evaluation; the resulting oracle call both sharpens the surrogate into a new hypothesis and anchors the generator in real reward. SEGO achieves sample efficiency from two mechanisms within this loop: SMILES augmentation in the surrogate training, and an anchoring mechanism for the surrogate-guided generation. Our method outperforms both pure BO and pure RL methods and attains state-of-the-art performance on the practical molecular optimization (PMO) benchmark using only a tenth of the oracle budget of other methods. On a multiparameter docking optimization task, SEGO identifies candidates in half or less of the oracle budget of other state-of-the-art methods. These gains could enable prospective campaigns beyond predefined libraries based on direct experimental feedback.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Background and related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 De Novo Molecular Design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"De novo molecular design seeks to generate molecules with desired properties from scratch, bypassing the constraints of fixed libraries (Sanchez-Lengeling and Aspuru-Guzik, 2018 ) . Generative approaches span a broad range of architectures, from SMILES-based recurrent networks and transformers to graph-based, diffusion, and flow-based models. Among these, language-based models operating on SMILES representations have proven to be particularly sample efficient, capable of satisfying even three-dimensional objectives such as docking despite working from one-dimensional string inputs (Gao et al. , 2022 ; Brown et al. , 2019 ) . A key advantage of SMILES is its non-injective nature: a single molecule admits many valid string representations, and this redundancy can be leveraged through augmentation to improve generalization in low-data regimes (Guo and Schwaller, 2024 ; Arús-Pous et al. , 2019 ) . As the field has shifted toward stricter oracle budgets that better reflect real-world constraints, sample efficiency has become a central evaluation criterion, and language-based methods consistently rank among the top performers under these conditions. Recent advances such as combining experience replay with SMILES augmentation have pushed sample efficiency further, establishing strong baselines for generative molecular optimization (Guo et al. , 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, based on the formulation of the loss, these systems do not learn much from negative examples early in the optimization (Loeffler et al. , 2024 ) . In challenging optimization scenarios, where early batches yield little to no reward, slow learning causes long lag times which erode sample efficiency. This motivates integrating generative models with principled selection strategies that extract signal from all evaluations, even from negative evaluations early in the optimization trajectory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Bayesian Optimization on Molecules.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bayesian optimization has been applied to molecular discovery across domains in chemistry. For example, it has been used to identify redox-active molecules from libraries of scaffolds enumerated with substituents and featurized with DFT descriptors (Agarwal et al. , 2021 ) , to identify candidate protein binders from an Enamine library using fingerprint representations (Graff et al. , 2021 ) , and for materials discovery efforts, where sets of predefined building blocks were evaluated using time-dependent DFT (Strieth-Kalthoff et al. , 2024 ) . More broadly, library-based BO has been applied to transition-metal complexes (Janet et al. , 2020 ) , metal-organic frameworks (Comlek et al. , 2023 ) , with surrogate models trained on small labeled subsets and acquisition functions used to iteratively select candidates for evaluation. These case studies illustrate the strength of BO for optimization in challenging reward landscapes, but also highlight two recurring limitations: optimization is constrained to expert-crafted libraries that cannot propose structures beyond their initial enumeration, and performance depends heavily on molecular representation. Bespoke descriptors such as DFT features can be expensive to compute, while general-purpose fingerprints, though cheap, may not capture the structure-property relationships relevant to a given task.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent work has begun to address the representation challenge through language-model-based molecular embeddings (Kristiadi et al. , 2024 ; Ranković and Schwaller, 2023 ; Ranković et al. , 2025 ) . Building on this direction, the GOLLuM framework fine-tunes language model embeddings using Gaussian Process (GP) marginal likelihood, yielding task-adaptive representations that outperform both fingerprints and bespoke descriptors across diverse benchmarks (Ranković et al. , 2025 ) . Crucially, because GOLLuM operates directly on SMILES strings without requiring additional descriptor computation, it can featurize novel molecules on-the-fly, making it feasible to apply BO over dynamically generated libraries rather than fixed, predefined candidate sets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Hybrid Generative BO methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To move beyond fixed libraries, several works have coupled generative models with BO. A common approach embeds discrete molecular representations into a continuous latent space, performs optimization there, and decodes the resulting vectors back into molecules (Filella-Merce et al. , 2025 ; Gómez-Bombarelli et al. , 2018 ) . However, this latent-optimize-then-decode paradigm faces practical challenges: decoded molecules are frequently invalid or redundant, and latent spaces can be high-dimensional and poorly structured, complicating surrogate modeling and uncertainty quantification. Alternative approaches bypass latent embeddings and optimize acquisition functions directly over discrete representations using grammar-constrained search (Moss et al. , 2020 ) , graph-based kernels (Oh et al. , 2019 ) , or learned generation policies (Swersky et al. , 2020 ) , though these methods face their own difficulties in maintaining chemical plausibility and scaling to large batch sizes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A more recent line of work decouples the generative and optimization stages entirely, first using a generative model to propose candidates and then applying BO to select the most informative subset for evaluation. Dodds et al. ( 2024 ) developed a hybrid framework combining REINVENT with a fingerprint-driven, random-forest-based active learning loop, demonstrating improved sample efficiency over baseline REINVENT, though the modest efficiency of the underlying generative model limits practical applicability. Tripp and Hernández-Lobato ( 2024 ) used a genetic algorithm as a molecule generation engine and a Tanimoto kernel on molecular fingerprints to achieve excellent results on the PMO benchmark. Muthyala et al. ( 2025 ) also proposed a generate-then-optimize system with a new acquisition function for batch selection. While SEGO shares this hybrid format, it introduces surrogate-guided generation and augmented optimization to improve sample efficiency substantially. Further, SEGO builds on frontier methods in both Bayesian optimization and generative design, enabling new levels of sample efficiency not achieved by prior hybrid approaches.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 SEGO Design\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SEGO is a framework for generative optimization that attains excellent sample efficiency through the intelligent allocation of oracle calls across dynamically generated libraries (Figure 1 ). Each oracle call drives updates to a generative agent, a GP surrogate model, as well as the surrogate’s learned featurization. SEGO is organized around the following components:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An anchor generative agent π θ Agent \\\\pi_{\\\\theta_{\\\\text{Agent}}} updated only on true-oracle feedback.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An inner-loop agent π θ Inner \\\\pi_{\\\\theta_{\\\\text{Inner}}} , re-instantiated at each BO iteration from the anchor and trained against the GP posterior mean to produce a dynamic candidate library.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A deep-kernel GP surrogate over LLM embeddings, which featurizes the dynamic library for selection via an acquisition function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SEGO’s sample efficiency rests on two contributions: an anchored surrogate-guided inner loop, enabling surrogate-driven generation while retaining information from true-oracle feedback; and SMILES-augmented surrogate updates, which exploits the string-based featurizer to extract additional signal from each oracle call.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Molecular Generation Engine.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As our generative engine, we take Saturn (Guo et al. , 2026 ) , a Mamba-based (Gu and Dao, 2024 ) autoregressive SMILES generator trained with reinforcement learning. A SMILES string x = ( a 1 , … , a T ) x=(a_{1},...,a_{T}) is generated as a Markov process,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where a t a_{t} is the token selected at step t t , s t s_{t} is the token sequence so far, and π θ Agent \\\\pi_{\\\\theta_{\\\\text{Agent}}} is the Mamba backbone. The aim in RL is to maximize the expected reward (equation 2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Augmented likelihood (equation 3 ) is defined, where the prior is a pretrained model with frozen weights, R R is a reward function, and σ \\\\sigma is a scalar modulating the reward’s effect:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Maximizing equation 2 is equivalent (up to a factor) to minimizing the squared difference between the Augmented and Agent likelihoods (equation 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where B B is a batch of SMILES sampled from the agent and A ∗ A^{*} denotes the action sequences across all timesteps in the batch. Sample efficiency in Saturn comes from Augmented Memory: a replay buffer of the top-100 rewarded SMILES is maintained, randomized into equivalent SMILES, and used for N a u g m e n t a t i o n N_{augmentation} additional update rounds per iteration. Mode collapse is mitigated by purging the buffer of scaffolds with low-diversity samples and assigning zero reward to repeated generations beyond a threshold.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Bayesian Optimization Engine.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As our BO engine, we take GOLLuM, a flexible framework based on SMILES that jointly trains LLM embeddings and GP hyperparameters (Ranković et al. , 2025 ) . A learned feature transformation is composed with the base kernel:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where g ϕ g_{\\\\phi} is a parameterized feature extractor with parameters ϕ \\\\phi , trained jointly with the GP hyperparameters through the marginal likelihood. Though multiple ways of constructing g ϕ g_{\\\\phi} were proposed in the original paper, we take the projection layer variant: g ϕ ( 𝐱 ) = 𝐏 LLM ( x ) g_{\\\\phi}(\\\\mathbf{x})=\\\\mathbf{P}\\\\text{LLM}(x) , where 𝐏 ∈ ℝ m × d \\\\mathbf{P}\\\\in\\\\mathbb{R}^{m\\\\times d} is a learned linear projection followed by ELU activation. The GP hyperparameters θ = ( ℓ , σ 2 , σ n 2 , c ) \\\\theta=(\\\\ell,\\\\sigma^{2},\\\\sigma_{n}^{2},c) and feature parameters ϕ \\\\phi are jointly optimized by maximizing the log marginal likelihood:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GOLLuM is particularly well suited as the BO engine in SEGO because it operates directly on SMILES strings, bypassing the need for hand-crafted molecular descriptors. Through joint optimization of the LLM and GP via the marginal likelihood, GOLLuM learns to smooth the representation space so that the GP can generalize effectively from very few observations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 SEGO Operation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At iteration i i , SEGO maintains the anchor agent π θ Agent ( i ) \\\\pi_{\\\\theta_{\\\\text{Agent}}}^{(i)} and a GP 𝒢 𝒫 ( i ) \\\\mathcal{GP}^{(i)} fit to the oracle-evaluated set 𝒟 ( i ) = { ( x j , y j ) } j ≤ i \\\\mathcal{D}^{(i)}=\\\\{(x_{j},y_{j})\\\\}_{j\\\\leq i} . The loop is initialized by sampling L L molecules from π θ Prior \\\\pi_{\\\\theta_{\\\\text{Prior}}} , scoring them under the acquisition function, and selecting n init n_{\\\\text{init}} for oracle evaluation; these seed 𝒟 ( 0 ) \\\\mathcal{D}^{(0)} and provide the first updates to π θ Agent \\\\pi_{\\\\theta_{\\\\text{Agent}}} and 𝒢 𝒫 \\\\mathcal{GP} . Each iteration:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Initializes π θ Inner \\\\pi_{\\\\theta_{\\\\text{Inner}}} from π θ Agent ( i ) \\\\pi_{\\\\theta_{\\\\text{Agent}}}^{(i)} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Trains π θ Inner \\\\pi_{\\\\theta_{\\\\text{Inner}}} for K inner K_{\\\\text{inner}} Saturn updates against R surr ( x ) = μ ( i ) ( x ) R_{\\\\text{surr}}(x)=\\\\mu^{(i)}(x) , the GP posterior mean.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Samples a candidate library 𝒞 ( i ) \\\\mathcal{C}^{(i)} of size L L from π θ Inner \\\\pi_{\\\\theta_{\\\\text{Inner}}} , then discards π θ Inner \\\\pi_{\\\\theta_{\\\\text{Inner}}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Selects x i ⋆ = arg max x ∈ 𝒞 ( i ) α ( x ) x^{\\\\star}_{i}=\\\\arg\\\\max_{x\\\\in\\\\mathcal{C}^{(i)}}\\\\alpha(x) under acquisition function α \\\\alpha (we use expected improvement).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Queries the oracle to obtain y i ⋆ = f oracle ( x i ⋆ ) y^{\\\\star}_{i}=f_{\\\\text{oracle}}(x^{\\\\star}_{i}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Constructs an augmented set 𝒜 i = { ( x i , k ⋆ , y i ⋆ ) } k = 1 N aug \\\\mathcal{A}_{i}=\\\\{(x^{\\\\star}_{i,k},y^{\\\\star}_{i})\\\\}_{k=1}^{N_{\\\\text{aug}}} from N aug N_{\\\\text{aug}} randomized SMILES variants of x i ⋆ x^{\\\\star}_{i} , updates 𝒟 ( i + 1 ) ← 𝒟 ( i ) ∪ 𝒜 i \\\\mathcal{D}^{(i+1)}\\\\leftarrow\\\\mathcal{D}^{(i)}\\\\cup\\\\mathcal{A}_{i} , refits ( θ , ϕ ) (\\\\theta,\\\\phi) via Eq. ( 3.2 ), and updates π θ Agent \\\\pi_{\\\\theta_{\\\\text{Agent}}} by one round of Saturn updates (Eq. ( 4 )) with reward y i ⋆ y^{\\\\star}_{i} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 SEGO architecture ablation experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We explore the impact of key hyperparameters in SEGO: the role of the inner loop in sample efficiency, the anchor’s role in training stability, and augmented optimization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experimental Setup. Following Guo and Schwaller ( 2024 ) , we optimize molecules according to a multi-property optimization (MPO) objective: molecular weight (MW) < < 350 Da, number of rings ≥ \\\\geq 2, and maximize the topological polar surface area (tPSA). To satisfy this objective, molecules must have rings saturated with heteroatoms. Such saturated molecules are dissimilar from training data, so this experiment also evaluates out-of-distribution optimization. We use an extremely stringent oracle budget of 100 calls, representing 10% of the already stringent budget employed in (Guo et al. , 2026 ) , which better reflects realistic budgets for experimental campaigns. All experiments were run across 10 seeds (0-9 inclusive) using a prior pretrained on the ChEMBL 33 dataset (Gaulton et al. , 2012 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metrics. As our sample efficiency metrics, we take Yield , the number of unique molecules generated above a reward threshold and Oracle Burden (OB), the number of oracle calls required to generate N molecules above a reward threshold. As our threshold, we take 0.7 as molecules start to possess saturated heteroatom rings. We also compute diversity metrics, as there is a tradeoff in most generative methods between yield and diversity. We compute IntDiv , the internal diversity (Polykovskiy et al. , 2020 ) , Scaffolds , the number of Bemis-Murcko scaffolds (Bemis and Murcko, 1996 ) , and #Circles , (Xie et al. , 2023 ) which measures chemical space coverage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inner loop anchor strategies. We explore three strategies for instantiating the inner loop, in which the RL agent receives reward from the GP predicted mean. The default strategy initializes each inner loop from an anchor, an RL agent previously trained on true oracle rewards, giving the inner loop a warm start in a promising region of chemical space. To test whether this warm start is necessary, we ablate with two alternatives: a prior strategy, which re-initializes from the pretrained prior at each iteration, testing whether the inner loop can learn to generate high-scoring molecules without any oracle-informed starting point; and a continuous strategy, which retains the inner loop agent across iterations, testing whether it is sufficiently flexible to track the changing GP landscape throughout the campaign.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prior initialization strategy fails to produce hits in most cases (Table 1 ). This is unsurprising: at early iterations the GP has been trained on as few as 10 molecules, so its reward landscape is poorly calibrated across chemical space, providing too faint a signal for the agent to learn from scratch. The continuous strategy yields competitive hit counts, but at the cost of reliability. Since the agent is never reset, it tends to commit to a narrow region of chemical space; if that region contains hits, oracle burden is low, but the agent struggles to transition elsewhere in the absence of hits. This is reflected in the high variance of OB1 across seeds. We adopt the anchored framework as default.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Surrogate-guided inner loop. Given the strong performance of the anchor strategy, we next assess the impact of the number of inner loop iterations. To isolate the effect of surrogate-informed training from the additional augmented memory updates that the inner loop provides, we include a further control: a variant with zero inner loops but with N anchor augmentation rounds set to 210, matching the total number of updates the 20-step inner loop would receive (20 inner loop steps × \\\\times 10 inner loop augmentations, plus 10 outer loop augmentations). Sample efficiency improves and variance across seeds decreases as the number of inner loop iterations increases, and diversity follows suit, reflecting the stronger optimization (Table A1 ). The augmentation-matched control, despite receiving the same number of gradient updates from the replay buffer, suffers in yield, oracle burden, and the number of seeds that successfully identify a hit, confirming that the benefit of the inner loop stems from training on GP-predicted rewards rather than from additional augmented memory updates alone. Though the difference between 0 and 20 inner loops does not reach statistical significance for most metrics, the consistent trends across all metrics suggest that the inner loop improves both the reliability and efficiency of SEGO.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Augmented Optimization. Inspired by Augmented Memory, in which a generative model is updated according to N N augmentation rounds of randomized, high-reward SMILES, we develop augmented optimization, in which library SMILES are augmented N N -fold and the GP is trained on all representations for each molecule. Because the GOLLuM framework learns its latent space through implicit contrastive learning — reorganizing so that similar molecules have more similar representations — we hypothesize that providing many representations of the same molecule will encourage the GP to align them, improving predictive accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Augmented optimization improves both oracle burden and yield without statistically significant impacts on internal diversity and #Circles (Table 2 ). The effect saturates at 5-fold augmentation; no statistically significant difference was observed between 5-, 10-, and 20-fold augmentation. To investigate the mechanism behind this improvement, we examined the impact of augmentation on both representational alignment and predictive accuracy. At each iteration we sampled 500 molecules from the prior and from the anchor and recorded the GP mean prediction across 10 augmentations of each molecule. SMILES augmentation narrows the range of predictions for different representations of the same molecule, and this alignment improves with additional augmentation rounds (Figure A1 ). Crucially, the effect is observed for the prior as well as the anchor, indicating that augmentation improves global representational alignment rather than only aligning the narrow region of chemical space being explored by the oracle. This corresponds to gains in predictive accuracy — measured by the correlation between mean predicted values and true oracle values — particularly early on, when the GP has very little training data (Figure A1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Isomer Collapse. The previous experiments showcase the powerful optimization behavior of SEGO, though as is typical of strong optimizers, this comes at the cost of diversity. The Bemis-Murcko scaffold metric even inflates diversity for this oracle: dense saturation means that molecules differing in only heteroatom placement are assigned different scaffolds. To probe diversity collapse for this oracle, we introduce the metric Isomers , defined as the number of unique hit molecular formulas. Despite generating 59 ( ± \\\\pm 25) unique molecules at this threshold, SEGO produces only 12 ( ± \\\\pm 9) unique isomers, revealing a form of mode collapse in which the agent repeatedly generates constitutional isomers of the same formula (Table A2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We address this isomer collapse by modulating the expected improvement acquisition function. Standard expected improvement can favor isomers predicted to score near the previous best over novel formulas whose higher uncertainty does not translate into a higher chance of exceeding the incumbent. Following Brochu et al. ( 2010 ) , we inflate the incumbent by a margin ξ \\\\xi and so require a candidate’s predicted improvement to clear this raised threshold. We observe that ξ = 0.05 \\\\xi=0.05 and ξ = 0.1 \\\\xi=0.1 significantly increase the number of unique hit isomers, with no decrease in the overall number of hits (Table A2 , implementation details in section A.3 ). We also introduce an isomer filter analogous to a Bemis-Murcko scaffold filter, setting a stringent threshold of 2. This filter offers additional gains when combined with the inflated expected improvement, showing significant improvements over the baseline in isomers and both diversity metrics. Both the ξ \\\\xi parameter and the isomer bucket size are hyperparameters that can be tuned to balance exploration and exploitation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hybrid versus individual approaches. SEGO is a hybrid architecture that enables sample-efficient search over open-ended chemical space. We compare SEGO to the individual generative and BO engines it is built from to showcase the benefits of the hybrid approach. To benchmark GOLLuM, we take a random sample of 500 molecules from ChEMBL 33 (Gaulton et al. , 2012 ) , the same database used to train the prior in Saturn and in SEGO, to mimic a real-world prospective campaign in which a desirable library is not known. Given the benefits of augmented optimization observed earlier, we also test GOLLuM with 10-fold SMILES augmentation (GOLLuM-10x). For SEGO, we use the anchored framework with 20 inner loop iterations and an augmented optimization factor of 10. We also use ξ = 0.05 \\\\xi=0.05 as well as an Isomer diversity filter with a bucket size of 2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Saturn fails to produce any hits at this strict oracle budget (Table 3 ). With the recommended batch size of 16 and augmentation rounds of 10, it undergoes only 6 rounds of RL updates, which is insufficient to shift the distribution toward hit molecules. Smaller batch sizes in Saturn do not lead to gains in sample efficiency due to the noise in approximating expected reward with small batch sizes (Guo et al. , 2026 ) . GOLLuM fails because the randomly sampled library does not contain hits. In SEGO, the surrogate-steered generation shifts the distribution toward high reward regions providing the selector with hit-containing libraries to yield 63 ( ± 9 \\\\pm 9 ) hits.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 PMO benchmark\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experimental Setup. We evaluate SEGO on the practical molecular optimization (PMO) benchmark, a suite of 23 oracles spanning model-based scoring (e.g., DRD2, JNK3, GSK3 β \\\\beta ), similarity-based objectives (e.g., Albuterol, Mestranol, Celecoxib rediscovery), and physicochemical and multi-property composites (e.g., QED, isomer matching, scaffold hopping) (Gao et al. , 2022 ) . PMO reports the sum of each trajectory’s area under the curve (AUC) of the top-10 average performance as a function of cumulative oracle queries, favoring methods that find high values in fewer calls. The PMO takes a relaxed budget of 10,000. For increased relevance to expensive oracles, following Nguyen and Grover ( 2025 ) , we instead report results for baselines on the PMO when restricted to an oracle budget of 1,000 (PMO-1k). For SEGO, we restrict oracle budget to just 100 oracle calls, after which the top-1 and top-10 scores are held fixed for computing metrics at the 1k horizon (Krüger et al. , 2026 ) to showcase sample efficiency. As many objectives are similarity based, and therefore diversity is not a priority, we run SEGO with vanilla expected improvement as the acquisition function. We report both top-1 AUC (consistent with 1% of oracle calls) and top-10 AUC, aggregated across 3 seeds (0–2 inclusive). We compare against published PMO leaderboard methods including REINVENT (Loeffler et al. , 2024 ) , Augmented Memory (Guo and Schwaller, 2024 ) , Genetic GFN (Kim et al. , 2024 ) , MOLLEO (Wang et al. , 2025 ) , Graph GA (Jensen, 2019 ) , and GP BO (Tripp and Hernández-Lobato, 2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the top-1 threshold, which mirrors the baselines by evaluating the top 1% of the oracle budget, SEGO achieves state-of-the-art performance despite consuming only a tenth of the oracle budget available to other methods (Table 4 ). SEGO attains competitive performance at the top-10 threshold which is particularly challenging because top-10 represents 10% of the total budget available to SEGO. Extended results for individual oracles are reported in the appendix (Table A3 ). Together, these results demonstrate SEGO’s strong optimization capability under tight budget constraints.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The most closely related method in this benchmark is GP BO (Tripp and Hernández-Lobato, 2024 ) , which uses BO to select candidates from a library generated by a genetic algorithm. SEGO outperforms GP BO at a fraction of the oracle budget, highlighting the strength of its underlying generative and optimization components. LICO (Nguyen and Grover, 2025 ) follows a similar framework but replaces the GP surrogate with an LLM. SEGO surpasses LICO with greater sample efficiency, though LICO’s strong results suggest that substituting SEGO’s GP with an LLM is a promising direction for future work. Another language-model approach is MOLLEO (Wang et al. , 2025 ) , which leverages an LLM to perform mutation and crossover operations within an evolutionary algorithm. While MOLLEO achieves competitive performance, it relies on detailed textual task descriptions provided to the LLM, raising concerns about data leakage and limiting its applicability to prospective campaigns where such descriptions may not be available.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Molecular Docking Task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experimental Setup. Following Lee et al. ( 2024b ) , we benchmark SEGO on multiparameter optimization docking tasks against five targets (parp1, fa7, 5ht1b, braf and jak2). We optimize the MPO objective function (equation 7 ), where DS ^ \\\\widehat{\\\\mathrm{DS}} is the normalized QuickVina 2 (Alhossary et al. , 2015 ) score and SA ^ \\\\widehat{\\\\mathrm{SA}} is the normalized synthetic accessibility score. We compare against the goal-aware fragment extraction, assembly and modification (GEAM) model (Lee et al. , 2024b ) and against Saturn.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metrics. Mirroring a campaign in which the aim is to find several promising candidates for further evaluation and in which experiments are expensive, we take the oracle burden at 10 hits ( OB@10 ). Here, we define a hit as a strict hit with stringent criteria requiring a molecule to have a docking score better than the median of known actives, a Q E D > 0.7 QED>0.7 and S A < 3 SA<3 (Guo et al. , 2026 ) . We take strict hits to test the optimization capacity of the model. SEGO was run for an oracle budget of 600 calls, Saturn and GEAM were run for 3,000 calls.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results. SEGO finds 10 strict hits in half or less of the oracles required by Saturn and GEAM, showcasing strong optimization performance on a pharmacologically relevant oracle (Table 5 ). Generated molecules for each target are shown in Figure A2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce SEGO, a hybrid generative optimizer for sample-efficient search in open-ended chemical space. Unlike pure generative methods that rely on slow updates to shift the distribution and conventional BO methods that operate on fixed libraries, SEGO leverages the principled exploration of BO across dynamically generated libraries. This design yields gains in hit-finding and sample efficiency compared to the individual parent methods on a toy oracle. SEGO also achieves excellent performance on the PMO benchmark, reaching state-of-the-art top 1% performance in only 100 oracle calls—one-tenth of the oracle budget of PMO-1k. On a more challenging MPO molecular docking oracle, SEGO identifies multiple hits much faster than current state-of-the-art methods.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our ablations show that our design decisions—namely, guiding an inner loop based on surrogate predictions and anchoring that inner loop—contribute to the sample efficiency of SEGO. Further, data augmentation, known to be beneficial for sample efficiency in generative models, can improve the predictive performance of BO as well and improve representational alignment between diverse SMILES. Together, this setup yields an effective optimizer that is prone to repeat isomer generation. We enhance diversity through the inclusion of an isomer-based diversity filter as well as by inflating the previous best in the expected improvement acquisition function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Excellent performance on the PMO benchmark and the MPO docking benchmark hint at powerful optimization ability; however, it may not reflect optimization behavior in the complex reward landscapes found in real-world campaigns where competing objectives like synthetic accessibility, solubility, and activity must be navigated simultaneously. Further work should assess SEGO on challenging oracles including physics-based simulations relevant to drug-discovery and catalysis as well as stringent synthesis constraints before considering deployment to wet-lab prospective campaigns (Lee et al. , 2024a ) . Finally, practical high-throughput campaigns require multiobjective optimization over Pareto fronts and batched acquisition (Sin et al. , 2025 ) which could be introduced into SEGO via alternative acquisition functions (Muthyala et al. , 2025 ) . Despite these limitations, the sample efficiency and hit-finding ability demonstrated here are a promising step toward generative molecular design driven by direct experimental feedback.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Software and Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Software to run SEGO can be found at https://github.com/schwallergroup/sego .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As AI-driven methods become increasingly integrated into chemical discovery, it is important to acknowledge the dual-use risks inherent in molecular generation, which become more relevant as engines become more powerful. The risks of SEGO are in line with existing performant molecular design engines. Such risks can be mitigated through controlled access to oracles and established safeguards at the institutional level.\"\n }\n]","isConversational":false}}},{"rowIdx":296,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.13101"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"TSSM: Triaxial State Space Model for Global Station Weather Forecasting with Temporal-Variable-Historical Modeling"},"authors":{"kind":"list like","value":["Songru Yang","Zili Liu","Tao Han","Ben Fei","Fenghua Ling","Lei Bai","Chang Liu","Xiangyang Ji","Zhenwei Shi","Zhengxia Zou"],"string":"[\n \"Songru Yang\",\n \"Zili Liu\",\n \"Tao Han\",\n \"Ben Fei\",\n \"Fenghua Ling\",\n \"Lei Bai\",\n \"Chang Liu\",\n \"Xiangyang Ji\",\n \"Zhenwei Shi\",\n \"Zhengxia Zou\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.13101v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Global Station Weather Forecasting (GSWF) plays a vital role in delivering timely, localized weather predictions worldwide [ 56 , 59 , 18 ] . By improving average accuracy, capturing extreme events, and extending effective forecasting horizons, GSWF supports critical applications such as disaster management and transportation through precise, site-specific forecasts. This capability is increasingly important for mitigating climate-change impacts and local-scale extremes that grid-based approaches often fail to resolve [ 52 , 72 ] . Existing methods have already supported key scenarios such as the Winter Olympics [ 59 ] ."},{"type":"para","text":"Existing data-driven approaches to GSWF generally formulate the task as multivariate time series forecasting. They use historical observations to predict future dynamics with advanced backbones, including MLP-based models [ 66 , 31 , 39 ] , LSTM variants [ 23 , 21 ] , Transformers [ 71 , 37 , 36 ] , and Mamba-based architectures [ 43 , 15 , 40 ] . Recent studies have incorporated regional context [ 22 , 39 , 40 ] . These methods aim to learn complex temporal-variable dependencies and nonlinear weather patterns, offering a compelling complement to traditional physics-based forecasting [ 3 ] . Nevertheless, even increasingly sophisticated designs [ 59 , 62 ] often provide limited gains, while average accuracy, extreme event prediction, and long-horizon stability remain unsatisfactory due to severe error accumulation in iterative forecasting."},{"type":"para","text":"We attribute these limitations to an overreliance on temporal look-back windows inherited from conventional time series forecasting. Under this paradigm, although previous methods build trend-seasonal-residual decomposition or retrieve memorized patterns [ 31 , 58 ] , they operate under stationarity and myopic assumptions [ 5 , 11 ] , presuming that future patterns remain consistent with, and depend mainly on, a limited historical window. Such assumptions conflict with the inherently non-stationary, cross-scale periodic nature of atmospheric dynamics [ 44 , 24 ] , thereby constraining the model’s ability to capture long-range dependencies and the underlying physical structures essential for accurate forecasting."},{"type":"para","text":"Beyond the short-term temporal look-back window, we revisit GSWF from a long-term, cross-scale perspective. Inspired by same-period climatological analysis in traditional forecasting practice [ 41 ] , we observe that although weather series are highly chaotic along the current temporal dimension, they exhibit consistent and separable patterns in historical evolution, i.e., month/day/hour-aligned observations across years at the same station, as illustrated in Fig. 1 ."},{"type":"para","text":"Building on this insight, we expand temporal-variable modeling into a historical dimension and propose a novel Triaxial State Space Model (TSSM) with a temporal-variable-historical forecasting pipeline. As shown in Fig. 2 , TSSM leverages the directed scanning capability of state space models for axial modeling, anchors short-term hourly dynamics on the temporal and variable axes to long-term yearly evolution through historical-axis modeling, and causally learns evolutionary patterns from the same month, day, and hour across historical years. This design provides long-term references for forecasting, improves extreme event capture especially for long-horizon settings, and calibrates predictions to mitigate accumulated errors in iterative forecasting. Moreover, TSSM extends the existing GSWF setting to support forecasting under missing observations [ 49 ] , thereby improving robustness in real-world scenarios where missing data are unavoidable."},{"type":"para","text":"Specifically, our pipeline contains three key designs: triaxial data reorganization , causal forecasting , and the TSSM architecture . First, for triaxial data reorganization , we establish a historical axis without extra inputs by stacking multivariate weather series from previous years at the same month/day/hour into a T × V × H T\\times V\\times H tensor, corresponding to temporal, variable, and historical axes, as shown in Fig. 2 (a). Second, for causal forecasting , each current prediction is supported by historical windows from years 0 to H H in the look-back window and years 0 to H − 1 H-1 in the forecast window, while temporal-variable patterns flow causally from look-back to forecast windows, no future leakage as Fig. 2 (c). Third, for the TSSM architecture , we introduce temporal, variable, and historical scanning modules, i.e., T-Scan, V-Scan, and H-Scan, to efficiently model yearly expanded inputs. T-Scan captures short-term dynamics, V-Scan models inter-variable correlations, and H-Scan captures long-term historical evolution and deviations from historical averages toward recent years, as shown in Fig. 2 (b). These modules can be independently combined and hierarchically shared to construct weather states from seasonal patterns to extremes. When available history decreases, TSSM naturally degenerates to temporal-variable prediction anchored by last-year references, and shorter cycles such as weekly aligned data can also be adopted."},{"type":"para","text":"We conduct extensive experiments on Weather-5K [ 19 ] , our main testbed, currently the largest GSWF-specific station weather dataset that covers and extends existing station datasets, and further validate generalizability on hydrological data under human regulation and general time series datasets spanning human behavioral patterns and energy dynamics [ 28 , 70 , 26 ] . It contains hourly temperature, dew point, wind direction, wind rate, and sea-level pressure records from 5k+ surface weather stations worldwide, spanning six continents over a 10-year period. In both qualitative and quantitative evaluations, TSSM significantly outperforms existing methods in terms of both average forecasting accuracy and extreme event prediction accuracy as shown in Fig. 2 (d). For Weather-5K, our method achieves the best or second-best performance on 88% of the 72 results across four metrics, with average gains of 10% in overall accuracy and 61% in extreme event prediction. For other 2 datasets, TSSM also achieves in total 95% best or second-best results. Moreover, our advantages become even more pronounced in long-term and robust forecasting. It reaches a 37.5% overall gain at 240h and up to 103.5% under 48h × \\times 5 iterative manner with cumulative iterative error < < 4.3%, even outperforming one-step forecasting, and exhibits strong robustness to missing data, with less than 10% degradation compared with > > 50% for baselines, at an 80% missing rate."},{"type":"para","text":"Our contributions can be summarized as follows:"},{"type":"para","text":"We introduce TSSM that extends GSWF to a triaxial temporal-variable-historical paradigm for jointly modeling short-term dynamics and long-term evolution while compensating for full-window stationarity and cross-scale periodicity beyond look-back windows, offering a nearly free lunch for comprehensive performance gains."},{"type":"para","text":"We propose triaxial scanning mechanisms, including T-Scan, V-Scan, and H-Scan. Based on short-term dynamics and variable correlations, we extend historical-axis modeling in an isomorphic state space model manner to learn the long-term evolution patterns of period-aligned historical weather and design a history-enhanced pipeline with triaxial data reorganization and causal forecasting."},{"type":"para","text":"Extensive experiments on Weather-5K, the largest up-to-date GSWF dataset and human-involved datasets, validate that TSSM not only significantly outperforms existing methods in average and extreme event prediction but also provides an unprecedented solution to cumulative errors in iterative forecasting and missing observations without extra information or high complexity."},{"type":"heading","level":1,"text":"II Related Works"},{"type":"heading","level":2,"text":"II-A Data-driven Weather Prediction"},{"type":"para","text":"Since 2022, the AI and atmospheric science communities have shown rapidly growing interest in data-driven Numerical Weather Prediction (NWP) models, such as Pangu-Weather [ 4 ] , GraphCast [ 27 ] , GenCast [ 10 ] , and related methods [ 50 , 6 , 12 ] , which operate on gridded reanalysis data (e.g., ERA5 [ 20 ] ) at resolutions such as 0.25 ∘ 0.25^{\\circ} and 0.09 ∘ 0.09^{\\circ} . In gridded weather-field forecasting and nowcasting, recent studies have emphasized precipitation/rainfall dynamics and multi-source fusion [ 55 , 29 , 64 , 7 , 33 , 67 , 34 ] . However, these large-scale, region-averaged methods may not match station-level weather systems and can overlook highly destructive local extreme weather. In contrast to weather-field forecasting, station-level forecasting focuses on point-wise in-situ observations and localized physical regimes, some initial attempts, such as Corrformer and WSSM [ 59 , 62 , 22 ] , have treated station weather forecasting as an independent task and achieved promising results. However, these methods still rely on short-term temporal look-back windows, which greatly limits their capability, robustness and real-world applicability."},{"type":"heading","level":2,"text":"II-B Multivariate Time Series Forecasting"},{"type":"para","text":"Multivariate Time Series Forecasting aims to predict future dynamics from short-term temporal-variable dependencies within a look-back window, implicitly assuming local stationarity and dominant dependence on recent history [ 66 , 47 , 57 , 31 , 14 , 17 , 38 , 35 , 42 , 45 , 51 , 54 , 58 , 60 , 68 , 69 ] . Early explorations mainly relied on CNN/RNN backbones to extract local temporal patterns, while simple pooling-like operations were used to model variable correlations [ 23 , 21 ] . More recently, Transformer-based methods have enabled more flexible factorized modeling of time-variable interactions via attention mechanisms [ 71 , 37 , 36 , 9 , 1 ] . In parallel, SSM-based methods with directional scanning have improved long-range dependency modeling and computational efficiency [ 16 , 15 , 40 , 43 ] . A recent shared trend across both Transformer and SSM methods is the shift toward time series foundation models and LLM-enhanced pipelines, aiming for better transferability, robustness, and long-horizon stability [ 9 , 1 , 8 ] . Existing GSWF methods inherit this framework and have achieved promising results [ 59 , 62 ] . However, although incorporating geographical information or multi-station modeling can bring marginal improvements, forecasting solely from a look-back window is inherently limited in capturing long-term weather structures, such as annual or seasonal backgrounds. These methods also lack physical structure [ 44 , 24 ] : even a simple historical average can provide significant stationarity support for understanding extreme weather. As a result, predictions often degenerate into repetitions of look-back-window patterns, largely due to stationarity and myopic assumptions. This motivates our preliminary extension of GSWF beyond the look-back window into a historical dimension."},{"type":"heading","level":2,"text":"II-C Period-aware and Memory-enhanced Forecasting"},{"type":"para","text":"Periodicity has been extensively studied in time series forecasting, as it provides a useful prior for modeling long-range temporal dependencies in a simplified and robust manner. Existing methods, such as [ 58 , 48 , 65 ] , usually establish cross-period dependencies by associating positions with the same phase across different cycles. Nevertheless, unlike our method, these approaches still extract periodic patterns only from the limited look-back window. When the observation window is insufficient or corrupted, such locally mined periodic features can become unreliable, leading to degraded performance."},{"type":"para","text":"Taking a step further, memory mechanisms are an effective strategy for improving long-horizon forecasting performance, but they have only been sparsely explored in time series forecasting. Existing methods typically treat periodicity as a mechanism for memory pattern extraction, where sequences are segmented based on periods and representative patterns are stored to enhance forecasting, such as [ 31 , 30 ] . More recent work either augments recurrent models with long-memory filters [ 61 , 25 ] or adopts retrieval-augmented LLMs for memory-enhanced forecasting [ 53 ] . Compared with our method, however, these approaches typically require explicit memory-bank design and maintenance, they often merge or select memory items using statistical criteria, such as feature similarity or attention, which may discard physically meaningful structures and cause pattern mismatch, such as injecting winter memories into summer forecasts."},{"type":"heading","level":1,"text":"III Method"},{"type":"para","text":"An overview of the Triaxial State Space Model (TSSM) is shown in Fig. 3 . We pioneer the modeling of historical evolution as a state-transition process along the historical axis, describe coexisting short-long-term weather dynamics in an isomorphic SSM manner, and maintain axial linear complexity under multiplied data dimensions. TSSM extends the GSWF pipeline through two co-designed components: the triaxially reorganized data structure and the causal forecasting strategy."},{"type":"heading","level":2,"text":"III-A Triaxial Reorganized Data"},{"type":"para","text":"Incorporating a historical axis requires reorganizing conventional time-variable 2D data into a time-variable-history 3D representation, as shown in Fig. 2 (a). Rather than introducing additional external inputs, we extract intrinsic historical context through period-aligned analysis, as illustrated in Fig. 3 (b), multivariate records V i V_{i} from different years Y i Y_{i} in absolute time are stacked along the historical dimension on a relative temporal axis aligned by the month, day, and hour, which can be readily integrated into existing pipelines."},{"type":"para","text":"Let 𝐗 all ∈ ℝ T all × V \\mathbf{X}_{\\text{all}}\\in\\mathbb{R}^{T_{\\text{all}}\\times V} denote the complete weather series of a station, where T all T_{\\text{all}} is the total number of time steps and V V is the number of variables. Through period-aligned alignment, the temporal axis is folded so that samples with identical sub-timestamps (i.e., the same month/day/hour) are aligned across H H years. This reshaping yields a historically augmented tensor 𝐗 ch ∈ ℝ H × T all / H × V \\mathbf{X}_{\\text{ch}}\\in\\mathbb{R}^{H\\times T_{\\text{all}}/H\\times V} . To generate the final triaxial input 𝐗 ∈ ℝ H × T × V \\mathbf{X}\\in\\mathbb{R}^{H\\times T\\times V} , we apply a sliding window of length T T along the intra-year time axis. Among the three axes, the temporal and historical axes are ordered, whereas the variable axis is unordered; therefore, we randomly shuffle the variable order for each sample. The construction is defined as:"},{"type":"para","text":"where h ∈ { 0 , 1 , … , H − 1 } h\\in\\{0,1,\\dots,H-1\\} denotes the year index, t ∈ { 0 , 1 , … , T − 1 } t\\in\\{0,1,\\dots,T-1\\} denotes the time index within the sliding window, i i denotes the start index along the intra-year time axis, and π ( ⋅ ) \\pi(\\cdot) is a permutation over variable indices."},{"type":"heading","level":2,"text":"III-B Causal Forecasting"},{"type":"para","text":"Introducing a historical dimension redefines the forecasting task by requiring both historical and temporal causality to be modeled. To predict the future window of the h h -th year, we use the look-back windows from years 0 to h h and the future windows from years 0 to h − 1 h-1 as conditions, as briefly shown in Fig. 2 (c) and illustrated across the full pipeline in Fig. 3 (a)."},{"type":"para","text":"Consequently, information flows unidirectionally along two axes: historically, look-back information propagates from year 0 to h h , while period-aligned history in the future window propagates from year 0 to h − 1 h-1 ; temporally, information propagates from the look-back window to the future, as in existing GSWF settings. This dual-causality mechanism has a two-fold interpretation: long-term historical evolution is dynamically calibrated by short-term patterns within the look-back window, while these short-term patterns are constrained by the long-term background. In fact, this long–short-term correlation reflects multi-periodic weather processes."},{"type":"para","text":"Samples from all years h ∈ [ 0 , H ) h\\in[0,H) are processed simultaneously during training, thereby maintaining overall computational efficiency. The index of current year is H − 1 H-1 The computation and optimization are formulated as follows:"},{"type":"para","text":"where 𝐘 ^ h \\hat{\\mathbf{Y}}_{h} denotes the predicted future for the h h -th year, and ℱ Θ \\mathcal{F}_{\\Theta} denotes TSSM parameterized by Θ \\Theta and optimized with ℒ MSE \\mathcal{L}_{\\text{MSE}} . The term 𝐗 0 : h = { 𝐗 0 , … , 𝐗 h − 1 } \\mathbf{X}_{0:h}=\\{\\mathbf{X}_{0},\\dots,\\mathbf{X}_{h-1}\\} represents look-back windows from years 0 to h h . The term 𝐘 0 : h − 1 = { 𝐘 0 , … , 𝐘 h − 2 } \\mathbf{Y}_{0:h-1}=\\{\\mathbf{Y}_{0},\\dots,\\mathbf{Y}_{h-2}\\} represents future windows from preceding years. During inference, the forecast 𝐘 ^ H − 1 \\hat{\\mathbf{Y}}_{H-1} is causally conditioned on full observations 𝐗 0 : H \\mathbf{X}_{0:H} and 𝐘 0 : H − 1 \\mathbf{Y}_{0:H-1} ."},{"type":"heading","level":2,"text":"III-C Triaxial State Space Model"},{"type":"para","text":"With historically stacked data, modeling global correlations leads to a dramatic increase in complexity. Our Triaxial State Space Model efficiently fuses temporal, variable, and historical information through dedicated scanning branches along each axis, leveraging the directed scanning mechanism of state space models. These branches can be fused independently and shared hierarchically to construct weather-state representations ranging from seasonal patterns to extreme events. The detailed design of TSSM and its operation within the forecasting pipeline are shown in Fig. 3 (a)."},{"type":"para","text":"For clarity, we present the mechanisms in the order of historical scanning (H-Scan), variable scanning (V-Scan), and temporal scanning (T-Scan). We denote the observation encoding of 𝐗 0 : H \\mathbf{X}_{0:H} within the look-back window T o T_{o} as 𝐎 0 : H \\mathbf{O}_{0:H} and the historical future encoding of 𝐘 0 : H − 1 \\mathbf{Y}_{0:H-1} as 𝐅 0 : H − 1 \\mathbf{F}_{0:H-1} . Both are processed by a time-step-wise Tokenizer ( ⋅ ) \\text{Tokenizer}(\\cdot) with latent dimension d d as follows:"},{"type":"heading","level":3,"text":"III-C 1 Historical Scanning"},{"type":"para","text":"Along the historical axis, the historical scanning branch constructs an anomaly-enhanced unidirectional SSM starting from year 0 . After reshaping the features into ℝ ( T × C ) × H × d \\mathbb{R}^{(T\\times C)\\times H\\times d} to explicitly decouple the historical axis, we first scan historical values to extract evolution patterns 𝐟 o h \\mathbf{f}_{o}^{h} and 𝐟 f h \\mathbf{f}_{f}^{h} . The computations are as follows:"},{"type":"para","text":"More importantly, we scan the deviations between each historical value and the their average to uncover extreme patterns 𝐟 o e \\mathbf{f}_{o}^{e} and 𝐟 f e \\mathbf{f}_{f}^{e} . This operation is directly motivated by how climatological analysis evaluates extreme weather: anomaly values that deviate significantly from the historical average Avg-h ( ⋅ ) \\text{Avg-h}(\\cdot) indicate potential extreme events. The evolution and extreme patterns are then fused by MLPs to obtain historical features 𝐙 O H \\mathbf{Z}_{O}^{H} and 𝐙 F H \\mathbf{Z}_{F}^{H} . The computations are as follows:"},{"type":"heading","level":3,"text":"III-C 2 Variable Scanning"},{"type":"para","text":"Along the variable axis, because unordered modeling has already been handled during data reorganization, the variable scanning branch adopts a simple SSM. After reshaping into ℝ ( T × H ) × C × d \\mathbb{R}^{(T\\times H)\\times C\\times d} to explicitly decouple the variable axis, we capture inter-variable correlations 𝐙 O V \\mathbf{Z}_{O}^{V} and 𝐙 F V \\mathbf{Z}_{F}^{V} . Computations are as follows:"},{"type":"para","text":"Notably, shuffling the variable order forces the model to distinguish and interact with different variables based on their content rather than their relative positions along the axis."},{"type":"heading","level":3,"text":"III-C 3 Temporal Scanning"},{"type":"para","text":"Along the temporal axis, the temporal scanning branch fuses features from the other branches and generates the final prediction. It includes a bidirectional SSM to extract temporal dependencies in the look-back window and a forward and half-backward SSM to model temporal dependencies from preceding years while injecting triaxial look-back conditions into the current future window. The inputs are reshaped into ℝ ( H × C ) × T × d \\mathbb{R}^{(H\\times C)\\times T\\times d} to explicitly decouple the temporal axis."},{"type":"para","text":"For 𝐎 0 : h \\mathbf{O}_{0:h} in the look-back window, we perform bidirectional scanning to extract the temporal dependency 𝐙 O T \\mathbf{Z}_{O}^{T} . Together with the historical feature 𝐙 O H \\mathbf{Z}_{O}^{H} and variable correlation 𝐙 O V \\mathbf{Z}_{O}^{V} , it forms the triaxial look-back condition 𝐙 O \\mathbf{Z}_{O} . The computations are as follows:"},{"type":"para","text":"For 𝐅 0 : H − 1 \\mathbf{F}_{0:H-1} in the future window, we first concatenate 𝐙 O \\mathbf{Z}_{O} with 𝐅 0 : H − 1 \\mathbf{F}_{0:H-1} , then perform a forward scan to inject conditions unidirectionally. Its output 𝐟 f t \\mathbf{f}_{f}^{t} is fused with a half-backward scan output 𝐟 b t \\mathbf{f}_{b}^{t} limited to the future window to generate the future temporal dependency 𝐙 F T \\mathbf{Z}_{F}^{T} conditioned on look-back observations. Computations are as follows:"},{"type":"para","text":"By combining the historical feature 𝐙 F H \\mathbf{Z}_{F}^{H} and variable correlation 𝐙 F V \\mathbf{Z}_{F}^{V} , we generate the triaxial future latent 𝐙 F \\mathbf{Z}_{F} and the final prediction for the current year, 𝐘 ^ H − 1 \\hat{\\mathbf{Y}}_{H-1} , as follows:"},{"type":"para","text":"The fusion module Fus ( ⋅ ) \\text{Fus}(\\cdot) above employs frequency and variable enhancement. Given input 𝐮 ∈ ℝ M × N × d \\mathbf{u}\\in\\mathbb{R}^{M\\times N\\times d} with sequence length N N and channel size M M , learnable filters 𝐅 \\mathbf{F} and MLP modulators M ( ⋅ ) \\text{M}(\\cdot) are used to adaptively select crucial frequency components and identify variable-specific patterns. We denote the corresponding frequency- and variable-enhanced latents as 𝐮 f \\mathbf{u}_{f} and 𝐮 v \\mathbf{u}_{v} . Computations are as follows:"},{"type":"heading","level":3,"text":"III-C 4 Hierarchically Shared Structure"},{"type":"para","text":"We organize TSSM into a hierarchical structure to process weather-series features at multiple temporal resolutions. This design is motivated by two considerations: (1) weather series exhibit multi-scale temporal dynamics, and hierarchical processing enables coarse-to-fine representations ranging from global periodic patterns to local extreme events; and (2) although lie on the same temporal axis, patterns from different years are difficult to align precisely in time. Hierarchical processing therefore expands the historical receptive field and better captures shared patterns in historical evolution."},{"type":"para","text":"As shown in Fig. 3 (b), we first perform hierarchical patching on the weather series to obtain a set of micro-to-macro features, denoted as [ 𝐩 N 0 \\mathbf{p}_{N_{0}} , 𝐩 N 1 \\mathbf{p}_{N_{1}} , …, 𝐩 N l \\mathbf{p}_{N_{l}} ], where N i N_{i} is the number of patches. We then use TSSM to process these features sequentially from N l N_{l} to N 0 N_{0} . Between two successive scales, we insert average-pooled features from the previous scale at the beginning and end of the current sequence, enabling the encoder to explicitly exploit the previous scale’s global prior when processing the current scale. Across all scales, we bilinearly upsample lower-resolution features to the highest resolution to form the final hierarchically encoded output 𝐳 \\mathbf{z} . Given the input embedding 𝐮 ∈ ℝ M × N × d \\mathbf{u}\\in\\mathbb{R}^{M\\times N\\times d} , the computations are as follows:"},{"type":"para","text":"where 𝐳 N i \\mathbf{z}_{N_{i}} denotes the feature at scale N i N_{i} , and 𝐡𝐩 N i \\mathbf{hp}_{N_{i}} denotes the average-pooled head patch at scale N i N_{i} . Through this design, we encode multi-scale dynamics with shared weights, improving prediction accuracy while maintaining simplicity."},{"type":"heading","level":1,"text":"IV Experiments"},{"type":"heading","level":2,"text":"IV-A Dataset and Setup"},{"type":"para","text":"Dataset: We curate and benchmark Global Station Weather Forecasting (GSWF) tasks on Weather-5K , the largest up-to-date global station weather benchmark. It records hourly in-situ measurements of temperature, dew point, wind direction, wind rate, and sea-level pressure from 5k stations worldwide from 2014 to 2024. Although only Weather-5K is used in our experiments, it integrates and is quality-controlled against major station weather datasets, including HadISD, ICOADS, and others, making it sufficient to support reliable validation of model performance."},{"type":"para","text":"Beyond natural weather patterns, local weather systems are often highly coupled with human activities. We further validate our method on long-term explicit coupling patterns between weather systems and human water-regulation behaviors using Hydrological Data . This dataset records hydrological observations from five reservoirs in the United States over 46 years. It contains both natural precipitation and anthropogenic regulation signals. Because extreme events are regulated and extreme patterns are often absent in the test years, only average accuracy is adopted for comparison."},{"type":"para","text":"Furthermore, to verify generalizability, we conduct evaluations on five General Time Series Data datasets that include human activities and energy dynamics. Since existing public datasets generally lack sufficiently long records with annual periodicity, we adopt a week-aligned historical structure with a historical range covering the past four weeks."},{"type":"para","text":"Evaluation Protocol: We comprehensively evaluate model performance in terms of average accuracy and extreme event capture using four metrics. For accuracy , we employ MAE/MSE to assess average deviations. For extreme event prediction , we use SEDI99.5th/SEDI90.0th to evaluate the prediction coverage of real-world extreme events. The computations are as follows:"},{"type":"para","text":"where 𝐪 low p \\mathbf{q}^{p}_{\\text{low}} and 𝐪 up p \\mathbf{q}^{p}_{\\text{up}} are lower and upper p p th percentiles, respectively, and p p is set to 99.5 / 90.0 99.5/90.0 . 𝕀 ( ⋅ ) \\mathbb{I}(\\cdot) is indicator function."},{"type":"para","text":"Baselines: To comprehensively validate the superiority of our method, in addition to GSWF-specific models marked by ⋆ \\star , we collect state-of-the-art multivariate time series forecasting methods and retrain them on the GSWF task. Notably, cross-parameter-scale foundation model (Chronos2 120M), period-aware and memory-enhanced models which are relevant to our method are specifically included in this comparasion, marked by ♡ \\heartsuit , ◆ \\lozenge and † {\\dagger} , respectively. Moreover, we also introduce domain-specific methods, as marked by ♠ \\spadesuit and ♣ \\clubsuit . For the missing-observation scenario, we compare with Merlin [ 63 ] , a recent noise-robust forecasting method."},{"type":"heading","level":2,"text":"IV-B Overall performance"},{"type":"heading","level":3,"text":"IV-B 1 Quantitative Results"},{"type":"para","text":"We quantitatively compare the averaged accuracy and extreme weather event prediction performance of comprehensive baselines with a 48 hour look-back window for prediction horizons of 24, 48, 120, and 240 hours."},{"type":"para","text":"Weather-5K Data : As shown in Tab. I , our method significantly outperforms the competing methods, achieving average advantages of 10% in accuracy and 61% in SEDI with 88% of all metrics ranked as either the best or second-best highlighted in red / blue . In particular, for SEDI at the 99.5th percentile, which reflects the capture capability for rare and extreme weather events, our method achieves an improvement of over 90%. For longer forecasting windows, where atmospheric chaos becomes stronger and observational support becomes weaker, the advantage of our method is even more pronounced, reaching an average improvement of 36% for 240h prediction. The results verify the effectiveness of history-enhanced design."},{"type":"para","text":"However, the period-aware and memory-enhanced methods, which are more closely related to ours, do not achieve superior performance. We attribute this to the insufficiency of reliable short-term periodic patterns and the memory confusion caused by chaotic dynamics under the short-sightedness of limited look-back windows."},{"type":"para","text":"Furthermore, despite a substantial disadvantage in parameter scale (5M vs. 120M), our method still overwhelmingly outperforms the fine-tuned foundation model Chronos2, achieving a 61% overall improvement. On the one hand, this suggests that station-level weather sequences exhibit more intricate and domain-specific patterns than general time series, such that universal temporal knowledge does not necessarily translate into substantial performance gains. On the other hand, it demonstrates that the effectiveness of GSWF methods stems primarily from task-specific model design rather than scale."},{"type":"para","text":"Compared with operational physics-based methods [ 13 ] , which assimilate massive multi-source observations through supercomputer-scale computation on dense grids, our lightweight method uses only sparse station observations yet achieves a 32.4% average gain on 34.7% of the metrics (25/72, marked in green ). The gains are especially evident for transient wind variables and extreme events, highlighting TSSM’s advantage in resolving local-scale dynamics and its potential as an alternative for localized weather systems."},{"type":"para","text":"Hydrology and General Data : As shown in Tab. II and Tab. III , TSSM achieves a 35% improvement in hydrological forecasting accuracy, as well as 45% and > > 100% gains in accuracy and SEDI on general time series datasets, respectively. Moreover, 95% of all metrics are ranked as either the best or second-best. These results further demonstrate the effectiveness of our approach for spatiotemporal sequences with varying degrees of human activities and energy dynamics, and reveal that empirical human behaviors and energy consumption also exhibit strong long-term evolutionary characteristics."},{"type":"para","text":"Notably, extreme event capture, measured by SEDI, is of critical importance in real-world scenarios, yet it remains underexplored in existing studies on general forecasting tasks. Our results show that mainstream methods exhibit limited extreme value modeling capability and often fail to capture extreme events in long-term forecasting. In contrast, TSSM maintains robust extreme event identification ability, further highlighting the importance of historical evolution modeling for reliable forecasting."},{"type":"heading","level":3,"text":"IV-B 2 Qualitative Results"},{"type":"para","text":"To verify the reliability of the qualitative results, we further visualize the GSWF outputs on challenging samples across all variables, as illustrated in Fig. 4 . It can be observed in gray boxes that our method substantially improves prediction accuracy and yields high-frequency patterns that are more consistent with extreme values. Meanwhile, we visualized history under noisy conditions, as indicated by the green box. Historically robust patterns that cover both look-back and forecasting window provide a strong reference and help correct noise contamination. More importantly, as shown in the blue boxes, our method does not blindly mimic history: even when historical information is misleading, it can still identify and output the correct forecast. In contrast, baselines tend to only capture low-frequency patterns and collapse toward mean values. This limitation stems from insufficient observations within limited look-back windows to support long-range prediction, which also explains the poor SEDI performance of existing approaches."},{"type":"para","text":"In summary, this qualitative comparison further demonstrates the effectiveness of TSSM and highlights the supportive role of historical information in forecasting, which validates the rationality of our design insight."},{"type":"heading","level":2,"text":"IV-C Ensemble Forecasting"},{"type":"para","text":"By producing multiple candidate predictions supported by different historical observation anchors, TSSM naturally supports ensemble forecasting without relying on complex generative frameworks."},{"type":"para","text":"We compare deterministic methods, which add noise to features before decoding to generate diverse predictions, with probabilistic generative methods that natively support ensemble forecasting. Since the number of past years is H − 1 = 9 H-1=9 , we set the number of candidate predictions to k = 9 k=9 ."},{"type":"para","text":"As shown in Tab. IV , our method substantially improves ensemble prediction accuracy, even surpassing the advanced probabilistic method. However, because the Best-in-9 results are selected according to accuracy, no obvious gain is observed in SEDI. This result demonstrates that multi-year aligned historical data can serve as effective anchors for ensemble forecasting. In contrast, the other deterministic baselines show no evident improvement, indicating that simply increasing the number of candidate forecasts is insufficient to improve ensemble prediction performance."},{"type":"heading","level":2,"text":"IV-D Long-Horizon Forecasting"},{"type":"para","text":"For long-horizon forecasting, existing paradigms rely primarily on short-term observations and are therefore unable to perform reliable one-step long-horizon forecasting. Moreover, during iterative long-term prediction, subsequent forecasts can only be based on inaccurate predictions from the previous step, leading to severe error accumulation and pattern drift."},{"type":"para","text":"By introducing the historical dimension, our method not only utilizes historical information to achieve stronger performance in one-step long-horizon forecasting, but also calibrates errors at each step with sufficient historical support, thereby greatly alleviating error accumulation and pattern drift in iterative forecasting."},{"type":"para","text":"As shown in Tab. V , TSSM not only achieves 37.5% substantial improvements in 240h long-horizon forecasting, but also shows that iterative short-term forecasting, such as 120h × \\times 2 and 48h × \\times 5, outperforms direct 240h × \\times 1 long-horizon forecasting, reaching a 103.5% gain. Owing to limited error accumulation, compared with direct 120h forecasting, iterative 120h × \\times 2 forecasting leads to only a marginal 4.3% degradation in accuracy and SEDI. This property is particularly valuable in weather forecasting, where extending the effective forecasting window is highly desirable."},{"type":"heading","level":2,"text":"IV-E Forecasting under Observation Missing"},{"type":"para","text":"Due to sensor failure, transmission packet loss, et al., global weather stations suffer from widespread missing observations. Existing methods typically employ an independent preprocessing step to impute missing values before model forecast, which introduces additional systematic and cascading errors."},{"type":"para","text":"By introducing the historical dimension, our method can compensate for the missing temporal observations using period-aligned information. Therefore, we take a step further to perform forecasting directly from incomplete observations."},{"type":"para","text":"As shown in Fig. 5 and Tab. VI , when exposed to varying degrees (20% to 80%) of random data missing, TSSM exhibits strong robustness against missing observations in the look-back window with only averaged 4.3% and 8% degradation on accuracy and SEDI. Even with 80% of observations missing, it can still retain 95.7% and 92% of its performance on accuracy and SEDI by relying on historical support. In contrast, for baseline methods, introducing missing noise severely degrades performance. Although state-of-the-art noise-robust forecasting paradigms can significantly alleviate this issue, their performance still drops considerably."},{"type":"para","text":"Notably, since we employ historical data, missing observations in historical records from at least one year ago have been sufficiently recovered. Therefore, we do not introduce observation missing in the historical dimension."},{"type":"heading","level":2,"text":"IV-F Forecasting under Limited Data"},{"type":"para","text":"The rapid accumulation of station weather data does not guarantee that sufficient data are naturally available across all regions. Data imbalance remains severe, requiring models to adapt to limited-data scenarios."},{"type":"para","text":"Although station observations provide global coverage, weather stations remain notably sparse in regions such as Africa (AFR) and South America (SAM) due to uneven regional development. Despite accounting for approximately 30% of global land area and population, these two regions host only 331 and 123 stations, respectively, together representing merely 1/14 of global observations."},{"type":"para","text":"As shown in Tab. VII , when trained on only a very small number of stations, the advantage of our method over baseline approaches becomes even more significant, with improvements of 38% in average accuracy and > > 100% in extreme value capture. By contrast, the competing methods can hardly learn weather dynamics from limited data."},{"type":"para","text":"Benefiting from strong support from historical information, TSSM can model weather dynamics more effectively, leading to superior performance when trained from scratch under limited-sample conditions in these two regions."},{"type":"heading","level":2,"text":"IV-G Ablation Study"},{"type":"heading","level":3,"text":"IV-G 1 Model Structure Analysis"},{"type":"para","text":"As shown in Tab. VIII , we permute and combine the three branches (T-Scan, V-Scan, H-Scan) while retaining at least one of the temporal and historical branches. When the temporal branch is removed, information from the look-back window is injected into the prediction window through modulation. We also ablate several other design choices, including the hierarchical structure level and variable-dimension shuffling. Although our default setting does not explicitly use historical observations, we further explore an explicit initialization strategy that uses the historical average from the previous year."},{"type":"para","text":"The results reveal distinct and meaningful functional roles across different dimensions. The historical dimension contributes more prominently to extreme event capture, and its benefit becomes more pronounced as the hierarchical receptive field expands. The temporal dimension, in contrast, uses historical information as an anchor and contributes more to prediction accuracy. The variable dimension models inter-variable dependencies, improving both accuracy and extreme event capture, while shuffling the variable dimension further enhances model performance."},{"type":"para","text":"Notably, joint modeling of the temporal and historical dimensions introduces a trade-off between prediction accuracy and extreme event capture. Temporal modeling tends to produce more accurate but smoother predictions, whereas historical modeling sacrifices part of the accuracy to better preserve extreme values. Incorporating both dimensions enables the model to achieve a balanced performance across the two objectives. In our analysis of the historical dimension Sec. 6 , we further show that this trade-off can be continuously adjusted by controlling the historical scope, allowing the model to adapt to different forecasting scenarios."},{"type":"para","text":"Finally, when historical averaging is used for prediction initialization, the role of the historical dimension is largely weakened, leading to overly smoothed predictions. When the same initialization strategy is applied to baseline methods such as iTransformer, it fails to bridge long-term historical evolution with short-term temporal dynamics, causing model failure. This further highlights the essential capability of our design in integrating short-long-term weather dynamics for GSWF."},{"type":"heading","level":3,"text":"IV-G 2 Historical Dimension Analysis"},{"type":"para","text":"By expanding the historical dimension, we effectively improve GSWF performance. In the structural ablation study, the historical and temporal dimensions exhibit a clear division of labor, which motivates us to further analyze the historical dimension and investigate how this phenomenon can be regulated."},{"type":"para","text":"As shown in Fig. 6 and Tab. IX , we conduct an ablation analysis using historical training data from different starting years h s t h_{st} . As the historical scope gradually expands, the model performance undergoes a continuous transition from last-year-anchored time-variable forecasting to history-enhanced forecasting. This gradual shift indicates that the trade-off between prediction accuracy and extreme event capture can be controlled by adjusting the scope of the historical dimension."},{"type":"para","text":"Furthermore, we investigate to what extent existing methods that do not explicitly model historical data can benefit from simply concatenating historical records into the temporal dimension, forming a longer look-back window."},{"type":"para","text":"As shown in Tab. X , we flatten eight years history into the time axis to construct a 384h look-back window. Most baseline methods achieve moderate performance improvements, but still significantly fall short of TSSM in effectively utilizing historical information. This demonstrates that transmitting historical information through the temporal dimension is far less efficient than modeling it with a dedicated historical dimension, further validating the effectiveness of our design."},{"type":"heading","level":1,"text":"V Limitation and Future Work"},{"type":"para","text":"As a preliminary attempt to extend GSWF along the historical dimension, this work adopts a simple and necessary design to incorporate period-aligned historical information. Although TSSM demonstrates clear effectiveness, we observe that historical information does not consistently improve all metrics, but instead introduces a trade-off between average accuracy and extreme event forecasting. Although continuous adjustment is achieved, replacing the fusion strategy between historical and temporal features with attention, gating, or direct interpolation does not fundamentally resolve this issue, nor does it produce continuous performance variations. These observations suggest that more effective and controllable historical-temporal fusion remains an important direction for future work."},{"type":"heading","level":1,"text":"VI Conclusion"},{"type":"para","text":"In this work, we propose the Triaxial State Space Model, a history-enhanced temporal-variable-historical forecasting framework that extends GSWF beyond short-term temporal look-back windows. Inspired by climatological analysis in traditional meteorology, TSSM aligns station observations by the same month, day, and hour across years, reorganizing weather series into a time-variable-history structure. Based on this representation, we design hierarchically shared temporal, variable, and historical scanning modules to model short-term dynamics, inter-variable correlations, and long-term historical evolution. A causal forecasting framework further integrates current observations and historical references while maintaining training efficiency."},{"type":"para","text":"Comprehensive experiments verify the superiority of TSSM on Weather-5K and other human-involved datasets. TSSM achieves 10% average accuracy gain and 61% extreme event gain on Weather-5K, obtains 95% best or second-best results on other datasets, and shows stronger advantages in long-term and robust forecasting. It reaches a 37.5% overall gain at 240h and up to 103.5% under iterative manner with cumulative iterative error < < 4.3%, and exhibits strong robustness to missing data, with < < 10% degradation compared with > > 50% for baselines, at an 80% missing rate. These results demonstrate that TSSM with historical-dimensional modeling provides a promising direction for more accurate, robust, and reliable global station weather forecasting."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Global Station Weather Forecasting (GSWF) plays a vital role in delivering timely, localized weather predictions worldwide [ 56 , 59 , 18 ] . By improving average accuracy, capturing extreme events, and extending effective forecasting horizons, GSWF supports critical applications such as disaster management and transportation through precise, site-specific forecasts. This capability is increasingly important for mitigating climate-change impacts and local-scale extremes that grid-based approaches often fail to resolve [ 52 , 72 ] . Existing methods have already supported key scenarios such as the Winter Olympics [ 59 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Existing data-driven approaches to GSWF generally formulate the task as multivariate time series forecasting. They use historical observations to predict future dynamics with advanced backbones, including MLP-based models [ 66 , 31 , 39 ] , LSTM variants [ 23 , 21 ] , Transformers [ 71 , 37 , 36 ] , and Mamba-based architectures [ 43 , 15 , 40 ] . Recent studies have incorporated regional context [ 22 , 39 , 40 ] . These methods aim to learn complex temporal-variable dependencies and nonlinear weather patterns, offering a compelling complement to traditional physics-based forecasting [ 3 ] . Nevertheless, even increasingly sophisticated designs [ 59 , 62 ] often provide limited gains, while average accuracy, extreme event prediction, and long-horizon stability remain unsatisfactory due to severe error accumulation in iterative forecasting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We attribute these limitations to an overreliance on temporal look-back windows inherited from conventional time series forecasting. Under this paradigm, although previous methods build trend-seasonal-residual decomposition or retrieve memorized patterns [ 31 , 58 ] , they operate under stationarity and myopic assumptions [ 5 , 11 ] , presuming that future patterns remain consistent with, and depend mainly on, a limited historical window. Such assumptions conflict with the inherently non-stationary, cross-scale periodic nature of atmospheric dynamics [ 44 , 24 ] , thereby constraining the model’s ability to capture long-range dependencies and the underlying physical structures essential for accurate forecasting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond the short-term temporal look-back window, we revisit GSWF from a long-term, cross-scale perspective. Inspired by same-period climatological analysis in traditional forecasting practice [ 41 ] , we observe that although weather series are highly chaotic along the current temporal dimension, they exhibit consistent and separable patterns in historical evolution, i.e., month/day/hour-aligned observations across years at the same station, as illustrated in Fig. 1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on this insight, we expand temporal-variable modeling into a historical dimension and propose a novel Triaxial State Space Model (TSSM) with a temporal-variable-historical forecasting pipeline. As shown in Fig. 2 , TSSM leverages the directed scanning capability of state space models for axial modeling, anchors short-term hourly dynamics on the temporal and variable axes to long-term yearly evolution through historical-axis modeling, and causally learns evolutionary patterns from the same month, day, and hour across historical years. This design provides long-term references for forecasting, improves extreme event capture especially for long-horizon settings, and calibrates predictions to mitigate accumulated errors in iterative forecasting. Moreover, TSSM extends the existing GSWF setting to support forecasting under missing observations [ 49 ] , thereby improving robustness in real-world scenarios where missing data are unavoidable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Specifically, our pipeline contains three key designs: triaxial data reorganization , causal forecasting , and the TSSM architecture . First, for triaxial data reorganization , we establish a historical axis without extra inputs by stacking multivariate weather series from previous years at the same month/day/hour into a T × V × H T\\\\times V\\\\times H tensor, corresponding to temporal, variable, and historical axes, as shown in Fig. 2 (a). Second, for causal forecasting , each current prediction is supported by historical windows from years 0 to H H in the look-back window and years 0 to H − 1 H-1 in the forecast window, while temporal-variable patterns flow causally from look-back to forecast windows, no future leakage as Fig. 2 (c). Third, for the TSSM architecture , we introduce temporal, variable, and historical scanning modules, i.e., T-Scan, V-Scan, and H-Scan, to efficiently model yearly expanded inputs. T-Scan captures short-term dynamics, V-Scan models inter-variable correlations, and H-Scan captures long-term historical evolution and deviations from historical averages toward recent years, as shown in Fig. 2 (b). These modules can be independently combined and hierarchically shared to construct weather states from seasonal patterns to extremes. When available history decreases, TSSM naturally degenerates to temporal-variable prediction anchored by last-year references, and shorter cycles such as weekly aligned data can also be adopted.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct extensive experiments on Weather-5K [ 19 ] , our main testbed, currently the largest GSWF-specific station weather dataset that covers and extends existing station datasets, and further validate generalizability on hydrological data under human regulation and general time series datasets spanning human behavioral patterns and energy dynamics [ 28 , 70 , 26 ] . It contains hourly temperature, dew point, wind direction, wind rate, and sea-level pressure records from 5k+ surface weather stations worldwide, spanning six continents over a 10-year period. In both qualitative and quantitative evaluations, TSSM significantly outperforms existing methods in terms of both average forecasting accuracy and extreme event prediction accuracy as shown in Fig. 2 (d). For Weather-5K, our method achieves the best or second-best performance on 88% of the 72 results across four metrics, with average gains of 10% in overall accuracy and 61% in extreme event prediction. For other 2 datasets, TSSM also achieves in total 95% best or second-best results. Moreover, our advantages become even more pronounced in long-term and robust forecasting. It reaches a 37.5% overall gain at 240h and up to 103.5% under 48h × \\\\times 5 iterative manner with cumulative iterative error < < 4.3%, even outperforming one-step forecasting, and exhibits strong robustness to missing data, with less than 10% degradation compared with > > 50% for baselines, at an 80% missing rate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions can be summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce TSSM that extends GSWF to a triaxial temporal-variable-historical paradigm for jointly modeling short-term dynamics and long-term evolution while compensating for full-window stationarity and cross-scale periodicity beyond look-back windows, offering a nearly free lunch for comprehensive performance gains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose triaxial scanning mechanisms, including T-Scan, V-Scan, and H-Scan. Based on short-term dynamics and variable correlations, we extend historical-axis modeling in an isomorphic state space model manner to learn the long-term evolution patterns of period-aligned historical weather and design a history-enhanced pipeline with triaxial data reorganization and causal forecasting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Extensive experiments on Weather-5K, the largest up-to-date GSWF dataset and human-involved datasets, validate that TSSM not only significantly outperforms existing methods in average and extreme event prediction but also provides an unprecedented solution to cumulative errors in iterative forecasting and missing observations without extra information or high complexity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Related Works\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Data-driven Weather Prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since 2022, the AI and atmospheric science communities have shown rapidly growing interest in data-driven Numerical Weather Prediction (NWP) models, such as Pangu-Weather [ 4 ] , GraphCast [ 27 ] , GenCast [ 10 ] , and related methods [ 50 , 6 , 12 ] , which operate on gridded reanalysis data (e.g., ERA5 [ 20 ] ) at resolutions such as 0.25 ∘ 0.25^{\\\\circ} and 0.09 ∘ 0.09^{\\\\circ} . In gridded weather-field forecasting and nowcasting, recent studies have emphasized precipitation/rainfall dynamics and multi-source fusion [ 55 , 29 , 64 , 7 , 33 , 67 , 34 ] . However, these large-scale, region-averaged methods may not match station-level weather systems and can overlook highly destructive local extreme weather. In contrast to weather-field forecasting, station-level forecasting focuses on point-wise in-situ observations and localized physical regimes, some initial attempts, such as Corrformer and WSSM [ 59 , 62 , 22 ] , have treated station weather forecasting as an independent task and achieved promising results. However, these methods still rely on short-term temporal look-back windows, which greatly limits their capability, robustness and real-world applicability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Multivariate Time Series Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multivariate Time Series Forecasting aims to predict future dynamics from short-term temporal-variable dependencies within a look-back window, implicitly assuming local stationarity and dominant dependence on recent history [ 66 , 47 , 57 , 31 , 14 , 17 , 38 , 35 , 42 , 45 , 51 , 54 , 58 , 60 , 68 , 69 ] . Early explorations mainly relied on CNN/RNN backbones to extract local temporal patterns, while simple pooling-like operations were used to model variable correlations [ 23 , 21 ] . More recently, Transformer-based methods have enabled more flexible factorized modeling of time-variable interactions via attention mechanisms [ 71 , 37 , 36 , 9 , 1 ] . In parallel, SSM-based methods with directional scanning have improved long-range dependency modeling and computational efficiency [ 16 , 15 , 40 , 43 ] . A recent shared trend across both Transformer and SSM methods is the shift toward time series foundation models and LLM-enhanced pipelines, aiming for better transferability, robustness, and long-horizon stability [ 9 , 1 , 8 ] . Existing GSWF methods inherit this framework and have achieved promising results [ 59 , 62 ] . However, although incorporating geographical information or multi-station modeling can bring marginal improvements, forecasting solely from a look-back window is inherently limited in capturing long-term weather structures, such as annual or seasonal backgrounds. These methods also lack physical structure [ 44 , 24 ] : even a simple historical average can provide significant stationarity support for understanding extreme weather. As a result, predictions often degenerate into repetitions of look-back-window patterns, largely due to stationarity and myopic assumptions. This motivates our preliminary extension of GSWF beyond the look-back window into a historical dimension.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Period-aware and Memory-enhanced Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Periodicity has been extensively studied in time series forecasting, as it provides a useful prior for modeling long-range temporal dependencies in a simplified and robust manner. Existing methods, such as [ 58 , 48 , 65 ] , usually establish cross-period dependencies by associating positions with the same phase across different cycles. Nevertheless, unlike our method, these approaches still extract periodic patterns only from the limited look-back window. When the observation window is insufficient or corrupted, such locally mined periodic features can become unreliable, leading to degraded performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taking a step further, memory mechanisms are an effective strategy for improving long-horizon forecasting performance, but they have only been sparsely explored in time series forecasting. Existing methods typically treat periodicity as a mechanism for memory pattern extraction, where sequences are segmented based on periods and representative patterns are stored to enhance forecasting, such as [ 31 , 30 ] . More recent work either augments recurrent models with long-memory filters [ 61 , 25 ] or adopts retrieval-augmented LLMs for memory-enhanced forecasting [ 53 ] . Compared with our method, however, these approaches typically require explicit memory-bank design and maintenance, they often merge or select memory items using statistical criteria, such as feature similarity or attention, which may discard physically meaningful structures and cause pattern mismatch, such as injecting winter memories into summer forecasts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An overview of the Triaxial State Space Model (TSSM) is shown in Fig. 3 . We pioneer the modeling of historical evolution as a state-transition process along the historical axis, describe coexisting short-long-term weather dynamics in an isomorphic SSM manner, and maintain axial linear complexity under multiplied data dimensions. TSSM extends the GSWF pipeline through two co-designed components: the triaxially reorganized data structure and the causal forecasting strategy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Triaxial Reorganized Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Incorporating a historical axis requires reorganizing conventional time-variable 2D data into a time-variable-history 3D representation, as shown in Fig. 2 (a). Rather than introducing additional external inputs, we extract intrinsic historical context through period-aligned analysis, as illustrated in Fig. 3 (b), multivariate records V i V_{i} from different years Y i Y_{i} in absolute time are stacked along the historical dimension on a relative temporal axis aligned by the month, day, and hour, which can be readily integrated into existing pipelines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝐗 all ∈ ℝ T all × V \\\\mathbf{X}_{\\\\text{all}}\\\\in\\\\mathbb{R}^{T_{\\\\text{all}}\\\\times V} denote the complete weather series of a station, where T all T_{\\\\text{all}} is the total number of time steps and V V is the number of variables. Through period-aligned alignment, the temporal axis is folded so that samples with identical sub-timestamps (i.e., the same month/day/hour) are aligned across H H years. This reshaping yields a historically augmented tensor 𝐗 ch ∈ ℝ H × T all / H × V \\\\mathbf{X}_{\\\\text{ch}}\\\\in\\\\mathbb{R}^{H\\\\times T_{\\\\text{all}}/H\\\\times V} . To generate the final triaxial input 𝐗 ∈ ℝ H × T × V \\\\mathbf{X}\\\\in\\\\mathbb{R}^{H\\\\times T\\\\times V} , we apply a sliding window of length T T along the intra-year time axis. Among the three axes, the temporal and historical axes are ordered, whereas the variable axis is unordered; therefore, we randomly shuffle the variable order for each sample. The construction is defined as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where h ∈ { 0 , 1 , … , H − 1 } h\\\\in\\\\{0,1,\\\\dots,H-1\\\\} denotes the year index, t ∈ { 0 , 1 , … , T − 1 } t\\\\in\\\\{0,1,\\\\dots,T-1\\\\} denotes the time index within the sliding window, i i denotes the start index along the intra-year time axis, and π ( ⋅ ) \\\\pi(\\\\cdot) is a permutation over variable indices.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Causal Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Introducing a historical dimension redefines the forecasting task by requiring both historical and temporal causality to be modeled. To predict the future window of the h h -th year, we use the look-back windows from years 0 to h h and the future windows from years 0 to h − 1 h-1 as conditions, as briefly shown in Fig. 2 (c) and illustrated across the full pipeline in Fig. 3 (a).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequently, information flows unidirectionally along two axes: historically, look-back information propagates from year 0 to h h , while period-aligned history in the future window propagates from year 0 to h − 1 h-1 ; temporally, information propagates from the look-back window to the future, as in existing GSWF settings. This dual-causality mechanism has a two-fold interpretation: long-term historical evolution is dynamically calibrated by short-term patterns within the look-back window, while these short-term patterns are constrained by the long-term background. In fact, this long–short-term correlation reflects multi-periodic weather processes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Samples from all years h ∈ [ 0 , H ) h\\\\in[0,H) are processed simultaneously during training, thereby maintaining overall computational efficiency. The index of current year is H − 1 H-1 The computation and optimization are formulated as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝐘 ^ h \\\\hat{\\\\mathbf{Y}}_{h} denotes the predicted future for the h h -th year, and ℱ Θ \\\\mathcal{F}_{\\\\Theta} denotes TSSM parameterized by Θ \\\\Theta and optimized with ℒ MSE \\\\mathcal{L}_{\\\\text{MSE}} . The term 𝐗 0 : h = { 𝐗 0 , … , 𝐗 h − 1 } \\\\mathbf{X}_{0:h}=\\\\{\\\\mathbf{X}_{0},\\\\dots,\\\\mathbf{X}_{h-1}\\\\} represents look-back windows from years 0 to h h . The term 𝐘 0 : h − 1 = { 𝐘 0 , … , 𝐘 h − 2 } \\\\mathbf{Y}_{0:h-1}=\\\\{\\\\mathbf{Y}_{0},\\\\dots,\\\\mathbf{Y}_{h-2}\\\\} represents future windows from preceding years. During inference, the forecast 𝐘 ^ H − 1 \\\\hat{\\\\mathbf{Y}}_{H-1} is causally conditioned on full observations 𝐗 0 : H \\\\mathbf{X}_{0:H} and 𝐘 0 : H − 1 \\\\mathbf{Y}_{0:H-1} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Triaxial State Space Model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With historically stacked data, modeling global correlations leads to a dramatic increase in complexity. Our Triaxial State Space Model efficiently fuses temporal, variable, and historical information through dedicated scanning branches along each axis, leveraging the directed scanning mechanism of state space models. These branches can be fused independently and shared hierarchically to construct weather-state representations ranging from seasonal patterns to extreme events. The detailed design of TSSM and its operation within the forecasting pipeline are shown in Fig. 3 (a).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For clarity, we present the mechanisms in the order of historical scanning (H-Scan), variable scanning (V-Scan), and temporal scanning (T-Scan). We denote the observation encoding of 𝐗 0 : H \\\\mathbf{X}_{0:H} within the look-back window T o T_{o} as 𝐎 0 : H \\\\mathbf{O}_{0:H} and the historical future encoding of 𝐘 0 : H − 1 \\\\mathbf{Y}_{0:H-1} as 𝐅 0 : H − 1 \\\\mathbf{F}_{0:H-1} . Both are processed by a time-step-wise Tokenizer ( ⋅ ) \\\\text{Tokenizer}(\\\\cdot) with latent dimension d d as follows:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-C 1 Historical Scanning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Along the historical axis, the historical scanning branch constructs an anomaly-enhanced unidirectional SSM starting from year 0 . After reshaping the features into ℝ ( T × C ) × H × d \\\\mathbb{R}^{(T\\\\times C)\\\\times H\\\\times d} to explicitly decouple the historical axis, we first scan historical values to extract evolution patterns 𝐟 o h \\\\mathbf{f}_{o}^{h} and 𝐟 f h \\\\mathbf{f}_{f}^{h} . The computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"More importantly, we scan the deviations between each historical value and the their average to uncover extreme patterns 𝐟 o e \\\\mathbf{f}_{o}^{e} and 𝐟 f e \\\\mathbf{f}_{f}^{e} . This operation is directly motivated by how climatological analysis evaluates extreme weather: anomaly values that deviate significantly from the historical average Avg-h ( ⋅ ) \\\\text{Avg-h}(\\\\cdot) indicate potential extreme events. The evolution and extreme patterns are then fused by MLPs to obtain historical features 𝐙 O H \\\\mathbf{Z}_{O}^{H} and 𝐙 F H \\\\mathbf{Z}_{F}^{H} . The computations are as follows:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-C 2 Variable Scanning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Along the variable axis, because unordered modeling has already been handled during data reorganization, the variable scanning branch adopts a simple SSM. After reshaping into ℝ ( T × H ) × C × d \\\\mathbb{R}^{(T\\\\times H)\\\\times C\\\\times d} to explicitly decouple the variable axis, we capture inter-variable correlations 𝐙 O V \\\\mathbf{Z}_{O}^{V} and 𝐙 F V \\\\mathbf{Z}_{F}^{V} . Computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, shuffling the variable order forces the model to distinguish and interact with different variables based on their content rather than their relative positions along the axis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-C 3 Temporal Scanning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Along the temporal axis, the temporal scanning branch fuses features from the other branches and generates the final prediction. It includes a bidirectional SSM to extract temporal dependencies in the look-back window and a forward and half-backward SSM to model temporal dependencies from preceding years while injecting triaxial look-back conditions into the current future window. The inputs are reshaped into ℝ ( H × C ) × T × d \\\\mathbb{R}^{(H\\\\times C)\\\\times T\\\\times d} to explicitly decouple the temporal axis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For 𝐎 0 : h \\\\mathbf{O}_{0:h} in the look-back window, we perform bidirectional scanning to extract the temporal dependency 𝐙 O T \\\\mathbf{Z}_{O}^{T} . Together with the historical feature 𝐙 O H \\\\mathbf{Z}_{O}^{H} and variable correlation 𝐙 O V \\\\mathbf{Z}_{O}^{V} , it forms the triaxial look-back condition 𝐙 O \\\\mathbf{Z}_{O} . The computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For 𝐅 0 : H − 1 \\\\mathbf{F}_{0:H-1} in the future window, we first concatenate 𝐙 O \\\\mathbf{Z}_{O} with 𝐅 0 : H − 1 \\\\mathbf{F}_{0:H-1} , then perform a forward scan to inject conditions unidirectionally. Its output 𝐟 f t \\\\mathbf{f}_{f}^{t} is fused with a half-backward scan output 𝐟 b t \\\\mathbf{f}_{b}^{t} limited to the future window to generate the future temporal dependency 𝐙 F T \\\\mathbf{Z}_{F}^{T} conditioned on look-back observations. Computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By combining the historical feature 𝐙 F H \\\\mathbf{Z}_{F}^{H} and variable correlation 𝐙 F V \\\\mathbf{Z}_{F}^{V} , we generate the triaxial future latent 𝐙 F \\\\mathbf{Z}_{F} and the final prediction for the current year, 𝐘 ^ H − 1 \\\\hat{\\\\mathbf{Y}}_{H-1} , as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fusion module Fus ( ⋅ ) \\\\text{Fus}(\\\\cdot) above employs frequency and variable enhancement. Given input 𝐮 ∈ ℝ M × N × d \\\\mathbf{u}\\\\in\\\\mathbb{R}^{M\\\\times N\\\\times d} with sequence length N N and channel size M M , learnable filters 𝐅 \\\\mathbf{F} and MLP modulators M ( ⋅ ) \\\\text{M}(\\\\cdot) are used to adaptively select crucial frequency components and identify variable-specific patterns. We denote the corresponding frequency- and variable-enhanced latents as 𝐮 f \\\\mathbf{u}_{f} and 𝐮 v \\\\mathbf{u}_{v} . Computations are as follows:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-C 4 Hierarchically Shared Structure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We organize TSSM into a hierarchical structure to process weather-series features at multiple temporal resolutions. This design is motivated by two considerations: (1) weather series exhibit multi-scale temporal dynamics, and hierarchical processing enables coarse-to-fine representations ranging from global periodic patterns to local extreme events; and (2) although lie on the same temporal axis, patterns from different years are difficult to align precisely in time. Hierarchical processing therefore expands the historical receptive field and better captures shared patterns in historical evolution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Fig. 3 (b), we first perform hierarchical patching on the weather series to obtain a set of micro-to-macro features, denoted as [ 𝐩 N 0 \\\\mathbf{p}_{N_{0}} , 𝐩 N 1 \\\\mathbf{p}_{N_{1}} , …, 𝐩 N l \\\\mathbf{p}_{N_{l}} ], where N i N_{i} is the number of patches. We then use TSSM to process these features sequentially from N l N_{l} to N 0 N_{0} . Between two successive scales, we insert average-pooled features from the previous scale at the beginning and end of the current sequence, enabling the encoder to explicitly exploit the previous scale’s global prior when processing the current scale. Across all scales, we bilinearly upsample lower-resolution features to the highest resolution to form the final hierarchically encoded output 𝐳 \\\\mathbf{z} . Given the input embedding 𝐮 ∈ ℝ M × N × d \\\\mathbf{u}\\\\in\\\\mathbb{R}^{M\\\\times N\\\\times d} , the computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝐳 N i \\\\mathbf{z}_{N_{i}} denotes the feature at scale N i N_{i} , and 𝐡𝐩 N i \\\\mathbf{hp}_{N_{i}} denotes the average-pooled head patch at scale N i N_{i} . Through this design, we encode multi-scale dynamics with shared weights, improving prediction accuracy while maintaining simplicity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Dataset and Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dataset: We curate and benchmark Global Station Weather Forecasting (GSWF) tasks on Weather-5K , the largest up-to-date global station weather benchmark. It records hourly in-situ measurements of temperature, dew point, wind direction, wind rate, and sea-level pressure from 5k stations worldwide from 2014 to 2024. Although only Weather-5K is used in our experiments, it integrates and is quality-controlled against major station weather datasets, including HadISD, ICOADS, and others, making it sufficient to support reliable validation of model performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond natural weather patterns, local weather systems are often highly coupled with human activities. We further validate our method on long-term explicit coupling patterns between weather systems and human water-regulation behaviors using Hydrological Data . This dataset records hydrological observations from five reservoirs in the United States over 46 years. It contains both natural precipitation and anthropogenic regulation signals. Because extreme events are regulated and extreme patterns are often absent in the test years, only average accuracy is adopted for comparison.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Furthermore, to verify generalizability, we conduct evaluations on five General Time Series Data datasets that include human activities and energy dynamics. Since existing public datasets generally lack sufficiently long records with annual periodicity, we adopt a week-aligned historical structure with a historical range covering the past four weeks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluation Protocol: We comprehensively evaluate model performance in terms of average accuracy and extreme event capture using four metrics. For accuracy , we employ MAE/MSE to assess average deviations. For extreme event prediction , we use SEDI99.5th/SEDI90.0th to evaluate the prediction coverage of real-world extreme events. The computations are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝐪 low p \\\\mathbf{q}^{p}_{\\\\text{low}} and 𝐪 up p \\\\mathbf{q}^{p}_{\\\\text{up}} are lower and upper p p th percentiles, respectively, and p p is set to 99.5 / 90.0 99.5/90.0 . 𝕀 ( ⋅ ) \\\\mathbb{I}(\\\\cdot) is indicator function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baselines: To comprehensively validate the superiority of our method, in addition to GSWF-specific models marked by ⋆ \\\\star , we collect state-of-the-art multivariate time series forecasting methods and retrain them on the GSWF task. Notably, cross-parameter-scale foundation model (Chronos2 120M), period-aware and memory-enhanced models which are relevant to our method are specifically included in this comparasion, marked by ♡ \\\\heartsuit , ◆ \\\\lozenge and † {\\\\dagger} , respectively. Moreover, we also introduce domain-specific methods, as marked by ♠ \\\\spadesuit and ♣ \\\\clubsuit . For the missing-observation scenario, we compare with Merlin [ 63 ] , a recent noise-robust forecasting method.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Overall performance\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"IV-B 1 Quantitative Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We quantitatively compare the averaged accuracy and extreme weather event prediction performance of comprehensive baselines with a 48 hour look-back window for prediction horizons of 24, 48, 120, and 240 hours.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Weather-5K Data : As shown in Tab. I , our method significantly outperforms the competing methods, achieving average advantages of 10% in accuracy and 61% in SEDI with 88% of all metrics ranked as either the best or second-best highlighted in red / blue . In particular, for SEDI at the 99.5th percentile, which reflects the capture capability for rare and extreme weather events, our method achieves an improvement of over 90%. For longer forecasting windows, where atmospheric chaos becomes stronger and observational support becomes weaker, the advantage of our method is even more pronounced, reaching an average improvement of 36% for 240h prediction. The results verify the effectiveness of history-enhanced design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, the period-aware and memory-enhanced methods, which are more closely related to ours, do not achieve superior performance. We attribute this to the insufficiency of reliable short-term periodic patterns and the memory confusion caused by chaotic dynamics under the short-sightedness of limited look-back windows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Furthermore, despite a substantial disadvantage in parameter scale (5M vs. 120M), our method still overwhelmingly outperforms the fine-tuned foundation model Chronos2, achieving a 61% overall improvement. On the one hand, this suggests that station-level weather sequences exhibit more intricate and domain-specific patterns than general time series, such that universal temporal knowledge does not necessarily translate into substantial performance gains. On the other hand, it demonstrates that the effectiveness of GSWF methods stems primarily from task-specific model design rather than scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Compared with operational physics-based methods [ 13 ] , which assimilate massive multi-source observations through supercomputer-scale computation on dense grids, our lightweight method uses only sparse station observations yet achieves a 32.4% average gain on 34.7% of the metrics (25/72, marked in green ). The gains are especially evident for transient wind variables and extreme events, highlighting TSSM’s advantage in resolving local-scale dynamics and its potential as an alternative for localized weather systems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hydrology and General Data : As shown in Tab. II and Tab. III , TSSM achieves a 35% improvement in hydrological forecasting accuracy, as well as 45% and > > 100% gains in accuracy and SEDI on general time series datasets, respectively. Moreover, 95% of all metrics are ranked as either the best or second-best. These results further demonstrate the effectiveness of our approach for spatiotemporal sequences with varying degrees of human activities and energy dynamics, and reveal that empirical human behaviors and energy consumption also exhibit strong long-term evolutionary characteristics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, extreme event capture, measured by SEDI, is of critical importance in real-world scenarios, yet it remains underexplored in existing studies on general forecasting tasks. Our results show that mainstream methods exhibit limited extreme value modeling capability and often fail to capture extreme events in long-term forecasting. In contrast, TSSM maintains robust extreme event identification ability, further highlighting the importance of historical evolution modeling for reliable forecasting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"IV-B 2 Qualitative Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To verify the reliability of the qualitative results, we further visualize the GSWF outputs on challenging samples across all variables, as illustrated in Fig. 4 . It can be observed in gray boxes that our method substantially improves prediction accuracy and yields high-frequency patterns that are more consistent with extreme values. Meanwhile, we visualized history under noisy conditions, as indicated by the green box. Historically robust patterns that cover both look-back and forecasting window provide a strong reference and help correct noise contamination. More importantly, as shown in the blue boxes, our method does not blindly mimic history: even when historical information is misleading, it can still identify and output the correct forecast. In contrast, baselines tend to only capture low-frequency patterns and collapse toward mean values. This limitation stems from insufficient observations within limited look-back windows to support long-range prediction, which also explains the poor SEDI performance of existing approaches.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In summary, this qualitative comparison further demonstrates the effectiveness of TSSM and highlights the supportive role of historical information in forecasting, which validates the rationality of our design insight.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C Ensemble Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By producing multiple candidate predictions supported by different historical observation anchors, TSSM naturally supports ensemble forecasting without relying on complex generative frameworks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare deterministic methods, which add noise to features before decoding to generate diverse predictions, with probabilistic generative methods that natively support ensemble forecasting. Since the number of past years is H − 1 = 9 H-1=9 , we set the number of candidate predictions to k = 9 k=9 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Tab. IV , our method substantially improves ensemble prediction accuracy, even surpassing the advanced probabilistic method. However, because the Best-in-9 results are selected according to accuracy, no obvious gain is observed in SEDI. This result demonstrates that multi-year aligned historical data can serve as effective anchors for ensemble forecasting. In contrast, the other deterministic baselines show no evident improvement, indicating that simply increasing the number of candidate forecasts is insufficient to improve ensemble prediction performance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-D Long-Horizon Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For long-horizon forecasting, existing paradigms rely primarily on short-term observations and are therefore unable to perform reliable one-step long-horizon forecasting. Moreover, during iterative long-term prediction, subsequent forecasts can only be based on inaccurate predictions from the previous step, leading to severe error accumulation and pattern drift.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By introducing the historical dimension, our method not only utilizes historical information to achieve stronger performance in one-step long-horizon forecasting, but also calibrates errors at each step with sufficient historical support, thereby greatly alleviating error accumulation and pattern drift in iterative forecasting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Tab. V , TSSM not only achieves 37.5% substantial improvements in 240h long-horizon forecasting, but also shows that iterative short-term forecasting, such as 120h × \\\\times 2 and 48h × \\\\times 5, outperforms direct 240h × \\\\times 1 long-horizon forecasting, reaching a 103.5% gain. Owing to limited error accumulation, compared with direct 120h forecasting, iterative 120h × \\\\times 2 forecasting leads to only a marginal 4.3% degradation in accuracy and SEDI. This property is particularly valuable in weather forecasting, where extending the effective forecasting window is highly desirable.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-E Forecasting under Observation Missing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Due to sensor failure, transmission packet loss, et al., global weather stations suffer from widespread missing observations. Existing methods typically employ an independent preprocessing step to impute missing values before model forecast, which introduces additional systematic and cascading errors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By introducing the historical dimension, our method can compensate for the missing temporal observations using period-aligned information. Therefore, we take a step further to perform forecasting directly from incomplete observations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Fig. 5 and Tab. VI , when exposed to varying degrees (20% to 80%) of random data missing, TSSM exhibits strong robustness against missing observations in the look-back window with only averaged 4.3% and 8% degradation on accuracy and SEDI. Even with 80% of observations missing, it can still retain 95.7% and 92% of its performance on accuracy and SEDI by relying on historical support. In contrast, for baseline methods, introducing missing noise severely degrades performance. Although state-of-the-art noise-robust forecasting paradigms can significantly alleviate this issue, their performance still drops considerably.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, since we employ historical data, missing observations in historical records from at least one year ago have been sufficiently recovered. Therefore, we do not introduce observation missing in the historical dimension.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-F Forecasting under Limited Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The rapid accumulation of station weather data does not guarantee that sufficient data are naturally available across all regions. Data imbalance remains severe, requiring models to adapt to limited-data scenarios.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although station observations provide global coverage, weather stations remain notably sparse in regions such as Africa (AFR) and South America (SAM) due to uneven regional development. Despite accounting for approximately 30% of global land area and population, these two regions host only 331 and 123 stations, respectively, together representing merely 1/14 of global observations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Tab. VII , when trained on only a very small number of stations, the advantage of our method over baseline approaches becomes even more significant, with improvements of 38% in average accuracy and > > 100% in extreme value capture. By contrast, the competing methods can hardly learn weather dynamics from limited data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Benefiting from strong support from historical information, TSSM can model weather dynamics more effectively, leading to superior performance when trained from scratch under limited-sample conditions in these two regions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-G Ablation Study\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"IV-G 1 Model Structure Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Tab. VIII , we permute and combine the three branches (T-Scan, V-Scan, H-Scan) while retaining at least one of the temporal and historical branches. When the temporal branch is removed, information from the look-back window is injected into the prediction window through modulation. We also ablate several other design choices, including the hierarchical structure level and variable-dimension shuffling. Although our default setting does not explicitly use historical observations, we further explore an explicit initialization strategy that uses the historical average from the previous year.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results reveal distinct and meaningful functional roles across different dimensions. The historical dimension contributes more prominently to extreme event capture, and its benefit becomes more pronounced as the hierarchical receptive field expands. The temporal dimension, in contrast, uses historical information as an anchor and contributes more to prediction accuracy. The variable dimension models inter-variable dependencies, improving both accuracy and extreme event capture, while shuffling the variable dimension further enhances model performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, joint modeling of the temporal and historical dimensions introduces a trade-off between prediction accuracy and extreme event capture. Temporal modeling tends to produce more accurate but smoother predictions, whereas historical modeling sacrifices part of the accuracy to better preserve extreme values. Incorporating both dimensions enables the model to achieve a balanced performance across the two objectives. In our analysis of the historical dimension Sec. 6 , we further show that this trade-off can be continuously adjusted by controlling the historical scope, allowing the model to adapt to different forecasting scenarios.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, when historical averaging is used for prediction initialization, the role of the historical dimension is largely weakened, leading to overly smoothed predictions. When the same initialization strategy is applied to baseline methods such as iTransformer, it fails to bridge long-term historical evolution with short-term temporal dynamics, causing model failure. This further highlights the essential capability of our design in integrating short-long-term weather dynamics for GSWF.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"IV-G 2 Historical Dimension Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By expanding the historical dimension, we effectively improve GSWF performance. In the structural ablation study, the historical and temporal dimensions exhibit a clear division of labor, which motivates us to further analyze the historical dimension and investigate how this phenomenon can be regulated.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Fig. 6 and Tab. IX , we conduct an ablation analysis using historical training data from different starting years h s t h_{st} . As the historical scope gradually expands, the model performance undergoes a continuous transition from last-year-anchored time-variable forecasting to history-enhanced forecasting. This gradual shift indicates that the trade-off between prediction accuracy and extreme event capture can be controlled by adjusting the scope of the historical dimension.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Furthermore, we investigate to what extent existing methods that do not explicitly model historical data can benefit from simply concatenating historical records into the temporal dimension, forming a longer look-back window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Tab. X , we flatten eight years history into the time axis to construct a 384h look-back window. Most baseline methods achieve moderate performance improvements, but still significantly fall short of TSSM in effectively utilizing historical information. This demonstrates that transmitting historical information through the temporal dimension is far less efficient than modeling it with a dedicated historical dimension, further validating the effectiveness of our design.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Limitation and Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As a preliminary attempt to extend GSWF along the historical dimension, this work adopts a simple and necessary design to incorporate period-aligned historical information. Although TSSM demonstrates clear effectiveness, we observe that historical information does not consistently improve all metrics, but instead introduces a trade-off between average accuracy and extreme event forecasting. Although continuous adjustment is achieved, replacing the fusion strategy between historical and temporal features with attention, gating, or direct interpolation does not fundamentally resolve this issue, nor does it produce continuous performance variations. These observations suggest that more effective and controllable historical-temporal fusion remains an important direction for future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"VI Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we propose the Triaxial State Space Model, a history-enhanced temporal-variable-historical forecasting framework that extends GSWF beyond short-term temporal look-back windows. Inspired by climatological analysis in traditional meteorology, TSSM aligns station observations by the same month, day, and hour across years, reorganizing weather series into a time-variable-history structure. Based on this representation, we design hierarchically shared temporal, variable, and historical scanning modules to model short-term dynamics, inter-variable correlations, and long-term historical evolution. A causal forecasting framework further integrates current observations and historical references while maintaining training efficiency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Comprehensive experiments verify the superiority of TSSM on Weather-5K and other human-involved datasets. TSSM achieves 10% average accuracy gain and 61% extreme event gain on Weather-5K, obtains 95% best or second-best results on other datasets, and shows stronger advantages in long-term and robust forecasting. It reaches a 37.5% overall gain at 240h and up to 103.5% under iterative manner with cumulative iterative error < < 4.3%, and exhibits strong robustness to missing data, with < < 10% degradation compared with > > 50% for baselines, at an 80% missing rate. These results demonstrate that TSSM with historical-dimensional modeling provides a promising direction for more accurate, robust, and reliable global station weather forecasting.\"\n }\n]","isConversational":false}}},{"rowIdx":297,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12447"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"The Computational Basis of Confidence in Large Language Models"},"authors":{"kind":"list like","value":["Dharshan Kumaran","Viorica Patraucean","Maks Ovsanikov","Petar Veličković","Nathaniel Daw"],"string":"[\n \"Dharshan Kumaran\",\n \"Viorica Patraucean\",\n \"Maks Ovsanikov\",\n \"Petar Veličković\",\n \"Nathaniel Daw\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12447v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"Introduction"},{"type":"para","text":"Confidence is usually understood as a system’s estimate of whether its own answer is correct (Pouget et al. , 2016 , Fleming and Daw, 2017 , Kepecs and Mainen, 2012 , Mamassian, 2016 ) . For large language models (LLMs), several candidate confidence signals have been proposed, including answer-token probabilities, consistency across repeated samples, and direct verbal reports such as “I am 80% confident” (Xiong et al. , 2023 , Kadavath et al. , 2022 , Steyvers and Peters, 2025 , Tian et al. , 2023 ) . Among these, answer logits are perhaps the most fundamental, as they directly determine which response the model selects. Existing work has largely evaluated such signals according to a single criterion—how well they predict correctness—and under this empirical definition, answer logits are widely regarded as confidence signals."},{"type":"para","text":"However, predictive accuracy alone does not establish what these signals represent. A large answer-logit difference indicates that the model strongly favours one answer over competing alternatives — and specifies a probability distribution over options — but the computation underlying this preference is largely unspecified. One possibility is that answer logits are a monotonic readout of a latent decision variable encoding noisy task-relevant evidence—a variable that, under a normative process model defined by the task, is sufficient in principle to compute the posterior probability that the current choice is correct (the posterior confidence). Alternatively, answer logits may instead reflect a heuristic score. This score may incorporate task-irrelevant information, or combine task-relevant information in a manner that deviates from an ideal observer, and so need not read out this decision variable at all. Consequently, although such a score may predict correctness empirically, it need not carry the information from which the posterior confidence could, in principle, be computed."},{"type":"para","text":"We address this question through the framework of statistical decision confidence (SDC), developed in computational neuroscience (Kepecs et al. , 2008 , Kepecs and Mainen, 2012 , Hangya et al. , 2016 , Sanders et al. , 2016 ) (see Methods for description of the ideal observer model and Figure 1 ). Under this framework, confidence is defined normatively as the posterior probability that the chosen option is correct given the latent decision variable—a noisy internal representation of the evidence in the stimulus favouring one alternative over the other. Critically, under the SDC framework this latent decision variable is a sufficient statistic for the posterior: once it is known, the posterior probability of correctness is fully determined. Consequently, any signal that faithfully reads out this variable carries the information that is, in principle, sufficient to compute the posterior confidence—rather than merely indicating how strongly the model prefers one answer over another."},{"type":"para","text":"Our central question is therefore whether the observable answer-logit difference behaves like such a latent decision variable. If it does, answer logits are more than empirical predictors of correctness: they read out a quantity that is, in principle, sufficient to compute the posterior probability that the answer is correct. The statistical decision confidence framework provides a concrete way to test this hypothesis, making qualitative predictions about the relationship between the decision variable, confidence, and behaviour that can be evaluated empirically."},{"type":"para","text":"To investigate this hypothesis, we turn to simple perceptual decision tasks, where the experimenter explicitly specifies the task-relevant stimulus-evidence axis and thereby the normative process model linking evidence, choice, and confidence (Figure 1 ) (Kiani and Shadlen, 2009 , Kepecs et al. , 2008 , Lak et al. , 2014 ) . We use binary visual discrimination tasks inspired by classic experimental paradigms in animals (Kepecs et al. , 2008 , Kiani and Shadlen, 2009 , Masset et al. , 2020 ) . On each trial, the stimulus favours one of two possible responses, and the strength of this evidence is varied systematically from ambiguous to easy, yielding a one-dimensional stimulus-strength axis analogous to classic perceptual decision experiments. This provides a normative benchmark against which the computational role of answer logits can be evaluated. Throughout the paper, we use the term stimulus strength to refer to this experimenter-defined evidence favouring one response over the other."},{"type":"para","text":"The latent decision variable is not directly observable. Instead, we observe the logits assigned to the candidate answer tokens. We therefore use the answer-logit difference, LD = ℓ A − ℓ B \\mathrm{LD}=\\ell_{A}-\\ell_{B} , as a candidate readout of the latent decision variable. The sign of LD indicates which answer the model favours (positive favours (A), negative favours (B), and values near zero indicate indifference), while its magnitude indicates how strongly the model favours the chosen answer over the alternative. The central question of this study is whether LD merely reflects a heuristic score that specifies the model’s preference between competing responses, or whether it behaves as a readout of the latent decision variable—one that is sufficient in principle to compute the posterior confidence."},{"type":"para","text":"We test four signatures of statistical decision confidence, ordered from weakest to strongest, each illustrated by the normative model in Figure 1 (Kepecs et al. , 2008 , Hangya et al. , 2016 , Sanders et al. , 2016 ) ."},{"type":"para","text":"(1) Psychometric function. As the stimulus increasingly favours answer (A), the probability of choosing (A) should increase monotonically, establishing that the model’s choices are governed by the experimenter-defined stimulus evidence."},{"type":"para","text":"(2) Signed evidence. Signed LD should vary monotonically with signed stimulus strength. This asks whether LD is a graded readout of the evidence favouring one answer over the other, with stimuli that increasingly favour one option producing progressively more positive LD and stimuli favouring the other producing progressively more negative LD. The relationship need not be linear. Together, these first two signatures concern the encoding of evidence: they establish whether objective stimulus evidence is systematically represented in the model’s decision variable. Importantly, these signatures are consistent both with a normative latent decision variable and with heuristic decision signals that preserve the ordering of stimulus evidence."},{"type":"para","text":"(3) Trial-specific confidence. LD magnitude should predict correctness even when stimulus strength is held fixed. This is a key test: among trials of equal objective difficulty, choices made with larger | LD | |\\mathrm{LD}| should be more likely to be correct. If LD is a readout of the latent decision variable, statistical decision confidence predicts this positive within-strength relationship because trial-to-trial fluctuations in the decision variable determine the probability that the current choice is correct."},{"type":"para","text":"(4) Folded-X pattern. As stimulus strength increases, confidence should increase on correct trials but decrease on error trials, producing the characteristic folded-X pattern observed in perceptual decision studies (Kepecs et al. , 2008 , Sanders et al. , 2016 , Hirokawa et al. , 2019 ) . Under statistical decision confidence, strong stimuli usually push the noisy decision variable far onto the correct side of the boundary, yielding high-confidence correct choices. However, when a strong stimulus nonetheless produces an error, the noisy decision variable must have crossed the decision boundary by only a small margin, yielding low-confidence errors."},{"type":"para","text":"Together, these final two signatures ask whether LD behaves as the latent decision variable prescribed by the normative model, rather than merely encoding stimulus evidence. They test whether trial-to-trial fluctuations in LD have the structure expected if LD is a readout of the latent decision variable—the variable that is sufficient in principle to compute the posterior probability that the current choice is correct. Such behaviour is not generally expected of heuristic decision signals that incorporate task-irrelevant information or combine task-relevant information in a manner that departs from the normative process model, since in these cases trial-to-trial variation in LD no longer reflects only the latent task-relevant evidence specified by the normative model. Collectively, the four signatures therefore progress from establishing that LD systematically encodes stimulus evidence to testing whether it reads out the latent decision variable prescribed by statistical decision confidence."},{"type":"para","text":"The final two signatures require trial-to-trial variation in the latent decision variable while objective stimulus strength is held constant. In biological observers, this variation arises naturally because repeated presentations of the same stimulus produce different internal sensory measurements, and therefore different values of the latent decision variable. Deterministic multimodal LLMs, however, do not exhibit this form of variability: identical inputs produce identical internal computations and identical answer logits. To recover the analogous regime, we generate multiple stimulus exemplars at each stimulus-strength level. Each exemplar preserves the same experimenter-controlled stimulus evidence favouring one response over the other while randomizing task-irrelevant (“nuisance”) properties of the image, such as the precise spatial arrangement of the visual elements. This parallels classic perceptual paradigms such as random-dot motion, where motion coherence fixes objective stimulus strength while the particular dot realization is resampled on every trial (Kiani and Shadlen, 2009 ) . Although these nuisance manipulations preserve objective stimulus strength, they induce trial-to-trial variation in its latent decision signal. Under the normative model, this variation plays the same role as sensory noise in the ideal observer, producing a distribution of latent decision values at fixed stimulus strength. Consequently, any relationship between | LD | |\\mathrm{LD}| and correctness within a stimulus-strength level reflects trial-specific variation in the model’s internal decision signal rather than differences in objective difficulty, allowing signatures (3) and (4) to be tested in deterministic multimodal LLMs."},{"type":"para","text":"Pinning down the computational basis of confidence in multimodal LLMs is a fundamental step toward understanding what model confidence represents. Rather than asking simply whether confidence signals predict correctness, we ask whether they behave as readouts of the latent decision variable—one that is sufficient in principle to compute the posterior confidence. Simple perceptual decisions provide a uniquely tractable setting in which to answer this question because the evidence supporting each alternative is experimentally controlled and the predictions of statistical decision confidence are well defined. Establishing whether answer logits satisfy these predictions in this simplest setting provides a baseline against which confidence in reasoning, factual recall, and complex visual question answering can be interpreted."},{"type":"heading","level":1,"text":"Results"},{"type":"para","text":"We organize the results around the four signatures of statistical decision confidence introduced above and illustrated in Figure 1 (Kepecs et al. , 2008 , Hangya et al. , 2016 , Sanders et al. , 2016 ) . The first two signatures concern the encoding of evidence: whether choice and signed LD vary monotonically with the experimenter-defined stimulus-strength axis. The latter two test whether LD behaves as the latent decision variable prescribed by the normative model. Specifically, we ask whether, at fixed objective stimulus strength, larger | LD | |\\mathrm{LD}| predicts a higher probability that the current choice is correct, and whether correct and error trials exhibit the characteristic folded-X geometry. For the fixed-strength analysis, the primary quantitative result is the pooled logistic regression controlling for stimulus-strength bin; the binned plots are provided as visualizations of the same effect."},{"type":"heading","level":2,"text":"Qwen: Evidence Encoding and Latent Decision Variable Signatures in Perceptual Tasks"},{"type":"heading","level":4,"text":"Encoding of Evidence."},{"type":"para","text":"Across the three synthetic perceptual decision tasks (see example stimuli in Figure 7 ), the model demonstrated orderly psychometric behavior, satisfying the first core signature of evidence encoding. Choice probability increased monotonically with signed stimulus strength, and logistic fits effectively captured the behavioral transitions (Figure 2 ). While objective performance was robust across the board, the model exhibited varying degrees of task-specific response bias—most notably a pronounced bias against the BELOW response in the size task—which are quantified by the shifts in the point of subjective equality (PSE) detailed in Table 1 (A)."},{"type":"para","text":"Crucially, the model also satisfied the second encoding signature: the signed answer-logit difference ( L D LD ) showed a strong linear correlation with signed stimulus strength. Mean signed L D LD varied linearly with the physical evidence, crossing near the psychometric boundary and reversing sign across the two stimulus classes. This confirms that the model’s answer logits carry a graded, signed representation of task-relevant evidence reflecting both the direction and strength of the sensory evidence. As a basic check, a single-temperature transformation of | LD | |\\mathrm{LD}| was both discriminative and well-calibrated on held-out trials across all three tasks (Figure 2 , middle)—demonstrating that LD is an accurate empirical predictor of correctness, but not yet establishing that it reads out the latent decision variable in the sense required by statistical decision confidence."},{"type":"heading","level":4,"text":"Latent decision variable signatures."},{"type":"para","text":"This signed evidence code provides the necessary substrate to test the stricter readout predictions of statistical decision confidence in the subsequent analyses: specifically, whether | L D | |LD| predicts correctness when objective stimulus difficulty is held constant, and whether it exhibits the signature folded-X correct/error interaction."},{"type":"heading","level":4,"text":"Answer-logit confidence predicts correctness at fixed stimulus strength."},{"type":"para","text":"Even before controlling for difficulty, | L D | |LD| predicted correctness beyond stimulus strength: across all three tasks it was the stronger predictor, and adding it to a stimulus-strength model significantly improved fit (contrast AUROC 0.864 → 0.907 0.864\\to 0.907 , combined 0.926 0.926 ; all likelihood-ratio tests p < 10 − 100 p<10^{-100} ). The stronger latent decision variable signature asks whether LD carries trial-specific decision evidence, rather than simply varying with objective stimulus strength. Because easier stimuli produce both larger | L D | |LD| and higher accuracy, we asked whether | L D | |LD| predicts correctness with stimulus strength held fixed, fitting logistic regressions of correctness on | L D | |LD| with fixed effects for stimulus-strength bin, correct ∼ C ( | stimulus | , bin ) + | L D | \\mathrm{correct}\\sim C(|\\mathrm{stimulus}|,\\mathrm{bin})+|LD| . This removes the main effect of difficulty; the residual trial-to-trial variation arises from nuisance variation in the stimuli (independent noise realizations and positional jitter), which perturbs the model’s internal evidence even at fixed nominal strength—the analogue of the internal noise exploited by fixed-stimulus confidence tests in animals (Kepecs et al. , 2008 ) ."},{"type":"para","text":"Across all three tasks, | L D | |LD| remained a strong predictor of correctness within fixed stimulus-strength bins (Figure 3 ; Table 1 (B)). The effect was large and highly significant in every task, strongest for contrast ( β = 4.12 \\beta=4.12 , z = 53.42 z=53.42 , p < 10 − 300 p<10^{-300} ): even with objective difficulty held fixed, trials with larger answer-logit differences were substantially more likely to be correct—establishing that Qwen’s LD carries trial-specific decision evidence beyond objective stimulus strength."},{"type":"heading","level":4,"text":"Correct and error trials show the folded-X."},{"type":"para","text":"The strictest signature separates correct and error trials: statistical decision confidence predicts that, as stimulus strength increases, | L D | |LD| rises on correct trials but falls on error trials, producing the folded-X (intuition and schematic in Figure 1 ). We tested this by regressing | L D | |LD| on absolute stimulus strength, correctness, and their interaction, | L D | ∼ | Δ | × Correct |LD|\\sim|\\Delta|\\times\\mathrm{Correct} ."},{"type":"para","text":"Across all three tasks the interaction was strongly positive (Figure 4 ; Table 1 (C)): the slope of | L D | |LD| on stimulus strength was positive for correct trials and negative for error trials (contrast, + 8.08 +8.08 vs. − 1.11 -1.11 ; the same pattern held for size and shape). These opposing slopes yield the folded-X. The folded-X is not a generic consequence of a signal predicting correctness or being well calibrated. Rather, it is the specific geometry predicted when confidence is a monotonic function of the latent decision variable. Across the perceptual tasks, Qwen satisfies all four signatures, indicating that answer-logit difference behaves as a readout of the latent decision variable—one that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence. This provides the baseline against which we interpret the more complex tasks that follow."},{"type":"heading","level":4,"text":"Replication across models."},{"type":"para","text":"The same signatures held in two further non-reasoning models, Gemma 3 12B (Figures 8 , 9 , 10 ; Table 2 ) and Gemma 4 12B (Figure 12 ; Table 5 ), and in a reasoning model, Gemini Flash 3 (Figure 13 ; Table 7 ; see Supplemental Results). In each, | L D | |LD| predicted correctness within fixed stimulus-strength bins, and the folded-X was present in every model–task cell with a single exception; the Gemma models additionally reproduced the signed-LD evidence code. That exception was the Gemma 3 12B contrast error branch, which was flat rather than falling (Table 2 (D))—not, by itself, evidence against an SDC-like signal, as we take up in the Supplemental Results. For Gemma 4 12B the size task was excluded because objective performance was near chance, dominated by a response bias rather than graded uncertainty."},{"type":"heading","level":4,"text":"Summary."},{"type":"para","text":"Across every model–task combination but one—three non-reasoning models and one reasoning model—answer-logit difference satisfied the discriminating signatures of statistical decision confidence: it predicted correctness at fixed stimulus strength and, in all but one case, exhibited the normative folded-X. The sole exception, Gemma 3 on the contrast task, departed only in the strictest signature, showing a flat rather than inverted error branch while still exhibiting strong within-strength prediction of correctness. Taken together, these results show that, in simple perceptual decisions, answer logits behave not merely as heuristic preference scores that predict correctness empirically, but as monotonic readouts of a latent decision variable. Consequently, they read out a quantity that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence."},{"type":"para","text":"This conclusion concerns computational function rather than implementation. The normative model in Figure 1 is not intended as a literal description of how multimodal LLMs compute their decisions. Rather, it specifies the functional properties of any signal that represents the posterior probability that the current choice is correct. Satisfying the four signatures therefore does not imply that models implement the particular computations of the normative observer. Instead, it establishes that the observable answer-logit difference behaves as though it were a readout of a latent decision variable with the same computational role: a representation from which the posterior probability that the current choice is correct can, in principle, be computed."},{"type":"heading","level":4,"text":"Abstention selectively identifies high-confidence decisions."},{"type":"para","text":"We next asked whether the LD signal, which the preceding analyses identified as a readout of the latent decision variable, is behaviourally consequential—building on evidence that language models causally use confidence to drive behaviour (Kumaran et al. , 2026b ) . After making the initial forced-choice contrast judgment, the model was given a second-stage decision: either commit its previous answer or abstain, and was instructed to abstain unless it believed its answer had at least a 60% chance of being correct (see Methods). The resulting policy was conservative and highly selective. The model abstained on 87.7% of trials, committing only its highest-confidence responses. Accuracy on these committed trials increased from the baseline forced-choice accuracy of 92.2% to 99.95%, while the original first-stage answers on abstained trials remained close to baseline accuracy (91.2%; Table 3 ). Thus, abstention selectively retained the most reliable decisions while rejecting those with low answer-logit confidence (low | L D | |LD| ), which were disproportionately error-prone."},{"type":"para","text":"Critically, abstention tracked the internal confidence signal rather than external difficulty. Answer-logit confidence predicted abstention better than physical stimulus strength (AUROC 0.840 0.840 vs. 0.796 0.796 ), and adding stimulus strength barely improved this ( 0.843 0.843 ), so | L D | |LD| carried most of the information the policy used (Figure 15 ). The effect survived difficulty controls: within fixed stimulus-strength bins, lower | L D | |LD| still strongly predicted abstention (odds ratio = 2.38 =2.38 , p < 10 − 300 p<10^{-300} ). And the model’s policy closely approximated an idealized rule that abstains on the lowest- | L D | |LD| trials—the two overlapped on 91.6% of trials, and the optimal threshold improved committed accuracy by only 0.05 0.05 percentage points. Abstention is therefore governed by the same answer-logit signal characterized above, and acts on it almost exactly as a simple confidence threshold would."},{"type":"heading","level":4,"text":"Generalization to memory-based decisions: city-population comparisons."},{"type":"para","text":"To test whether these signatures extend beyond perception, we turned to a memory-based decision—judging which of two cities has the larger population, which requires retrieving and comparing stored world knowledge rather than reading a perceptual stimulus (the perceptual-to-memory extension studied in humans by Sanders et al. 2016 ). We drew 5 , 000 5{,}000 cities (population > 1 >1 M) from the SimpleMaps database, giving 50 , 000 50{,}000 binary comparisons. Here the experimenter’s stimulus-strength axis—the ground-truth population ratio—is only a noisy proxy for the evidence the model brings to the choice, since recalled city size, retrieval strength, and semantic salience vary from trial to trial."},{"type":"para","text":"This shows up in the encoding signatures. The psychometric function was shallow (accuracy 61.8 % 61.8\\% ; slope 0.90 0.90 , pseudo- R 2 = 0.069 R^{2}=0.069 ), with a response bias toward option A (Table 4 (A)): objective population evidence was a weak trial-level predictor of choice. But this reflects low signal-to-noise, not an absent mapping—signed L D LD still varied approximately linearly with signed population evidence (Figure 5 ), so the second encoding signature, the mean mapping from objective evidence onto the decision variable, was intact. Encoding here is weak at the level of individual choices but faithful in the mean, the gap being large trial-to-trial noise in the recalled evidence."},{"type":"para","text":"Despite the shallow psychometric, both latent decision variable signatures held. Within fixed population-evidence bins, | L D | |LD| predicted correctness (odds ratio = 1.36 =1.36 , p < 10 − 180 p<10^{-180} ; Table 4 (B))—LD carried trial-specific decision evidence—and the full folded-X was present, with | L D | |LD| rising with evidence on correct trials and falling on error trials (strongly positive interaction; Table 4 (C)). The same trial-to-trial noise that flattens the psychometric is what makes these latent decision variable signatures informative, since it is the within-strength fluctuation in evidence that they exploit."},{"type":"para","text":"The pattern replicated in Gemma 3 12B, where the psychometric was shallower still (pseudo- R 2 = 0.049 R^{2}=0.049 ) yet the folded-X was if anything sharper—a clearly negative error branch ( − 0.589 -0.589 ) against a positive correct branch ( + 0.678 +0.678 ; Figure 11 ; Table 6 ). This makes the memory task the clearest dissociation between objective stimulus evidence and the model’s internal decision evidence. Here the experimenter-defined stimulus-strength axis predicts individual choices only weakly, yet LD continues to satisfy both latent decision variable signatures, indicating that its interpretation as a readout of the latent decision variable does not depend on a tightly coupled objective evidence axis."},{"type":"heading","level":2,"text":"CLEVR: Qwen 2.5 7B, Gemini Flash 3, and complex visual reasoning"},{"type":"para","text":"The perceptual tasks establish that answer logits can behave like a readout of the latent decision variable in settings where the normative process model can be well specified. We next ask whether this interpretation generalizes to complex visual reasoning, where the underlying decision process is no longer described by a simple perceptual evidence model."},{"type":"para","text":"We evaluated Qwen 2.5 7B and Gemini Flash 3 on CLEVR count-equality and object-existence questions while degrading images with parametric Gaussian blur (larger σ \\sigma , lower stimulus strength; Figure 6 ). One feature of this paradigm fundamentally changes what the SDC signatures can tell us. Unlike contrast or size discrimination, blur does not define a signed evidence axis: it removes information rather than favouring one answer over the other, and the mapping from blur to the model’s latent decision variable is neither monotonic nor known. Consequently, the first two signatures—which rely on a well-defined evidence axis—do not transfer directly. What remains well defined is whether | LD | |\\mathrm{LD}| carries trial-specific information about whether the current choice is correct beyond objective image difficulty (signature 3), and how | LD | |\\mathrm{LD}| behaves on correct and error trials (signature 4)."},{"type":"para","text":"The third latent decision variable signature remained robust. Adding | LD | |\\mathrm{LD}| to blur strength in nested logistic models improved discrimination of correct from error trials in every dataset, with AUROC gains of + 0.029 +0.029 to + 0.072 +0.072 , confirmed by likelihood-ratio tests throughout (Table 8 ), and the same conclusion was reached non-parametrically within fixed blur bins (Figure 6 ). We next asked whether this effect could be explained by other measurable properties of the input. Adding scene and question controls—including the number of unique objects and colours, and spatial crowding (Methods)—improved prediction, particularly for counting, yet | LD | |\\mathrm{LD}| continued to add significant information beyond blur and these controls in all estimable nested models. In the fixed-blur analyses with the full control set, the | LD | |\\mathrm{LD}| effect remained significant for Qwen count, Qwen existence, and Gemini Flash 3 existence; the Gemini Flash 3 count model was not estimable owing to collinearity among the control variables, so we make no claim there. Thus, across all estimable models, answer-logit difference continued to carry information about whether the current choice was correct beyond the observable properties of the input."},{"type":"para","text":"The correct/error structure was more nuanced. On correct trials, | LD | |\\mathrm{LD}| increased with stimulus strength in all four analyses. On error trials, however, the slope was approximately flat (Gemini count, Qwen existence, Gemini existence) or modestly positive (Qwen count), never negative, although the correct × \\times error interaction remained significant throughout (Table 9 ; Figure 6 , row D). Thus, | LD | |\\mathrm{LD}| continued to behave differently on correct and error trials, but did not exhibit the canonical folded-X."},{"type":"para","text":"Whether this constitutes a failure of statistical decision confidence cannot be determined from these data alone. The folded-X is a prediction of a specific normative process model in which the experimenter specifies both the task-relevant evidence axis and its mapping to the latent decision variable. In CLEVR, neither is known. Consequently, the normative posterior cannot be derived, and the folded-X is no longer a diagnostic prediction. We therefore cannot conclude that answer logits in CLEVR either satisfy or violate statistical decision confidence. The secure conclusion is the weaker one: answer-logit difference continues to carry trial-specific information about whether the current choice is correct beyond objective properties of the input, but whether it behaves as a normative latent decision variable remains unresolved until an appropriate process model is established (see Discussion)."},{"type":"heading","level":1,"text":"Discussion"},{"type":"para","text":"We asked whether answer logits in multimodal LLMs merely provide useful predictors of correctness, or whether they warrant the stronger interpretation as readouts of the latent decision variable prescribed by statistical decision confidence (SDC). We addressed this question in three multimodal non-reasoning models (Qwen2.5-VL, Gemma 3, Gemma 4) and one multimodal reasoning model (Gemini Flash 3) across three decision regimes: low-level perceptual discrimination (contrast, size, shape), a memory-based decision (city-population comparison), and complex visual reasoning (CLEVR count and existence tasks under blur). Across the perceptual and memory tasks, and consistently across models, answer-logit difference (LD) satisfied all four qualitative signatures of SDC. LD encoded the signed stimulus evidence, predicted correctness beyond objective stimulus strength, and—with the single exception of Gemma 3 on the contrast task—exhibited the characteristic folded-X geometry predicted when confidence is a monotonic function of the latent decision variable. The same signal was also behaviourally consequential: in a second-stage abstention task, LD predicted the model’s decision to commit or abstain better than objective stimulus strength, and the resulting policy closely approximated a simple threshold on LD. Together, these findings show that, in simple perceptual and memory-based settings, answer logits are more than heuristic preference scores that predict correctness empirically: they behave as monotonic readouts of the latent decision variable—a quantity that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence."},{"type":"para","text":"A central issue is that statistical decision confidence is defined relative to a specified normative process model. In simple perceptual tasks, the experimenter defines both the task-relevant evidence variable and the process by which noisy evidence generates a latent decision variable. In more complex tasks, neither is generally known. Consequently, an apparent failure of the SDC signatures does not, by itself, imply that answer logits are non-normative. We believe the CLEVR results should be interpreted in this light. Crucially, answer-logit difference continued to carry trial-specific information about whether the model’s current choice was correct beyond objective properties of the input, even after controlling for blur and measurable scene properties. What was absent was the folded-X. We do not interpret this as evidence against statistical decision confidence. Rather, the folded-X is a prediction of a specific normative process model in which the experimenter specifies both the task-relevant evidence axis and its mapping onto a latent decision variable. In CLEVR, neither is known. Consequently, the absence of the folded-X does not by itself distinguish between a failure of statistical decision confidence and a mismatch between the experimenter-defined process model and the computation actually performed by the model."},{"type":"para","text":"More generally, apparent departures from the experimenter-defined normative model may arise in at least three ways. First, the model may incorporate task-irrelevant information into its decision variable. A simple thought experiment illustrates how this can arise. Consider a recognition-memory task in which the model must decide which of two items has been encountered more often. The experimenter defines the relevant evidence as familiarity acquired during the experiment. Suppose, however, that the model cannot distinguish this from pre-existing real-world familiarity, so that its latent decision variable combines both sources of evidence (see Supplemental Discussion for details). The model’s effective evidence variable therefore differs from the experimenter-defined evidence axis, and behaviour analysed with respect to experimental familiarity need no longer exhibit the signatures predicted by the normative model, even if confidence is computed normatively with respect to the model’s own effective evidence variable. An analogous mismatch may underlie the CLEVR results if the model’s effective evidence depends upon latent variables not captured by the experimenter-defined blur axis."},{"type":"para","text":"Second, the model may ignore task-relevant information specified by the experimenter. Third, it may combine the available evidence in a non-Bayesian (heuristic) fashion rather than computing the normative posterior. Any of these possibilities could account for the absence of the folded-X in CLEVR. Distinguishing between them requires developing explicit computational process models for complex reasoning tasks, analogous to those available for simple perceptual decisions."},{"type":"para","text":"It is also worth distinguishing the aim of the present framework from the focus of confidence calibration (Guo et al. , 2017 , Steyvers and Peters, 2025 , Xiong et al. , 2023 , Tian et al. , 2023 , Kadavath et al. , 2022 ) . Calibration seeks a mapping from a model’s raw confidence signal to empirical correctness, so that reported confidence better matches the observed frequency of correct responses. This is valuable for prediction and deployment, but it does not by itself establish what the underlying signal represents. A well-calibrated confidence score may arise from a latent decision variable supporting the normative posterior, but it may equally arise from a heuristic preference signal that predicts correctness empirically. Statistical decision confidence asks a different question: whether the underlying signal itself behaves as a monotonic readout of the latent decision variable prescribed by a normative process model. Calibration therefore concerns the mapping between a confidence score and empirical accuracy, whereas the SDC framework concerns the computational interpretation of the confidence signal itself."},{"type":"para","text":"One qualification concerns the level of inference these results support, and it connects directly to a long-standing debate about human confidence: whether confidence reflects the posterior probability of being correct (the Bayesian confidence hypothesis, BCH; Pouget et al. , 2016 , Kepecs and Mainen, 2012 , Adler and Ma, 2018 ) or the strength of the raw evidence itself, without an explicit probability computation (confidence in raw evidence space, CRES; Xue et al. , 2024 ). Because our tasks use boundary-defined, non-overlapping response categories, they fall in the regime where the folded-X is a robust rather than fragile prediction of statistical decision confidence (Adler and Ma, 2018 ) . Within this regime the four signatures establish that answer-logit difference behaves as a readout of the latent decision variable, a quantity sufficient in principle to compute the posterior. What they do not fix is whether | L D | |LD| is that posterior or a monotone transform of it that does not renormalise for evidence reliability. Because the normative posterior is itself a monotonic (sigmoidal) function of the decision variable, any faithful readout of that variable—whether the calibrated posterior or the raw evidence—imposes the same ordering on trials and so reproduces all four signatures at any fixed level of reliability; what the signatures cannot see is whether the slope of that mapping tracks reliability. BCH and CRES therefore diverge only when reliability is manipulated independently of stimulus strength—under BCH the mapping from | L D | |LD| to accuracy is reliability-invariant, because a calibrated posterior already absorbs reliability, whereas under CRES a fixed | L D | |LD| corresponds to progressively lower accuracy as reliability falls, because raw evidence is not renormalised. Our single signed-strength axis varies evidence and reliability together and therefore cannot separate the two. Adjudicating them requires independently manipulating stimulus strength and evidence reliability, together with quantitative model comparison rather than qualitative signatures alone (Adler and Ma, 2018 , Xue et al. , 2024 ) —an important direction that the framework developed here newly makes tractable for artificial systems."},{"type":"para","text":"Our results also bear on the distinction between first- and second-order theories of confidence (Pouget et al. , 2016 , Fleming and Daw, 2017 , Mamassian and de Gardelle, 2022 ) . The present work concerns the confidence carried by answer logits themselves. We find that, in simple perceptual and memory-based decisions, answer-logit difference behaves as a readout of the latent decision variable prescribed by statistical decision confidence. This does not preclude the existence of additional confidence computations. Recent work suggests that verbally reported confidence and answer-logit confidence are dissociable signals that independently influence behaviour (Kumaran et al. , 2026a , b , Kumaran, 2026 ) , implying that models may construct a second-stage evaluation of their own answers beyond the decision signal studied here. Rather than competing explanations, these signals may occupy different levels of the confidence hierarchy: answer-logit difference provides the decision-linked signal supporting statistical decision confidence, whereas verbal confidence may incorporate additional computations reflecting broader uncertainty about the answer. Understanding how these signals interact, when they diverge, and which ultimately governs behaviour remains an important direction for future work."},{"type":"para","text":"More broadly, our results suggest that statistical decision confidence provides a principled framework for understanding what model confidence represents, rather than simply how well it predicts correctness. Although we have focused primarily on visual perceptual decisions, the framework is not inherently visual. We further showed that the same qualitative signatures extend to a memory-based language task, and answer-logit differences are available whenever a model commits to a discrete response. The framework therefore provides a general approach for studying confidence in language models, multimodal agents, and more complex reasoning systems, provided an appropriate normative process model can be specified. Finally, the signatures studied here are behavioural. Identifying the internal representations that give rise to them would provide a mechanistic account of confidence analogous to the progression in neuroscience from behavioural signatures of confidence to neural representations of the underlying decision variable."},{"type":"heading","level":1,"text":"Methods"},{"type":"heading","level":2,"text":"Normative model"},{"type":"para","text":"The four signatures we test derive from a standard normative (ideal-observer) model of two-alternative decision making, schematised in Figure 1 . The experimenter specifies a signed stimulus-evidence variable, Δ \\Delta , which quantifies the evidence favouring one alternative over the other. The observer does not access Δ \\Delta directly, but instead forms a noisy internal representation,"},{"type":"para","text":"where ϵ \\epsilon is internal decision noise. The latent decision variable d d therefore fluctuates from trial to trial even when the objective stimulus evidence is held fixed. These trial-to-trial fluctuations are essential: they make a fixed stimulus strength map to a distribution of latent decision values rather than a single value, and it is this variation that the latent decision variable signatures (3 and 4) exploit."},{"type":"para","text":"A deterministic VLM produces identical logits for identical inputs, so such variability does not arise from repeated presentation of the same image. Instead, it is supplied across stimulus exemplars: images with the same objective stimulus strength but differing in task-irrelevant (nuisance) features produce different internal evidence. The nuisance manipulations used for each task are described below."},{"type":"para","text":"The sign of d d determines the observer’s choice. Confidence is defined by statistical decision confidence (SDC) as the posterior probability that this choice is correct,"},{"type":"para","text":"Computing this posterior requires combining the prior distribution over stimulus states with the likelihood of observing the latent decision value under each possible stimulus state,"},{"type":"para","text":"where the denominator marginalises over all possible stimulus states. Thus, the latent decision variable is a sufficient statistic for confidence: once d d is known, the posterior probability that the current choice is correct is fully determined. The posterior increases monotonically with | d | |d| , so a single latent variable determines both the observer’s choice (its sign) and its normative confidence (its magnitude)."},{"type":"para","text":"Neither d d nor the observer’s implied likelihood function are directly observable. Rather than attempting to recover the posterior itself, we ask whether the observable answer-logit difference behaves as a monotonic readout of the latent decision variable,"},{"type":"para","text":"where f f is monotonic. If so, LD inherits the qualitative signatures predicted for the latent decision variable—the psychometric function, signed-evidence encoding, fixed-strength correctness prediction, and the folded-X pattern—without requiring explicit recovery of the posterior."},{"type":"heading","level":2,"text":"Tasks"},{"type":"heading","level":4,"text":"Simple perceptual decision tasks"},{"type":"para","text":"We constructed three simple perceptual tasks that allowed precise experimental control over stimulus strength while providing large numbers of trials near perceptual decision boundaries. All tasks were evaluated using the same multimodal LLM and analysis pipeline. Each task generated a signed stimulus-strength variable, Δ \\Delta , where positive and negative values favoured opposite response categories. Model choices were analysed using response-token logits, allowing extraction of both categorical decisions and continuous decision variables."},{"type":"heading","level":4,"text":"Contrast discrimination."},{"type":"para","text":"Stimuli consisted of two independent band-pass filtered noise patches presented side-by-side on a uniform grey background. Each patch was 256 × 256 256\\times 256 pixels and the two patches were separated by a 24-pixel gap. Noise images were generated by sampling Gaussian white noise and applying a log-Gaussian band-pass filter in the Fourier domain. The filter was centred at 4 cycles/degree with a bandwidth of 1 octave. The filtered image was transformed back to the spatial domain, normalised to zero mean and unit variance, and scaled to the desired root-mean-square (RMS) contrast."},{"type":"para","text":"The baseline RMS contrast was fixed at c 0 = 0.28 c_{0}=0.28 . On each trial, left and right contrasts were determined by a signed contrast difference parameter Δ \\Delta ,"},{"type":"para","text":"Positive values of Δ \\Delta therefore indicated that the right patch had higher contrast, whereas negative values indicated that the left patch had higher contrast."},{"type":"para","text":"Stimulus strengths ranged approximately from -0.40 to +0.40. Sampling density was highest near Δ = 0 \\Delta=0 , with additional stimulus levels extending into easier discrimination regimes. Independent random seeds were used for the left and right noise patches on every trial."},{"type":"para","text":"The model was asked:"},{"type":"para","text":"“This image shows two noise patches, LEFT and RIGHT. Which has HIGHER CONTRAST?”"},{"type":"para","text":"Responses were given using two response letters. To eliminate confounding between perceptual side preferences and token preferences, the mapping between response letters and spatial locations was counterbalanced across trials. Half of trials used M M for left and V V for right, whereas the remaining trials used the reverse mapping."},{"type":"para","text":"Let ℓ R \\ell_{R} and ℓ L \\ell_{L} denote the total logit mass assigned to the tokens currently mapped to the right and left alternatives. The signed decision variable was"},{"type":"heading","level":4,"text":"Object-size discrimination."},{"type":"para","text":"Stimuli consisted of two filled circles arranged vertically. Each circle was rendered within a 256 × 256 256\\times 256 pixel image region, with the upper and lower regions separated by a 48-pixel gap. Circle luminance was fixed at 0.12 on a background luminance of 0.78. Shapes were rendered using 4 × 4\\times supersampling followed by Lanczos downsampling to produce anti-aliased edges."},{"type":"para","text":"The baseline circle radius was r 0 = 52 r_{0}=52 pixels. Stimulus strength was parameterised using a signed log-radius ratio,"},{"type":"para","text":"Circle radii were generated according to"},{"type":"para","text":"This parameterisation preserved the geometric mean radius across trials while manipulating relative size. Positive values of Δ \\Delta indicated that the lower circle was larger."},{"type":"para","text":"To prevent exploitation of trivial spatial cues, each circle received an independent random positional jitter sampled uniformly from ± 6 \\pm 6 pixels in both horizontal and vertical directions. Stimulus strengths ranged approximately from -0.40 to +0.60 log-radius units, with oversampling concentrated near the psychometric boundary."},{"type":"para","text":"The model was instructed:"},{"type":"para","text":"“Two circles are shown vertically, one higher and one lower. Which circle is larger? Answer with exactly one word: ABOVE or BELOW.”"},{"type":"para","text":"The signed decision variable was defined as"},{"type":"para","text":"where ℓ below \\ell_{\\mathrm{below}} and ℓ above \\ell_{\\mathrm{above}} denote the logit masses assigned to the corresponding response tokens."},{"type":"heading","level":4,"text":"Shape categorization."},{"type":"para","text":"Stimuli consisted of a single centrally presented rounded rectangle rendered within a 256 × 256 256\\times 256 pixel image. Shapes were displayed using a background luminance of 0.78 and object luminance of 0.12. Anti-aliasing was implemented using 4 × 4\\times supersampling followed by Lanczos downsampling."},{"type":"para","text":"The base side length was 120 pixels. Shape category was manipulated using a continuous morph parameter m ∈ [ 0 , 1 ] m\\in[0,1] , where m = 0 m=0 corresponded to a square-like angular shape and m = 1 m=1 corresponded to a circle. The morph continuum was implemented by varying the corner radius of a rounded rectangle according to"},{"type":"para","text":"where s s denotes side length."},{"type":"para","text":"Stimulus strength was defined relative to each model’s empirically estimated angular–round boundary m ∗ m^{\\ast} , as Δ = m − m ∗ \\Delta=m-m^{\\ast} , so that Δ = 0 \\Delta=0 marks the model’s point of subjective equality ( m ∗ = 0.73 m^{\\ast}=0.73 for Qwen, 0.38 0.38 for Gemma 3). Negative values of Δ \\Delta corresponded to shapes categorised as angular, whereas positive values corresponded to shapes categorised as round."},{"type":"para","text":"To introduce nuisance variation, side length was independently jittered by up to ± 6 % \\pm 6\\% on each trial and stimulus position was jittered by up to ± 6 \\pm 6 pixels. Stimuli were sampled densely in the vicinity of the empirical category boundary, with Δ \\Delta values extending approximately from -0.12 to +0.12."},{"type":"para","text":"The model was instructed:"},{"type":"para","text":"“A single shape is shown. Is the shape more ANGULAR or more ROUND? Answer with exactly one word: ANGULAR or ROUND.”"},{"type":"para","text":"The signed decision variable was"},{"type":"heading","level":4,"text":"Abstention task: Qwen 2.5 7B"},{"type":"para","text":"To test whether decision-confidence signals predicted subsequent behavior, we conducted a second-stage abstention experiment using the contrast-discrimination task. Each abstention trial began with a previously generated forced-choice response from the model. The original image, question, and model answer were then presented back to the model in a multi-turn chat format. The model was asked to decide whether to output its previous answer to the user ( A ) or abstain ( B )."},{"type":"para","text":"The abstention prompt was:"},{"type":"para","text":"“You now need to decide whether to output your previous answer to the user (A) or abstain (B). If you don’t think your answer has 60% chance of being correct, choose B although this will disappoint the user. Respond with exactly one letter. A = output answer B = abstain Decision:”"},{"type":"para","text":"The 60% threshold was selected empirically. Preliminary experiments revealed that more permissive prompts resulted in near-universal commitment, whereas stricter prompts resulted in near-universal abstention. Our goal was not to optimize abstention performance but to place the model in a regime where both commitment and abstention occurred frequently enough to permit analysis of the relationship between internal confidence signals and overt behavior. The chosen threshold yielded substantial variability in abstention decisions, allowing us to test whether answer-logit confidence predicted the model’s willingness to stand behind its previous answer. The generated abstention decision ( A or B ) was used for all behavioural analyses."},{"type":"heading","level":4,"text":"City-population binary-choice task."},{"type":"para","text":"To test whether answer-logit confidence generalized beyond perceptual discrimination, we constructed a factual memory-based decision task using the SimpleMaps World Cities Database. The database provides city names, country information, geographic coordinates, and population estimates for cities and towns worldwide. We restricted the stimulus set to cities with population greater than one million people. This yielded 5 , 000 5,000 unique cities."},{"type":"para","text":"Trials consisted of binary population comparisons between two cities. On each trial, the model was asked which of two cities had the larger population. The signed stimulus-strength variable was defined as"},{"type":"para","text":"so that Δ > 0 \\Delta>0 indicated that option A was correct and Δ < 0 \\Delta<0 indicated that option B was correct. The absolute stimulus strength, | Δ | |\\Delta| , therefore measured the population-ratio difference between the two cities. City pairs were sampled from the eligible city set to generate 50,000 binary-choice trials. Cities were allowed to appear in multiple different pairs, but each trial contained two distinct cities."},{"type":"para","text":"The prompt was:"},{"type":"para","text":"“Which city has the higher population? A. [city A] B. [city B] Answer with exactly one letter: A or B.”"},{"type":"para","text":"The model’s first-token logits were extracted at the answer position. Let ℓ A \\ell_{A} and ℓ B \\ell_{B} denote the logit masses assigned to the answer tokens A and B . We defined the signed decision variable as"},{"type":"para","text":"Thus, positive values indicated evidence favouring option A and negative values indicated evidence favouring option B. Accuracy was computed by comparing the sign of L D LD with the sign of Δ \\Delta , and answer-logit confidence was defined as | L D | |LD| ."},{"type":"heading","level":4,"text":"CLEVR count and existence tasks"},{"type":"para","text":"CLEVR (Johnson et al. , 2017 ) is a synthetic visual-question-answering benchmark of rendered 3D scenes whose objects vary in shape, colour, material, and size, paired with compositional questions and ground-truth scene graphs and functional programs—a controlled, fully annotated structure that lets us select well-defined count and existence questions and compute objective scene-difficulty measures while degrading only the rendered image. We drew questions from the v1.0 training and validation splits. Scene annotations and programs were used only to select questions and, for the count task, to compute the two queried object counts; the model was always evaluated on the rendered image and natural-language question, never on the scene graph or program."},{"type":"para","text":"Both tasks used a two-letter forced-choice format, with the mapping between letters and response categories counterbalanced within each image–question pair (each letter assigned to each category once across the two presentations). We defined the semantic logit difference as the log-probability of the letter denoting the target category ( equal for count, yes for existence) minus that of the alternative, averaged across the two letter mappings to remove fixed letter-identity biases. Questions were sampled deterministically (seed 42), and each image was evaluated across a range of Gaussian blur levels parameterised by filter radius σ \\sigma ."},{"type":"para","text":"For Qwen2.5-VL, which showed a strong affirmative-response bias that counterbalancing does not remove, analyses were restricted to yes-truth trials (equal counts in the count task; queried object present in existence), avoiding the regime in which responding was bias-dominated. Gemini Flash 3 showed no comparable asymmetry, so both truth arms were retained; its answer logits were read with the reasoning-cut procedure described in the Models section."},{"type":"heading","level":3,"text":"Count-comparison task"},{"type":"para","text":"Candidate questions were those whose program terminated in equal_integer (whether two object counts are equal), restricted to one or two relational operations so that the counted sets were defined relative to other objects in the scene. We kept questions with at least one non-zero queried count and removed duplicate image–question pairs. Qwen2.5-VL was run on 1 , 000 1{,}000 yes-truth questions at σ = 0 , 2 , 4 \\sigma=0,2,4 , a fine grid from 4 4 to 10 10 in steps of 0.25 0.25 , and coarser levels at 12 , 14 , 16 , 18 , 20 , 25 , 30 12,14,16,18,20,25,30 ; Gemini Flash 3 used the D/N letter pair on 500 500 questions per truth arm at σ = 0 \\sigma=0 – 10 10 in unit steps."},{"type":"heading","level":3,"text":"Simple-existence task"},{"type":"para","text":"Candidate questions were those terminating in exist , restricted to simple two-attribute filter-only programs (two filter_* operations, e.g. colour and shape; questions with relational or same_* operations were excluded). Both models used the M/V letter pair. Qwen2.5-VL was run on 800 800 yes-truth questions at σ = 0 , 1 \\sigma=0,1 then a fine grid from 2 2 to 10 10 in steps of 0.25 0.25 ; Gemini Flash 3 was evaluated on both truth arms over the same grid."},{"type":"heading","level":4,"text":"Difficulty controls."},{"type":"para","text":"To test whether answer-logit confidence merely proxied visual or question difficulty, we augmented each trial with scene and question metadata from the CLEVR annotations: scene-level controls (total objects; unique colours, shapes, sizes, and materials; maximum objects sharing an attribute; maximum identical object types; and spatial-crowding measures from pairwise object distances) and, where applicable, program controls (program length and counts of filter, relation, same-attribute, count, existence, union, and intersection operations), plus the queried counts and their absolute difference for count questions. After removing highly collinear controls, we tested whether | LD | |\\mathrm{LD}| improved prediction beyond blur strength and these controls (nested likelihood-ratio tests) and whether it predicted correctness within coarse blur bins after including them (fixed-blur-bin analyses)."},{"type":"heading","level":2,"text":"Models"},{"type":"heading","level":4,"text":"Non-reasoning models."},{"type":"para","text":"We evaluated three non-reasoning multimodal language models, all run locally with the HuggingFace transformers library: Qwen2.5-VL-7B-Instruct ( Qwen/Qwen2.5-VL-7B-Instruct ), Gemma 3 12B ( google/gemma-3-12b-it ), and Gemma 4 12B ( google/gemma-4-12B-it ). Each was prompted to answer with a single letter or word (see the task descriptions), so the model returned a direct answer with no reasoning trace and the answer logits could be read from the first response token. Although Gemma 4 is reasoning-capable, this single-token response format elicits a direct answer, placing it in the same non-reasoning regime as the other two models."},{"type":"heading","level":4,"text":"Reasoning model."},{"type":"para","text":"We evaluated one reasoning model, Gemini Flash 3 preview, accessed through the Beyond API on Google infrastructure. Confidence was read using the reasoning-cut procedure below."},{"type":"heading","level":4,"text":"Reasoning-cut readout (Gemini Flash 3)."},{"type":"para","text":"A reasoning model does not expose a well-defined answer logit until it commits to an answer, and the signal saturates once the answer is verbalized. On each trial the model was prompted to reason step by step and to end with a one-letter answer; the generated trace was truncated at 90 % 90\\% of its length, and the model was then asked to answer the original question from the reasoning produced so far, yielding a graded answer-logit difference before saturation. We applied this 90 % 90\\% reasoning-cut readout to Gemini Flash 3 on both the contrast discrimination task and the CLEVR count and existence tasks. At the 90 % 90\\% cut, accuracy did not differ significantly from accuracy after the complete reasoning trace ( p > 0.1 p>0.1 )."},{"type":"heading","level":1,"text":"Appendix"},{"type":"heading","level":2,"text":"Supplemental Results: Gemma 3 12B"},{"type":"para","text":"Gemma 3 12B reproduced the encoding and readout signatures across the three perceptual tasks (Figures 8 , 9 , 10 ; Table 2 ). Two task-specific points: the size task carried a strong response bias, so we additionally confirmed the fixed-strength result in a model-relative analysis defining correctness relative to the model’s own PSE (Figure 9 ); and the contrast error branch was flat rather than falling (Table 2 (D)), the only departure from the full folded-X. We treat the flat branch as neutral: it is neither the falling branch predicted by SDC nor the rising branch sometimes observed when confidence fails to follow the SDC geometry. LD nevertheless predicted correctness within fixed stimulus-strength bins, with | L D | |LD| predicting correctness (Table 2 (C)). The signatures also held in the population QA task, with a clean folded-X carrying a strongly negative error branch despite a shallow psychometric (Figure 11 ; Table 6 )."},{"type":"heading","level":2,"text":"Supplemental Results: Gemma 4 12B"},{"type":"para","text":"Gemma 4 12B reproduced the signatures on contrast and shape, including the full folded-X (Figure 12 ; Table 5 ). The size task was excluded for near-chance objective performance (accuracy 0.509 0.509 ), driven by a large response bias rather than graded uncertainty."},{"type":"heading","level":4,"text":"Gemini Flash 3 (reasoning model)."},{"type":"para","text":"On the contrast task at the 90% reasoning cut, Gemini Flash 3 showed the same signatures—a steep psychometric, fixed-strength prediction of correctness, and a clear folded-X (Figure 13 ; Table 7 )."},{"type":"heading","level":2,"text":"Supplemental Discussion"},{"type":"heading","level":4,"text":"Recognition-memory example."},{"type":"para","text":"Consider a recognition-memory task in which the experimenter defines the relevant evidence as the difference in familiarity acquired during the experiment,"},{"type":"para","text":"Suppose, however, that the model cannot distinguish experimentally acquired familiarity from pre-existing real-world familiarity,"},{"type":"para","text":"so that its latent decision variable is"},{"type":"para","text":"The experimenter-defined correctness depends only on Δ expt \\Delta_{\\rm expt} , but the observed decision variable depends on both sources of familiarity. Consequently, the normative posterior probability that the current choice is experimentally correct would be"},{"type":"para","text":"rather than the simpler expression obtained under the experimenter-defined model alone. That is, because Δ rw \\Delta_{\\rm rw} also contributes to the generation of d d , the posterior defined over the experimenter’s evidence axis no longer coincides with the posterior defined over the model’s effective evidence variable."},{"type":"para","text":"If the experimenter analyses behaviour only with respect to Δ expt \\Delta_{\\rm expt} , the fixed-strength and folded-X signatures need not hold, since trial-to-trial variation in the decision variable now reflects both experimental and real-world familiarity. This does not by itself imply that the model departs from the normative model defined over its own effective evidence variable; it illustrates how the experimenter-defined evidence axis may differ from the model’s effective evidence variable."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Confidence is usually understood as a system’s estimate of whether its own answer is correct (Pouget et al. , 2016 , Fleming and Daw, 2017 , Kepecs and Mainen, 2012 , Mamassian, 2016 ) . For large language models (LLMs), several candidate confidence signals have been proposed, including answer-token probabilities, consistency across repeated samples, and direct verbal reports such as “I am 80% confident” (Xiong et al. , 2023 , Kadavath et al. , 2022 , Steyvers and Peters, 2025 , Tian et al. , 2023 ) . Among these, answer logits are perhaps the most fundamental, as they directly determine which response the model selects. Existing work has largely evaluated such signals according to a single criterion—how well they predict correctness—and under this empirical definition, answer logits are widely regarded as confidence signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, predictive accuracy alone does not establish what these signals represent. A large answer-logit difference indicates that the model strongly favours one answer over competing alternatives — and specifies a probability distribution over options — but the computation underlying this preference is largely unspecified. One possibility is that answer logits are a monotonic readout of a latent decision variable encoding noisy task-relevant evidence—a variable that, under a normative process model defined by the task, is sufficient in principle to compute the posterior probability that the current choice is correct (the posterior confidence). Alternatively, answer logits may instead reflect a heuristic score. This score may incorporate task-irrelevant information, or combine task-relevant information in a manner that deviates from an ideal observer, and so need not read out this decision variable at all. Consequently, although such a score may predict correctness empirically, it need not carry the information from which the posterior confidence could, in principle, be computed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We address this question through the framework of statistical decision confidence (SDC), developed in computational neuroscience (Kepecs et al. , 2008 , Kepecs and Mainen, 2012 , Hangya et al. , 2016 , Sanders et al. , 2016 ) (see Methods for description of the ideal observer model and Figure 1 ). Under this framework, confidence is defined normatively as the posterior probability that the chosen option is correct given the latent decision variable—a noisy internal representation of the evidence in the stimulus favouring one alternative over the other. Critically, under the SDC framework this latent decision variable is a sufficient statistic for the posterior: once it is known, the posterior probability of correctness is fully determined. Consequently, any signal that faithfully reads out this variable carries the information that is, in principle, sufficient to compute the posterior confidence—rather than merely indicating how strongly the model prefers one answer over another.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our central question is therefore whether the observable answer-logit difference behaves like such a latent decision variable. If it does, answer logits are more than empirical predictors of correctness: they read out a quantity that is, in principle, sufficient to compute the posterior probability that the answer is correct. The statistical decision confidence framework provides a concrete way to test this hypothesis, making qualitative predictions about the relationship between the decision variable, confidence, and behaviour that can be evaluated empirically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To investigate this hypothesis, we turn to simple perceptual decision tasks, where the experimenter explicitly specifies the task-relevant stimulus-evidence axis and thereby the normative process model linking evidence, choice, and confidence (Figure 1 ) (Kiani and Shadlen, 2009 , Kepecs et al. , 2008 , Lak et al. , 2014 ) . We use binary visual discrimination tasks inspired by classic experimental paradigms in animals (Kepecs et al. , 2008 , Kiani and Shadlen, 2009 , Masset et al. , 2020 ) . On each trial, the stimulus favours one of two possible responses, and the strength of this evidence is varied systematically from ambiguous to easy, yielding a one-dimensional stimulus-strength axis analogous to classic perceptual decision experiments. This provides a normative benchmark against which the computational role of answer logits can be evaluated. Throughout the paper, we use the term stimulus strength to refer to this experimenter-defined evidence favouring one response over the other.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The latent decision variable is not directly observable. Instead, we observe the logits assigned to the candidate answer tokens. We therefore use the answer-logit difference, LD = ℓ A − ℓ B \\\\mathrm{LD}=\\\\ell_{A}-\\\\ell_{B} , as a candidate readout of the latent decision variable. The sign of LD indicates which answer the model favours (positive favours (A), negative favours (B), and values near zero indicate indifference), while its magnitude indicates how strongly the model favours the chosen answer over the alternative. The central question of this study is whether LD merely reflects a heuristic score that specifies the model’s preference between competing responses, or whether it behaves as a readout of the latent decision variable—one that is sufficient in principle to compute the posterior confidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We test four signatures of statistical decision confidence, ordered from weakest to strongest, each illustrated by the normative model in Figure 1 (Kepecs et al. , 2008 , Hangya et al. , 2016 , Sanders et al. , 2016 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(1) Psychometric function. As the stimulus increasingly favours answer (A), the probability of choosing (A) should increase monotonically, establishing that the model’s choices are governed by the experimenter-defined stimulus evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(2) Signed evidence. Signed LD should vary monotonically with signed stimulus strength. This asks whether LD is a graded readout of the evidence favouring one answer over the other, with stimuli that increasingly favour one option producing progressively more positive LD and stimuli favouring the other producing progressively more negative LD. The relationship need not be linear. Together, these first two signatures concern the encoding of evidence: they establish whether objective stimulus evidence is systematically represented in the model’s decision variable. Importantly, these signatures are consistent both with a normative latent decision variable and with heuristic decision signals that preserve the ordering of stimulus evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(3) Trial-specific confidence. LD magnitude should predict correctness even when stimulus strength is held fixed. This is a key test: among trials of equal objective difficulty, choices made with larger | LD | |\\\\mathrm{LD}| should be more likely to be correct. If LD is a readout of the latent decision variable, statistical decision confidence predicts this positive within-strength relationship because trial-to-trial fluctuations in the decision variable determine the probability that the current choice is correct.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(4) Folded-X pattern. As stimulus strength increases, confidence should increase on correct trials but decrease on error trials, producing the characteristic folded-X pattern observed in perceptual decision studies (Kepecs et al. , 2008 , Sanders et al. , 2016 , Hirokawa et al. , 2019 ) . Under statistical decision confidence, strong stimuli usually push the noisy decision variable far onto the correct side of the boundary, yielding high-confidence correct choices. However, when a strong stimulus nonetheless produces an error, the noisy decision variable must have crossed the decision boundary by only a small margin, yielding low-confidence errors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Together, these final two signatures ask whether LD behaves as the latent decision variable prescribed by the normative model, rather than merely encoding stimulus evidence. They test whether trial-to-trial fluctuations in LD have the structure expected if LD is a readout of the latent decision variable—the variable that is sufficient in principle to compute the posterior probability that the current choice is correct. Such behaviour is not generally expected of heuristic decision signals that incorporate task-irrelevant information or combine task-relevant information in a manner that departs from the normative process model, since in these cases trial-to-trial variation in LD no longer reflects only the latent task-relevant evidence specified by the normative model. Collectively, the four signatures therefore progress from establishing that LD systematically encodes stimulus evidence to testing whether it reads out the latent decision variable prescribed by statistical decision confidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The final two signatures require trial-to-trial variation in the latent decision variable while objective stimulus strength is held constant. In biological observers, this variation arises naturally because repeated presentations of the same stimulus produce different internal sensory measurements, and therefore different values of the latent decision variable. Deterministic multimodal LLMs, however, do not exhibit this form of variability: identical inputs produce identical internal computations and identical answer logits. To recover the analogous regime, we generate multiple stimulus exemplars at each stimulus-strength level. Each exemplar preserves the same experimenter-controlled stimulus evidence favouring one response over the other while randomizing task-irrelevant (“nuisance”) properties of the image, such as the precise spatial arrangement of the visual elements. This parallels classic perceptual paradigms such as random-dot motion, where motion coherence fixes objective stimulus strength while the particular dot realization is resampled on every trial (Kiani and Shadlen, 2009 ) . Although these nuisance manipulations preserve objective stimulus strength, they induce trial-to-trial variation in its latent decision signal. Under the normative model, this variation plays the same role as sensory noise in the ideal observer, producing a distribution of latent decision values at fixed stimulus strength. Consequently, any relationship between | LD | |\\\\mathrm{LD}| and correctness within a stimulus-strength level reflects trial-specific variation in the model’s internal decision signal rather than differences in objective difficulty, allowing signatures (3) and (4) to be tested in deterministic multimodal LLMs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pinning down the computational basis of confidence in multimodal LLMs is a fundamental step toward understanding what model confidence represents. Rather than asking simply whether confidence signals predict correctness, we ask whether they behave as readouts of the latent decision variable—one that is sufficient in principle to compute the posterior confidence. Simple perceptual decisions provide a uniquely tractable setting in which to answer this question because the evidence supporting each alternative is experimentally controlled and the predictions of statistical decision confidence are well defined. Establishing whether answer logits satisfy these predictions in this simplest setting provides a baseline against which confidence in reasoning, factual recall, and complex visual question answering can be interpreted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We organize the results around the four signatures of statistical decision confidence introduced above and illustrated in Figure 1 (Kepecs et al. , 2008 , Hangya et al. , 2016 , Sanders et al. , 2016 ) . The first two signatures concern the encoding of evidence: whether choice and signed LD vary monotonically with the experimenter-defined stimulus-strength axis. The latter two test whether LD behaves as the latent decision variable prescribed by the normative model. Specifically, we ask whether, at fixed objective stimulus strength, larger | LD | |\\\\mathrm{LD}| predicts a higher probability that the current choice is correct, and whether correct and error trials exhibit the characteristic folded-X geometry. For the fixed-strength analysis, the primary quantitative result is the pooled logistic regression controlling for stimulus-strength bin; the binned plots are provided as visualizations of the same effect.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Qwen: Evidence Encoding and Latent Decision Variable Signatures in Perceptual Tasks\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Encoding of Evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across the three synthetic perceptual decision tasks (see example stimuli in Figure 7 ), the model demonstrated orderly psychometric behavior, satisfying the first core signature of evidence encoding. Choice probability increased monotonically with signed stimulus strength, and logistic fits effectively captured the behavioral transitions (Figure 2 ). While objective performance was robust across the board, the model exhibited varying degrees of task-specific response bias—most notably a pronounced bias against the BELOW response in the size task—which are quantified by the shifts in the point of subjective equality (PSE) detailed in Table 1 (A).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Crucially, the model also satisfied the second encoding signature: the signed answer-logit difference ( L D LD ) showed a strong linear correlation with signed stimulus strength. Mean signed L D LD varied linearly with the physical evidence, crossing near the psychometric boundary and reversing sign across the two stimulus classes. This confirms that the model’s answer logits carry a graded, signed representation of task-relevant evidence reflecting both the direction and strength of the sensory evidence. As a basic check, a single-temperature transformation of | LD | |\\\\mathrm{LD}| was both discriminative and well-calibrated on held-out trials across all three tasks (Figure 2 , middle)—demonstrating that LD is an accurate empirical predictor of correctness, but not yet establishing that it reads out the latent decision variable in the sense required by statistical decision confidence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Latent decision variable signatures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This signed evidence code provides the necessary substrate to test the stricter readout predictions of statistical decision confidence in the subsequent analyses: specifically, whether | L D | |LD| predicts correctness when objective stimulus difficulty is held constant, and whether it exhibits the signature folded-X correct/error interaction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Answer-logit confidence predicts correctness at fixed stimulus strength.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Even before controlling for difficulty, | L D | |LD| predicted correctness beyond stimulus strength: across all three tasks it was the stronger predictor, and adding it to a stimulus-strength model significantly improved fit (contrast AUROC 0.864 → 0.907 0.864\\\\to 0.907 , combined 0.926 0.926 ; all likelihood-ratio tests p < 10 − 100 p<10^{-100} ). The stronger latent decision variable signature asks whether LD carries trial-specific decision evidence, rather than simply varying with objective stimulus strength. Because easier stimuli produce both larger | L D | |LD| and higher accuracy, we asked whether | L D | |LD| predicts correctness with stimulus strength held fixed, fitting logistic regressions of correctness on | L D | |LD| with fixed effects for stimulus-strength bin, correct ∼ C ( | stimulus | , bin ) + | L D | \\\\mathrm{correct}\\\\sim C(|\\\\mathrm{stimulus}|,\\\\mathrm{bin})+|LD| . This removes the main effect of difficulty; the residual trial-to-trial variation arises from nuisance variation in the stimuli (independent noise realizations and positional jitter), which perturbs the model’s internal evidence even at fixed nominal strength—the analogue of the internal noise exploited by fixed-stimulus confidence tests in animals (Kepecs et al. , 2008 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across all three tasks, | L D | |LD| remained a strong predictor of correctness within fixed stimulus-strength bins (Figure 3 ; Table 1 (B)). The effect was large and highly significant in every task, strongest for contrast ( β = 4.12 \\\\beta=4.12 , z = 53.42 z=53.42 , p < 10 − 300 p<10^{-300} ): even with objective difficulty held fixed, trials with larger answer-logit differences were substantially more likely to be correct—establishing that Qwen’s LD carries trial-specific decision evidence beyond objective stimulus strength.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Correct and error trials show the folded-X.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The strictest signature separates correct and error trials: statistical decision confidence predicts that, as stimulus strength increases, | L D | |LD| rises on correct trials but falls on error trials, producing the folded-X (intuition and schematic in Figure 1 ). We tested this by regressing | L D | |LD| on absolute stimulus strength, correctness, and their interaction, | L D | ∼ | Δ | × Correct |LD|\\\\sim|\\\\Delta|\\\\times\\\\mathrm{Correct} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across all three tasks the interaction was strongly positive (Figure 4 ; Table 1 (C)): the slope of | L D | |LD| on stimulus strength was positive for correct trials and negative for error trials (contrast, + 8.08 +8.08 vs. − 1.11 -1.11 ; the same pattern held for size and shape). These opposing slopes yield the folded-X. The folded-X is not a generic consequence of a signal predicting correctness or being well calibrated. Rather, it is the specific geometry predicted when confidence is a monotonic function of the latent decision variable. Across the perceptual tasks, Qwen satisfies all four signatures, indicating that answer-logit difference behaves as a readout of the latent decision variable—one that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence. This provides the baseline against which we interpret the more complex tasks that follow.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Replication across models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same signatures held in two further non-reasoning models, Gemma 3 12B (Figures 8 , 9 , 10 ; Table 2 ) and Gemma 4 12B (Figure 12 ; Table 5 ), and in a reasoning model, Gemini Flash 3 (Figure 13 ; Table 7 ; see Supplemental Results). In each, | L D | |LD| predicted correctness within fixed stimulus-strength bins, and the folded-X was present in every model–task cell with a single exception; the Gemma models additionally reproduced the signed-LD evidence code. That exception was the Gemma 3 12B contrast error branch, which was flat rather than falling (Table 2 (D))—not, by itself, evidence against an SDC-like signal, as we take up in the Supplemental Results. For Gemma 4 12B the size task was excluded because objective performance was near chance, dominated by a response bias rather than graded uncertainty.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Summary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across every model–task combination but one—three non-reasoning models and one reasoning model—answer-logit difference satisfied the discriminating signatures of statistical decision confidence: it predicted correctness at fixed stimulus strength and, in all but one case, exhibited the normative folded-X. The sole exception, Gemma 3 on the contrast task, departed only in the strictest signature, showing a flat rather than inverted error branch while still exhibiting strong within-strength prediction of correctness. Taken together, these results show that, in simple perceptual decisions, answer logits behave not merely as heuristic preference scores that predict correctness empirically, but as monotonic readouts of a latent decision variable. Consequently, they read out a quantity that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This conclusion concerns computational function rather than implementation. The normative model in Figure 1 is not intended as a literal description of how multimodal LLMs compute their decisions. Rather, it specifies the functional properties of any signal that represents the posterior probability that the current choice is correct. Satisfying the four signatures therefore does not imply that models implement the particular computations of the normative observer. Instead, it establishes that the observable answer-logit difference behaves as though it were a readout of a latent decision variable with the same computational role: a representation from which the posterior probability that the current choice is correct can, in principle, be computed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Abstention selectively identifies high-confidence decisions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next asked whether the LD signal, which the preceding analyses identified as a readout of the latent decision variable, is behaviourally consequential—building on evidence that language models causally use confidence to drive behaviour (Kumaran et al. , 2026b ) . After making the initial forced-choice contrast judgment, the model was given a second-stage decision: either commit its previous answer or abstain, and was instructed to abstain unless it believed its answer had at least a 60% chance of being correct (see Methods). The resulting policy was conservative and highly selective. The model abstained on 87.7% of trials, committing only its highest-confidence responses. Accuracy on these committed trials increased from the baseline forced-choice accuracy of 92.2% to 99.95%, while the original first-stage answers on abstained trials remained close to baseline accuracy (91.2%; Table 3 ). Thus, abstention selectively retained the most reliable decisions while rejecting those with low answer-logit confidence (low | L D | |LD| ), which were disproportionately error-prone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Critically, abstention tracked the internal confidence signal rather than external difficulty. Answer-logit confidence predicted abstention better than physical stimulus strength (AUROC 0.840 0.840 vs. 0.796 0.796 ), and adding stimulus strength barely improved this ( 0.843 0.843 ), so | L D | |LD| carried most of the information the policy used (Figure 15 ). The effect survived difficulty controls: within fixed stimulus-strength bins, lower | L D | |LD| still strongly predicted abstention (odds ratio = 2.38 =2.38 , p < 10 − 300 p<10^{-300} ). And the model’s policy closely approximated an idealized rule that abstains on the lowest- | L D | |LD| trials—the two overlapped on 91.6% of trials, and the optimal threshold improved committed accuracy by only 0.05 0.05 percentage points. Abstention is therefore governed by the same answer-logit signal characterized above, and acts on it almost exactly as a simple confidence threshold would.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Generalization to memory-based decisions: city-population comparisons.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether these signatures extend beyond perception, we turned to a memory-based decision—judging which of two cities has the larger population, which requires retrieving and comparing stored world knowledge rather than reading a perceptual stimulus (the perceptual-to-memory extension studied in humans by Sanders et al. 2016 ). We drew 5 , 000 5{,}000 cities (population > 1 >1 M) from the SimpleMaps database, giving 50 , 000 50{,}000 binary comparisons. Here the experimenter’s stimulus-strength axis—the ground-truth population ratio—is only a noisy proxy for the evidence the model brings to the choice, since recalled city size, retrieval strength, and semantic salience vary from trial to trial.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This shows up in the encoding signatures. The psychometric function was shallow (accuracy 61.8 % 61.8\\\\% ; slope 0.90 0.90 , pseudo- R 2 = 0.069 R^{2}=0.069 ), with a response bias toward option A (Table 4 (A)): objective population evidence was a weak trial-level predictor of choice. But this reflects low signal-to-noise, not an absent mapping—signed L D LD still varied approximately linearly with signed population evidence (Figure 5 ), so the second encoding signature, the mean mapping from objective evidence onto the decision variable, was intact. Encoding here is weak at the level of individual choices but faithful in the mean, the gap being large trial-to-trial noise in the recalled evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite the shallow psychometric, both latent decision variable signatures held. Within fixed population-evidence bins, | L D | |LD| predicted correctness (odds ratio = 1.36 =1.36 , p < 10 − 180 p<10^{-180} ; Table 4 (B))—LD carried trial-specific decision evidence—and the full folded-X was present, with | L D | |LD| rising with evidence on correct trials and falling on error trials (strongly positive interaction; Table 4 (C)). The same trial-to-trial noise that flattens the psychometric is what makes these latent decision variable signatures informative, since it is the within-strength fluctuation in evidence that they exploit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pattern replicated in Gemma 3 12B, where the psychometric was shallower still (pseudo- R 2 = 0.049 R^{2}=0.049 ) yet the folded-X was if anything sharper—a clearly negative error branch ( − 0.589 -0.589 ) against a positive correct branch ( + 0.678 +0.678 ; Figure 11 ; Table 6 ). This makes the memory task the clearest dissociation between objective stimulus evidence and the model’s internal decision evidence. Here the experimenter-defined stimulus-strength axis predicts individual choices only weakly, yet LD continues to satisfy both latent decision variable signatures, indicating that its interpretation as a readout of the latent decision variable does not depend on a tightly coupled objective evidence axis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"CLEVR: Qwen 2.5 7B, Gemini Flash 3, and complex visual reasoning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The perceptual tasks establish that answer logits can behave like a readout of the latent decision variable in settings where the normative process model can be well specified. We next ask whether this interpretation generalizes to complex visual reasoning, where the underlying decision process is no longer described by a simple perceptual evidence model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated Qwen 2.5 7B and Gemini Flash 3 on CLEVR count-equality and object-existence questions while degrading images with parametric Gaussian blur (larger σ \\\\sigma , lower stimulus strength; Figure 6 ). One feature of this paradigm fundamentally changes what the SDC signatures can tell us. Unlike contrast or size discrimination, blur does not define a signed evidence axis: it removes information rather than favouring one answer over the other, and the mapping from blur to the model’s latent decision variable is neither monotonic nor known. Consequently, the first two signatures—which rely on a well-defined evidence axis—do not transfer directly. What remains well defined is whether | LD | |\\\\mathrm{LD}| carries trial-specific information about whether the current choice is correct beyond objective image difficulty (signature 3), and how | LD | |\\\\mathrm{LD}| behaves on correct and error trials (signature 4).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The third latent decision variable signature remained robust. Adding | LD | |\\\\mathrm{LD}| to blur strength in nested logistic models improved discrimination of correct from error trials in every dataset, with AUROC gains of + 0.029 +0.029 to + 0.072 +0.072 , confirmed by likelihood-ratio tests throughout (Table 8 ), and the same conclusion was reached non-parametrically within fixed blur bins (Figure 6 ). We next asked whether this effect could be explained by other measurable properties of the input. Adding scene and question controls—including the number of unique objects and colours, and spatial crowding (Methods)—improved prediction, particularly for counting, yet | LD | |\\\\mathrm{LD}| continued to add significant information beyond blur and these controls in all estimable nested models. In the fixed-blur analyses with the full control set, the | LD | |\\\\mathrm{LD}| effect remained significant for Qwen count, Qwen existence, and Gemini Flash 3 existence; the Gemini Flash 3 count model was not estimable owing to collinearity among the control variables, so we make no claim there. Thus, across all estimable models, answer-logit difference continued to carry information about whether the current choice was correct beyond the observable properties of the input.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The correct/error structure was more nuanced. On correct trials, | LD | |\\\\mathrm{LD}| increased with stimulus strength in all four analyses. On error trials, however, the slope was approximately flat (Gemini count, Qwen existence, Gemini existence) or modestly positive (Qwen count), never negative, although the correct × \\\\times error interaction remained significant throughout (Table 9 ; Figure 6 , row D). Thus, | LD | |\\\\mathrm{LD}| continued to behave differently on correct and error trials, but did not exhibit the canonical folded-X.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Whether this constitutes a failure of statistical decision confidence cannot be determined from these data alone. The folded-X is a prediction of a specific normative process model in which the experimenter specifies both the task-relevant evidence axis and its mapping to the latent decision variable. In CLEVR, neither is known. Consequently, the normative posterior cannot be derived, and the folded-X is no longer a diagnostic prediction. We therefore cannot conclude that answer logits in CLEVR either satisfy or violate statistical decision confidence. The secure conclusion is the weaker one: answer-logit difference continues to carry trial-specific information about whether the current choice is correct beyond objective properties of the input, but whether it behaves as a normative latent decision variable remains unresolved until an appropriate process model is established (see Discussion).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We asked whether answer logits in multimodal LLMs merely provide useful predictors of correctness, or whether they warrant the stronger interpretation as readouts of the latent decision variable prescribed by statistical decision confidence (SDC). We addressed this question in three multimodal non-reasoning models (Qwen2.5-VL, Gemma 3, Gemma 4) and one multimodal reasoning model (Gemini Flash 3) across three decision regimes: low-level perceptual discrimination (contrast, size, shape), a memory-based decision (city-population comparison), and complex visual reasoning (CLEVR count and existence tasks under blur). Across the perceptual and memory tasks, and consistently across models, answer-logit difference (LD) satisfied all four qualitative signatures of SDC. LD encoded the signed stimulus evidence, predicted correctness beyond objective stimulus strength, and—with the single exception of Gemma 3 on the contrast task—exhibited the characteristic folded-X geometry predicted when confidence is a monotonic function of the latent decision variable. The same signal was also behaviourally consequential: in a second-stage abstention task, LD predicted the model’s decision to commit or abstain better than objective stimulus strength, and the resulting policy closely approximated a simple threshold on LD. Together, these findings show that, in simple perceptual and memory-based settings, answer logits are more than heuristic preference scores that predict correctness empirically: they behave as monotonic readouts of the latent decision variable—a quantity that, under the experimenter-defined normative process model, is sufficient in principle to compute the posterior confidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central issue is that statistical decision confidence is defined relative to a specified normative process model. In simple perceptual tasks, the experimenter defines both the task-relevant evidence variable and the process by which noisy evidence generates a latent decision variable. In more complex tasks, neither is generally known. Consequently, an apparent failure of the SDC signatures does not, by itself, imply that answer logits are non-normative. We believe the CLEVR results should be interpreted in this light. Crucially, answer-logit difference continued to carry trial-specific information about whether the model’s current choice was correct beyond objective properties of the input, even after controlling for blur and measurable scene properties. What was absent was the folded-X. We do not interpret this as evidence against statistical decision confidence. Rather, the folded-X is a prediction of a specific normative process model in which the experimenter specifies both the task-relevant evidence axis and its mapping onto a latent decision variable. In CLEVR, neither is known. Consequently, the absence of the folded-X does not by itself distinguish between a failure of statistical decision confidence and a mismatch between the experimenter-defined process model and the computation actually performed by the model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"More generally, apparent departures from the experimenter-defined normative model may arise in at least three ways. First, the model may incorporate task-irrelevant information into its decision variable. A simple thought experiment illustrates how this can arise. Consider a recognition-memory task in which the model must decide which of two items has been encountered more often. The experimenter defines the relevant evidence as familiarity acquired during the experiment. Suppose, however, that the model cannot distinguish this from pre-existing real-world familiarity, so that its latent decision variable combines both sources of evidence (see Supplemental Discussion for details). The model’s effective evidence variable therefore differs from the experimenter-defined evidence axis, and behaviour analysed with respect to experimental familiarity need no longer exhibit the signatures predicted by the normative model, even if confidence is computed normatively with respect to the model’s own effective evidence variable. An analogous mismatch may underlie the CLEVR results if the model’s effective evidence depends upon latent variables not captured by the experimenter-defined blur axis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, the model may ignore task-relevant information specified by the experimenter. Third, it may combine the available evidence in a non-Bayesian (heuristic) fashion rather than computing the normative posterior. Any of these possibilities could account for the absence of the folded-X in CLEVR. Distinguishing between them requires developing explicit computational process models for complex reasoning tasks, analogous to those available for simple perceptual decisions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is also worth distinguishing the aim of the present framework from the focus of confidence calibration (Guo et al. , 2017 , Steyvers and Peters, 2025 , Xiong et al. , 2023 , Tian et al. , 2023 , Kadavath et al. , 2022 ) . Calibration seeks a mapping from a model’s raw confidence signal to empirical correctness, so that reported confidence better matches the observed frequency of correct responses. This is valuable for prediction and deployment, but it does not by itself establish what the underlying signal represents. A well-calibrated confidence score may arise from a latent decision variable supporting the normative posterior, but it may equally arise from a heuristic preference signal that predicts correctness empirically. Statistical decision confidence asks a different question: whether the underlying signal itself behaves as a monotonic readout of the latent decision variable prescribed by a normative process model. Calibration therefore concerns the mapping between a confidence score and empirical accuracy, whereas the SDC framework concerns the computational interpretation of the confidence signal itself.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One qualification concerns the level of inference these results support, and it connects directly to a long-standing debate about human confidence: whether confidence reflects the posterior probability of being correct (the Bayesian confidence hypothesis, BCH; Pouget et al. , 2016 , Kepecs and Mainen, 2012 , Adler and Ma, 2018 ) or the strength of the raw evidence itself, without an explicit probability computation (confidence in raw evidence space, CRES; Xue et al. , 2024 ). Because our tasks use boundary-defined, non-overlapping response categories, they fall in the regime where the folded-X is a robust rather than fragile prediction of statistical decision confidence (Adler and Ma, 2018 ) . Within this regime the four signatures establish that answer-logit difference behaves as a readout of the latent decision variable, a quantity sufficient in principle to compute the posterior. What they do not fix is whether | L D | |LD| is that posterior or a monotone transform of it that does not renormalise for evidence reliability. Because the normative posterior is itself a monotonic (sigmoidal) function of the decision variable, any faithful readout of that variable—whether the calibrated posterior or the raw evidence—imposes the same ordering on trials and so reproduces all four signatures at any fixed level of reliability; what the signatures cannot see is whether the slope of that mapping tracks reliability. BCH and CRES therefore diverge only when reliability is manipulated independently of stimulus strength—under BCH the mapping from | L D | |LD| to accuracy is reliability-invariant, because a calibrated posterior already absorbs reliability, whereas under CRES a fixed | L D | |LD| corresponds to progressively lower accuracy as reliability falls, because raw evidence is not renormalised. Our single signed-strength axis varies evidence and reliability together and therefore cannot separate the two. Adjudicating them requires independently manipulating stimulus strength and evidence reliability, together with quantitative model comparison rather than qualitative signatures alone (Adler and Ma, 2018 , Xue et al. , 2024 ) —an important direction that the framework developed here newly makes tractable for artificial systems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our results also bear on the distinction between first- and second-order theories of confidence (Pouget et al. , 2016 , Fleming and Daw, 2017 , Mamassian and de Gardelle, 2022 ) . The present work concerns the confidence carried by answer logits themselves. We find that, in simple perceptual and memory-based decisions, answer-logit difference behaves as a readout of the latent decision variable prescribed by statistical decision confidence. This does not preclude the existence of additional confidence computations. Recent work suggests that verbally reported confidence and answer-logit confidence are dissociable signals that independently influence behaviour (Kumaran et al. , 2026a , b , Kumaran, 2026 ) , implying that models may construct a second-stage evaluation of their own answers beyond the decision signal studied here. Rather than competing explanations, these signals may occupy different levels of the confidence hierarchy: answer-logit difference provides the decision-linked signal supporting statistical decision confidence, whereas verbal confidence may incorporate additional computations reflecting broader uncertainty about the answer. Understanding how these signals interact, when they diverge, and which ultimately governs behaviour remains an important direction for future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"More broadly, our results suggest that statistical decision confidence provides a principled framework for understanding what model confidence represents, rather than simply how well it predicts correctness. Although we have focused primarily on visual perceptual decisions, the framework is not inherently visual. We further showed that the same qualitative signatures extend to a memory-based language task, and answer-logit differences are available whenever a model commits to a discrete response. The framework therefore provides a general approach for studying confidence in language models, multimodal agents, and more complex reasoning systems, provided an appropriate normative process model can be specified. Finally, the signatures studied here are behavioural. Identifying the internal representations that give rise to them would provide a mechanistic account of confidence analogous to the progression in neuroscience from behavioural signatures of confidence to neural representations of the underlying decision variable.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Methods\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Normative model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The four signatures we test derive from a standard normative (ideal-observer) model of two-alternative decision making, schematised in Figure 1 . The experimenter specifies a signed stimulus-evidence variable, Δ \\\\Delta , which quantifies the evidence favouring one alternative over the other. The observer does not access Δ \\\\Delta directly, but instead forms a noisy internal representation,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ϵ \\\\epsilon is internal decision noise. The latent decision variable d d therefore fluctuates from trial to trial even when the objective stimulus evidence is held fixed. These trial-to-trial fluctuations are essential: they make a fixed stimulus strength map to a distribution of latent decision values rather than a single value, and it is this variation that the latent decision variable signatures (3 and 4) exploit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A deterministic VLM produces identical logits for identical inputs, so such variability does not arise from repeated presentation of the same image. Instead, it is supplied across stimulus exemplars: images with the same objective stimulus strength but differing in task-irrelevant (nuisance) features produce different internal evidence. The nuisance manipulations used for each task are described below.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sign of d d determines the observer’s choice. Confidence is defined by statistical decision confidence (SDC) as the posterior probability that this choice is correct,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Computing this posterior requires combining the prior distribution over stimulus states with the likelihood of observing the latent decision value under each possible stimulus state,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the denominator marginalises over all possible stimulus states. Thus, the latent decision variable is a sufficient statistic for confidence: once d d is known, the posterior probability that the current choice is correct is fully determined. The posterior increases monotonically with | d | |d| , so a single latent variable determines both the observer’s choice (its sign) and its normative confidence (its magnitude).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither d d nor the observer’s implied likelihood function are directly observable. Rather than attempting to recover the posterior itself, we ask whether the observable answer-logit difference behaves as a monotonic readout of the latent decision variable,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where f f is monotonic. If so, LD inherits the qualitative signatures predicted for the latent decision variable—the psychometric function, signed-evidence encoding, fixed-strength correctness prediction, and the folded-X pattern—without requiring explicit recovery of the posterior.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Tasks\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Simple perceptual decision tasks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We constructed three simple perceptual tasks that allowed precise experimental control over stimulus strength while providing large numbers of trials near perceptual decision boundaries. All tasks were evaluated using the same multimodal LLM and analysis pipeline. Each task generated a signed stimulus-strength variable, Δ \\\\Delta , where positive and negative values favoured opposite response categories. Model choices were analysed using response-token logits, allowing extraction of both categorical decisions and continuous decision variables.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contrast discrimination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stimuli consisted of two independent band-pass filtered noise patches presented side-by-side on a uniform grey background. Each patch was 256 × 256 256\\\\times 256 pixels and the two patches were separated by a 24-pixel gap. Noise images were generated by sampling Gaussian white noise and applying a log-Gaussian band-pass filter in the Fourier domain. The filter was centred at 4 cycles/degree with a bandwidth of 1 octave. The filtered image was transformed back to the spatial domain, normalised to zero mean and unit variance, and scaled to the desired root-mean-square (RMS) contrast.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline RMS contrast was fixed at c 0 = 0.28 c_{0}=0.28 . On each trial, left and right contrasts were determined by a signed contrast difference parameter Δ \\\\Delta ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Positive values of Δ \\\\Delta therefore indicated that the right patch had higher contrast, whereas negative values indicated that the left patch had higher contrast.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stimulus strengths ranged approximately from -0.40 to +0.40. Sampling density was highest near Δ = 0 \\\\Delta=0 , with additional stimulus levels extending into easier discrimination regimes. Independent random seeds were used for the left and right noise patches on every trial.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model was asked:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“This image shows two noise patches, LEFT and RIGHT. Which has HIGHER CONTRAST?”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Responses were given using two response letters. To eliminate confounding between perceptual side preferences and token preferences, the mapping between response letters and spatial locations was counterbalanced across trials. Half of trials used M M for left and V V for right, whereas the remaining trials used the reverse mapping.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let ℓ R \\\\ell_{R} and ℓ L \\\\ell_{L} denote the total logit mass assigned to the tokens currently mapped to the right and left alternatives. The signed decision variable was\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Object-size discrimination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stimuli consisted of two filled circles arranged vertically. Each circle was rendered within a 256 × 256 256\\\\times 256 pixel image region, with the upper and lower regions separated by a 48-pixel gap. Circle luminance was fixed at 0.12 on a background luminance of 0.78. Shapes were rendered using 4 × 4\\\\times supersampling followed by Lanczos downsampling to produce anti-aliased edges.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline circle radius was r 0 = 52 r_{0}=52 pixels. Stimulus strength was parameterised using a signed log-radius ratio,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Circle radii were generated according to\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This parameterisation preserved the geometric mean radius across trials while manipulating relative size. Positive values of Δ \\\\Delta indicated that the lower circle was larger.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To prevent exploitation of trivial spatial cues, each circle received an independent random positional jitter sampled uniformly from ± 6 \\\\pm 6 pixels in both horizontal and vertical directions. Stimulus strengths ranged approximately from -0.40 to +0.60 log-radius units, with oversampling concentrated near the psychometric boundary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model was instructed:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“Two circles are shown vertically, one higher and one lower. Which circle is larger? Answer with exactly one word: ABOVE or BELOW.”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The signed decision variable was defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ℓ below \\\\ell_{\\\\mathrm{below}} and ℓ above \\\\ell_{\\\\mathrm{above}} denote the logit masses assigned to the corresponding response tokens.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Shape categorization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stimuli consisted of a single centrally presented rounded rectangle rendered within a 256 × 256 256\\\\times 256 pixel image. Shapes were displayed using a background luminance of 0.78 and object luminance of 0.12. Anti-aliasing was implemented using 4 × 4\\\\times supersampling followed by Lanczos downsampling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base side length was 120 pixels. Shape category was manipulated using a continuous morph parameter m ∈ [ 0 , 1 ] m\\\\in[0,1] , where m = 0 m=0 corresponded to a square-like angular shape and m = 1 m=1 corresponded to a circle. The morph continuum was implemented by varying the corner radius of a rounded rectangle according to\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where s s denotes side length.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stimulus strength was defined relative to each model’s empirically estimated angular–round boundary m ∗ m^{\\\\ast} , as Δ = m − m ∗ \\\\Delta=m-m^{\\\\ast} , so that Δ = 0 \\\\Delta=0 marks the model’s point of subjective equality ( m ∗ = 0.73 m^{\\\\ast}=0.73 for Qwen, 0.38 0.38 for Gemma 3). Negative values of Δ \\\\Delta corresponded to shapes categorised as angular, whereas positive values corresponded to shapes categorised as round.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To introduce nuisance variation, side length was independently jittered by up to ± 6 % \\\\pm 6\\\\% on each trial and stimulus position was jittered by up to ± 6 \\\\pm 6 pixels. Stimuli were sampled densely in the vicinity of the empirical category boundary, with Δ \\\\Delta values extending approximately from -0.12 to +0.12.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model was instructed:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“A single shape is shown. Is the shape more ANGULAR or more ROUND? Answer with exactly one word: ANGULAR or ROUND.”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The signed decision variable was\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Abstention task: Qwen 2.5 7B\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether decision-confidence signals predicted subsequent behavior, we conducted a second-stage abstention experiment using the contrast-discrimination task. Each abstention trial began with a previously generated forced-choice response from the model. The original image, question, and model answer were then presented back to the model in a multi-turn chat format. The model was asked to decide whether to output its previous answer to the user ( A ) or abstain ( B ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The abstention prompt was:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“You now need to decide whether to output your previous answer to the user (A) or abstain (B). If you don’t think your answer has 60% chance of being correct, choose B although this will disappoint the user. Respond with exactly one letter. A = output answer B = abstain Decision:”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 60% threshold was selected empirically. Preliminary experiments revealed that more permissive prompts resulted in near-universal commitment, whereas stricter prompts resulted in near-universal abstention. Our goal was not to optimize abstention performance but to place the model in a regime where both commitment and abstention occurred frequently enough to permit analysis of the relationship between internal confidence signals and overt behavior. The chosen threshold yielded substantial variability in abstention decisions, allowing us to test whether answer-logit confidence predicted the model’s willingness to stand behind its previous answer. The generated abstention decision ( A or B ) was used for all behavioural analyses.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"City-population binary-choice task.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether answer-logit confidence generalized beyond perceptual discrimination, we constructed a factual memory-based decision task using the SimpleMaps World Cities Database. The database provides city names, country information, geographic coordinates, and population estimates for cities and towns worldwide. We restricted the stimulus set to cities with population greater than one million people. This yielded 5 , 000 5,000 unique cities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Trials consisted of binary population comparisons between two cities. On each trial, the model was asked which of two cities had the larger population. The signed stimulus-strength variable was defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so that Δ > 0 \\\\Delta>0 indicated that option A was correct and Δ < 0 \\\\Delta<0 indicated that option B was correct. The absolute stimulus strength, | Δ | |\\\\Delta| , therefore measured the population-ratio difference between the two cities. City pairs were sampled from the eligible city set to generate 50,000 binary-choice trials. Cities were allowed to appear in multiple different pairs, but each trial contained two distinct cities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prompt was:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"“Which city has the higher population? A. [city A] B. [city B] Answer with exactly one letter: A or B.”\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model’s first-token logits were extracted at the answer position. Let ℓ A \\\\ell_{A} and ℓ B \\\\ell_{B} denote the logit masses assigned to the answer tokens A and B . We defined the signed decision variable as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, positive values indicated evidence favouring option A and negative values indicated evidence favouring option B. Accuracy was computed by comparing the sign of L D LD with the sign of Δ \\\\Delta , and answer-logit confidence was defined as | L D | |LD| .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"CLEVR count and existence tasks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CLEVR (Johnson et al. , 2017 ) is a synthetic visual-question-answering benchmark of rendered 3D scenes whose objects vary in shape, colour, material, and size, paired with compositional questions and ground-truth scene graphs and functional programs—a controlled, fully annotated structure that lets us select well-defined count and existence questions and compute objective scene-difficulty measures while degrading only the rendered image. We drew questions from the v1.0 training and validation splits. Scene annotations and programs were used only to select questions and, for the count task, to compute the two queried object counts; the model was always evaluated on the rendered image and natural-language question, never on the scene graph or program.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Both tasks used a two-letter forced-choice format, with the mapping between letters and response categories counterbalanced within each image–question pair (each letter assigned to each category once across the two presentations). We defined the semantic logit difference as the log-probability of the letter denoting the target category ( equal for count, yes for existence) minus that of the alternative, averaged across the two letter mappings to remove fixed letter-identity biases. Questions were sampled deterministically (seed 42), and each image was evaluated across a range of Gaussian blur levels parameterised by filter radius σ \\\\sigma .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Qwen2.5-VL, which showed a strong affirmative-response bias that counterbalancing does not remove, analyses were restricted to yes-truth trials (equal counts in the count task; queried object present in existence), avoiding the regime in which responding was bias-dominated. Gemini Flash 3 showed no comparable asymmetry, so both truth arms were retained; its answer logits were read with the reasoning-cut procedure described in the Models section.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Count-comparison task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Candidate questions were those whose program terminated in equal_integer (whether two object counts are equal), restricted to one or two relational operations so that the counted sets were defined relative to other objects in the scene. We kept questions with at least one non-zero queried count and removed duplicate image–question pairs. Qwen2.5-VL was run on 1 , 000 1{,}000 yes-truth questions at σ = 0 , 2 , 4 \\\\sigma=0,2,4 , a fine grid from 4 4 to 10 10 in steps of 0.25 0.25 , and coarser levels at 12 , 14 , 16 , 18 , 20 , 25 , 30 12,14,16,18,20,25,30 ; Gemini Flash 3 used the D/N letter pair on 500 500 questions per truth arm at σ = 0 \\\\sigma=0 – 10 10 in unit steps.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"Simple-existence task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Candidate questions were those terminating in exist , restricted to simple two-attribute filter-only programs (two filter_* operations, e.g. colour and shape; questions with relational or same_* operations were excluded). Both models used the M/V letter pair. Qwen2.5-VL was run on 800 800 yes-truth questions at σ = 0 , 1 \\\\sigma=0,1 then a fine grid from 2 2 to 10 10 in steps of 0.25 0.25 ; Gemini Flash 3 was evaluated on both truth arms over the same grid.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Difficulty controls.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether answer-logit confidence merely proxied visual or question difficulty, we augmented each trial with scene and question metadata from the CLEVR annotations: scene-level controls (total objects; unique colours, shapes, sizes, and materials; maximum objects sharing an attribute; maximum identical object types; and spatial-crowding measures from pairwise object distances) and, where applicable, program controls (program length and counts of filter, relation, same-attribute, count, existence, union, and intersection operations), plus the queried counts and their absolute difference for count questions. After removing highly collinear controls, we tested whether | LD | |\\\\mathrm{LD}| improved prediction beyond blur strength and these controls (nested likelihood-ratio tests) and whether it predicted correctness within coarse blur bins after including them (fixed-blur-bin analyses).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Models\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Non-reasoning models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated three non-reasoning multimodal language models, all run locally with the HuggingFace transformers library: Qwen2.5-VL-7B-Instruct ( Qwen/Qwen2.5-VL-7B-Instruct ), Gemma 3 12B ( google/gemma-3-12b-it ), and Gemma 4 12B ( google/gemma-4-12B-it ). Each was prompted to answer with a single letter or word (see the task descriptions), so the model returned a direct answer with no reasoning trace and the answer logits could be read from the first response token. Although Gemma 4 is reasoning-capable, this single-token response format elicits a direct answer, placing it in the same non-reasoning regime as the other two models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Reasoning model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated one reasoning model, Gemini Flash 3 preview, accessed through the Beyond API on Google infrastructure. Confidence was read using the reasoning-cut procedure below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Reasoning-cut readout (Gemini Flash 3).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A reasoning model does not expose a well-defined answer logit until it commits to an answer, and the signal saturates once the answer is verbalized. On each trial the model was prompted to reason step by step and to end with a one-letter answer; the generated trace was truncated at 90 % 90\\\\% of its length, and the model was then asked to answer the original question from the reasoning produced so far, yielding a graded answer-logit difference before saturation. We applied this 90 % 90\\\\% reasoning-cut readout to Gemini Flash 3 on both the contrast discrimination task and the CLEVR count and existence tasks. At the 90 % 90\\\\% cut, accuracy did not differ significantly from accuracy after the complete reasoning trace ( p > 0.1 p>0.1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Appendix\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Supplemental Results: Gemma 3 12B\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gemma 3 12B reproduced the encoding and readout signatures across the three perceptual tasks (Figures 8 , 9 , 10 ; Table 2 ). Two task-specific points: the size task carried a strong response bias, so we additionally confirmed the fixed-strength result in a model-relative analysis defining correctness relative to the model’s own PSE (Figure 9 ); and the contrast error branch was flat rather than falling (Table 2 (D)), the only departure from the full folded-X. We treat the flat branch as neutral: it is neither the falling branch predicted by SDC nor the rising branch sometimes observed when confidence fails to follow the SDC geometry. LD nevertheless predicted correctness within fixed stimulus-strength bins, with | L D | |LD| predicting correctness (Table 2 (C)). The signatures also held in the population QA task, with a clean folded-X carrying a strongly negative error branch despite a shallow psychometric (Figure 11 ; Table 6 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Supplemental Results: Gemma 4 12B\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gemma 4 12B reproduced the signatures on contrast and shape, including the full folded-X (Figure 12 ; Table 5 ). The size task was excluded for near-chance objective performance (accuracy 0.509 0.509 ), driven by a large response bias rather than graded uncertainty.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Gemini Flash 3 (reasoning model).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the contrast task at the 90% reasoning cut, Gemini Flash 3 showed the same signatures—a steep psychometric, fixed-strength prediction of correctness, and a clear folded-X (Figure 13 ; Table 7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"Supplemental Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Recognition-memory example.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a recognition-memory task in which the experimenter defines the relevant evidence as the difference in familiarity acquired during the experiment,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose, however, that the model cannot distinguish experimentally acquired familiarity from pre-existing real-world familiarity,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so that its latent decision variable is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The experimenter-defined correctness depends only on Δ expt \\\\Delta_{\\\\rm expt} , but the observed decision variable depends on both sources of familiarity. Consequently, the normative posterior probability that the current choice is experimentally correct would be\"\n },\n {\n \"type\": \"para\",\n \"text\": \"rather than the simpler expression obtained under the experimenter-defined model alone. That is, because Δ rw \\\\Delta_{\\\\rm rw} also contributes to the generation of d d , the posterior defined over the experimenter’s evidence axis no longer coincides with the posterior defined over the model’s effective evidence variable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If the experimenter analyses behaviour only with respect to Δ expt \\\\Delta_{\\\\rm expt} , the fixed-strength and folded-X signatures need not hold, since trial-to-trial variation in the decision variable now reflects both experimental and real-world familiarity. This does not by itself imply that the model departs from the normative model defined over its own effective evidence variable; it illustrates how the experimenter-defined evidence axis may differ from the model’s effective evidence variable.\"\n }\n]","isConversational":false}}},{"rowIdx":298,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.12443"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Language Identification with Succinct Machine-Independent Traces"},"authors":{"kind":"list like","value":["Moses Charikar","Jon Kleinberg","Chirag Pabbaraju"],"string":"[\n \"Moses Charikar\",\n \"Jon Kleinberg\",\n \"Chirag Pabbaraju\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.12443v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Moses Charikar \\Email moses@cs.stanford.edu \\addr Stanford University and \\Name Jon Kleinberg \\Email kleinberg@cornell.edu \\addr Cornell University and \\Name Chirag Pabbaraju \\Email cpabbara@cs.stanford.edu \\addr Stanford University"},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"The success of large language models has motivated several lines of recent work exploring theoretical models that might provide insight into how they are achieving strong performance from the types of training data and training pipelines that they are built from. One direction this research has taken has been to start from a classical model of Gold on language identification in the limit (Gold, 1967 ) , in which we view the language learning task as a game played between an adversary and an algorithm: the adversary thinks of a secret language K K known only to come from a countable list of candidates L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},... ; the adversary enumerates the strings of K K in an arbitrary order 1 1 1 A valid enumeration x 1 , x 2 , … x_{1},x_{2},\\dots of K K satisfies: (1) x i ∈ K x_{i}\\in K , for all i i , and (2) ∀ x ∈ K \\forall x\\in K , there exists i i such that x i = x x_{i}=x . ; in each step t t the algorithm guesses the index i t i_{t} of the true language; and the algorithm wins the game (it has identified K K in the limit ) if for some time t ∗ t^{*} , and all t ≥ t ∗ t\\geq t^{*} , we have L i t = K L_{i_{t}}=K — the algorithm is correct in every step beginning with t ∗ t^{*} ."},{"type":"para","text":"The main results in Gold’s framework, beginning with Gold’s original theorem about his model, are negative (Gold, 1967 ) ; for all but highly constrained families of languages (much more restricted than the class of regular languages), the algorithm cannot win this game (Angluin, 1980 ) . Yet real language models seem much more powerful than what these sweeping negative results suggest, and so the question has turned to finding reasonable variations on this model in which we are able to obtain positive results more consistent with the successes we see in practice."},{"type":"heading","level":4,"text":"Computational traces."},{"type":"para","text":"One promising line of recent work has explored the power of annotation or computational traces in helping solve the problem of language identification in the limit (Papazov and Flammarion, 2025 ; Bhattamishra et al., 2026 ; Peng et al., 2026 ) . These models are motivated by the observation that training corpora often contains text or sequence data that is enriched by some kind of meta-data, thinking traces, or domain-relevant annotation. This can include cases in which the data itself has this type of supporting information, as in the way that learning from corpora of source code can be more effective if the code is commented, or learning from corpora of mathematics can be more effective if the proofs contain step-by-step arguments. It also includes cases in which the data has been annotated algorithmically, as in the success of chain-of-thought and related methods (Wei et al., 2022 ) . This range of examples suggests that learners can be more effective on a corpus of training data if that data is annotated with some kind of computational trace."},{"type":"para","text":"These recent models have begun from the assumption that there is an automata-theoretic underpinning for the candidate languages — for example, that they are produced by an unknown finite-state machine, pushdown automaton, or Turing machine — and that the annotation in question is produced from the computational trace of the machine itself. Prior work obtains possibility results with this approach. This line of work leaves open two fundamental classes of questions, however. First, the kind of annotation they describe is fine-grained and expansive: in one case it is represented in terms of the states of the underlying machine (Peng et al., 2026 ) , and in the other case it is represented by the power-set of the vocabulary, with an exponential blow-up as a result (Bhattamishra et al., 2026 ) . Second, the premise of these models is that annotation is directly tied to the automaton that produced the language; it leaves open the question of how we might usefully annotate for identification when we have no access to the underlying machine models."},{"type":"para","text":"In contrast, the kinds of annotations or computational trace we see in practice tend not to come from an underlying machine model, but tend to be more designed as objects in their own right, with their own vocabularies, and much coarser than the full state set of the machine architecture. For example, in commented code, it is the comments that form the additional tokens helping learn the corpus; and with chain-of-thought, it is the thinking tokens that provide this help. Neither of these corresponds to the state of the underlying machine that produced the training data, to the extent that there is such a machine at all. What would a theory of computational traces look like if we take the tokens in the trace to have this coarse-grained structure, and independent of any specific machine model? Could we still obtain possibility results for language identification in the limit?"},{"type":"heading","level":4,"text":"The present work: Compact, machine-independent traces."},{"type":"para","text":"In this paper, we develop such a theory, and we show that annotation with a small set of tokens — independent of the size of the state set of the underlying machine — is sufficient to achieve language identification in the limit. Moreover, our construction of the annotation works at the level of the language itself, and does not assume knowledge of any underlying machine producing the language."},{"type":"para","text":"We will describe the model in detail in the subsequent sections, but we begin here with a high-level overview of it. As a starting point, let’s consider how earlier machine-based models defined computational traces in the case of a finite automaton: as the automaton processed the string one symbol at a time, from left to right, the trace reported the state of the automaton at each symbol. This kind of trace has the property if x = x 1 x 2 … x n x=x_{1}x_{2}\\ldots x_{n} is the input string, then the trace annotates each input symbol x i x_{i} with an annotation symbol c i c_{i} (corresponding to the state of the automaton at that point); and because c i c_{i} is the state, it depends only on the prefix x 1 x 2 … x i x_{1}x_{2}\\ldots x_{i} up to that point, and not any subsequent symbols."},{"type":"para","text":"We use this as the defining property of our traces: we say that a trace coloring function c c over a finite set of colors P P (the “palette”) is a function that maps finite strings to elements of P P , and we define the color trace of a string x = x 1 x 2 … x n x=x_{1}x_{2}\\ldots x_{n} to be the value of c c applied to each of its prefixes:"},{"type":"para","text":"where x ≤ i x_{\\leq i} denotes the prefix x 1 x 2 … x i x_{1}x_{2}\\ldots x_{i} and ε \\varepsilon denotes the empty string. A coloring of x x by the states of a finite automaton that produces it is one way to produce such a trace coloring, but our definition here makes clear that we can define this object as an annotation scheme for any language, even if we don’t know the underlying machine model, or even if there isn’t one. (This is the same sense in which comments or thinking tokens can annotate a string in a training corpus even though they don’t correspond in any respect to the generative mechanism that originally produced the string.)"},{"type":"heading","level":4,"text":"Identification using traces."},{"type":"para","text":"We would like to achieve identification in the limit using these traces, and further, with small palettes, so that we do not need to provide annotations using voluminous alphabets like fine-grained state sets. Formally, given an instance of language identification in the limit with candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\ldots , we consider a process in which each language L i L_{i} is first annotated by its own trace coloring function c L i c_{L_{i}} , and then an adversary enumerates the strings x ∈ K x\\in K , each as an annotated pair consisting of x x and its color trace, ( x , trace c K ( x ) ) (x,{\\mathrm{trace}}_{c_{K}}(x)) ."},{"type":"para","text":"Is this sufficient to enable language identification in the limit, even with no underlying machine model? We show that it is: there is an algorithm 𝒜 \\mathcal{A} such that for any collection of candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\ldots , there exists a choice of trace colorings c L 1 , c L 2 , c L 3 , … c_{L_{1}},c_{L_{2}},c_{L_{3}},\\ldots for which 𝒜 \\mathcal{A} is able to achieve identification in the limit when presented with the ordered pairs ( x , trace c K ( x ) ) (x,{\\mathrm{trace}}_{c_{K}}(x)) arising from an enumeration of the adversary’s language K K ."},{"type":"para","text":"Moreover, the trace colorings required can be constructed using very few colors: for problem instances with an alphabet of size k k , there is an algorithm that achieves this using trace colorings with palettes of size at most k + 1 k+1 . Even in the case where the languages come from an underlying machine model, this is a bound that is independent of the number of states of these machines, which must grow unboundedly for an infinite family of languages. As such, it is a much more resource-efficient construction of computational traces — and one that applies much more broadly — than the computational traces proposed by earlier approaches."},{"type":"para","text":"We prove this result by establishing an exact combinatorial characterization of when a trace coloring permits identification in the limit, and we relax this to a related sufficient condition for identification that we show how to achieve. This sufficient condition has interesting combinatorial interpretations in its own right, and we show matching upper and lower bounds for this condition in the case of languages over a binary alphabet (when k = 2 k=2 ): our result shows how to achieve the required condition with k + 1 = 3 k+1=3 colors, and we prove that 2 colors are not sufficient."},{"type":"para","text":"We explore further extensions of our result, including the fact that identification in the limit is still possible (using a larger palette) when the trace can be corrupted to a bounded extent; and the fact that when the candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\ldots are regular, we can achieve identification in the limit with trace coloring functions that use only two colors."},{"type":"heading","level":2,"text":"1.1 Main Results"},{"type":"para","text":"Given a language collection 2 2 2 All languages in the collection are assumed to be non-empty. 𝒞 \\mathcal{C} , our objective is to associate every language L L in 𝒞 \\mathcal{C} with a suitable trace coloring function c L c_{L} , such that when every x ∈ L x\\in L is accompanied with trace c L ( x ) {\\mathrm{trace}}_{c_{L}}(x) , the collection 𝒞 \\mathcal{C} becomes identifiable in the limit. Towards this, as our first contribution, we derive an exact characterization of when identification in the limit is possible with color traces. Our characterization is structurally similar to Angluin’s condition (Angluin, 1980 ) for identification in the limit, but additionally takes into account the color traces seen in the input; the proof is given in \\Cref sec:characterization-appendix."},{"type":"para","text":"Let 𝒞 \\mathcal{C} be a countable language collection. Then, 𝒞 \\mathcal{C} is identifiable in the limit with a color trace given by the trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} if and only if for every language L ∈ 𝒞 L\\in\\mathcal{C} , there exists a finite “tell-tale” subset T L ⊆ L T_{L}\\subseteq L , such that for every language L ′ ∈ 𝒞 L^{\\prime}\\in\\mathcal{C} that is a proper subset of L L , either (1) L ′ L^{\\prime} does not contain T L T_{L} , or (2) there exists x ∈ L ′ x\\in L^{\\prime} such that trace c L ′ ( x ) ≠ trace c L ( x ) {\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\neq{\\mathrm{trace}}_{c_{L}}(x) ."},{"type":"para","text":"The characterization above gives a precise objective for constructing trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} . Perhaps surprisingly, our next result, which is one of the main structural results of the paper, shows that for every language collection, we can construct trace coloring functions that use a small palette, and have the stronger property that they always satisfy requirement (2) above."},{"type":"para","text":"Let 𝒞 \\mathcal{C} be a language collection over a finite alphabet Σ \\Sigma of size k k . There exist trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 , that satisfy the following “distinguishable coloring condition”: for every L , L ′ ∈ 𝒞 L,L^{\\prime}\\in\\mathcal{C} that satisfy L ⊊ L ′ L\\subsetneq L^{\\prime} , there exists x ∈ L x\\in L for which trace c L ( x ) ≠ trace c L ′ ( x ) {\\mathrm{trace}}_{c_{L}}(x)\\neq{\\mathrm{trace}}_{c_{L^{\\prime}}}(x) ."},{"type":"para","text":"We also show that our trace coloring functions use an optimal palette size for the purposes of satisfying the distinguishable coloring condition over a binary alphabet ( \\Cref prop:2-coloring-lower-bound-for-sufficient-condition): namely, there exists a (finite) language collection over a binary alphabet for which any set of trace coloring functions that satisfies the distinguishable coloring condition necessarily requires three colors."},{"type":"para","text":"Given that our trace coloring functions ensure that requirement (2) of the characterizing condition in \\Cref thm:characterization-identification-with-traces always holds, tell-tale sets are not needed and we get our main result:"},{"type":"para","text":"Let 𝒞 \\mathcal{C} be a countable language collection over a finite alphabet Σ \\Sigma of size k k . There exist trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 that make 𝒞 \\mathcal{C} identifiable in the limit with color traces."},{"type":"para","text":"In order to specify a trace coloring function for a language, in general, one needs to specify the color that it maps every string in the universe to. However, our trace-coloring functions from \\Cref thm:main-theorem above have a particularly special structure for regular language collections. Concretely, consider a regular language L L , and let M M be any finite automaton that recognizes L L . Denote its set of states by Q ( M ) Q(M) . For any state q q in Q ( M ) Q(M) , all strings that arrive at this state q q (starting from the initial state) are assigned the same color by our state coloring function c L c_{L} for language L L . Now, associate the state q q with this common color. Then, for any x ∈ L x\\in L , the sequence of colors associated with the sequence of states it traverses in M M would exactly equal the color trace generated by our trace coloring function on x x . Thus, for a regular language, our trace coloring functions c L c_{L} can be described very compactly, by simply specifying the unique color associated with every state in any automaton recognizing L L ."},{"type":"para","text":"This structural property of our trace coloring functions motivates studying a natural class of trace coloring functions that come from directly coloring the states of automata for regular languages — we denote these as state coloring functions . Concretely, a state coloring function c M c_{M} maps every state in Q ( M ) Q(M) to a palette P P . Denoting L ( M ) L(M) to be the regular language recognized by M M , every x ∈ L ( M ) x\\in L(M) has a well-defined state color trace associated with it: if x x traverses the sequence of states q 0 , … , q | x | q_{0},\\dots,q_{|x|} in M M , then the state color trace is simply"},{"type":"para","text":"The work of Peng et al. ( 2026 ) and Bhattamishra et al. ( 2026 ) can be viewed in this framework (see Section 1.2 for a discussion). For the computational traces used by Peng et al. ( 2026 ) , the ‘color’ of a state is the name of the state itself. Our notion is a coarsening of this state information."},{"type":"para","text":"With this definition ( 2 ), we can now study the distinguishable coloring condition for any given collection 𝒞 = { M 1 , M 2 , … } \\mathcal{C}=\\{M_{1},M_{2},\\dots\\} of finite automata. By the reasoning above, we can show that state coloring functions derived from our trace coloring functions satisfy the condition with k + 1 k+1 colors. Recall that for arbitrary language collections, we could show that our trace coloring functions achieved an optimal palette size for the distinguishable coloring condition only in the case of a binary alphabet. Interestingly, for arbitrary automaton collections, we show that k + 1 k+1 is the optimal palette size achievable by state coloring functions for the distinguishable coloring condition for any alphabet size k k ."},{"type":"para","text":"Let 𝒞 \\mathcal{C} be an automata collection over an alphabet of size k k . There exist state coloring functions { c M } M ∈ 𝒞 \\{c_{M}\\}_{M\\in\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 , such that for any M , M ′ ∈ 𝒞 M,M^{\\prime}\\in\\mathcal{C} that satisfy L ( M ) ⊊ L ( M ′ ) L(M)\\subsetneq L(M^{\\prime}) , there exists x ∈ L ( M ) x\\in L(M) for which trace c M ( x ) ≠ trace c M ′ ( x ) {\\mathrm{trace}}_{c_{M}}(x)\\neq{\\mathrm{trace}}_{c_{M^{\\prime}}}(x) . Moreover, for any k k , there exists a collection of automata over an alphabet of size k k such that any set of state coloring functions requires k + 1 k+1 colors to satisfy this condition."},{"type":"para","text":"lemma:state-coloring-optimal-bound above is primarily about the tightness of k + 1 k+1 colors for state coloring functions to satisfy the distinguishable coloring condition for any given collection of automata. We now return to our main question of identification in the limit with arbitrary trace coloring functions, but for the special class of regular language collections. In particular, we ask: what is the optimal palette size necessary for these collections? Instead of the general-purpose trace coloring functions given by \\Cref thm:main-theorem, which use k + 1 k+1 colors, can we perhaps use different trace coloring functions that use fewer colors? Our final result shows that any collection of regular languages, where every language is infinite, is in fact identifiable in the limit using only two colors!"},{"type":"para","text":"Let 𝒞 = { L 1 , L 2 , … } \\mathcal{C}=\\{L_{1},L_{2},\\dots\\} be a countable collection of regular languages, where every L i L_{i} is infinite. Then, 𝒞 \\mathcal{C} is identifiable in the limit with color traces given by trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} that use just two colors."},{"type":"para","text":"We note that a palette size of two is optimal above, since using a single color is equivalent to having no trace information, and we know by Gold’s results that there are simple regular language collections, where every language is infinite, that are not identifiable in the limit without any traces."},{"type":"para","text":"Finally, we also study identification in the limit with traces that are potentially corrupted ( \\Cref sec:corruptions); in this setting, we derive a robust version of the coloring lemma, and analogously show optimality of the palette size in the case of state coloring functions for regular languages."},{"type":"para","text":"We note that our primary focus in this paper is information-theoretic. Namely, for our upper bounds for identification, we seek to specify a function that maps inputs and color traces to a guess for the target language. Nevertheless, for implementing the identification algorithm of \\Cref thm:main-theorem, we would require access to the coloring functions c L c_{L} for every language L L in the collection, and the ability to determine membership of any string x x in any language L L in the collection."},{"type":"heading","level":2,"text":"1.2 Related Work"},{"type":"para","text":"We discuss two recent works most closely related to our work, that demonstrate the power of annotation or computational traces for language identification."},{"type":"para","text":"Peng et al. ( 2026 ) take a machine-centric view, studying identification for languages accepted by a particular machine model (deterministic finite automata, deterministic push down automata, and Turing machines). For a machine M M and input x x , they consider x x augmented by a computational trace given by the sequence of computational steps of M M on x x . In the case of a DFA D D , the computational trace is simply the sequence of states visited by D D while processing input x x . In the case of a Turing machine M M , the computational trace is the sequence of ‘states’ of the Turing machine while processing x x , where each ‘state’ includes the state of the finite state control, but also the contents of the memory tape, the position of the head, and so on. Given an enumeration of inputs accepted by machine M M , augmented by their computational traces, they show that one can converge to the correct representation of M M , and hence identify the language accepted by M M . We note that this notion of computational traces requires very detailed, fine-grained information to facilitate language identification. The computational trace for input x x can be significantly longer than x x . They also study the question of robust identification, i.e., language identification from corrupted traces, establishing results in different error regimes."},{"type":"para","text":"Another recent work (Bhattamishra et al., 2026 ) also takes a machine-centric view, studying learnability of regular languages in the Next Symbol Prediction (NSP) setting. Here, the learner receives strings from the language, as well as rich annotation for every prefix of the string: whether the prefix belongs to the language, and which next symbols can lead to a string in the language, i.e., k + 1 k+1 bits for alphabet size k k . They show that this annotation is information-theoretically sufficient to identify minimal DFAs; however the problem of improper learning remains computationally intractable. We note that the annotation considered in this work is closely related to, but significantly richer that the annotation we use in our work. If the input alphabet is of size k k , each prefix of the input string is annotated with a symbol from an alphabet size of 2 k + 1 2^{k+1} . In contrast to our work, this work is focussed on regular languages. Much of their focus is on computationally efficient learning – an issue that we (mostly) do not study in our work."},{"type":"heading","level":1,"text":"2 Background and Motivating Examples"},{"type":"para","text":"Language identification with a color trace can be related to language identification with a list (Charikar, Pabbaraju, and Tewari, 2025 ) , where the algorithm guesses a list of k k languages at every step, and the algorithm identifies in the limit if the list eventually always contains the target language. In particular, Charikar et al. ( 2025 , Theorem 2) shows that a language collection 𝒞 \\mathcal{C} is identifiable in the limit with a list of size k k if and only if it can be expressed as a union of k k collections, each of which is identifiable in the limit with a single guess. By this characterization, any collection 𝒞 \\mathcal{C} that is k k -list identifiable decomposes as 𝒞 = ∪ i = 1 k 𝒞 i \\mathcal{C}=\\cup_{i=1}^{k}\\mathcal{C}_{i} , where each 𝒞 i \\mathcal{C}_{i} is identifiable. In this case, we can assign a constant color trace corresponding to a unique color i i to every language in 𝒞 i \\mathcal{C}_{i} . Then, the color traces seen along with the input reveal the specific 𝒞 i \\mathcal{C}_{i} that the target language K K belongs to. Once this is known, one can simply apply any known algorithm that identifies in the limit to the collection 𝒞 i \\mathcal{C}_{i} ; since 𝒞 i \\mathcal{C}_{i} is identifiable in the limit, the algorithm will correctly identify the target language. Thus, k k -list identifiability always implies identifiability with k k colors."},{"type":"para","text":"Interestingly, the converse is not true: identifiability with a color trace does not always imply identifiability with a finite-size list. To see this, fix any alphabet Σ \\Sigma . For any finite F ⊆ Σ ∗ F\\subseteq\\Sigma^{*} , let L F = Σ ∗ ∖ F L_{F}=\\Sigma^{*}\\setminus F . Now, consider the collection 𝒞 = { L F : F is a finite subset of Σ ∗ } \\mathcal{C}=\\{L_{F}:F\\text{ is a finite subset of }\\Sigma^{*}\\} . This collection is not identifiable with a list of any finite size (see Remark 3 in Charikar et al. ( 2025 ) ). However, there is a simple trace coloring scheme with just two colors that makes the collection identifiable: associate every L ∈ 𝒞 L\\in\\mathcal{C} with the trace coloring function c L ( x ) = 𝟙 [ x ∈ L ] c_{L}(x)=\\mathds{1}[x\\in L] . We call this the “accept-reject” coloring. Note then that c L F ( x ) = 0 c_{L_{F}}(x)=0 only if x ∈ F x\\in F . Thus, an algorithm may initialize F = ∅ F=\\emptyset , and initialize its guess for the target language K K to be L ∅ L_{\\emptyset} . Whenever it sees an input string x x whose color trace has 0 in it, it obtains the prefix w w of x x at which the color 0 was seen (i.e., c K ( w ) = 0 c_{K}(w)=0 ). The algorithm then adds w w to F F , and updates its guess L F L_{F} accordingly. If the target language K K was L ∅ L_{\\emptyset} , the algorithm will never see a 0 in any color trace, and its initial guess remains correct throughout. If the target language K K was L F L_{F} for some F ≠ ∅ F\\neq\\emptyset , then for every w ∈ F w\\in F , the algorithm is guaranteed to see an input x x that contains w w as its prefix; at this point, the algorithm correctly infers w w ’s membership in F F . Since F F is finite, the algorithm identifies L F L_{F} in the limit."},{"type":"para","text":"The example above motivates studying the simple accept-reject coloring scheme further. Notice a key property about the collection in the example above: for every L ∈ 𝒞 L\\in\\mathcal{C} , if a string w w was not in L L , it could be extended by some suffix y y such that w y ∈ L wy\\in L . Indeed, this “reject-extendability” property is sufficient for a collection to be identifiable in the limit with the accept-reject coloring."},{"type":"para","text":"Let 𝒞 \\mathcal{C} be a countable language collection over the alphabet Σ \\Sigma , where every L ∈ 𝒞 L\\in\\mathcal{C} satisfies the property: for every w ∈ Σ ∗ w\\in\\Sigma^{*} , if w ∉ L w\\notin L , then there exists a suffix y ∈ Σ ∗ y\\in\\Sigma^{*} such that w y ∈ L wy\\in L . Then, the collection 𝒞 \\mathcal{C} is identifiable in the limit with a color trace given by the accept-reject trace coloring."},{"type":"para","text":"Let K ∈ 𝒞 = { L 1 , L 2 , … } K\\in\\mathcal{C}=\\{L_{1},L_{2},\\dots\\} be the target language. For every L ∈ 𝒞 L\\in\\mathcal{C} , consider the accept-reject coloring given by the trace function c L ( x ) = 𝟙 [ x ∈ L ] c_{L}(x)=\\mathds{1}[x\\in L] . Consider the algorithm, which at time step t t , outputs the smallest index i t i_{t} such that the language L i t L_{i_{t}} is consistent with all the information seen in the input so far. Namely, L i t L_{i_{t}} satisfies that { x 1 , … , x t } ⊆ L i t \\{x_{1},\\dots,x_{t}\\}\\subseteq L_{i_{t}} , and furthermore, trace c K ( x b ) = trace c L i t ( x b ) {\\mathrm{trace}}_{c_{K}}(x_{b})={\\mathrm{trace}}_{c_{L_{i_{t}}}}(x_{b}) for all 1 ≤ b ≤ t 1\\leq b\\leq t . Furthermore, for every j < i t j<i_{t} , either { x 1 , … , x t } ⊈ L j \\{x_{1},\\dots,x_{t}\\}\\nsubseteq L_{j} , or trace c K ( x b ) ≠ trace c L j ( x b ) {\\mathrm{trace}}_{c_{K}}(x_{b})\\neq{\\mathrm{trace}}_{c_{L_{j}}}(x_{b}) for some 1 ≤ b ≤ t 1\\leq b\\leq t ."},{"type":"para","text":"Now, let z z be the smallest index in the collection for which L z = K L_{z}=K . We claim that the index i t i_{t} output by the algorithm converges to z z . To see this, note that the index z z is always a valid candidate for the algorithm to output, since every string in the input is from L z = K L_{z}=K , and furthermore, c L z c_{L_{z}} and c K c_{K} are identical. Now, consider any j < z j<z for which L j ⊉ L z L_{j}\\nsupseteq L_{z} . For each such L j L_{j} , there exists some x ∈ L z ∖ L j x\\in L_{z}\\setminus L_{j} which shows up in the input at some finite time. At this time, the language L j L_{j} becomes inconsistent. Since there are finitely many j < z j<z , we have that beyond some finite time, every L j L_{j} satisfying j < z j<z and L j ⊉ L z L_{j}\\nsupseteq L_{z} is rendered inconsistent by the algorithm. So, we restrict our attention to languages L j L_{j} that satisfy j < z j<z and L j ⊋ L z L_{j}\\supsetneq L_{z} (note that L j L_{j} cannot equal L z L_{z} , by our choice of z z ). Fix any such L j L_{j} , and consider any w ∈ L j ∖ L z w\\in L_{j}\\setminus L_{z} . By the assumed reject-extendability property, there exists some suffix y ∈ Σ ∗ y\\in\\Sigma^{*} such that w y ∈ L z wy\\in L_{z} ; furthermore, this string w y wy is guaranteed to show up in the input at some finite time. But now, notice that c K ( w ) = c L z ( w ) = 0 c_{K}(w)=c_{L_{z}}(w)=0 , but c L j ( w ) = 1 c_{L_{j}}(w)=1 . Thus, trace c L j ( w y ) ≠ trace c K ( w y ) {\\mathrm{trace}}_{c_{L_{j}}}(wy)\\neq{\\mathrm{trace}}_{c_{K}}(wy) , and hence, the algorithm will declare L j L_{j} to have an inconsistent color trace on w y wy . In this manner, every proper superset of L z L_{z} occurring before it gets invalidated at some finite time, and we conclude that the algorithm’s guess eventually converges to z z ."},{"type":"para","text":"The sufficient condition above for which the accept-reject coloring works may be deemed to be fairly weak: it only stipulates that every rejected string may be extended in some way to acceptance. Could it not be necessary? Namely, could every language collection be identifiable with the simple accept-reject coloring? The following example shows that this is not the case."},{"type":"para","text":"Consider a collection 𝒞 \\mathcal{C} over the binary alphabet Σ = { 0 , 1 } \\Sigma=\\{0,1\\} , which comprises of the regular language L ∞ = { 0 , 1 } ∗ L_{\\infty}=\\{0,1\\}^{*} , together with the regular languages L i = { 0 , 1 } ∗ ∖ 1 i { 0 , 1 } ∗ L_{i}=\\{0,1\\}^{*}\\setminus 1^{i}\\{0,1\\}^{*} for i ≥ 1 i\\geq 1 . Namely, L i L_{i} excludes strings in { 0 , 1 } ∗ \\{0,1\\}^{*} that start with i i ones. We have that for every i ≥ 1 i\\geq 1 , L i ⊊ L i + 1 L_{i}\\subsetneq L_{i+1} , L i ⊊ L ∞ L_{i}\\subsetneq L_{\\infty} , and furthermore, ∪ i L i = L ∞ \\cup_{i}L_{i}=L_{\\infty} . Observe that for any i ∈ ℕ ∪ { ∞ } i\\in\\mathds{N}\\cup\\{\\infty\\} and x ∈ L i x\\in L_{i} , the accept-reject coloring satisfies trace c L i ( x ) = ( 1 , 1 , 1 , … , 1 ) ⏟ | x | + 1 times {\\mathrm{trace}}_{c_{L_{i}}}(x)=\\underbrace{(1,1,1,\\dots,1)}_{|x|+1\\text{ times}} . This is because each L i L_{i} is prefix-closed: every prefix of x ∈ L i x\\in L_{i} is also contained in L i L_{i} . To see this, note that any x ∈ L i x\\in L_{i} does not start with 1 i 1^{i} , and hence no prefix of x x starts with 1 i 1^{i} as well. In particular, no string ever has 0 in its color trace. Furthermore, for any i > j i>j , trace c L i ( x ) = trace c L j ( x ) {\\mathrm{trace}}_{c_{L_{i}}}(x)={\\mathrm{trace}}_{c_{L_{j}}}(x) for every x ∈ L i x\\in L_{i} ."},{"type":"para","text":"Now, fix any identification algorithm, and consider an adversary that starts enumerating the strings in L 1 L_{1} in the order that they appear in a fixed background ordering of L ∞ L_{\\infty} , together with their color traces (according to c L 1 c_{L_{1}} , which is always the constant-ones trace). Then, at some finite time, the algorithm must guess L 1 L_{1} to be the target language. At this time, the adversary switches to enumerating the strings in L 2 L_{2} , also in the fixed background ordering of L ∞ L_{\\infty} , and starting from the leftmost string in L 2 L_{2} that has not been enumerated as yet. The adversary continues giving color traces according to L 2 L_{2} , which they can legally do, since L 1 ⊊ L 2 L_{1}\\subsetneq L_{2} , and the traces of L 2 L_{2} and L 1 L_{1} matched on all strings in L 1 L_{1} . Now, the algorithm must guess the target language to be L 2 L_{2} at some subsequent finite time, at which time the adversary switches to L 3 L_{3} . The adversary repeats this switching process endlessly, enumerating each L i L_{i} for a finite phase. As the adversary begins enumerating L i L_{i} from the leftmost string that has not yet been enumerated in the fixed background ordering of L ∞ L_{\\infty} , the adversary eventually enumerates every string in L ∞ L_{\\infty} , together also with its correct color trace according to c L ∞ c_{L_{\\infty}} . But the algorithm guesses an incorrect language infinitely often, and hence does not identify L ∞ L_{\\infty} in the limit."},{"type":"para","text":"The structure that enabled the adversarial strategy in \\Cref example:accept-reject-failure distills precisely into the characterizing condition of \\Cref thm:characterization-identification-with-traces. Indeed, one direction of the characterization is a generalization of the adversarial strategy employed above. In the other direction, we appropriately modify the algorithm used in \\Cref prop:reject-extendability-sufficiency to account for trace discrepancies, and tell-tales associated with languages."},{"type":"heading","level":1,"text":"3 The Coloring Lemma"},{"type":"para","text":"In this section, we restate and prove our main coloring lemma. We also discuss an extension to the setting where the color traces observed have a bounded number of corruptions. See 1.2"},{"type":"para","text":"For any language L L , define the set Pref ( L ) \\mathrm{Pref}(L) to be the set of all strings in Σ ∗ \\Sigma^{*} that can be extended to form a string contained in L L . Namely,"},{"type":"para","text":"Now, for any L ∈ 𝒞 L\\in\\mathcal{C} , define IsAccept L : Σ ∗ → { 0 , 1 } \\mathrm{IsAccept}_{L}:\\Sigma^{*}\\to\\{0,1\\} and NextLive L : Σ ∗ → { 0 , 1 , … , k } \\mathrm{NextLive}_{L}:\\Sigma^{*}\\to\\{0,1,\\dots,k\\} as:"},{"type":"para","text":"In words, IsAccept L ( x ) \\mathrm{IsAccept}_{L}(x) is the indicator for x x being in L L , whereas NextLive L ( x ) \\mathrm{NextLive}_{L}(x) counts the number of letters a ∈ Σ a\\in\\Sigma such that x a xa can be further extended to form a string contained in L L . Now fix the palette P = { 1 , 2 , … , k + 1 } P=\\{1,2,\\dots,k+1\\} , and define the trace coloring function c L : Σ ∗ → P c_{L}:\\Sigma^{*}\\to P as follows:"},{"type":"para","text":"If x ∉ Pref ( L ) x\\notin\\mathrm{Pref}(L) , c L ( x ) = 1 c_{L}(x)=1 ; otherwise, if x ∈ Pref ( L ) x\\in\\mathrm{Pref}(L) , then either IsAccept L ( x ) = 1 \\mathrm{IsAccept}_{L}(x)=1 , or NextLive L ( x ) > 0 \\mathrm{NextLive}_{L}(x)>0 ; hence c L ( x ) ≥ 1 c_{L}(x)\\geq 1 . Therefore, c L ( x ) ∈ { 1 , 2 , … , k + 1 } c_{L}(x)\\in\\{1,2,\\dots,k+1\\} for all x ∈ Σ ∗ x\\in\\Sigma^{*} . We now observe that the functions c L c_{L} satisfy monotonicity on any string x x with respect to inclusion."},{"type":"para","text":"For any x ∈ Σ ∗ x\\in\\Sigma^{*} , and L ⊆ L ′ L\\subseteq L^{\\prime} , it holds that"},{"type":"para","text":"Now, fix any L , L ′ ∈ 𝒞 L,L^{\\prime}\\in\\mathcal{C} for which L ⊊ L ′ L\\subsetneq L^{\\prime} , and fix any z ∈ L ′ ∖ L z\\in L^{\\prime}\\setminus L ."},{"type":"heading","level":4,"text":"Case 1: z ∈ Pref ( L ) z\\in\\mathrm{Pref}(L) ."},{"type":"para","text":"In this case, let t ∈ Σ ∗ t\\in\\Sigma^{*} be such that z t ∈ L zt\\in L (note that t ≠ ε t\\neq\\varepsilon , since z ∉ L z\\notin L ). We claim that x := z t x:=zt satisfies trace c L ( x ) ≠ trace c L ′ ( x ) {\\mathrm{trace}}_{c_{L}}(x)\\neq{\\mathrm{trace}}_{c_{L^{\\prime}}}(x) . To see this, note that IsAccept L ( z ) = 0 \\mathrm{IsAccept}_{L}(z)=0 , but IsAccept L ′ ( z ) = 1 \\mathrm{IsAccept}_{L^{\\prime}}(z)=1 ; together with \\Cref observation:monotonicity-trace-coloring, this implies that c L ( z ) < c L ′ ( z ) c_{L}(z)<c_{L^{\\prime}}(z) ."},{"type":"heading","level":4,"text":"Case 2: z ∉ Pref ( L ) z\\notin\\mathrm{Pref}(L) ."},{"type":"para","text":"In this case, let w ∈ Σ ∗ w\\in\\Sigma^{*} be the maximal prefix of z z that satisfies w ∈ Pref ( L ) w\\in\\mathrm{Pref}(L) , and let z = w a t z=wat , where a ∈ Σ a\\in\\Sigma , and t ∈ Σ ∗ t\\in\\Sigma^{*} . Here, by maximal, we mean that w a ∉ Pref ( L ) wa\\notin\\mathrm{Pref}(L) . Note that w w can be ε \\varepsilon , but its existence is guaranteed since L L is non-empty."},{"type":"para","text":"Now, since w ∈ Pref ( L ) w\\in\\mathrm{Pref}(L) , there exists y ∈ Σ ∗ y\\in\\Sigma^{*} such that w y ∈ L wy\\in L . We claim that x := w y x:=wy satisfies that trace c L ( x ) ≠ trace c L ′ ( x ) {\\mathrm{trace}}_{c_{L}}(x)\\neq{\\mathrm{trace}}_{c_{L^{\\prime}}}(x) . To see this, observe that w a ∉ Pref ( L ) wa\\notin\\mathrm{Pref}(L) , since w w was chosen to be the maximal prefix of z z satisfying w ∈ Pref ( L ) w\\in\\mathrm{Pref}(L) . However, w a ∈ Pref ( L ′ ) wa\\in\\mathrm{Pref}(L^{\\prime}) , since z = w a t ∈ L ′ ∖ L z=wat\\in L^{\\prime}\\setminus L . Thus, NextLive L ( w ) < NextLive L ′ ( w ) \\mathrm{NextLive}_{L}(w)<\\mathrm{NextLive}_{L^{\\prime}}(w) . By \\Cref observation:monotonicity-trace-coloring, this implies that c L ( w ) < c L ′ ( w ) c_{L}(w)<c_{L^{\\prime}}(w) ."},{"type":"para","text":"In both cases, we have shown the existence of an x ∈ L x\\in L satisfying trace c L ( x ) ≠ trace c L ′ ( x ) {\\mathrm{trace}}_{c_{L}}(x)\\neq{\\mathrm{trace}}_{c_{L^{\\prime}}}(x) . The next claim shows that at least for k = 2 k=2 , the bound achieved by our construction is tight."},{"type":"para","text":"There exists a finite collection 𝒞 \\mathcal{C} of non-empty languages over the binary alphabet Σ = { 0 , 1 } \\Sigma=\\{0,1\\} , such that any family of trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} that satisfies the distinguishable coloring condition must necessarily use 3 3 colors."},{"type":"para","text":"Consider the finite collection 𝒞 = { L 1 , L 2 , L 3 , L 4 , L 5 } \\mathcal{C}=\\{L_{1},L_{2},L_{3},L_{4},L_{5}\\} , where"},{"type":"para","text":"Fix any trace coloring functions c L 1 , c L 2 , c L 3 , c L 4 , c L 5 c_{L_{1}},c_{L_{2}},c_{L_{3}},c_{L_{4}},c_{L_{5}} that map to the palette { red , blue } \\{\\mathrm{red},\\mathrm{blue}\\} , and satisfy the distinguishable coloring condition."},{"type":"para","text":"Since L 1 L_{1} only contains ε \\varepsilon , it must hold that c L 1 ( ε ) ≠ c L 2 ( ε ) = c L 3 ( ε ) = c L 4 ( ε ) = c L 5 ( ε ) c_{L_{1}}(\\varepsilon)\\neq c_{L_{2}}(\\varepsilon)=c_{L_{3}}(\\varepsilon)=c_{L_{4}}(\\varepsilon)=c_{L_{5}}(\\varepsilon) . Without loss of generality, let c L 1 ( ε ) = red c_{L_{1}}(\\varepsilon)=\\mathrm{red} , so that c L 2 ( ε ) = c L 3 ( ε ) = c L 4 ( ε ) = c L 5 ( ε ) = blue c_{L_{2}}(\\varepsilon)=c_{L_{3}}(\\varepsilon)=c_{L_{4}}(\\varepsilon)=c_{L_{5}}(\\varepsilon)=\\mathrm{blue} ."},{"type":"para","text":"Then, for L 2 L_{2} , it must hold that c L 2 ( 0 ) ≠ c L 4 ( 0 ) = c L 5 ( 0 ) c_{L_{2}}(0)\\neq c_{L_{4}}(0)=c_{L_{5}}(0) . Similarly, for L 3 L_{3} , it must be the case that c L 3 ( 1 ) ≠ c L 4 ( 1 ) = c L 5 ( 1 ) c_{L_{3}}(1)\\neq c_{L_{4}}(1)=c_{L_{5}}(1) . Without loss of generality, suppose c L 2 ( 0 ) = red c_{L_{2}}(0)=\\mathrm{red} and c L 3 ( 1 ) = red c_{L_{3}}(1)=\\mathrm{red} , so that c L 4 ( 0 ) = c L 5 ( 0 ) = blue c_{L_{4}}(0)=c_{L_{5}}(0)=\\mathrm{blue} , and c L 4 ( 1 ) = c L 5 ( 1 ) = blue c_{L_{4}}(1)=c_{L_{5}}(1)=\\mathrm{blue} ."},{"type":"para","text":"But now, consider the traces of L 4 L_{4} . By the previous assignments, we have that"},{"type":"para","text":"So, even though L 4 ⊊ L 5 L_{4}\\subsetneq L_{5} , every x ∈ L 4 x\\in L_{4} satisfies trace c L 4 ( x ) = trace c L 5 ( x ) {\\mathrm{trace}}_{c_{L_{4}}}(x)={\\mathrm{trace}}_{c_{L_{5}}}(x) . This contradicts the trace coloring functions satisfying the distinguishable coloring condition."},{"type":"para","text":"We remark that the lower bound above does not, in general, preclude identifiability in the limit with trace coloring functions that use only 2 colors. Rather, it merely shows that the distinguishable coloring condition may not be achieved for all collections using 2 colors. Indeed, the collection above is finite, and every finite collection is identifiable without any color traces (Angluin, 1980 ) ."},{"type":"heading","level":2,"text":"3.1 Extension to Corrupted Traces"},{"type":"para","text":"We also obtain the guarantee of \\Cref lemma:main-coloring-upper-bound in the setting where the color traces seen may be corrupted . Formally, we consider a notion of corrupted traces similar to Peng et al. ( 2026 ) , where the trace seen along with an input string x x is trace ~ c K ( x ) \\widetilde{{\\mathrm{trace}}}_{c_{K}}(x) , a corrupted version of trace c K ( x ) {\\mathrm{trace}}_{c_{K}}(x) . We restrict the nature of allowed corruptions as follows. For any x x , we require | trace ~ c K ( x ) | = | trace c K ( x ) | |\\widetilde{{\\mathrm{trace}}}_{c_{K}}(x)|=|{\\mathrm{trace}}_{c_{K}}(x)| (i.e., the corrupted trace must have the same length as the uncorrupted trace), and that every element in trace ~ c K ( x ) \\widetilde{{\\mathrm{trace}}}_{c_{K}}(x) belongs to the same palette P P used by c K c_{K} (i.e., the corrupted trace cannot introduce colors outside the palette). With these restrictions, we consider traces containing a bounded number of corruptions, as measured by the Hamming distance between the corrupted and uncorrupted traces. Namely, if trace c K ( x ) = ( a 1 , a 2 , … , a n ) {\\mathrm{trace}}_{c_{K}}(x)=(a_{1},a_{2},\\dots,a_{n}) and trace ~ c K ( x ) = ( b 1 , b 2 , … , b n ) \\widetilde{{\\mathrm{trace}}}_{c_{K}}(x)=(b_{1},b_{2},\\dots,b_{n}) , we restrict d ( a , b ) = | { i : a i ≠ b i , 1 ≤ i ≤ n } | ≤ ℓ d(a,b)=|\\{i:a_{i}\\neq b_{i},1\\leq i\\leq n\\}|\\leq\\ell , where ℓ > 0 \\ell>0 is an a priori known corruption budget."},{"type":"para","text":"In this setting, we wish to correctly identify the target language K K in the limit for any target language K K , and for any enumeration ( x 1 , trace ~ c K ( x 1 ) ) , ( x 2 , trace ~ c K ( x 2 ) ) , … (x_{1},\\widetilde{{\\mathrm{trace}}}_{c_{K}}(x_{1})),(x_{2},\\widetilde{{\\mathrm{trace}}}_{c_{K}}(x_{2})),\\dots of K K with corrupted traces satisfying a corrupted budget ℓ \\ell . From the proof of \\Cref thm:characterization-identification-with-traces, it is straightforward to verify that a sufficient condition for identifiability of a collection 𝒞 \\mathcal{C} in this setting is the following: for every L ∈ 𝒞 L\\in\\mathcal{C} , there exists a finite tell-tale T L ⊆ L T_{L}\\subseteq L , such that for every L ′ ∈ 𝒞 L^{\\prime}\\in\\mathcal{C} satisfying L ′ ⊊ L L^{\\prime}\\subsetneq L , either (1) L ′ L^{\\prime} does not contain T L T_{L} , or (2) there exists x ∈ L ′ x\\in L^{\\prime} such that d ( trace c L ′ ( x ) , trace c L ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L^{\\prime}}}(x),{\\mathrm{trace}}_{c_{L}}(x)\\right)>2\\ell ."},{"type":"para","text":"Then, just like \\Cref lemma:main-coloring-upper-bound, we seek trace coloring functions that always satisfy (2) above. This will guarantee identifiability with corrupted traces having at most ℓ \\ell corruptions. Indeed, the next lemma constructs trace coloring functions that always satisfy (2) with a palette of size O ( k 2 ℓ + 1 ) O(k^{2\\ell+1}) ."},{"type":"para","text":"Let 𝒞 \\mathcal{C} be a language collection comprising of non-empty languages over a finite alphabet Σ \\Sigma of size k k , and fix ℓ > 0 \\ell>0 . There exist trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} mapping to a palette P P of size max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) \\max(8\\ell+4,8k^{2\\ell+1}) , such that for any L , L ′ ∈ 𝒞 L,L^{\\prime}\\in\\mathcal{C} that satisfy L ⊊ L ′ L\\subsetneq L^{\\prime} , and L L contains some string x x having length at least 2 ℓ 2\\ell , there exists x ∈ L x\\in L for which d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell ."},{"type":"para","text":"The proof of this lemma is given in \\Cref sec:robust-coloring-appendix. The high-level idea is to propagate the discrepancy at a single location implied by the construction in \\Cref lemma:main-coloring-upper-bound to all the 2 ℓ 2\\ell locations preceding it."},{"type":"heading","level":1,"text":"4 Traces for Regular Languages"},{"type":"para","text":"The trace coloring functions that we constructed in \\Cref sec:main-coloring satisfy a special property for regular languages. Consider any regular language L L , and let M M be any Deterministic Finite-State Automaton (DFA) 3 3 3 We give a formal definition of a DFA in \\Cref sec:state-traces-appendix. that recognizes L L . Let Q ( M ) Q(M) be its set of states, δ M \\delta_{M} be the associated state transition function, q 0 ( M ) q_{0}(M) be its initial state and F ( M ) ⊆ Q ( M ) F(M)\\subseteq Q(M) its accepting states. For any state q q in Q ( M ) Q(M) , let W q W_{q} be the set of all the strings that arrive at state q q , starting from the initial state."},{"type":"para","text":"Recall the definition of our trace coloring function c L c_{L} from ( 6 ). We claim that for any state q q , for any w , w ′ ∈ W q w,w^{\\prime}\\in W_{q} , c L ( w ) = c L ( w ′ ) c_{L}(w)=c_{L}(w^{\\prime}) . To see this, note that since w w and w ′ w^{\\prime} both end up at the state q q , for any suffix y y , w y wy and w ′ y w^{\\prime}y also end up in the same state in Q ( M ) Q(M) . This implies that w ∈ Pref ( L ) ⇔ w ′ ∈ Pref ( L ) w\\in\\mathrm{Pref}(L)\\iff w^{\\prime}\\in\\mathrm{Pref}(L) , and also that IsAccept L ( w ) = IsAccept L ( w ′ ) , NextLive L ( w ) = NextLive L ( w ′ ) \\mathrm{IsAccept}_{L}(w)=\\mathrm{IsAccept}_{L}(w^{\\prime}),\\mathrm{NextLive}_{L}(w)=\\mathrm{NextLive}_{L}(w^{\\prime}) . We conclude that c L ( w ) = c L ( w ′ ) c_{L}(w)=c_{L}(w^{\\prime}) ."},{"type":"para","text":"We now restate and prove \\Cref lemma:state-coloring-optimal-bound, which constructs state coloring functions using k + 1 k+1 colors so as to satisfy the distinguishable coloring condition, and also show that this palette size is optimal."},{"type":"para","text":"See 1.4"},{"type":"para","text":"For any state q ∈ Q ( M ) q\\in Q(M) , we say that q q is a live state if there exists a suffix y ∈ Σ ∗ y\\in\\Sigma^{*} such that δ M ( q , y ) ∈ F ( M ) \\delta_{M}(q,y)\\in F(M) ; otherwise, we call q q a dead state. Now, similar to the definitions in ( 4 ) and ( 5 ), define IsAccept M : Q ( M ) → { 0 , 1 } \\mathrm{IsAccept}_{M}:Q(M)\\to\\{0,1\\} and NextLive M : Q ( M ) → { 0 , 1 , … , k } \\mathrm{NextLive}_{M}:Q(M)\\to\\{0,1,\\dots,k\\} as:"},{"type":"para","text":"Then, define the state coloring function c M : Q ( M ) → P c_{M}:Q(M)\\to P as follows:"},{"type":"para","text":"We can then verify that for any x = x 0 x 1 … x n ∈ L ( M ) x=x_{0}x_{1}\\dots x_{n}\\in L(M) (where x 0 = ε x_{0}=\\varepsilon ) that traverses the states q 0 , q 1 , … , q n q_{0},q_{1},\\dots,q_{n} in M M , and for any q i q_{i} , it holds that c M ( q i ) = c L ( M ) ( x 0 x 1 … x i ) c_{M}(q_{i})=c_{L(M)}(x_{0}x_{1}\\dots x_{i}) , where c L ( M ) c_{L(M)} is exactly the trace coloring function defined in ( 6 ) in the proof of \\Cref lemma:main-coloring-upper-bound. The conclusion of that lemma implies that { c M } M ∈ 𝒞 \\{c_{M}\\}_{M\\in\\mathcal{C}} satisfy the required distinguishable coloring condition."},{"type":"para","text":"We now show that k + 1 k+1 colors are optimal. Fix k ≥ 1 k\\geq 1 , and consider 𝒞 = { M 0 , M 1 , … , M k } \\mathcal{C}=\\{M_{0},M_{1},\\dots,M_{k}\\} . Each M i M_{i} has just two states q 0 ( M i ) q_{0}(M_{i}) and q 1 ( M i ) q_{1}(M_{i}) , where q 0 ( M i ) q_{0}(M_{i}) is the initial state, and also the only accepting state, and q 1 ( M i ) q_{1}(M_{i}) is a rejecting sink state. Namely, ∀ i ∀ a : \\forall i\\forall a: δ M i ( q 1 ( M i ) , a ) = q 1 ( M i ) \\delta_{M_{i}}(q_{1}(M_{i}),a)=q_{1}(M_{i}) ."},{"type":"para","text":"Now, in M 0 M_{0} , we further set δ M 0 ( q 0 ( M 0 ) , a ) = q 1 ( M 0 ) \\delta_{M_{0}}(q_{0}(M_{0}),a)=q_{1}(M_{0}) for every a ∈ Σ a\\in\\Sigma . So, it holds that L ( M 0 ) = { ε } L(M_{0})=\\{\\varepsilon\\} . Next, for i ∈ { 1 , … , k } i\\in\\{1,\\dots,k\\} , we set"},{"type":"para","text":"Thus, L ( M i ) = { 1 , … , i } ∗ L(M_{i})=\\{1,\\dots,i\\}^{*} . Observe that L ( M 0 ) ⊊ L ( M 1 ) ⊊ ⋯ ⊊ L ( M k ) L(M_{0})\\subsetneq L(M_{1})\\subsetneq\\dots\\subsetneq L(M_{k}) ."},{"type":"para","text":"Now, fix any state coloring functions c M 0 , c M 1 , … , c M k c_{M_{0}},c_{M_{1}},\\dots,c_{M_{k}} that use a palette of size k k , and assume that they satisfy: for every M i , M j ∈ 𝒞 M_{i},M_{j}\\in\\mathcal{C} with L ( M i ) ⊊ L ( M j ) L(M_{i})\\subsetneq L(M_{j}) , there exists x ∈ L ( M i ) x\\in L(M_{i}) for which trace c M i ( x ) ≠ trace c M j ( x ) {\\mathrm{trace}}_{c_{M_{i}}}(x)\\neq{\\mathrm{trace}}_{c_{M_{j}}}(x) . Fix any i , j ∈ { 0 , 1 , … , k } i,j\\in\\{0,1,\\dots,k\\} satisfying i < j i<j . Observe that any x ∈ L ( M i ) x\\in L(M_{i}) simply loops on the state q 0 ( M i ) q_{0}(M_{i}) , giving trace c M i ( x ) = ( c M i ( q 0 ( M i ) ) , … , c M i ( q 0 ( M i ) ) ) ⏟ | x | + 1 times {\\mathrm{trace}}_{c_{M_{i}}}(x)=\\underbrace{(c_{M_{i}}(q_{0}(M_{i})),\\dots,c_{M_{i}}(q_{0}(M_{i})))}_{|x|+1\\text{ times}} . Thus, for the assumed guarantee to hold, it must be the case that for every i < j i<j , c M i ( q 0 ( M i ) ) ≠ c M j ( q 0 ( M j ) ) c_{M_{i}}(q_{0}(M_{i}))\\neq c_{M_{j}}(q_{0}(M_{j})) . But this is not possible if the state coloring functions use only k k colors."},{"type":"para","text":"In \\Cref sec:corrupted-state-traces-appendix, we extend the construction of the state coloring functions above to the setting with corrupted traces, similar to \\Cref sec:corruptions. Here, we show that a palette of size O ( k 2 ℓ + 1 ) O(k^{2\\ell+1}) suffices to achieve a similar guarantee as \\Cref lemma:corruptions-coloring-upper-bound, and that k 2 ℓ − 1 k^{2\\ell-1} colors are necessary in general."},{"type":"para","text":"Finally, we restate and prove \\Cref thm:regular-languages-2-colors. See 1.5"},{"type":"para","text":"Before stating the formal proof, we give some intuition, specifically since the requirement that every L i L_{i} be infinite might appear counterintuitive at first glance for this positive result. The trace coloring functions that we build interestingly arise via state coloring functions. That is, we carefully choose a sequence of automata M 1 , M 2 , … M_{1},M_{2},\\dots that accept L 1 , L 2 , … L_{1},L_{2},\\dots , prescribe state coloring functions for these automata, and then directly argue that they achieve the goal of identification with just two colors. This is in contrast to \\Cref lemma:state-coloring-optimal-bound, where the automata were fixed and provided to us, and we were concerned with the (stronger) distinguishable coloring condition; here, we choose suitable automata as a tool for constructing trace coloring functions for identification."},{"type":"para","text":"While there are several automata that recognize L i L_{i} , we want to choose M i M_{i} with a view to satisfying the following property: for each previous automaton M j M_{j} where j < i j<i and L j L_{j} is a superset of L i L_{i} , we can designate a particular state in M i M_{i} with a particular color, which helps distinguish it from M j M_{j} . In order to choose such a designated state for every j < i j<i , we need enough states (at least i − 1 i-1 ) in L i L_{i} to begin with. A sufficient condition to ensure this is that every L i L_{i} is infinite (intuitively, since there are long enough strings in the language, these must cause a loop, which can be “disentangled” to enlarge the state space). This ensures that with just two colors, we can distinguish the color traces of every language L i L_{i} from the color traces of every superset of L i L_{i} that appears before L i L_{i} . Note that this is a weaker property than the distinguishable coloring condition (which requires such a property for all supersets in the collection, not just the ones appearing before), but is sufficient for identification in the limit (see \\Cref remark:weaker-condition-sufficient)."},{"type":"para","text":"We now give the formal proof."},{"type":"para","text":"Fix M 1 , M 2 , … M_{1},M_{2},\\dots to be any arbitrary DFAs that recognize L 1 , L 2 , … L_{1},L_{2},\\dots . Now recall the definition of a live state in a DFA: a state q ∈ Q ( M ) q\\in Q(M) is live if ∃ \\exists a suffix y ∈ Σ ∗ y\\in\\Sigma^{*} such that δ M ( q , y ) ∈ F ( M ) \\delta_{M}(q,y)\\in F(M) . Now call a state q q reachable if ∃ w ∈ Σ ∗ \\exists w\\in\\Sigma^{*} such that δ M ( q 0 ( M ) , w ) = q \\delta_{M}(q_{0}(M),w)=q . By \\Cref lemma:increasing-live-states proved in \\Cref sec:state-traces-appendix, we can assume that every M i M_{i} has at least i − 1 i-1 live and reachable states. We will now inductively determine a binary state coloring function c M i c_{M_{i}} for every i ≥ 1 i\\geq 1 ."},{"type":"para","text":"To begin with, fix c M 1 c_{M_{1}} arbitrarily. Now, for any i > 1 i>1 , let us assume that c M 1 , … , c M i − 1 c_{M_{1}},\\dots,c_{M_{i-1}} have been determined. Consider M i M_{i} : there are at least i − 1 i-1 live and reachable states in M i M_{i} . Pick any i − 1 i-1 of these, and denote the set as { q b } b = 1 , … , i − 1 \\{q_{b}\\}_{b=1,\\dots,i-1} . Associate each q b q_{b} with a prefix w b ∈ Σ ∗ w_{b}\\in\\Sigma^{*} and a suffix y b ∈ Σ ∗ y_{b}\\in\\Sigma^{*} , such that δ M i ( q 0 ( M i ) , w b ) = q b \\delta_{M_{i}}(q_{0}(M_{i}),w_{b})=q_{b} , and δ M i ( q b , y b ) ∈ F ( M i ) \\delta_{M_{i}}(q_{b},y_{b})\\in F(M_{i}) , so that x b = w b y b ∈ L i x_{b}=w_{b}y_{b}\\in L_{i} . Now, consider the set { j < i : L j ⊋ L i } \\{j<i:L_{j}\\supsetneq L_{i}\\} : this set has size at most i − 1 i-1 . Thus, we can map every j j in this set to a distinct live q b q_{b} in M i M_{i} . Consider traversing the prefix w b w_{b} associated with q b q_{b} in M j M_{j} , and let q b ′ = δ M j ( q 0 ( M j ) , w b ) q^{\\prime}_{b}=\\delta_{M_{j}}(q_{0}(M_{j}),w_{b}) . We will set c M i ( q b ) = 1 − c M j ( q b ′ ) c_{M_{i}}(q_{b})=1-c_{M_{j}}(q^{\\prime}_{b}) . After doing this for every j j , we assign c M i ( q ) c_{M_{i}}(q) for the remaining states in M i M_{i} arbitrarily. This completes the specification of c M i ( q ) c_{M_{i}}(q) ."},{"type":"para","text":"Now consider any i ≥ 1 i\\geq 1 . We claim that: ( ⋆ ) (\\star) for every j < i j<i , if L j ⊋ L i L_{j}\\supsetneq L_{i} , then there exists x x in L i L_{i} for which trace c M i ( x ) ≠ trace c M j ( x ) {\\mathrm{trace}}_{c_{M_{i}}}(x)\\neq{\\mathrm{trace}}_{c_{M_{j}}}(x) . Indeed, fix any such L j L_{j} , and let q b q_{b} be the distinct live state in M i M_{i} that we had associated L j L_{j} with in the inductive construction of c M i c_{M_{i}} above. Now consider the string x b = w b y b ∈ L i x_{b}=w_{b}y_{b}\\in L_{i} associated with the state q b q_{b} in M i M_{i} . Note that the prefix w b w_{b} reaches the state q b ′ = δ M j ( q 0 ( M j ) , w b ) q^{\\prime}_{b}=\\delta_{M_{j}}(q_{0}(M_{j}),w_{b}) in M j M_{j} , whereas it reaches the state q b q_{b} in M i M_{i} ; our construction above guarantees that c M i ( q b ) = 1 − c M j ( q b ′ ) c_{M_{i}}(q_{b})=1-c_{M_{j}}(q^{\\prime}_{b}) . Thus, trace c M i ( x b ) ≠ trace c M j ( x b ) {\\mathrm{trace}}_{c_{M_{i}}}(x_{b})\\neq{\\mathrm{trace}}_{c_{M_{j}}}(x_{b}) , as claimed."},{"type":"para","text":"Finally, by \\Cref remark:weaker-condition-sufficient in the proof of \\Cref thm:characterization-identification-with-traces, the property ( ⋆ ) (\\star) is sufficient for the algorithm considered in that proof to identify the collection in the limit with traces."},{"type":"heading","level":1,"text":"5 Conclusion and Discussion"},{"type":"para","text":"We studied the problem of language identification with coarse-grained annotations, independent of any underlying machine implementation of the target language. Our main result shows that every countable language collection can be identified in the limit with annotations the same size as the input strings, with alphabet size one more than the input alphabet size."},{"type":"para","text":"One open problem is to prove a lower bound on the number of colors required for language identification with a color trace. For regular language collections, our result shows that two colors always suffice, independent of their alphabet size k k . Do constantly many colors always suffice for any arbitrary collection? Or must the number of colors grow linearly with k k ? Another open problem is to establish a lower bound on the number of colors required to achieve the distinguishable coloring condition — going beyond our lower bound of 3 colors seems challenging."},{"type":"para","text":"Regarding the question about the number of colors necessary for identification, very recently, we have been able to show that the result of \\Cref thm:regular-languages-2-colors, that trace coloring functions with just two colors suffice for identifying collections of regular languages that are all infinite, can in fact be generalized to collections of arbitrary languages that are all infinite. In fact, while trace coloring functions color every character in each string, this strengthened result can even be obtained with “terminal” coloring functions, where one only needs to assign a color to the entire string at once."},{"type":"para","text":"However, the terminal coloring functions for each language in this 2-coloring result are collection-dependent (as are the trace coloring functions from \\Cref thm:regular-languages-2-colors). Note, in contrast, that our general-purpose trace-coloring functions from \\Cref thm:main-theorem which use k + 1 k+1 colors, are collection-independent . This raises the natural question: do there exist collection-independent terminal coloring functions that use only two colors, and suffice for identification in the limit for all countable language collections (possibly with the restriction that every language is infinite)?"},{"type":"para","text":"It turns out that the answer to this question is yes; however, the required 2-coloring functions turn out to be highly non-constructive (and, we can prove, necessarily so in a precise sense), involving reasoning with uncountable ordinals and transfinite recursion. As a result, this new result is in a sense not directly comparable to the general trace coloring result in \\Cref thm:main-theorem of the present paper: the new result is quantitatively stronger (using 2 colors instead of k + 1 k+1 ), but it markedly lacks the constructive, natural formulation of \\Cref thm:main-theorem. Because the premise, techniques and analyses used to obtain these results largely depart from the present results in this paper, we defer them to a different manuscript (Charikar et al., 2026 ) ."},{"type":"para","text":"This work was supported by Moses Charikar’s and Gregory Valiant’s Simons Investigator Awards, a Google PhD Fellowship, a Simons Collaboration grant and a grant from the MacArthur Foundation."},{"type":"para","text":"sectionappendix"},{"type":"heading","level":2,"text":"A.1 Characterization of Identification in the Limit with Color Traces"},{"type":"para","text":"See 1.1"},{"type":"heading","level":4,"text":"If direction."},{"type":"para","text":"Suppose that the stated condition holds for the collection 𝒞 = { L 1 , L 2 , … } \\mathcal{C}=\\{L_{1},L_{2},\\dots\\} and the associated trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} . Furthermore, let T L T_{L} be the finite tell-tale subset associated with language L L in the collection. Denote the unknown target language by K K , and consider the following identification algorithm: at time step t t , upon seeing ( x 1 , trace c K ( x 1 ) ) , … , ( x t , trace c K ( x t ) ) (x_{1},{\\mathrm{trace}}_{c_{K}}(x_{1})),\\dots,(x_{t},{\\mathrm{trace}}_{c_{K}}(x_{t})) , the algorithm outputs the smallest index i t i_{t} , which satisfies: (1) { x 1 , … , x t } ⊆ L i t \\{x_{1},\\dots,x_{t}\\}\\subseteq L_{i_{t}} , (2) T L i t ⊆ { x 1 , … , x t } T_{L_{i_{t}}}\\subseteq\\{x_{1},\\dots,x_{t}\\} , and (3) trace c L i t ( x j ) = trace c K ( x j ) {\\mathrm{trace}}_{c_{L_{i_{t}}}}(x_{j})={\\mathrm{trace}}_{c_{K}}(x_{j}) for every 1 ≤ j ≤ t 1\\leq j\\leq t . We remark that without the check in (3), the algorithm would be identical to Angluin’s algorithm for identification in the limit ( Angluin , 1980 ) ."},{"type":"para","text":"Let z z be any index in the collection for which L z = K L_{z}=K , and trace c L z ( x ) = trace c K ( x ) {\\mathrm{trace}}_{c_{L_{z}}}(x)={\\mathrm{trace}}_{c_{K}}(x) for every x ∈ K x\\in K . Note that such an index exists since K ∈ 𝒞 K\\in\\mathcal{C} . Note also that requirements (1) and (3) hold for index z z at all time steps. Furthermore, since T L z T_{L_{z}} is a finite subset of K K , and the input eventually enumerates every string in K K , there exists a finite time at which requirement (2) gets satisfied as well. Thus, beyond this time, z z is a valid candidate index for the algorithm to output. We proceed assuming that we are beyond this time."},{"type":"para","text":"Now, consider any j < z j<z for which L j ⊉ L z L_{j}\\nsupseteq L_{z} . For each such L j L_{j} , there exists some x ∈ L z ∖ L j x\\in L_{z}\\setminus L_{j} which is guaranteed to show up in the input at some finite time. At this time and beyond, index j j violates (1). Since there are only finitely many j < z j<z , we have that beyond some finite time, every j j satisfying j < z j<z and L j ⊉ L z L_{j}\\nsupseteq L_{z} violates (1). So, we assume being beyond this time as well, and next restrict our attention to indices j j that satisfy j < z j<z and L j ⊋ L z L_{j}\\supsetneq L_{z} . Suppose in one case that T L j ⊆ L z T_{L_{j}}\\subseteq L_{z} holds. Then, at some finite time, all of T L j T_{L_{j}} will show up in the input, and hence (2) will always be satisfied beyond this time by index j j . But now observe that we are in a position where L z L_{z} and L j L_{j} satisfy: L z ⊊ L j L_{z}\\subsetneq L_{j} and T L j ⊆ L z T_{L_{j}}\\subseteq L_{z} . By the condition in the theorem, it must then hold that there exists x ∈ L z x\\in L_{z} for which trace c L z ( x ) = trace c K ( x ) ≠ trace c L j ( x ) {\\mathrm{trace}}_{c_{L_{z}}}(x)={\\mathrm{trace}}_{c_{K}}(x)\\neq{\\mathrm{trace}}_{c_{L_{j}}}(x) . Such an x x is guaranteed to show up in the input at some finite time, together with trace c K ( x ) {\\mathrm{trace}}_{c_{K}}(x) ; at this time and beyond, index j j violates (3). In the other case, suppose that T L j ⊈ L z T_{L_{j}}\\nsubseteq L_{z} . In this case, at least one element in T L j T_{L_{j}} will never show up in the input, and hence (2) will never be satisfied by index j j . In all cases, we have shown that there exists a finite time beyond which index j j stops being a candidate for the algorithm. Since there are only finitely many j < z j<z , we conclude that the algorithm eventually always outputs some j ≤ z j\\leq z for which L j = L z = K L_{j}=L_{z}=K ."},{"type":"para","text":"We remark that the analysis of the algorithm requires only the following weaker condition on the the countable collection 𝒞 = { L 1 , L 2 , … } \\mathcal{C}=\\{L_{1},L_{2},\\dots\\} and trace coloring functions { c L j } j ∈ ℕ \\{c_{L_{j}}\\}_{j\\in\\mathds{N}} : for every language L j ∈ 𝒞 L_{j}\\in\\mathcal{C} , there exists a finite tell-tale subset T L j ⊆ L j T_{L_{j}}\\subseteq L_{j} such that for every language L i ∈ 𝒞 L_{i}\\in\\mathcal{C} that is a proper subset of L j L_{j} and i > j i>j , either (1) L i L_{i} does not contain T L j T_{L_{j}} , or (2) there exists x ∈ L i x\\in L_{i} such that trace c L i ( x ) ≠ trace c L j ( x ) {\\mathrm{trace}}_{c_{L_{i}}}(x)\\neq{\\mathrm{trace}}_{c_{L_{j}}}(x) ."},{"type":"heading","level":4,"text":"Only if direction."},{"type":"para","text":"Fix a set of trace coloring functions { c L } L ∈ 𝒞 \\{c_{L}\\}_{L\\in\\mathcal{C}} , and suppose that there exists a language L ∈ 𝒞 L\\in\\mathcal{C} such that for every finite T ⊆ L T\\subseteq L , there exists a language L ′ ∈ 𝒞 L^{\\prime}\\in\\mathcal{C} that is a proper subset of L L , such that L ′ L^{\\prime} contains T T and also trace c L ′ ( x ) = trace c L ( x ) {\\mathrm{trace}}_{c_{L^{\\prime}}}(x)={\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′ x\\in L^{\\prime} . We will show that the collection is not identifiable in the limit with color traces given by these trace coloring functions."},{"type":"para","text":"Towards this, fix any identification algorithm, and consider an adversary that starts enumerating the strings in L L , together with their color traces according to c L c_{L} , in the order that they appear in a fixed background ordering of L L . Then, at some finite time, the algorithm must guess an index i i such that L i = L L_{i}=L . Suppose that the input strings enumerated so far by the adversary comprise of the set T T . Then, by assumption, there exists a language L ′ ∈ 𝒞 L^{\\prime}\\in\\mathcal{C} that is a proper subset of L L , such that L ′ L^{\\prime} contains T T , and furthermore, trace c L ′ ( x ) = trace c L ( x ) {\\mathrm{trace}}_{c_{L^{\\prime}}}(x)={\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′ x\\in L^{\\prime} (including T T ). Thus, the adversary switches to enumerating strings in L ′ L^{\\prime} with their color traces (which they can legally do, since T ⊆ L ′ T\\subseteq L^{\\prime} , and the traces of L ′ L^{\\prime} and L L match on all strings in L ′ L^{\\prime} ). Now, the algorithm must guess an index i i satisfying L i = L ′ L_{i}=L^{\\prime} at some subsequent finite time. At this time, the adversary switches back to enumerating L L (they can do this since traces of L ′ L^{\\prime} and L L match at all strings in L ′ L^{\\prime} ), starting from the leftmost string that has not yet been enumerated in the fixed background ordering of L L . Again, at some subsequent finite time, the algorithm must guess an index i i such that L i = L L_{i}=L . Suppose that the input strings enumerated so far by the adversary comprise of the set T ′ T^{\\prime} . Then, by assumption again, there exists a language L ′′ ∈ 𝒞 L^{\\prime\\prime}\\in\\mathcal{C} that is a proper subset of L L , such that L ′′ L^{\\prime\\prime} contains T ′ T^{\\prime} , and furthermore, trace c L ′′ ( x ) = trace c L ( x ) {\\mathrm{trace}}_{c_{L^{\\prime\\prime}}}(x)={\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′′ x\\in L^{\\prime\\prime} (including T ′ T^{\\prime} ). Thus, the adversary switches to enumerating strings in L ′′ L^{\\prime\\prime} . The adversary thus continues switching back and forth between L L and some other language L ~ ⊊ L \\tilde{L}\\subsetneq L endlessly. Since at each switch back to L L , the adversary begins with the leftmost string in L L that has not yet been enumerated in its fixed background ordering, the adversary eventually enumerates every string in L L along with its trace according to c L c_{L} . But also, each switch back to L L follows a time step at which the algorithm guesses an index i i for which L i = L ~ ⊊ L L_{i}=\\tilde{L}\\subsetneq L , and hence the algorithm guesses an incorrect index infinitely often. Thus, the algorithm does not identify L L in the limit."},{"type":"heading","level":2,"text":"A.2 Robust Coloring Lemma"},{"type":"para","text":"See 3.5"},{"type":"para","text":"For any language L ∈ 𝒞 L\\in\\mathcal{C} , we extend the definitions of IsAccept L \\mathrm{IsAccept}_{L} and NextLive L \\mathrm{NextLive}_{L} from the proof of \\Cref lemma:main-coloring-upper-bound, and also consider some global quantities for the language. Namely, consider the functions IsAccept L \\mathrm{IsAccept}_{L} and NextLive L \\mathrm{NextLive}_{L} that map Σ ∗ \\Sigma^{*} to a non-negative integer, and the quantities ShortAccept L \\mathrm{ShortAccept}_{L} and ShortPrefix L \\mathrm{ShortPrefix}_{L} , defined as follows:"},{"type":"para","text":"Here, Σ { i , i + 1 , … , j } \\Sigma^{\\{i,i+1,\\dots,j}\\} denotes the set { y ∈ Σ ∗ : i ≤ | y | ≤ j } \\{y\\in\\Sigma^{*}:i\\leq|y|\\leq j\\} , with | y | = 0 |y|=0 implying the empty string y = ε y=\\varepsilon ."},{"type":"para","text":"In words, IsAccept L ( x ) \\mathrm{IsAccept}_{L}(x) counts the number of suffixes y y having length | y | ∈ { 0 , 1 , … , 2 ℓ } |y|\\in\\{0,1,\\dots,2\\ell\\} for which x y xy is in L L , NextLive L ( x ) \\mathrm{NextLive}_{L}(x) counts the number of suffixes y y having length | y | ∈ { 1 , … , 2 ℓ + 1 } |y|\\in\\{1,\\dots,2\\ell+1\\} such that x y xy can be extended further to form a string in L L , ShortAccept L \\mathrm{ShortAccept}_{L} counts the number of strings y y having length | y | ∈ { 0 , … , 2 ℓ − 1 } |y|\\in\\{0,\\dots,2\\ell-1\\} such that y y belongs to L L , and ShortPrefix L \\mathrm{ShortPrefix}_{L} counts the number of prefixes y y having length | y | ∈ { 0 , … , 2 ℓ } |y|\\in\\{0,\\dots,2\\ell\\} such that y y can be extended further to form a string in L L . Note that ShortAccept L \\mathrm{ShortAccept}_{L} and ShortPrefix L \\mathrm{ShortPrefix}_{L} are global properties of the language L L . Furthermore, for any x x ,"},{"type":"para","text":"Now fix the palette P = { 1 , 2 , … , B } P=\\{1,2,\\dots,B\\} , where B = max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) B=\\max(8\\ell+4,8k^{2\\ell+1}) , and define the trace coloring function c L : Σ ∗ → P c_{L}:\\Sigma^{*}\\to P as follows:"},{"type":"para","text":"If x ∉ Pref ( L ) x\\notin\\mathrm{Pref}(L) , c L ( x ) = 1 c_{L}(x)=1 ; otherwise, if x ∈ Pref ( L ) x\\in\\mathrm{Pref}(L) , then either IsAccept L ( x ) > 0 \\mathrm{IsAccept}_{L}(x)>0 , or NextLive L ( x ) > 0 \\mathrm{NextLive}_{L}(x)>0 , and hence c L ( x ) ≥ 1 c_{L}(x)\\geq 1 . Together with ( 12 ), this implies that c L ( x ) ∈ P c_{L}(x)\\in P for all x ∈ Σ ∗ x\\in\\Sigma^{*} . Similar to the proof of \\Cref lemma:main-coloring-upper-bound, we now observe that the quantities defined above satisfy monotonicity on any string x x with respect to inclusion."},{"type":"para","text":"Let L , L ′ L,L^{\\prime} be languages satisfying L ⊆ L ′ L\\subseteq L^{\\prime} . Then, for any x ∈ Σ ∗ x\\in\\Sigma^{*} , it holds that:"},{"type":"para","text":"Now, fix any L , L ′ ∈ 𝒞 L,L^{\\prime}\\in\\mathcal{C} for which L ⊊ L ′ L\\subsetneq L^{\\prime} , and L L contains some string x x having length at least 2 ℓ 2\\ell ."},{"type":"heading","level":4,"text":"Case 1: ShortAccept L < ShortAccept L ′ \\mathrm{ShortAccept}_{L}<\\mathrm{ShortAccept}_{L^{\\prime}} or ShortPrefix L < ShortPrefix L ′ \\mathrm{ShortPrefix}_{L}<\\mathrm{ShortPrefix}_{L^{\\prime}} ."},{"type":"para","text":"In this case, using \\Cref observation:monotonicity-trace-coloring-corruptions, note that c L ( x ) < c L ′ ( x ) c_{L}(x)<c_{L^{\\prime}}(x) for every x ∈ L x\\in L . Then, choose any x ∈ L x\\in L having length at least 2 ℓ 2\\ell , which is guaranteed to exist by assumption. We have that each of c L ( ε ) , c L ( x ≤ 1 ) , … , c L ( x ) c_{L}(\\varepsilon),c_{L}(x_{\\leq 1}),\\dots,c_{L}(x) is strictly smaller than c L ′ ( ε ) , c L ′ ( x ≤ 1 ) , … , c L ′ ( x ) c_{L^{\\prime}}(\\varepsilon),c_{L^{\\prime}}(x_{\\leq 1}),\\dots,c_{L^{\\prime}}(x) respectively, and hence, d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell as required."},{"type":"heading","level":4,"text":"Case 2: ShortAccept L = ShortAccept L ′ \\mathrm{ShortAccept}_{L}=\\mathrm{ShortAccept}_{L^{\\prime}} and ShortPrefix L = ShortPrefix L ′ \\mathrm{ShortPrefix}_{L}=\\mathrm{ShortPrefix}_{L^{\\prime}} ."},{"type":"para","text":"Since L ⊊ L ′ L\\subsetneq L^{\\prime} , y ∈ L ⟹ y ∈ L ′ y\\in L\\implies y\\in L^{\\prime} , and y ∈ Pref ( L ) ⟹ y ∈ Pref ( L ′ ) y\\in\\mathrm{Pref}(L)\\implies y\\in\\mathrm{Pref}(L^{\\prime}) . Therefore, this case implies the stronger properties that"},{"type":"para","text":"Now, fix any z ∈ L ′ ∖ L z\\in L^{\\prime}\\setminus L . Note that by ( 14 ), it must be the case that | z | ≥ 2 ℓ |z|\\geq 2\\ell . We then have the following two subcases:"},{"type":"heading","level":4,"text":"Subcase 2a: z ∈ Pref ( L ) z\\in\\mathrm{Pref}(L) ."},{"type":"para","text":"In this case, let t ∈ Σ ∗ t\\in\\Sigma^{*} be such that z t ∈ L zt\\in L (note that t ≠ ε t\\neq\\varepsilon , since z ∉ L z\\notin L ). We claim that x := z t x:=zt satisfies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell . To see this, let z = z 0 z 1 , … , z n z=z_{0}z_{1},\\dots,z_{n} , where z 0 = ε z_{0}=\\varepsilon and n ≥ 2 ℓ n\\geq 2\\ell . For any p ≤ q p\\leq q , denote z p : q := z p z p + 1 … , z q z_{p:q}:=z_{p}z_{p+1}\\dots,z_{q} . We then observe that for any i ∈ { 0 , 1 , … , 2 ℓ } i\\in\\{0,1,\\dots,2\\ell\\} , IsAccept L ( z 0 : n − i ) \\mathrm{IsAccept}_{L}(z_{0:n-i}) is strictly smaller than IsAccept L ′ ( z 0 : n − i ) \\mathrm{IsAccept}_{L^{\\prime}}(z_{0:n-i}) ; this is because there exists a suffix y = z n − i + 1 : n y=z_{n-i+1:n} having length | y | ∈ { 0 , … , 2 ℓ } |y|\\in\\{0,\\dots,2\\ell\\} , for which z 0 : n − i y ∉ L z_{0:n-i}y\\notin L but z 0 : n − i y ∈ L ′ z_{0:n-i}y\\in L^{\\prime} (note that z 0 : n − i y z_{0:n-i}y is simply equal to z z ). This implies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell ."},{"type":"heading","level":4,"text":"Subcase 2b: z ∉ Pref ( L ) z\\notin\\mathrm{Pref}(L) ."},{"type":"para","text":"In this case, let w ∈ Σ ∗ w\\in\\Sigma^{*} be the maximal prefix of z z that satisfies w ∈ Pref ( L ) w\\in\\mathrm{Pref}(L) , and let z = w a t z=wat , where a ∈ Σ a\\in\\Sigma , and t ∈ Σ ∗ t\\in\\Sigma^{*} . Here, by maximal, we mean that w a ∉ Pref ( L ) wa\\notin\\mathrm{Pref}(L) . We first claim that | w | ≥ 2 ℓ |w|\\geq 2\\ell . To see this, suppose that | w | < 2 ℓ |w|<2\\ell . In this case, | w a | ≤ 2 ℓ |wa|\\leq 2\\ell . But it also holds that w a ∉ Pref ( L ) wa\\notin\\mathrm{Pref}(L) , and w a ∈ Pref ( L ′ ) wa\\in\\mathrm{Pref}(L^{\\prime}) (since z = w a t ∈ L ′ z=wat\\in L^{\\prime} ). This contradicts ( 15 ). Thus, | w | ≥ 2 ℓ |w|\\geq 2\\ell ."},{"type":"para","text":"So, let w = w 0 w 1 , … , w n w=w_{0}w_{1},\\dots,w_{n} , where w 0 = ε w_{0}=\\varepsilon . We now argue that for every i ∈ { 0 , 1 , … , 2 ℓ } i\\in\\{0,1,\\dots,2\\ell\\} , NextLive L ( w 0 : n − i ) \\mathrm{NextLive}_{L}(w_{0:n-i}) is strictly smaller than NextLive L ′ ( w 0 : n − i ) \\mathrm{NextLive}_{L^{\\prime}}(w_{0:n-i}) . Again, this holds because there exists a suffix y = w n − i + 1 : n a y=w_{n-i+1:n}a having length | y | ∈ { 1 , … , 2 ℓ + 1 } |y|\\in\\{1,\\dots,2\\ell+1\\} , for which w 0 : n − i y ∉ Pref ( L ) w_{0:n-i}y\\notin\\mathrm{Pref}(L) but w 0 : n − i y ∉ Pref ( L ′ ) w_{0:n-i}y\\notin\\mathrm{Pref}(L^{\\prime}) (note that w 0 : n − i y w_{0:n-i}y is simply equal to w a wa ). This then implies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell ."},{"type":"para","text":"In both cases, we have shown the existence of an x ∈ L x\\in L satisfying d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{L}}(x),{\\mathrm{trace}}_{c_{L^{\\prime}}}(x)\\right)>2\\ell , as required."},{"type":"para","text":"We give a formal definition of a DFA following the textbooks Sipser ( 1996 ); Hopcroft et al. ( 2001 ) :"},{"type":"para","text":"A DFA M M over a finite alphabet Σ \\Sigma is a 5-tuple M = ( Q ( M ) , Σ , δ M , q 0 ( M ) , F ( M ) ) M=(Q(M),\\Sigma,\\delta_{M},q_{0}(M),F(M)) , where Q ( M ) Q(M) is a finite set of states, δ M : Q ( M ) × Σ → Q ( M ) \\delta_{M}:Q(M)\\times\\Sigma\\to Q(M) is a state transition function, q 0 ( M ) ∈ Q ( M ) q_{0}(M)\\in Q(M) is the initial state, and F ( M ) ⊆ Q ( M ) F(M)\\subseteq Q(M) is the set of accepting states. The regular language L ( M ) L(M) that is accepted by the DFA M M is defined as the set L ( M ) = { x ∈ Σ ∗ : δ M ( q 0 ( M ) , x ) ∈ F ( M ) } L(M)=\\{x\\in\\Sigma^{*}:\\delta_{M}(q_{0}(M),x)\\in F(M)\\} ."},{"type":"para","text":"We can extend the domain of the transition function δ M \\delta_{M} from Σ \\Sigma to Σ ∗ \\Sigma^{*} by defining δ M ( q , ε ) = q \\delta_{M}(q,\\varepsilon)=q , and recursively defining δ M ( q , a y ) = δ M ( δ M ( q , a ) , y ) \\delta_{M}(q,ay)=\\delta_{M}(\\delta_{M}(q,a),y) , for every q ∈ Q ( M ) , a ∈ Σ q\\in Q(M),a\\in\\Sigma and y ∈ Σ ∗ y\\in\\Sigma^{*} ."},{"type":"heading","level":2,"text":"B.1 Corrupted State Traces"},{"type":"para","text":"As we did in \\Cref sec:state-machines, we describe how we can derive state coloring functions that are robust to corruptions in state traces from the trace coloring functions constructed in \\Cref lemma:corruptions-coloring-upper-bound above. Consider any DFA M M in a given collection 𝒞 = { M 1 , M 2 , … } \\mathcal{C}=\\{M_{1},M_{2},\\dots\\} , and fix ℓ > 0 \\ell>0 . For any q ∈ Q ( M ) q\\in Q(M) , define"},{"type":"para","text":"Note that ShortAccept M \\mathrm{ShortAccept}_{M} and ShortPrefix M \\mathrm{ShortPrefix}_{M} are global properties of the DFA M M ."},{"type":"para","text":"Now fix the palette P = { 1 , 2 , … , B } P=\\{1,2,\\dots,B\\} , where B = max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) B=\\max(8\\ell+4,8k^{2\\ell+1}) , and define the state coloring function c M : Q ( M ) → P c_{M}:Q(M)\\to P as follows:"},{"type":"para","text":"For any x = x 0 x 1 … x n ∈ L ( M ) x=x_{0}x_{1}\\dots x_{n}\\in L(M) (where x 0 = ε x_{0}=\\varepsilon ), which traverses the states q 0 , q 1 , … , q n q_{0},q_{1},\\dots,q_{n} in M M , we have that"},{"type":"para","text":"where c L ( M ) c_{L(M)} is exactly the trace coloring function defined in ( 13 ) in the proof of \\Cref lemma:corruptions-coloring-upper-bound. Thus, we have that for any M , M ′ ∈ 𝒞 M,M^{\\prime}\\in\\mathcal{C} satisfying L ( M ) ⊊ L ( M ′ ) L(M)\\subsetneq L(M^{\\prime}) , there exists an x ∈ L ( M ) x\\in L(M) for which d ( trace c M ( x ) , trace c M ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{M}}(x),{\\mathrm{trace}}_{c_{M^{\\prime}}}(x)\\right)>2\\ell , and we can identify 𝒞 \\mathcal{C} in the limit using state color traces having at most ℓ \\ell corruptions."},{"type":"para","text":"Analogous to \\Cref lemma:state-coloring-optimal-bound, we next show that the palette size used above is optimal in order for state coloring functions in general."},{"type":"para","text":"For any k ≥ 1 k\\geq 1 and ℓ > 0 \\ell>0 , there exists a finite collection 𝒞 \\mathcal{C} of DFAs over the alphabet Σ = { 1 , 2 , … , k } \\Sigma=\\{1,2,\\dots,k\\} , such that any family of state coloring functions { c M } M ∈ 𝒞 \\{c_{M}\\}_{M\\in\\mathcal{C}} which satisfies: for every pair of DFAs M , M ′ ∈ 𝒞 M,M^{\\prime}\\in\\mathcal{C} satisfying that L ( M ) ⊊ L ( M ′ ) L(M)\\subsetneq L(M^{\\prime}) , there exists x ∈ L ( M ) x\\in L(M) for which d ( trace c M ( x ) , trace c M ′ ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{M}}(x),{\\mathrm{trace}}_{c_{M^{\\prime}}}(x)\\right)>2\\ell , necessarily requires a palette of size k 2 ℓ − 1 k^{2\\ell-1} ."},{"type":"para","text":"Let N = k 2 ℓ − 1 N=k^{2\\ell-1} , and S = { s 1 , … , s N } S=\\{s_{1},\\dots,s_{N}\\} be an ordering of all strings of size 2 ℓ − 1 2\\ell-1 in Σ 2 ℓ − 1 \\Sigma^{2\\ell-1} . Consider the collection 𝒞 = { M 1 , … , M N } \\mathcal{C}=\\{M_{1},\\dots,M_{N}\\} , where each M i M_{i} accepts the regular language"},{"type":"para","text":"Namely, L ( M i ) L(M_{i}) has strings of the form w y wy , where w w is any of the first i i strings in S S , and y y is any suffix in Σ ∗ \\Sigma^{*} . We have that L ( M 1 ) ⊊ L ( M 2 ) ⊊ ⋯ ⊊ L ( M N ) L(M_{1})\\subsetneq L(M_{2})\\subsetneq\\dots\\subsetneq L(M_{N}) ."},{"type":"para","text":"Now observe that each M i M_{i} can be specified by a DFA that first builds a prefix tree over the set { s 1 , … , s i } \\{s_{1},\\dots,s_{i}\\} , followed by an accepting sink state. Namely, for any x x , the prefix tree part checks whether the first 2 ℓ − 1 2\\ell-1 characters of x x belong to { s 1 , … , s i } \\{s_{1},\\dots,s_{i}\\} —if they do, then the string is accepted. Let q 0 ( M i ) q_{0}(M_{i}) be the initial state of M i M_{i} (i.e., the root of the prefix tree), and let q F ( M i ) q_{F}(M_{i}) be the single accepting sink state. Note then that any x ∈ L ( M i ) x\\in L(M_{i}) traverses a sequence of states of the form"},{"type":"para","text":"Now, fix any state coloring functions c M 1 , … , c M N c_{M_{1}},\\dots,c_{M_{N}} that use a palette P P of size N − 1 N-1 , and satisfy that for every M i , M j M_{i},M_{j} with L ( M i ) ⊊ L ( M j ) L(M_{i})\\subsetneq L(M_{j}) , there exists x ∈ L ( M i ) x\\in L(M_{i}) for which d ( trace c M i ( x ) , trace c M j ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{M_{i}}}(x),{\\mathrm{trace}}_{c_{M_{j}}}(x)\\right)>2\\ell ."},{"type":"para","text":"Fix any i , j ∈ { 1 , … , N } i,j\\in\\{1,\\dots,N\\} satisfying i < j i<j . For any x ∈ L ( M i ) x\\in L(M_{i}) , by the characterization ( 22 ) of the sequence of states traversed by x x in M i M_{i} , we get that trace c M i ( x ) {\\mathrm{trace}}_{c_{M_{i}}}(x) is some sequence in P 2 ℓ P^{2\\ell} , followed by the sequence ( c M i ( q F ( M i ) ) , … , c M i ( q F ( M i ) ) ) ⏟ | x | − 2 ℓ + 1 times \\underbrace{(c_{M_{i}}(q_{F}(M_{i})),\\dots,c_{M_{i}}(q_{F}(M_{i})))}_{|x|-2\\ell+1\\text{ times}} . Similarly, we have that trace c M j ( x ) {\\mathrm{trace}}_{c_{M_{j}}}(x) is some sequence in P 2 ℓ P^{2\\ell} , followed by the sequence c M j ( q F ( M j ) , … , c M j ( q F ( M j ) ) ) ⏟ | x | − 2 ℓ + 1 times \\underbrace{c_{M_{j}}(q_{F}(M_{j}),\\dots,c_{M_{j}}(q_{F}(M_{j})))}_{|x|-2\\ell+1\\text{ times}} . Thus, in order for d ( trace c M i ( x ) , trace c M j ( x ) ) > 2 ℓ d\\left({\\mathrm{trace}}_{c_{M_{i}}}(x),{\\mathrm{trace}}_{c_{M_{j}}}(x)\\right)>2\\ell to hold, we necessarily require that c M i ( q F ( M i ) ) ≠ c M j ( q F ( M j ) ) c_{M_{i}}(q_{F}(M_{i}))\\neq c_{M_{j}}(q_{F}(M_{j})) . But this is not possible for every i < j i<j , if the state coloring functions use only N − 1 N-1 colors."},{"type":"heading","level":2,"text":"B.2 Identifying Regular Languages with Two Colors"},{"type":"para","text":"Let M M be a DFA satisfying | L ( M ) | = ∞ |L(M)|=\\infty , and let M M have n n states that are live and reachable. Then, for any k > 0 k>0 , there exists a DFA M ′ M^{\\prime} satisfying L ( M ′ ) = L ( M ) L(M^{\\prime})=L(M) , such that M ′ M^{\\prime} has n + k n+k states that are live and reachable."},{"type":"para","text":"Since L ( M ) L(M) is infinite, it must contain strings of arbitrarily large length. Let x = x 1 x 2 … x t x=x_{1}x_{2}\\dots x_{t} be any string in L ( M ) L(M) having length t ≥ n t\\geq n . Then, consider the sequence of states q 0 , q 1 , … , q t q_{0},q_{1},\\dots,q_{t} traversed by x x in M M : note that every state in this sequence is live and reachable. Then, since there are only n n live and reachable states in M M , and the number of states in this sequence is t + 1 > n t+1>n , by the pigeonhole principle, there exists at least one live and reachable state that appears twice in the sequence. Let s s be the smallest index for which there exists i < s i<s such that q i = q s q_{i}=q_{s} (i.e., this is the first time a state got repeated in the sequence), and consider the intermediate sequence of transitions q i → q i + 1 → … → q s − 1 → q s q_{i}\\to q_{i+1}\\to\\dots\\to q_{s-1}\\to q_{s} , together with the symbols x i + 1 , … , x s − 1 , x s x_{i+1},\\dots,x_{s-1},x_{s} that effected these transitions. Note that by the choice of s s , the states q 0 , … , q s − 1 q_{0},\\dots,q_{s-1} are all distinct."},{"type":"para","text":"Now, consider a DFA M ′ M^{\\prime} that operates on the same alphabet Σ \\Sigma . M ′ M^{\\prime} has a copy q ′ q^{\\prime} of every state q q in M M , plus an additional state q new ′ q^{\\prime}_{{\\mathrm{new}}} . The state transitions for all the states in M ′ M^{\\prime} are identical to their copies in M M , except for the states q s − 1 ′ q^{\\prime}_{s-1} and q new ′ q^{\\prime}_{{\\mathrm{new}}} , where q s − 1 q_{s-1} is the copy of the state q s − 1 q_{s-1} , and q new ′ q^{\\prime}_{{\\mathrm{new}}} is the additional new state. The state transitions for q s − 1 ′ q^{\\prime}_{s-1} and q new q_{{\\mathrm{new}}} are as follows. First, we make the transitions out of q new ′ q^{\\prime}_{\\mathrm{new}} to be identical to those out of q s q_{s} , i.e., set δ M ′ ( q new ′ , a ) \\delta_{M^{\\prime}}(q^{\\prime}_{{\\mathrm{new}}},a) to be the copy of δ M ( q s , a ) \\delta_{M}(q_{s},a) for every a ∈ Σ a\\in\\Sigma . Next, we redirect the single transition q s − 1 ′ → x s q s ′ q^{\\prime}_{s-1}\\xrightarrow{x_{s}}q^{\\prime}_{s} to be q s − 1 ′ → x s q new ′ q^{\\prime}_{s-1}\\xrightarrow{x_{s}}q^{\\prime}_{\\mathrm{new}} instead; namely, for any a ≠ x s a\\neq x_{s} , we set δ M ′ ( q s − 1 ′ , a ) \\delta_{M^{\\prime}}(q^{\\prime}_{s-1},a) to be the copy of the state δ M ( q s − 1 , a ) \\delta_{M}(q_{s-1},a) , and we make q new ′ q^{\\prime}_{{\\mathrm{new}}} reachable by setting δ M ′ ( q s − 1 ′ , x s ) = q new ′ \\delta_{M^{\\prime}}(q^{\\prime}_{s-1},x_{s})=q^{\\prime}_{{\\mathrm{new}}} . In essence, we have redirected the transition from q s − 1 ′ q^{\\prime}_{s-1} which was looping back into q s ′ q^{\\prime}_{s} to the new state q new ′ q^{\\prime}_{{\\mathrm{new}}} , whose outward transitions are then identical to q s q_{s} . The initial state in M ′ M^{\\prime} is set to the copy of the initial state in M M , and the accepting states in M ′ M^{\\prime} are set to the copies of the accepting states in M M , plus the state q new ′ q^{\\prime}_{{\\mathrm{new}}} if q s q_{s} was an accepting state. This completes the specification of M ′ M^{\\prime} ."},{"type":"para","text":"We now argue that M ′ M^{\\prime} has n + 1 n+1 live and reachable states, and that L ( M ′ ) = L ( M ) L(M^{\\prime})=L(M) ; the lemma is then established by inducting this k k times. First, it is clear that for any state q q in M M that was either not live or reachable in M M , its copy q ′ q^{\\prime} has the same status in M ′ M^{\\prime} . Now, by construction, the newly added state q new ′ q^{\\prime}_{{\\mathrm{new}}} and the state q s ′ q^{\\prime}_{s} are both live and reachable, as witnessed by the traversal of x x through M ′ M^{\\prime} , which passes through both q s ′ q^{\\prime}_{s} and q new ′ q^{\\prime}_{\\mathrm{new}} on the way to an accepting state. Finally, for every state q ≠ q s q\\neq q_{s} that was live and reachable in M M , its copy q ′ q^{\\prime} remains live and reachable in M ′ M^{\\prime} . This can be seen by tracing out the sequence of states traversed by any y ∈ L ( M ) y\\in L(M) that passed through q q on the way to an accepting state in M M . In its traversal through M ′ M^{\\prime} , the only notable change is that a transition of the form q s − 1 → x s q s q_{s-1}\\xrightarrow{x_{s}}q_{s} gets replaced by q s − 1 ′ → x s q new ′ q^{\\prime}_{s-1}\\xrightarrow{x_{s}}q^{\\prime}_{{\\mathrm{new}}} . But since the transitions out of q new ′ q^{\\prime}_{\\mathrm{new}} are identical to q s q_{s} , y y continues on its path to an accepting state, and any occurrences of q q in the traversal through M M overlap with occurrences of q ′ q^{\\prime} in the traversal through M ′ M^{\\prime} ; thus, q ′ q^{\\prime} remains live and reachable in M ′ M^{\\prime} . Thus, we have argued that M ′ M^{\\prime} has n + 1 n+1 live and reachable states."},{"type":"para","text":"It remains to argue that L ( M ′ ) = L ( M ) L(M^{\\prime})=L(M) . Again, this is made clear by tracing out the traversal of any x x through M M and M ′ M^{\\prime} : if the traversal never uses the single redirected transition q s − 1 ′ → x s q new ′ q^{\\prime}_{s-1}\\xrightarrow{x_{s}}q^{\\prime}_{{\\mathrm{new}}} , then the end outcomes are clearly identical; if it does use this transition, then in M M , the transition leads into the state q s q_{s} , whereas in M ′ M^{\\prime} , the transition leads into the state q new ′ q^{\\prime}_{\\mathrm{new}} ; however, since the transitions out of q new ′ q^{\\prime}_{\\mathrm{new}} and q s q_{s} are identical, the end outcome is identical as well."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Moses Charikar \\\\Email moses@cs.stanford.edu \\\\addr Stanford University and \\\\Name Jon Kleinberg \\\\Email kleinberg@cornell.edu \\\\addr Cornell University and \\\\Name Chirag Pabbaraju \\\\Email cpabbara@cs.stanford.edu \\\\addr Stanford University\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The success of large language models has motivated several lines of recent work exploring theoretical models that might provide insight into how they are achieving strong performance from the types of training data and training pipelines that they are built from. One direction this research has taken has been to start from a classical model of Gold on language identification in the limit (Gold, 1967 ) , in which we view the language learning task as a game played between an adversary and an algorithm: the adversary thinks of a secret language K K known only to come from a countable list of candidates L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},... ; the adversary enumerates the strings of K K in an arbitrary order 1 1 1 A valid enumeration x 1 , x 2 , … x_{1},x_{2},\\\\dots of K K satisfies: (1) x i ∈ K x_{i}\\\\in K , for all i i , and (2) ∀ x ∈ K \\\\forall x\\\\in K , there exists i i such that x i = x x_{i}=x . ; in each step t t the algorithm guesses the index i t i_{t} of the true language; and the algorithm wins the game (it has identified K K in the limit ) if for some time t ∗ t^{*} , and all t ≥ t ∗ t\\\\geq t^{*} , we have L i t = K L_{i_{t}}=K — the algorithm is correct in every step beginning with t ∗ t^{*} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The main results in Gold’s framework, beginning with Gold’s original theorem about his model, are negative (Gold, 1967 ) ; for all but highly constrained families of languages (much more restricted than the class of regular languages), the algorithm cannot win this game (Angluin, 1980 ) . Yet real language models seem much more powerful than what these sweeping negative results suggest, and so the question has turned to finding reasonable variations on this model in which we are able to obtain positive results more consistent with the successes we see in practice.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Computational traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One promising line of recent work has explored the power of annotation or computational traces in helping solve the problem of language identification in the limit (Papazov and Flammarion, 2025 ; Bhattamishra et al., 2026 ; Peng et al., 2026 ) . These models are motivated by the observation that training corpora often contains text or sequence data that is enriched by some kind of meta-data, thinking traces, or domain-relevant annotation. This can include cases in which the data itself has this type of supporting information, as in the way that learning from corpora of source code can be more effective if the code is commented, or learning from corpora of mathematics can be more effective if the proofs contain step-by-step arguments. It also includes cases in which the data has been annotated algorithmically, as in the success of chain-of-thought and related methods (Wei et al., 2022 ) . This range of examples suggests that learners can be more effective on a corpus of training data if that data is annotated with some kind of computational trace.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These recent models have begun from the assumption that there is an automata-theoretic underpinning for the candidate languages — for example, that they are produced by an unknown finite-state machine, pushdown automaton, or Turing machine — and that the annotation in question is produced from the computational trace of the machine itself. Prior work obtains possibility results with this approach. This line of work leaves open two fundamental classes of questions, however. First, the kind of annotation they describe is fine-grained and expansive: in one case it is represented in terms of the states of the underlying machine (Peng et al., 2026 ) , and in the other case it is represented by the power-set of the vocabulary, with an exponential blow-up as a result (Bhattamishra et al., 2026 ) . Second, the premise of these models is that annotation is directly tied to the automaton that produced the language; it leaves open the question of how we might usefully annotate for identification when we have no access to the underlying machine models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast, the kinds of annotations or computational trace we see in practice tend not to come from an underlying machine model, but tend to be more designed as objects in their own right, with their own vocabularies, and much coarser than the full state set of the machine architecture. For example, in commented code, it is the comments that form the additional tokens helping learn the corpus; and with chain-of-thought, it is the thinking tokens that provide this help. Neither of these corresponds to the state of the underlying machine that produced the training data, to the extent that there is such a machine at all. What would a theory of computational traces look like if we take the tokens in the trace to have this coarse-grained structure, and independent of any specific machine model? Could we still obtain possibility results for language identification in the limit?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The present work: Compact, machine-independent traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we develop such a theory, and we show that annotation with a small set of tokens — independent of the size of the state set of the underlying machine — is sufficient to achieve language identification in the limit. Moreover, our construction of the annotation works at the level of the language itself, and does not assume knowledge of any underlying machine producing the language.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We will describe the model in detail in the subsequent sections, but we begin here with a high-level overview of it. As a starting point, let’s consider how earlier machine-based models defined computational traces in the case of a finite automaton: as the automaton processed the string one symbol at a time, from left to right, the trace reported the state of the automaton at each symbol. This kind of trace has the property if x = x 1 x 2 … x n x=x_{1}x_{2}\\\\ldots x_{n} is the input string, then the trace annotates each input symbol x i x_{i} with an annotation symbol c i c_{i} (corresponding to the state of the automaton at that point); and because c i c_{i} is the state, it depends only on the prefix x 1 x 2 … x i x_{1}x_{2}\\\\ldots x_{i} up to that point, and not any subsequent symbols.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use this as the defining property of our traces: we say that a trace coloring function c c over a finite set of colors P P (the “palette”) is a function that maps finite strings to elements of P P , and we define the color trace of a string x = x 1 x 2 … x n x=x_{1}x_{2}\\\\ldots x_{n} to be the value of c c applied to each of its prefixes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where x ≤ i x_{\\\\leq i} denotes the prefix x 1 x 2 … x i x_{1}x_{2}\\\\ldots x_{i} and ε \\\\varepsilon denotes the empty string. A coloring of x x by the states of a finite automaton that produces it is one way to produce such a trace coloring, but our definition here makes clear that we can define this object as an annotation scheme for any language, even if we don’t know the underlying machine model, or even if there isn’t one. (This is the same sense in which comments or thinking tokens can annotate a string in a training corpus even though they don’t correspond in any respect to the generative mechanism that originally produced the string.)\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Identification using traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We would like to achieve identification in the limit using these traces, and further, with small palettes, so that we do not need to provide annotations using voluminous alphabets like fine-grained state sets. Formally, given an instance of language identification in the limit with candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\\\ldots , we consider a process in which each language L i L_{i} is first annotated by its own trace coloring function c L i c_{L_{i}} , and then an adversary enumerates the strings x ∈ K x\\\\in K , each as an annotated pair consisting of x x and its color trace, ( x , trace c K ( x ) ) (x,{\\\\mathrm{trace}}_{c_{K}}(x)) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Is this sufficient to enable language identification in the limit, even with no underlying machine model? We show that it is: there is an algorithm 𝒜 \\\\mathcal{A} such that for any collection of candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\\\ldots , there exists a choice of trace colorings c L 1 , c L 2 , c L 3 , … c_{L_{1}},c_{L_{2}},c_{L_{3}},\\\\ldots for which 𝒜 \\\\mathcal{A} is able to achieve identification in the limit when presented with the ordered pairs ( x , trace c K ( x ) ) (x,{\\\\mathrm{trace}}_{c_{K}}(x)) arising from an enumeration of the adversary’s language K K .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Moreover, the trace colorings required can be constructed using very few colors: for problem instances with an alphabet of size k k , there is an algorithm that achieves this using trace colorings with palettes of size at most k + 1 k+1 . Even in the case where the languages come from an underlying machine model, this is a bound that is independent of the number of states of these machines, which must grow unboundedly for an infinite family of languages. As such, it is a much more resource-efficient construction of computational traces — and one that applies much more broadly — than the computational traces proposed by earlier approaches.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We prove this result by establishing an exact combinatorial characterization of when a trace coloring permits identification in the limit, and we relax this to a related sufficient condition for identification that we show how to achieve. This sufficient condition has interesting combinatorial interpretations in its own right, and we show matching upper and lower bounds for this condition in the case of languages over a binary alphabet (when k = 2 k=2 ): our result shows how to achieve the required condition with k + 1 = 3 k+1=3 colors, and we prove that 2 colors are not sufficient.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We explore further extensions of our result, including the fact that identification in the limit is still possible (using a larger palette) when the trace can be corrupted to a bounded extent; and the fact that when the candidate languages L 1 , L 2 , L 3 , … L_{1},L_{2},L_{3},\\\\ldots are regular, we can achieve identification in the limit with trace coloring functions that use only two colors.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Main Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a language collection 2 2 2 All languages in the collection are assumed to be non-empty. 𝒞 \\\\mathcal{C} , our objective is to associate every language L L in 𝒞 \\\\mathcal{C} with a suitable trace coloring function c L c_{L} , such that when every x ∈ L x\\\\in L is accompanied with trace c L ( x ) {\\\\mathrm{trace}}_{c_{L}}(x) , the collection 𝒞 \\\\mathcal{C} becomes identifiable in the limit. Towards this, as our first contribution, we derive an exact characterization of when identification in the limit is possible with color traces. Our characterization is structurally similar to Angluin’s condition (Angluin, 1980 ) for identification in the limit, but additionally takes into account the color traces seen in the input; the proof is given in \\\\Cref sec:characterization-appendix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be a countable language collection. Then, 𝒞 \\\\mathcal{C} is identifiable in the limit with a color trace given by the trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} if and only if for every language L ∈ 𝒞 L\\\\in\\\\mathcal{C} , there exists a finite “tell-tale” subset T L ⊆ L T_{L}\\\\subseteq L , such that for every language L ′ ∈ 𝒞 L^{\\\\prime}\\\\in\\\\mathcal{C} that is a proper subset of L L , either (1) L ′ L^{\\\\prime} does not contain T L T_{L} , or (2) there exists x ∈ L ′ x\\\\in L^{\\\\prime} such that trace c L ′ ( x ) ≠ trace c L ( x ) {\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L}}(x) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The characterization above gives a precise objective for constructing trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} . Perhaps surprisingly, our next result, which is one of the main structural results of the paper, shows that for every language collection, we can construct trace coloring functions that use a small palette, and have the stronger property that they always satisfy requirement (2) above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be a language collection over a finite alphabet Σ \\\\Sigma of size k k . There exist trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 , that satisfy the following “distinguishable coloring condition”: for every L , L ′ ∈ 𝒞 L,L^{\\\\prime}\\\\in\\\\mathcal{C} that satisfy L ⊊ L ′ L\\\\subsetneq L^{\\\\prime} , there exists x ∈ L x\\\\in L for which trace c L ( x ) ≠ trace c L ′ ( x ) {\\\\mathrm{trace}}_{c_{L}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also show that our trace coloring functions use an optimal palette size for the purposes of satisfying the distinguishable coloring condition over a binary alphabet ( \\\\Cref prop:2-coloring-lower-bound-for-sufficient-condition): namely, there exists a (finite) language collection over a binary alphabet for which any set of trace coloring functions that satisfies the distinguishable coloring condition necessarily requires three colors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given that our trace coloring functions ensure that requirement (2) of the characterizing condition in \\\\Cref thm:characterization-identification-with-traces always holds, tell-tale sets are not needed and we get our main result:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be a countable language collection over a finite alphabet Σ \\\\Sigma of size k k . There exist trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 that make 𝒞 \\\\mathcal{C} identifiable in the limit with color traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In order to specify a trace coloring function for a language, in general, one needs to specify the color that it maps every string in the universe to. However, our trace-coloring functions from \\\\Cref thm:main-theorem above have a particularly special structure for regular language collections. Concretely, consider a regular language L L , and let M M be any finite automaton that recognizes L L . Denote its set of states by Q ( M ) Q(M) . For any state q q in Q ( M ) Q(M) , all strings that arrive at this state q q (starting from the initial state) are assigned the same color by our state coloring function c L c_{L} for language L L . Now, associate the state q q with this common color. Then, for any x ∈ L x\\\\in L , the sequence of colors associated with the sequence of states it traverses in M M would exactly equal the color trace generated by our trace coloring function on x x . Thus, for a regular language, our trace coloring functions c L c_{L} can be described very compactly, by simply specifying the unique color associated with every state in any automaton recognizing L L .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This structural property of our trace coloring functions motivates studying a natural class of trace coloring functions that come from directly coloring the states of automata for regular languages — we denote these as state coloring functions . Concretely, a state coloring function c M c_{M} maps every state in Q ( M ) Q(M) to a palette P P . Denoting L ( M ) L(M) to be the regular language recognized by M M , every x ∈ L ( M ) x\\\\in L(M) has a well-defined state color trace associated with it: if x x traverses the sequence of states q 0 , … , q | x | q_{0},\\\\dots,q_{|x|} in M M , then the state color trace is simply\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The work of Peng et al. ( 2026 ) and Bhattamishra et al. ( 2026 ) can be viewed in this framework (see Section 1.2 for a discussion). For the computational traces used by Peng et al. ( 2026 ) , the ‘color’ of a state is the name of the state itself. Our notion is a coarsening of this state information.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With this definition ( 2 ), we can now study the distinguishable coloring condition for any given collection 𝒞 = { M 1 , M 2 , … } \\\\mathcal{C}=\\\\{M_{1},M_{2},\\\\dots\\\\} of finite automata. By the reasoning above, we can show that state coloring functions derived from our trace coloring functions satisfy the condition with k + 1 k+1 colors. Recall that for arbitrary language collections, we could show that our trace coloring functions achieved an optimal palette size for the distinguishable coloring condition only in the case of a binary alphabet. Interestingly, for arbitrary automaton collections, we show that k + 1 k+1 is the optimal palette size achievable by state coloring functions for the distinguishable coloring condition for any alphabet size k k .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be an automata collection over an alphabet of size k k . There exist state coloring functions { c M } M ∈ 𝒞 \\\\{c_{M}\\\\}_{M\\\\in\\\\mathcal{C}} mapping to a palette P P of size k + 1 k+1 , such that for any M , M ′ ∈ 𝒞 M,M^{\\\\prime}\\\\in\\\\mathcal{C} that satisfy L ( M ) ⊊ L ( M ′ ) L(M)\\\\subsetneq L(M^{\\\\prime}) , there exists x ∈ L ( M ) x\\\\in L(M) for which trace c M ( x ) ≠ trace c M ′ ( x ) {\\\\mathrm{trace}}_{c_{M}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{M^{\\\\prime}}}(x) . Moreover, for any k k , there exists a collection of automata over an alphabet of size k k such that any set of state coloring functions requires k + 1 k+1 colors to satisfy this condition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"lemma:state-coloring-optimal-bound above is primarily about the tightness of k + 1 k+1 colors for state coloring functions to satisfy the distinguishable coloring condition for any given collection of automata. We now return to our main question of identification in the limit with arbitrary trace coloring functions, but for the special class of regular language collections. In particular, we ask: what is the optimal palette size necessary for these collections? Instead of the general-purpose trace coloring functions given by \\\\Cref thm:main-theorem, which use k + 1 k+1 colors, can we perhaps use different trace coloring functions that use fewer colors? Our final result shows that any collection of regular languages, where every language is infinite, is in fact identifiable in the limit using only two colors!\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 = { L 1 , L 2 , … } \\\\mathcal{C}=\\\\{L_{1},L_{2},\\\\dots\\\\} be a countable collection of regular languages, where every L i L_{i} is infinite. Then, 𝒞 \\\\mathcal{C} is identifiable in the limit with color traces given by trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} that use just two colors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We note that a palette size of two is optimal above, since using a single color is equivalent to having no trace information, and we know by Gold’s results that there are simple regular language collections, where every language is infinite, that are not identifiable in the limit without any traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, we also study identification in the limit with traces that are potentially corrupted ( \\\\Cref sec:corruptions); in this setting, we derive a robust version of the coloring lemma, and analogously show optimality of the palette size in the case of state coloring functions for regular languages.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We note that our primary focus in this paper is information-theoretic. Namely, for our upper bounds for identification, we seek to specify a function that maps inputs and color traces to a guess for the target language. Nevertheless, for implementing the identification algorithm of \\\\Cref thm:main-theorem, we would require access to the coloring functions c L c_{L} for every language L L in the collection, and the ability to determine membership of any string x x in any language L L in the collection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We discuss two recent works most closely related to our work, that demonstrate the power of annotation or computational traces for language identification.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Peng et al. ( 2026 ) take a machine-centric view, studying identification for languages accepted by a particular machine model (deterministic finite automata, deterministic push down automata, and Turing machines). For a machine M M and input x x , they consider x x augmented by a computational trace given by the sequence of computational steps of M M on x x . In the case of a DFA D D , the computational trace is simply the sequence of states visited by D D while processing input x x . In the case of a Turing machine M M , the computational trace is the sequence of ‘states’ of the Turing machine while processing x x , where each ‘state’ includes the state of the finite state control, but also the contents of the memory tape, the position of the head, and so on. Given an enumeration of inputs accepted by machine M M , augmented by their computational traces, they show that one can converge to the correct representation of M M , and hence identify the language accepted by M M . We note that this notion of computational traces requires very detailed, fine-grained information to facilitate language identification. The computational trace for input x x can be significantly longer than x x . They also study the question of robust identification, i.e., language identification from corrupted traces, establishing results in different error regimes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Another recent work (Bhattamishra et al., 2026 ) also takes a machine-centric view, studying learnability of regular languages in the Next Symbol Prediction (NSP) setting. Here, the learner receives strings from the language, as well as rich annotation for every prefix of the string: whether the prefix belongs to the language, and which next symbols can lead to a string in the language, i.e., k + 1 k+1 bits for alphabet size k k . They show that this annotation is information-theoretically sufficient to identify minimal DFAs; however the problem of improper learning remains computationally intractable. We note that the annotation considered in this work is closely related to, but significantly richer that the annotation we use in our work. If the input alphabet is of size k k , each prefix of the input string is annotated with a symbol from an alphabet size of 2 k + 1 2^{k+1} . In contrast to our work, this work is focussed on regular languages. Much of their focus is on computationally efficient learning – an issue that we (mostly) do not study in our work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Background and Motivating Examples\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Language identification with a color trace can be related to language identification with a list (Charikar, Pabbaraju, and Tewari, 2025 ) , where the algorithm guesses a list of k k languages at every step, and the algorithm identifies in the limit if the list eventually always contains the target language. In particular, Charikar et al. ( 2025 , Theorem 2) shows that a language collection 𝒞 \\\\mathcal{C} is identifiable in the limit with a list of size k k if and only if it can be expressed as a union of k k collections, each of which is identifiable in the limit with a single guess. By this characterization, any collection 𝒞 \\\\mathcal{C} that is k k -list identifiable decomposes as 𝒞 = ∪ i = 1 k 𝒞 i \\\\mathcal{C}=\\\\cup_{i=1}^{k}\\\\mathcal{C}_{i} , where each 𝒞 i \\\\mathcal{C}_{i} is identifiable. In this case, we can assign a constant color trace corresponding to a unique color i i to every language in 𝒞 i \\\\mathcal{C}_{i} . Then, the color traces seen along with the input reveal the specific 𝒞 i \\\\mathcal{C}_{i} that the target language K K belongs to. Once this is known, one can simply apply any known algorithm that identifies in the limit to the collection 𝒞 i \\\\mathcal{C}_{i} ; since 𝒞 i \\\\mathcal{C}_{i} is identifiable in the limit, the algorithm will correctly identify the target language. Thus, k k -list identifiability always implies identifiability with k k colors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Interestingly, the converse is not true: identifiability with a color trace does not always imply identifiability with a finite-size list. To see this, fix any alphabet Σ \\\\Sigma . For any finite F ⊆ Σ ∗ F\\\\subseteq\\\\Sigma^{*} , let L F = Σ ∗ ∖ F L_{F}=\\\\Sigma^{*}\\\\setminus F . Now, consider the collection 𝒞 = { L F : F is a finite subset of Σ ∗ } \\\\mathcal{C}=\\\\{L_{F}:F\\\\text{ is a finite subset of }\\\\Sigma^{*}\\\\} . This collection is not identifiable with a list of any finite size (see Remark 3 in Charikar et al. ( 2025 ) ). However, there is a simple trace coloring scheme with just two colors that makes the collection identifiable: associate every L ∈ 𝒞 L\\\\in\\\\mathcal{C} with the trace coloring function c L ( x ) = 𝟙 [ x ∈ L ] c_{L}(x)=\\\\mathds{1}[x\\\\in L] . We call this the “accept-reject” coloring. Note then that c L F ( x ) = 0 c_{L_{F}}(x)=0 only if x ∈ F x\\\\in F . Thus, an algorithm may initialize F = ∅ F=\\\\emptyset , and initialize its guess for the target language K K to be L ∅ L_{\\\\emptyset} . Whenever it sees an input string x x whose color trace has 0 in it, it obtains the prefix w w of x x at which the color 0 was seen (i.e., c K ( w ) = 0 c_{K}(w)=0 ). The algorithm then adds w w to F F , and updates its guess L F L_{F} accordingly. If the target language K K was L ∅ L_{\\\\emptyset} , the algorithm will never see a 0 in any color trace, and its initial guess remains correct throughout. If the target language K K was L F L_{F} for some F ≠ ∅ F\\\\neq\\\\emptyset , then for every w ∈ F w\\\\in F , the algorithm is guaranteed to see an input x x that contains w w as its prefix; at this point, the algorithm correctly infers w w ’s membership in F F . Since F F is finite, the algorithm identifies L F L_{F} in the limit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The example above motivates studying the simple accept-reject coloring scheme further. Notice a key property about the collection in the example above: for every L ∈ 𝒞 L\\\\in\\\\mathcal{C} , if a string w w was not in L L , it could be extended by some suffix y y such that w y ∈ L wy\\\\in L . Indeed, this “reject-extendability” property is sufficient for a collection to be identifiable in the limit with the accept-reject coloring.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be a countable language collection over the alphabet Σ \\\\Sigma , where every L ∈ 𝒞 L\\\\in\\\\mathcal{C} satisfies the property: for every w ∈ Σ ∗ w\\\\in\\\\Sigma^{*} , if w ∉ L w\\\\notin L , then there exists a suffix y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} such that w y ∈ L wy\\\\in L . Then, the collection 𝒞 \\\\mathcal{C} is identifiable in the limit with a color trace given by the accept-reject trace coloring.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let K ∈ 𝒞 = { L 1 , L 2 , … } K\\\\in\\\\mathcal{C}=\\\\{L_{1},L_{2},\\\\dots\\\\} be the target language. For every L ∈ 𝒞 L\\\\in\\\\mathcal{C} , consider the accept-reject coloring given by the trace function c L ( x ) = 𝟙 [ x ∈ L ] c_{L}(x)=\\\\mathds{1}[x\\\\in L] . Consider the algorithm, which at time step t t , outputs the smallest index i t i_{t} such that the language L i t L_{i_{t}} is consistent with all the information seen in the input so far. Namely, L i t L_{i_{t}} satisfies that { x 1 , … , x t } ⊆ L i t \\\\{x_{1},\\\\dots,x_{t}\\\\}\\\\subseteq L_{i_{t}} , and furthermore, trace c K ( x b ) = trace c L i t ( x b ) {\\\\mathrm{trace}}_{c_{K}}(x_{b})={\\\\mathrm{trace}}_{c_{L_{i_{t}}}}(x_{b}) for all 1 ≤ b ≤ t 1\\\\leq b\\\\leq t . Furthermore, for every j < i t j<i_{t} , either { x 1 , … , x t } ⊈ L j \\\\{x_{1},\\\\dots,x_{t}\\\\}\\\\nsubseteq L_{j} , or trace c K ( x b ) ≠ trace c L j ( x b ) {\\\\mathrm{trace}}_{c_{K}}(x_{b})\\\\neq{\\\\mathrm{trace}}_{c_{L_{j}}}(x_{b}) for some 1 ≤ b ≤ t 1\\\\leq b\\\\leq t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, let z z be the smallest index in the collection for which L z = K L_{z}=K . We claim that the index i t i_{t} output by the algorithm converges to z z . To see this, note that the index z z is always a valid candidate for the algorithm to output, since every string in the input is from L z = K L_{z}=K , and furthermore, c L z c_{L_{z}} and c K c_{K} are identical. Now, consider any j < z j<z for which L j ⊉ L z L_{j}\\\\nsupseteq L_{z} . For each such L j L_{j} , there exists some x ∈ L z ∖ L j x\\\\in L_{z}\\\\setminus L_{j} which shows up in the input at some finite time. At this time, the language L j L_{j} becomes inconsistent. Since there are finitely many j < z j<z , we have that beyond some finite time, every L j L_{j} satisfying j < z j<z and L j ⊉ L z L_{j}\\\\nsupseteq L_{z} is rendered inconsistent by the algorithm. So, we restrict our attention to languages L j L_{j} that satisfy j < z j<z and L j ⊋ L z L_{j}\\\\supsetneq L_{z} (note that L j L_{j} cannot equal L z L_{z} , by our choice of z z ). Fix any such L j L_{j} , and consider any w ∈ L j ∖ L z w\\\\in L_{j}\\\\setminus L_{z} . By the assumed reject-extendability property, there exists some suffix y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} such that w y ∈ L z wy\\\\in L_{z} ; furthermore, this string w y wy is guaranteed to show up in the input at some finite time. But now, notice that c K ( w ) = c L z ( w ) = 0 c_{K}(w)=c_{L_{z}}(w)=0 , but c L j ( w ) = 1 c_{L_{j}}(w)=1 . Thus, trace c L j ( w y ) ≠ trace c K ( w y ) {\\\\mathrm{trace}}_{c_{L_{j}}}(wy)\\\\neq{\\\\mathrm{trace}}_{c_{K}}(wy) , and hence, the algorithm will declare L j L_{j} to have an inconsistent color trace on w y wy . In this manner, every proper superset of L z L_{z} occurring before it gets invalidated at some finite time, and we conclude that the algorithm’s guess eventually converges to z z .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sufficient condition above for which the accept-reject coloring works may be deemed to be fairly weak: it only stipulates that every rejected string may be extended in some way to acceptance. Could it not be necessary? Namely, could every language collection be identifiable with the simple accept-reject coloring? The following example shows that this is not the case.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider a collection 𝒞 \\\\mathcal{C} over the binary alphabet Σ = { 0 , 1 } \\\\Sigma=\\\\{0,1\\\\} , which comprises of the regular language L ∞ = { 0 , 1 } ∗ L_{\\\\infty}=\\\\{0,1\\\\}^{*} , together with the regular languages L i = { 0 , 1 } ∗ ∖ 1 i { 0 , 1 } ∗ L_{i}=\\\\{0,1\\\\}^{*}\\\\setminus 1^{i}\\\\{0,1\\\\}^{*} for i ≥ 1 i\\\\geq 1 . Namely, L i L_{i} excludes strings in { 0 , 1 } ∗ \\\\{0,1\\\\}^{*} that start with i i ones. We have that for every i ≥ 1 i\\\\geq 1 , L i ⊊ L i + 1 L_{i}\\\\subsetneq L_{i+1} , L i ⊊ L ∞ L_{i}\\\\subsetneq L_{\\\\infty} , and furthermore, ∪ i L i = L ∞ \\\\cup_{i}L_{i}=L_{\\\\infty} . Observe that for any i ∈ ℕ ∪ { ∞ } i\\\\in\\\\mathds{N}\\\\cup\\\\{\\\\infty\\\\} and x ∈ L i x\\\\in L_{i} , the accept-reject coloring satisfies trace c L i ( x ) = ( 1 , 1 , 1 , … , 1 ) ⏟ | x | + 1 times {\\\\mathrm{trace}}_{c_{L_{i}}}(x)=\\\\underbrace{(1,1,1,\\\\dots,1)}_{|x|+1\\\\text{ times}} . This is because each L i L_{i} is prefix-closed: every prefix of x ∈ L i x\\\\in L_{i} is also contained in L i L_{i} . To see this, note that any x ∈ L i x\\\\in L_{i} does not start with 1 i 1^{i} , and hence no prefix of x x starts with 1 i 1^{i} as well. In particular, no string ever has 0 in its color trace. Furthermore, for any i > j i>j , trace c L i ( x ) = trace c L j ( x ) {\\\\mathrm{trace}}_{c_{L_{i}}}(x)={\\\\mathrm{trace}}_{c_{L_{j}}}(x) for every x ∈ L i x\\\\in L_{i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any identification algorithm, and consider an adversary that starts enumerating the strings in L 1 L_{1} in the order that they appear in a fixed background ordering of L ∞ L_{\\\\infty} , together with their color traces (according to c L 1 c_{L_{1}} , which is always the constant-ones trace). Then, at some finite time, the algorithm must guess L 1 L_{1} to be the target language. At this time, the adversary switches to enumerating the strings in L 2 L_{2} , also in the fixed background ordering of L ∞ L_{\\\\infty} , and starting from the leftmost string in L 2 L_{2} that has not been enumerated as yet. The adversary continues giving color traces according to L 2 L_{2} , which they can legally do, since L 1 ⊊ L 2 L_{1}\\\\subsetneq L_{2} , and the traces of L 2 L_{2} and L 1 L_{1} matched on all strings in L 1 L_{1} . Now, the algorithm must guess the target language to be L 2 L_{2} at some subsequent finite time, at which time the adversary switches to L 3 L_{3} . The adversary repeats this switching process endlessly, enumerating each L i L_{i} for a finite phase. As the adversary begins enumerating L i L_{i} from the leftmost string that has not yet been enumerated in the fixed background ordering of L ∞ L_{\\\\infty} , the adversary eventually enumerates every string in L ∞ L_{\\\\infty} , together also with its correct color trace according to c L ∞ c_{L_{\\\\infty}} . But the algorithm guesses an incorrect language infinitely often, and hence does not identify L ∞ L_{\\\\infty} in the limit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure that enabled the adversarial strategy in \\\\Cref example:accept-reject-failure distills precisely into the characterizing condition of \\\\Cref thm:characterization-identification-with-traces. Indeed, one direction of the characterization is a generalization of the adversarial strategy employed above. In the other direction, we appropriately modify the algorithm used in \\\\Cref prop:reject-extendability-sufficiency to account for trace discrepancies, and tell-tales associated with languages.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The Coloring Lemma\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we restate and prove our main coloring lemma. We also discuss an extension to the setting where the color traces observed have a bounded number of corruptions. See 1.2\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any language L L , define the set Pref ( L ) \\\\mathrm{Pref}(L) to be the set of all strings in Σ ∗ \\\\Sigma^{*} that can be extended to form a string contained in L L . Namely,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, for any L ∈ 𝒞 L\\\\in\\\\mathcal{C} , define IsAccept L : Σ ∗ → { 0 , 1 } \\\\mathrm{IsAccept}_{L}:\\\\Sigma^{*}\\\\to\\\\{0,1\\\\} and NextLive L : Σ ∗ → { 0 , 1 , … , k } \\\\mathrm{NextLive}_{L}:\\\\Sigma^{*}\\\\to\\\\{0,1,\\\\dots,k\\\\} as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In words, IsAccept L ( x ) \\\\mathrm{IsAccept}_{L}(x) is the indicator for x x being in L L , whereas NextLive L ( x ) \\\\mathrm{NextLive}_{L}(x) counts the number of letters a ∈ Σ a\\\\in\\\\Sigma such that x a xa can be further extended to form a string contained in L L . Now fix the palette P = { 1 , 2 , … , k + 1 } P=\\\\{1,2,\\\\dots,k+1\\\\} , and define the trace coloring function c L : Σ ∗ → P c_{L}:\\\\Sigma^{*}\\\\to P as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If x ∉ Pref ( L ) x\\\\notin\\\\mathrm{Pref}(L) , c L ( x ) = 1 c_{L}(x)=1 ; otherwise, if x ∈ Pref ( L ) x\\\\in\\\\mathrm{Pref}(L) , then either IsAccept L ( x ) = 1 \\\\mathrm{IsAccept}_{L}(x)=1 , or NextLive L ( x ) > 0 \\\\mathrm{NextLive}_{L}(x)>0 ; hence c L ( x ) ≥ 1 c_{L}(x)\\\\geq 1 . Therefore, c L ( x ) ∈ { 1 , 2 , … , k + 1 } c_{L}(x)\\\\in\\\\{1,2,\\\\dots,k+1\\\\} for all x ∈ Σ ∗ x\\\\in\\\\Sigma^{*} . We now observe that the functions c L c_{L} satisfy monotonicity on any string x x with respect to inclusion.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any x ∈ Σ ∗ x\\\\in\\\\Sigma^{*} , and L ⊆ L ′ L\\\\subseteq L^{\\\\prime} , it holds that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any L , L ′ ∈ 𝒞 L,L^{\\\\prime}\\\\in\\\\mathcal{C} for which L ⊊ L ′ L\\\\subsetneq L^{\\\\prime} , and fix any z ∈ L ′ ∖ L z\\\\in L^{\\\\prime}\\\\setminus L .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case 1: z ∈ Pref ( L ) z\\\\in\\\\mathrm{Pref}(L) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this case, let t ∈ Σ ∗ t\\\\in\\\\Sigma^{*} be such that z t ∈ L zt\\\\in L (note that t ≠ ε t\\\\neq\\\\varepsilon , since z ∉ L z\\\\notin L ). We claim that x := z t x:=zt satisfies trace c L ( x ) ≠ trace c L ′ ( x ) {\\\\mathrm{trace}}_{c_{L}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x) . To see this, note that IsAccept L ( z ) = 0 \\\\mathrm{IsAccept}_{L}(z)=0 , but IsAccept L ′ ( z ) = 1 \\\\mathrm{IsAccept}_{L^{\\\\prime}}(z)=1 ; together with \\\\Cref observation:monotonicity-trace-coloring, this implies that c L ( z ) < c L ′ ( z ) c_{L}(z)<c_{L^{\\\\prime}}(z) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case 2: z ∉ Pref ( L ) z\\\\notin\\\\mathrm{Pref}(L) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this case, let w ∈ Σ ∗ w\\\\in\\\\Sigma^{*} be the maximal prefix of z z that satisfies w ∈ Pref ( L ) w\\\\in\\\\mathrm{Pref}(L) , and let z = w a t z=wat , where a ∈ Σ a\\\\in\\\\Sigma , and t ∈ Σ ∗ t\\\\in\\\\Sigma^{*} . Here, by maximal, we mean that w a ∉ Pref ( L ) wa\\\\notin\\\\mathrm{Pref}(L) . Note that w w can be ε \\\\varepsilon , but its existence is guaranteed since L L is non-empty.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, since w ∈ Pref ( L ) w\\\\in\\\\mathrm{Pref}(L) , there exists y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} such that w y ∈ L wy\\\\in L . We claim that x := w y x:=wy satisfies that trace c L ( x ) ≠ trace c L ′ ( x ) {\\\\mathrm{trace}}_{c_{L}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x) . To see this, observe that w a ∉ Pref ( L ) wa\\\\notin\\\\mathrm{Pref}(L) , since w w was chosen to be the maximal prefix of z z satisfying w ∈ Pref ( L ) w\\\\in\\\\mathrm{Pref}(L) . However, w a ∈ Pref ( L ′ ) wa\\\\in\\\\mathrm{Pref}(L^{\\\\prime}) , since z = w a t ∈ L ′ ∖ L z=wat\\\\in L^{\\\\prime}\\\\setminus L . Thus, NextLive L ( w ) < NextLive L ′ ( w ) \\\\mathrm{NextLive}_{L}(w)<\\\\mathrm{NextLive}_{L^{\\\\prime}}(w) . By \\\\Cref observation:monotonicity-trace-coloring, this implies that c L ( w ) < c L ′ ( w ) c_{L}(w)<c_{L^{\\\\prime}}(w) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In both cases, we have shown the existence of an x ∈ L x\\\\in L satisfying trace c L ( x ) ≠ trace c L ′ ( x ) {\\\\mathrm{trace}}_{c_{L}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x) . The next claim shows that at least for k = 2 k=2 , the bound achieved by our construction is tight.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There exists a finite collection 𝒞 \\\\mathcal{C} of non-empty languages over the binary alphabet Σ = { 0 , 1 } \\\\Sigma=\\\\{0,1\\\\} , such that any family of trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} that satisfies the distinguishable coloring condition must necessarily use 3 3 colors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the finite collection 𝒞 = { L 1 , L 2 , L 3 , L 4 , L 5 } \\\\mathcal{C}=\\\\{L_{1},L_{2},L_{3},L_{4},L_{5}\\\\} , where\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix any trace coloring functions c L 1 , c L 2 , c L 3 , c L 4 , c L 5 c_{L_{1}},c_{L_{2}},c_{L_{3}},c_{L_{4}},c_{L_{5}} that map to the palette { red , blue } \\\\{\\\\mathrm{red},\\\\mathrm{blue}\\\\} , and satisfy the distinguishable coloring condition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since L 1 L_{1} only contains ε \\\\varepsilon , it must hold that c L 1 ( ε ) ≠ c L 2 ( ε ) = c L 3 ( ε ) = c L 4 ( ε ) = c L 5 ( ε ) c_{L_{1}}(\\\\varepsilon)\\\\neq c_{L_{2}}(\\\\varepsilon)=c_{L_{3}}(\\\\varepsilon)=c_{L_{4}}(\\\\varepsilon)=c_{L_{5}}(\\\\varepsilon) . Without loss of generality, let c L 1 ( ε ) = red c_{L_{1}}(\\\\varepsilon)=\\\\mathrm{red} , so that c L 2 ( ε ) = c L 3 ( ε ) = c L 4 ( ε ) = c L 5 ( ε ) = blue c_{L_{2}}(\\\\varepsilon)=c_{L_{3}}(\\\\varepsilon)=c_{L_{4}}(\\\\varepsilon)=c_{L_{5}}(\\\\varepsilon)=\\\\mathrm{blue} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, for L 2 L_{2} , it must hold that c L 2 ( 0 ) ≠ c L 4 ( 0 ) = c L 5 ( 0 ) c_{L_{2}}(0)\\\\neq c_{L_{4}}(0)=c_{L_{5}}(0) . Similarly, for L 3 L_{3} , it must be the case that c L 3 ( 1 ) ≠ c L 4 ( 1 ) = c L 5 ( 1 ) c_{L_{3}}(1)\\\\neq c_{L_{4}}(1)=c_{L_{5}}(1) . Without loss of generality, suppose c L 2 ( 0 ) = red c_{L_{2}}(0)=\\\\mathrm{red} and c L 3 ( 1 ) = red c_{L_{3}}(1)=\\\\mathrm{red} , so that c L 4 ( 0 ) = c L 5 ( 0 ) = blue c_{L_{4}}(0)=c_{L_{5}}(0)=\\\\mathrm{blue} , and c L 4 ( 1 ) = c L 5 ( 1 ) = blue c_{L_{4}}(1)=c_{L_{5}}(1)=\\\\mathrm{blue} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"But now, consider the traces of L 4 L_{4} . By the previous assignments, we have that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"So, even though L 4 ⊊ L 5 L_{4}\\\\subsetneq L_{5} , every x ∈ L 4 x\\\\in L_{4} satisfies trace c L 4 ( x ) = trace c L 5 ( x ) {\\\\mathrm{trace}}_{c_{L_{4}}}(x)={\\\\mathrm{trace}}_{c_{L_{5}}}(x) . This contradicts the trace coloring functions satisfying the distinguishable coloring condition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We remark that the lower bound above does not, in general, preclude identifiability in the limit with trace coloring functions that use only 2 colors. Rather, it merely shows that the distinguishable coloring condition may not be achieved for all collections using 2 colors. Indeed, the collection above is finite, and every finite collection is identifiable without any color traces (Angluin, 1980 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Extension to Corrupted Traces\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also obtain the guarantee of \\\\Cref lemma:main-coloring-upper-bound in the setting where the color traces seen may be corrupted . Formally, we consider a notion of corrupted traces similar to Peng et al. ( 2026 ) , where the trace seen along with an input string x x is trace ~ c K ( x ) \\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x) , a corrupted version of trace c K ( x ) {\\\\mathrm{trace}}_{c_{K}}(x) . We restrict the nature of allowed corruptions as follows. For any x x , we require | trace ~ c K ( x ) | = | trace c K ( x ) | |\\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x)|=|{\\\\mathrm{trace}}_{c_{K}}(x)| (i.e., the corrupted trace must have the same length as the uncorrupted trace), and that every element in trace ~ c K ( x ) \\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x) belongs to the same palette P P used by c K c_{K} (i.e., the corrupted trace cannot introduce colors outside the palette). With these restrictions, we consider traces containing a bounded number of corruptions, as measured by the Hamming distance between the corrupted and uncorrupted traces. Namely, if trace c K ( x ) = ( a 1 , a 2 , … , a n ) {\\\\mathrm{trace}}_{c_{K}}(x)=(a_{1},a_{2},\\\\dots,a_{n}) and trace ~ c K ( x ) = ( b 1 , b 2 , … , b n ) \\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x)=(b_{1},b_{2},\\\\dots,b_{n}) , we restrict d ( a , b ) = | { i : a i ≠ b i , 1 ≤ i ≤ n } | ≤ ℓ d(a,b)=|\\\\{i:a_{i}\\\\neq b_{i},1\\\\leq i\\\\leq n\\\\}|\\\\leq\\\\ell , where ℓ > 0 \\\\ell>0 is an a priori known corruption budget.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this setting, we wish to correctly identify the target language K K in the limit for any target language K K , and for any enumeration ( x 1 , trace ~ c K ( x 1 ) ) , ( x 2 , trace ~ c K ( x 2 ) ) , … (x_{1},\\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x_{1})),(x_{2},\\\\widetilde{{\\\\mathrm{trace}}}_{c_{K}}(x_{2})),\\\\dots of K K with corrupted traces satisfying a corrupted budget ℓ \\\\ell . From the proof of \\\\Cref thm:characterization-identification-with-traces, it is straightforward to verify that a sufficient condition for identifiability of a collection 𝒞 \\\\mathcal{C} in this setting is the following: for every L ∈ 𝒞 L\\\\in\\\\mathcal{C} , there exists a finite tell-tale T L ⊆ L T_{L}\\\\subseteq L , such that for every L ′ ∈ 𝒞 L^{\\\\prime}\\\\in\\\\mathcal{C} satisfying L ′ ⊊ L L^{\\\\prime}\\\\subsetneq L , either (1) L ′ L^{\\\\prime} does not contain T L T_{L} , or (2) there exists x ∈ L ′ x\\\\in L^{\\\\prime} such that d ( trace c L ′ ( x ) , trace c L ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x),{\\\\mathrm{trace}}_{c_{L}}(x)\\\\right)>2\\\\ell .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, just like \\\\Cref lemma:main-coloring-upper-bound, we seek trace coloring functions that always satisfy (2) above. This will guarantee identifiability with corrupted traces having at most ℓ \\\\ell corruptions. Indeed, the next lemma constructs trace coloring functions that always satisfy (2) with a palette of size O ( k 2 ℓ + 1 ) O(k^{2\\\\ell+1}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒞 \\\\mathcal{C} be a language collection comprising of non-empty languages over a finite alphabet Σ \\\\Sigma of size k k , and fix ℓ > 0 \\\\ell>0 . There exist trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} mapping to a palette P P of size max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) \\\\max(8\\\\ell+4,8k^{2\\\\ell+1}) , such that for any L , L ′ ∈ 𝒞 L,L^{\\\\prime}\\\\in\\\\mathcal{C} that satisfy L ⊊ L ′ L\\\\subsetneq L^{\\\\prime} , and L L contains some string x x having length at least 2 ℓ 2\\\\ell , there exists x ∈ L x\\\\in L for which d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof of this lemma is given in \\\\Cref sec:robust-coloring-appendix. The high-level idea is to propagate the discrepancy at a single location implied by the construction in \\\\Cref lemma:main-coloring-upper-bound to all the 2 ℓ 2\\\\ell locations preceding it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Traces for Regular Languages\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The trace coloring functions that we constructed in \\\\Cref sec:main-coloring satisfy a special property for regular languages. Consider any regular language L L , and let M M be any Deterministic Finite-State Automaton (DFA) 3 3 3 We give a formal definition of a DFA in \\\\Cref sec:state-traces-appendix. that recognizes L L . Let Q ( M ) Q(M) be its set of states, δ M \\\\delta_{M} be the associated state transition function, q 0 ( M ) q_{0}(M) be its initial state and F ( M ) ⊆ Q ( M ) F(M)\\\\subseteq Q(M) its accepting states. For any state q q in Q ( M ) Q(M) , let W q W_{q} be the set of all the strings that arrive at state q q , starting from the initial state.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recall the definition of our trace coloring function c L c_{L} from ( 6 ). We claim that for any state q q , for any w , w ′ ∈ W q w,w^{\\\\prime}\\\\in W_{q} , c L ( w ) = c L ( w ′ ) c_{L}(w)=c_{L}(w^{\\\\prime}) . To see this, note that since w w and w ′ w^{\\\\prime} both end up at the state q q , for any suffix y y , w y wy and w ′ y w^{\\\\prime}y also end up in the same state in Q ( M ) Q(M) . This implies that w ∈ Pref ( L ) ⇔ w ′ ∈ Pref ( L ) w\\\\in\\\\mathrm{Pref}(L)\\\\iff w^{\\\\prime}\\\\in\\\\mathrm{Pref}(L) , and also that IsAccept L ( w ) = IsAccept L ( w ′ ) , NextLive L ( w ) = NextLive L ( w ′ ) \\\\mathrm{IsAccept}_{L}(w)=\\\\mathrm{IsAccept}_{L}(w^{\\\\prime}),\\\\mathrm{NextLive}_{L}(w)=\\\\mathrm{NextLive}_{L}(w^{\\\\prime}) . We conclude that c L ( w ) = c L ( w ′ ) c_{L}(w)=c_{L}(w^{\\\\prime}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now restate and prove \\\\Cref lemma:state-coloring-optimal-bound, which constructs state coloring functions using k + 1 k+1 colors so as to satisfy the distinguishable coloring condition, and also show that this palette size is optimal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"See 1.4\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any state q ∈ Q ( M ) q\\\\in Q(M) , we say that q q is a live state if there exists a suffix y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} such that δ M ( q , y ) ∈ F ( M ) \\\\delta_{M}(q,y)\\\\in F(M) ; otherwise, we call q q a dead state. Now, similar to the definitions in ( 4 ) and ( 5 ), define IsAccept M : Q ( M ) → { 0 , 1 } \\\\mathrm{IsAccept}_{M}:Q(M)\\\\to\\\\{0,1\\\\} and NextLive M : Q ( M ) → { 0 , 1 , … , k } \\\\mathrm{NextLive}_{M}:Q(M)\\\\to\\\\{0,1,\\\\dots,k\\\\} as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, define the state coloring function c M : Q ( M ) → P c_{M}:Q(M)\\\\to P as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can then verify that for any x = x 0 x 1 … x n ∈ L ( M ) x=x_{0}x_{1}\\\\dots x_{n}\\\\in L(M) (where x 0 = ε x_{0}=\\\\varepsilon ) that traverses the states q 0 , q 1 , … , q n q_{0},q_{1},\\\\dots,q_{n} in M M , and for any q i q_{i} , it holds that c M ( q i ) = c L ( M ) ( x 0 x 1 … x i ) c_{M}(q_{i})=c_{L(M)}(x_{0}x_{1}\\\\dots x_{i}) , where c L ( M ) c_{L(M)} is exactly the trace coloring function defined in ( 6 ) in the proof of \\\\Cref lemma:main-coloring-upper-bound. The conclusion of that lemma implies that { c M } M ∈ 𝒞 \\\\{c_{M}\\\\}_{M\\\\in\\\\mathcal{C}} satisfy the required distinguishable coloring condition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now show that k + 1 k+1 colors are optimal. Fix k ≥ 1 k\\\\geq 1 , and consider 𝒞 = { M 0 , M 1 , … , M k } \\\\mathcal{C}=\\\\{M_{0},M_{1},\\\\dots,M_{k}\\\\} . Each M i M_{i} has just two states q 0 ( M i ) q_{0}(M_{i}) and q 1 ( M i ) q_{1}(M_{i}) , where q 0 ( M i ) q_{0}(M_{i}) is the initial state, and also the only accepting state, and q 1 ( M i ) q_{1}(M_{i}) is a rejecting sink state. Namely, ∀ i ∀ a : \\\\forall i\\\\forall a: δ M i ( q 1 ( M i ) , a ) = q 1 ( M i ) \\\\delta_{M_{i}}(q_{1}(M_{i}),a)=q_{1}(M_{i}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, in M 0 M_{0} , we further set δ M 0 ( q 0 ( M 0 ) , a ) = q 1 ( M 0 ) \\\\delta_{M_{0}}(q_{0}(M_{0}),a)=q_{1}(M_{0}) for every a ∈ Σ a\\\\in\\\\Sigma . So, it holds that L ( M 0 ) = { ε } L(M_{0})=\\\\{\\\\varepsilon\\\\} . Next, for i ∈ { 1 , … , k } i\\\\in\\\\{1,\\\\dots,k\\\\} , we set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, L ( M i ) = { 1 , … , i } ∗ L(M_{i})=\\\\{1,\\\\dots,i\\\\}^{*} . Observe that L ( M 0 ) ⊊ L ( M 1 ) ⊊ ⋯ ⊊ L ( M k ) L(M_{0})\\\\subsetneq L(M_{1})\\\\subsetneq\\\\dots\\\\subsetneq L(M_{k}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any state coloring functions c M 0 , c M 1 , … , c M k c_{M_{0}},c_{M_{1}},\\\\dots,c_{M_{k}} that use a palette of size k k , and assume that they satisfy: for every M i , M j ∈ 𝒞 M_{i},M_{j}\\\\in\\\\mathcal{C} with L ( M i ) ⊊ L ( M j ) L(M_{i})\\\\subsetneq L(M_{j}) , there exists x ∈ L ( M i ) x\\\\in L(M_{i}) for which trace c M i ( x ) ≠ trace c M j ( x ) {\\\\mathrm{trace}}_{c_{M_{i}}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{M_{j}}}(x) . Fix any i , j ∈ { 0 , 1 , … , k } i,j\\\\in\\\\{0,1,\\\\dots,k\\\\} satisfying i < j i<j . Observe that any x ∈ L ( M i ) x\\\\in L(M_{i}) simply loops on the state q 0 ( M i ) q_{0}(M_{i}) , giving trace c M i ( x ) = ( c M i ( q 0 ( M i ) ) , … , c M i ( q 0 ( M i ) ) ) ⏟ | x | + 1 times {\\\\mathrm{trace}}_{c_{M_{i}}}(x)=\\\\underbrace{(c_{M_{i}}(q_{0}(M_{i})),\\\\dots,c_{M_{i}}(q_{0}(M_{i})))}_{|x|+1\\\\text{ times}} . Thus, for the assumed guarantee to hold, it must be the case that for every i < j i<j , c M i ( q 0 ( M i ) ) ≠ c M j ( q 0 ( M j ) ) c_{M_{i}}(q_{0}(M_{i}))\\\\neq c_{M_{j}}(q_{0}(M_{j})) . But this is not possible if the state coloring functions use only k k colors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In \\\\Cref sec:corrupted-state-traces-appendix, we extend the construction of the state coloring functions above to the setting with corrupted traces, similar to \\\\Cref sec:corruptions. Here, we show that a palette of size O ( k 2 ℓ + 1 ) O(k^{2\\\\ell+1}) suffices to achieve a similar guarantee as \\\\Cref lemma:corruptions-coloring-upper-bound, and that k 2 ℓ − 1 k^{2\\\\ell-1} colors are necessary in general.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, we restate and prove \\\\Cref thm:regular-languages-2-colors. See 1.5\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before stating the formal proof, we give some intuition, specifically since the requirement that every L i L_{i} be infinite might appear counterintuitive at first glance for this positive result. The trace coloring functions that we build interestingly arise via state coloring functions. That is, we carefully choose a sequence of automata M 1 , M 2 , … M_{1},M_{2},\\\\dots that accept L 1 , L 2 , … L_{1},L_{2},\\\\dots , prescribe state coloring functions for these automata, and then directly argue that they achieve the goal of identification with just two colors. This is in contrast to \\\\Cref lemma:state-coloring-optimal-bound, where the automata were fixed and provided to us, and we were concerned with the (stronger) distinguishable coloring condition; here, we choose suitable automata as a tool for constructing trace coloring functions for identification.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While there are several automata that recognize L i L_{i} , we want to choose M i M_{i} with a view to satisfying the following property: for each previous automaton M j M_{j} where j < i j<i and L j L_{j} is a superset of L i L_{i} , we can designate a particular state in M i M_{i} with a particular color, which helps distinguish it from M j M_{j} . In order to choose such a designated state for every j < i j<i , we need enough states (at least i − 1 i-1 ) in L i L_{i} to begin with. A sufficient condition to ensure this is that every L i L_{i} is infinite (intuitively, since there are long enough strings in the language, these must cause a loop, which can be “disentangled” to enlarge the state space). This ensures that with just two colors, we can distinguish the color traces of every language L i L_{i} from the color traces of every superset of L i L_{i} that appears before L i L_{i} . Note that this is a weaker property than the distinguishable coloring condition (which requires such a property for all supersets in the collection, not just the ones appearing before), but is sufficient for identification in the limit (see \\\\Cref remark:weaker-condition-sufficient).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now give the formal proof.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix M 1 , M 2 , … M_{1},M_{2},\\\\dots to be any arbitrary DFAs that recognize L 1 , L 2 , … L_{1},L_{2},\\\\dots . Now recall the definition of a live state in a DFA: a state q ∈ Q ( M ) q\\\\in Q(M) is live if ∃ \\\\exists a suffix y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} such that δ M ( q , y ) ∈ F ( M ) \\\\delta_{M}(q,y)\\\\in F(M) . Now call a state q q reachable if ∃ w ∈ Σ ∗ \\\\exists w\\\\in\\\\Sigma^{*} such that δ M ( q 0 ( M ) , w ) = q \\\\delta_{M}(q_{0}(M),w)=q . By \\\\Cref lemma:increasing-live-states proved in \\\\Cref sec:state-traces-appendix, we can assume that every M i M_{i} has at least i − 1 i-1 live and reachable states. We will now inductively determine a binary state coloring function c M i c_{M_{i}} for every i ≥ 1 i\\\\geq 1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To begin with, fix c M 1 c_{M_{1}} arbitrarily. Now, for any i > 1 i>1 , let us assume that c M 1 , … , c M i − 1 c_{M_{1}},\\\\dots,c_{M_{i-1}} have been determined. Consider M i M_{i} : there are at least i − 1 i-1 live and reachable states in M i M_{i} . Pick any i − 1 i-1 of these, and denote the set as { q b } b = 1 , … , i − 1 \\\\{q_{b}\\\\}_{b=1,\\\\dots,i-1} . Associate each q b q_{b} with a prefix w b ∈ Σ ∗ w_{b}\\\\in\\\\Sigma^{*} and a suffix y b ∈ Σ ∗ y_{b}\\\\in\\\\Sigma^{*} , such that δ M i ( q 0 ( M i ) , w b ) = q b \\\\delta_{M_{i}}(q_{0}(M_{i}),w_{b})=q_{b} , and δ M i ( q b , y b ) ∈ F ( M i ) \\\\delta_{M_{i}}(q_{b},y_{b})\\\\in F(M_{i}) , so that x b = w b y b ∈ L i x_{b}=w_{b}y_{b}\\\\in L_{i} . Now, consider the set { j < i : L j ⊋ L i } \\\\{j<i:L_{j}\\\\supsetneq L_{i}\\\\} : this set has size at most i − 1 i-1 . Thus, we can map every j j in this set to a distinct live q b q_{b} in M i M_{i} . Consider traversing the prefix w b w_{b} associated with q b q_{b} in M j M_{j} , and let q b ′ = δ M j ( q 0 ( M j ) , w b ) q^{\\\\prime}_{b}=\\\\delta_{M_{j}}(q_{0}(M_{j}),w_{b}) . We will set c M i ( q b ) = 1 − c M j ( q b ′ ) c_{M_{i}}(q_{b})=1-c_{M_{j}}(q^{\\\\prime}_{b}) . After doing this for every j j , we assign c M i ( q ) c_{M_{i}}(q) for the remaining states in M i M_{i} arbitrarily. This completes the specification of c M i ( q ) c_{M_{i}}(q) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now consider any i ≥ 1 i\\\\geq 1 . We claim that: ( ⋆ ) (\\\\star) for every j < i j<i , if L j ⊋ L i L_{j}\\\\supsetneq L_{i} , then there exists x x in L i L_{i} for which trace c M i ( x ) ≠ trace c M j ( x ) {\\\\mathrm{trace}}_{c_{M_{i}}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{M_{j}}}(x) . Indeed, fix any such L j L_{j} , and let q b q_{b} be the distinct live state in M i M_{i} that we had associated L j L_{j} with in the inductive construction of c M i c_{M_{i}} above. Now consider the string x b = w b y b ∈ L i x_{b}=w_{b}y_{b}\\\\in L_{i} associated with the state q b q_{b} in M i M_{i} . Note that the prefix w b w_{b} reaches the state q b ′ = δ M j ( q 0 ( M j ) , w b ) q^{\\\\prime}_{b}=\\\\delta_{M_{j}}(q_{0}(M_{j}),w_{b}) in M j M_{j} , whereas it reaches the state q b q_{b} in M i M_{i} ; our construction above guarantees that c M i ( q b ) = 1 − c M j ( q b ′ ) c_{M_{i}}(q_{b})=1-c_{M_{j}}(q^{\\\\prime}_{b}) . Thus, trace c M i ( x b ) ≠ trace c M j ( x b ) {\\\\mathrm{trace}}_{c_{M_{i}}}(x_{b})\\\\neq{\\\\mathrm{trace}}_{c_{M_{j}}}(x_{b}) , as claimed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, by \\\\Cref remark:weaker-condition-sufficient in the proof of \\\\Cref thm:characterization-identification-with-traces, the property ( ⋆ ) (\\\\star) is sufficient for the algorithm considered in that proof to identify the collection in the limit with traces.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion and Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We studied the problem of language identification with coarse-grained annotations, independent of any underlying machine implementation of the target language. Our main result shows that every countable language collection can be identified in the limit with annotations the same size as the input strings, with alphabet size one more than the input alphabet size.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One open problem is to prove a lower bound on the number of colors required for language identification with a color trace. For regular language collections, our result shows that two colors always suffice, independent of their alphabet size k k . Do constantly many colors always suffice for any arbitrary collection? Or must the number of colors grow linearly with k k ? Another open problem is to establish a lower bound on the number of colors required to achieve the distinguishable coloring condition — going beyond our lower bound of 3 colors seems challenging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regarding the question about the number of colors necessary for identification, very recently, we have been able to show that the result of \\\\Cref thm:regular-languages-2-colors, that trace coloring functions with just two colors suffice for identifying collections of regular languages that are all infinite, can in fact be generalized to collections of arbitrary languages that are all infinite. In fact, while trace coloring functions color every character in each string, this strengthened result can even be obtained with “terminal” coloring functions, where one only needs to assign a color to the entire string at once.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, the terminal coloring functions for each language in this 2-coloring result are collection-dependent (as are the trace coloring functions from \\\\Cref thm:regular-languages-2-colors). Note, in contrast, that our general-purpose trace-coloring functions from \\\\Cref thm:main-theorem which use k + 1 k+1 colors, are collection-independent . This raises the natural question: do there exist collection-independent terminal coloring functions that use only two colors, and suffice for identification in the limit for all countable language collections (possibly with the restriction that every language is infinite)?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It turns out that the answer to this question is yes; however, the required 2-coloring functions turn out to be highly non-constructive (and, we can prove, necessarily so in a precise sense), involving reasoning with uncountable ordinals and transfinite recursion. As a result, this new result is in a sense not directly comparable to the general trace coloring result in \\\\Cref thm:main-theorem of the present paper: the new result is quantitatively stronger (using 2 colors instead of k + 1 k+1 ), but it markedly lacks the constructive, natural formulation of \\\\Cref thm:main-theorem. Because the premise, techniques and analyses used to obtain these results largely depart from the present results in this paper, we defer them to a different manuscript (Charikar et al., 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work was supported by Moses Charikar’s and Gregory Valiant’s Simons Investigator Awards, a Google PhD Fellowship, a Simons Collaboration grant and a grant from the MacArthur Foundation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"sectionappendix\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 Characterization of Identification in the Limit with Color Traces\"\n },\n {\n \"type\": \"para\",\n \"text\": \"See 1.1\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"If direction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose that the stated condition holds for the collection 𝒞 = { L 1 , L 2 , … } \\\\mathcal{C}=\\\\{L_{1},L_{2},\\\\dots\\\\} and the associated trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} . Furthermore, let T L T_{L} be the finite tell-tale subset associated with language L L in the collection. Denote the unknown target language by K K , and consider the following identification algorithm: at time step t t , upon seeing ( x 1 , trace c K ( x 1 ) ) , … , ( x t , trace c K ( x t ) ) (x_{1},{\\\\mathrm{trace}}_{c_{K}}(x_{1})),\\\\dots,(x_{t},{\\\\mathrm{trace}}_{c_{K}}(x_{t})) , the algorithm outputs the smallest index i t i_{t} , which satisfies: (1) { x 1 , … , x t } ⊆ L i t \\\\{x_{1},\\\\dots,x_{t}\\\\}\\\\subseteq L_{i_{t}} , (2) T L i t ⊆ { x 1 , … , x t } T_{L_{i_{t}}}\\\\subseteq\\\\{x_{1},\\\\dots,x_{t}\\\\} , and (3) trace c L i t ( x j ) = trace c K ( x j ) {\\\\mathrm{trace}}_{c_{L_{i_{t}}}}(x_{j})={\\\\mathrm{trace}}_{c_{K}}(x_{j}) for every 1 ≤ j ≤ t 1\\\\leq j\\\\leq t . We remark that without the check in (3), the algorithm would be identical to Angluin’s algorithm for identification in the limit ( Angluin , 1980 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let z z be any index in the collection for which L z = K L_{z}=K , and trace c L z ( x ) = trace c K ( x ) {\\\\mathrm{trace}}_{c_{L_{z}}}(x)={\\\\mathrm{trace}}_{c_{K}}(x) for every x ∈ K x\\\\in K . Note that such an index exists since K ∈ 𝒞 K\\\\in\\\\mathcal{C} . Note also that requirements (1) and (3) hold for index z z at all time steps. Furthermore, since T L z T_{L_{z}} is a finite subset of K K , and the input eventually enumerates every string in K K , there exists a finite time at which requirement (2) gets satisfied as well. Thus, beyond this time, z z is a valid candidate index for the algorithm to output. We proceed assuming that we are beyond this time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, consider any j < z j<z for which L j ⊉ L z L_{j}\\\\nsupseteq L_{z} . For each such L j L_{j} , there exists some x ∈ L z ∖ L j x\\\\in L_{z}\\\\setminus L_{j} which is guaranteed to show up in the input at some finite time. At this time and beyond, index j j violates (1). Since there are only finitely many j < z j<z , we have that beyond some finite time, every j j satisfying j < z j<z and L j ⊉ L z L_{j}\\\\nsupseteq L_{z} violates (1). So, we assume being beyond this time as well, and next restrict our attention to indices j j that satisfy j < z j<z and L j ⊋ L z L_{j}\\\\supsetneq L_{z} . Suppose in one case that T L j ⊆ L z T_{L_{j}}\\\\subseteq L_{z} holds. Then, at some finite time, all of T L j T_{L_{j}} will show up in the input, and hence (2) will always be satisfied beyond this time by index j j . But now observe that we are in a position where L z L_{z} and L j L_{j} satisfy: L z ⊊ L j L_{z}\\\\subsetneq L_{j} and T L j ⊆ L z T_{L_{j}}\\\\subseteq L_{z} . By the condition in the theorem, it must then hold that there exists x ∈ L z x\\\\in L_{z} for which trace c L z ( x ) = trace c K ( x ) ≠ trace c L j ( x ) {\\\\mathrm{trace}}_{c_{L_{z}}}(x)={\\\\mathrm{trace}}_{c_{K}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L_{j}}}(x) . Such an x x is guaranteed to show up in the input at some finite time, together with trace c K ( x ) {\\\\mathrm{trace}}_{c_{K}}(x) ; at this time and beyond, index j j violates (3). In the other case, suppose that T L j ⊈ L z T_{L_{j}}\\\\nsubseteq L_{z} . In this case, at least one element in T L j T_{L_{j}} will never show up in the input, and hence (2) will never be satisfied by index j j . In all cases, we have shown that there exists a finite time beyond which index j j stops being a candidate for the algorithm. Since there are only finitely many j < z j<z , we conclude that the algorithm eventually always outputs some j ≤ z j\\\\leq z for which L j = L z = K L_{j}=L_{z}=K .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We remark that the analysis of the algorithm requires only the following weaker condition on the the countable collection 𝒞 = { L 1 , L 2 , … } \\\\mathcal{C}=\\\\{L_{1},L_{2},\\\\dots\\\\} and trace coloring functions { c L j } j ∈ ℕ \\\\{c_{L_{j}}\\\\}_{j\\\\in\\\\mathds{N}} : for every language L j ∈ 𝒞 L_{j}\\\\in\\\\mathcal{C} , there exists a finite tell-tale subset T L j ⊆ L j T_{L_{j}}\\\\subseteq L_{j} such that for every language L i ∈ 𝒞 L_{i}\\\\in\\\\mathcal{C} that is a proper subset of L j L_{j} and i > j i>j , either (1) L i L_{i} does not contain T L j T_{L_{j}} , or (2) there exists x ∈ L i x\\\\in L_{i} such that trace c L i ( x ) ≠ trace c L j ( x ) {\\\\mathrm{trace}}_{c_{L_{i}}}(x)\\\\neq{\\\\mathrm{trace}}_{c_{L_{j}}}(x) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Only if direction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a set of trace coloring functions { c L } L ∈ 𝒞 \\\\{c_{L}\\\\}_{L\\\\in\\\\mathcal{C}} , and suppose that there exists a language L ∈ 𝒞 L\\\\in\\\\mathcal{C} such that for every finite T ⊆ L T\\\\subseteq L , there exists a language L ′ ∈ 𝒞 L^{\\\\prime}\\\\in\\\\mathcal{C} that is a proper subset of L L , such that L ′ L^{\\\\prime} contains T T and also trace c L ′ ( x ) = trace c L ( x ) {\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)={\\\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′ x\\\\in L^{\\\\prime} . We will show that the collection is not identifiable in the limit with color traces given by these trace coloring functions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Towards this, fix any identification algorithm, and consider an adversary that starts enumerating the strings in L L , together with their color traces according to c L c_{L} , in the order that they appear in a fixed background ordering of L L . Then, at some finite time, the algorithm must guess an index i i such that L i = L L_{i}=L . Suppose that the input strings enumerated so far by the adversary comprise of the set T T . Then, by assumption, there exists a language L ′ ∈ 𝒞 L^{\\\\prime}\\\\in\\\\mathcal{C} that is a proper subset of L L , such that L ′ L^{\\\\prime} contains T T , and furthermore, trace c L ′ ( x ) = trace c L ( x ) {\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)={\\\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′ x\\\\in L^{\\\\prime} (including T T ). Thus, the adversary switches to enumerating strings in L ′ L^{\\\\prime} with their color traces (which they can legally do, since T ⊆ L ′ T\\\\subseteq L^{\\\\prime} , and the traces of L ′ L^{\\\\prime} and L L match on all strings in L ′ L^{\\\\prime} ). Now, the algorithm must guess an index i i satisfying L i = L ′ L_{i}=L^{\\\\prime} at some subsequent finite time. At this time, the adversary switches back to enumerating L L (they can do this since traces of L ′ L^{\\\\prime} and L L match at all strings in L ′ L^{\\\\prime} ), starting from the leftmost string that has not yet been enumerated in the fixed background ordering of L L . Again, at some subsequent finite time, the algorithm must guess an index i i such that L i = L L_{i}=L . Suppose that the input strings enumerated so far by the adversary comprise of the set T ′ T^{\\\\prime} . Then, by assumption again, there exists a language L ′′ ∈ 𝒞 L^{\\\\prime\\\\prime}\\\\in\\\\mathcal{C} that is a proper subset of L L , such that L ′′ L^{\\\\prime\\\\prime} contains T ′ T^{\\\\prime} , and furthermore, trace c L ′′ ( x ) = trace c L ( x ) {\\\\mathrm{trace}}_{c_{L^{\\\\prime\\\\prime}}}(x)={\\\\mathrm{trace}}_{c_{L}}(x) for every x ∈ L ′′ x\\\\in L^{\\\\prime\\\\prime} (including T ′ T^{\\\\prime} ). Thus, the adversary switches to enumerating strings in L ′′ L^{\\\\prime\\\\prime} . The adversary thus continues switching back and forth between L L and some other language L ~ ⊊ L \\\\tilde{L}\\\\subsetneq L endlessly. Since at each switch back to L L , the adversary begins with the leftmost string in L L that has not yet been enumerated in its fixed background ordering, the adversary eventually enumerates every string in L L along with its trace according to c L c_{L} . But also, each switch back to L L follows a time step at which the algorithm guesses an index i i for which L i = L ~ ⊊ L L_{i}=\\\\tilde{L}\\\\subsetneq L , and hence the algorithm guesses an incorrect index infinitely often. Thus, the algorithm does not identify L L in the limit.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 Robust Coloring Lemma\"\n },\n {\n \"type\": \"para\",\n \"text\": \"See 3.5\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any language L ∈ 𝒞 L\\\\in\\\\mathcal{C} , we extend the definitions of IsAccept L \\\\mathrm{IsAccept}_{L} and NextLive L \\\\mathrm{NextLive}_{L} from the proof of \\\\Cref lemma:main-coloring-upper-bound, and also consider some global quantities for the language. Namely, consider the functions IsAccept L \\\\mathrm{IsAccept}_{L} and NextLive L \\\\mathrm{NextLive}_{L} that map Σ ∗ \\\\Sigma^{*} to a non-negative integer, and the quantities ShortAccept L \\\\mathrm{ShortAccept}_{L} and ShortPrefix L \\\\mathrm{ShortPrefix}_{L} , defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, Σ { i , i + 1 , … , j } \\\\Sigma^{\\\\{i,i+1,\\\\dots,j}\\\\} denotes the set { y ∈ Σ ∗ : i ≤ | y | ≤ j } \\\\{y\\\\in\\\\Sigma^{*}:i\\\\leq|y|\\\\leq j\\\\} , with | y | = 0 |y|=0 implying the empty string y = ε y=\\\\varepsilon .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In words, IsAccept L ( x ) \\\\mathrm{IsAccept}_{L}(x) counts the number of suffixes y y having length | y | ∈ { 0 , 1 , … , 2 ℓ } |y|\\\\in\\\\{0,1,\\\\dots,2\\\\ell\\\\} for which x y xy is in L L , NextLive L ( x ) \\\\mathrm{NextLive}_{L}(x) counts the number of suffixes y y having length | y | ∈ { 1 , … , 2 ℓ + 1 } |y|\\\\in\\\\{1,\\\\dots,2\\\\ell+1\\\\} such that x y xy can be extended further to form a string in L L , ShortAccept L \\\\mathrm{ShortAccept}_{L} counts the number of strings y y having length | y | ∈ { 0 , … , 2 ℓ − 1 } |y|\\\\in\\\\{0,\\\\dots,2\\\\ell-1\\\\} such that y y belongs to L L , and ShortPrefix L \\\\mathrm{ShortPrefix}_{L} counts the number of prefixes y y having length | y | ∈ { 0 , … , 2 ℓ } |y|\\\\in\\\\{0,\\\\dots,2\\\\ell\\\\} such that y y can be extended further to form a string in L L . Note that ShortAccept L \\\\mathrm{ShortAccept}_{L} and ShortPrefix L \\\\mathrm{ShortPrefix}_{L} are global properties of the language L L . Furthermore, for any x x ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now fix the palette P = { 1 , 2 , … , B } P=\\\\{1,2,\\\\dots,B\\\\} , where B = max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) B=\\\\max(8\\\\ell+4,8k^{2\\\\ell+1}) , and define the trace coloring function c L : Σ ∗ → P c_{L}:\\\\Sigma^{*}\\\\to P as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If x ∉ Pref ( L ) x\\\\notin\\\\mathrm{Pref}(L) , c L ( x ) = 1 c_{L}(x)=1 ; otherwise, if x ∈ Pref ( L ) x\\\\in\\\\mathrm{Pref}(L) , then either IsAccept L ( x ) > 0 \\\\mathrm{IsAccept}_{L}(x)>0 , or NextLive L ( x ) > 0 \\\\mathrm{NextLive}_{L}(x)>0 , and hence c L ( x ) ≥ 1 c_{L}(x)\\\\geq 1 . Together with ( 12 ), this implies that c L ( x ) ∈ P c_{L}(x)\\\\in P for all x ∈ Σ ∗ x\\\\in\\\\Sigma^{*} . Similar to the proof of \\\\Cref lemma:main-coloring-upper-bound, we now observe that the quantities defined above satisfy monotonicity on any string x x with respect to inclusion.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let L , L ′ L,L^{\\\\prime} be languages satisfying L ⊆ L ′ L\\\\subseteq L^{\\\\prime} . Then, for any x ∈ Σ ∗ x\\\\in\\\\Sigma^{*} , it holds that:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any L , L ′ ∈ 𝒞 L,L^{\\\\prime}\\\\in\\\\mathcal{C} for which L ⊊ L ′ L\\\\subsetneq L^{\\\\prime} , and L L contains some string x x having length at least 2 ℓ 2\\\\ell .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case 1: ShortAccept L < ShortAccept L ′ \\\\mathrm{ShortAccept}_{L}<\\\\mathrm{ShortAccept}_{L^{\\\\prime}} or ShortPrefix L < ShortPrefix L ′ \\\\mathrm{ShortPrefix}_{L}<\\\\mathrm{ShortPrefix}_{L^{\\\\prime}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this case, using \\\\Cref observation:monotonicity-trace-coloring-corruptions, note that c L ( x ) < c L ′ ( x ) c_{L}(x)<c_{L^{\\\\prime}}(x) for every x ∈ L x\\\\in L . Then, choose any x ∈ L x\\\\in L having length at least 2 ℓ 2\\\\ell , which is guaranteed to exist by assumption. We have that each of c L ( ε ) , c L ( x ≤ 1 ) , … , c L ( x ) c_{L}(\\\\varepsilon),c_{L}(x_{\\\\leq 1}),\\\\dots,c_{L}(x) is strictly smaller than c L ′ ( ε ) , c L ′ ( x ≤ 1 ) , … , c L ′ ( x ) c_{L^{\\\\prime}}(\\\\varepsilon),c_{L^{\\\\prime}}(x_{\\\\leq 1}),\\\\dots,c_{L^{\\\\prime}}(x) respectively, and hence, d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell as required.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Case 2: ShortAccept L = ShortAccept L ′ \\\\mathrm{ShortAccept}_{L}=\\\\mathrm{ShortAccept}_{L^{\\\\prime}} and ShortPrefix L = ShortPrefix L ′ \\\\mathrm{ShortPrefix}_{L}=\\\\mathrm{ShortPrefix}_{L^{\\\\prime}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since L ⊊ L ′ L\\\\subsetneq L^{\\\\prime} , y ∈ L ⟹ y ∈ L ′ y\\\\in L\\\\implies y\\\\in L^{\\\\prime} , and y ∈ Pref ( L ) ⟹ y ∈ Pref ( L ′ ) y\\\\in\\\\mathrm{Pref}(L)\\\\implies y\\\\in\\\\mathrm{Pref}(L^{\\\\prime}) . Therefore, this case implies the stronger properties that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any z ∈ L ′ ∖ L z\\\\in L^{\\\\prime}\\\\setminus L . Note that by ( 14 ), it must be the case that | z | ≥ 2 ℓ |z|\\\\geq 2\\\\ell . We then have the following two subcases:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Subcase 2a: z ∈ Pref ( L ) z\\\\in\\\\mathrm{Pref}(L) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this case, let t ∈ Σ ∗ t\\\\in\\\\Sigma^{*} be such that z t ∈ L zt\\\\in L (note that t ≠ ε t\\\\neq\\\\varepsilon , since z ∉ L z\\\\notin L ). We claim that x := z t x:=zt satisfies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell . To see this, let z = z 0 z 1 , … , z n z=z_{0}z_{1},\\\\dots,z_{n} , where z 0 = ε z_{0}=\\\\varepsilon and n ≥ 2 ℓ n\\\\geq 2\\\\ell . For any p ≤ q p\\\\leq q , denote z p : q := z p z p + 1 … , z q z_{p:q}:=z_{p}z_{p+1}\\\\dots,z_{q} . We then observe that for any i ∈ { 0 , 1 , … , 2 ℓ } i\\\\in\\\\{0,1,\\\\dots,2\\\\ell\\\\} , IsAccept L ( z 0 : n − i ) \\\\mathrm{IsAccept}_{L}(z_{0:n-i}) is strictly smaller than IsAccept L ′ ( z 0 : n − i ) \\\\mathrm{IsAccept}_{L^{\\\\prime}}(z_{0:n-i}) ; this is because there exists a suffix y = z n − i + 1 : n y=z_{n-i+1:n} having length | y | ∈ { 0 , … , 2 ℓ } |y|\\\\in\\\\{0,\\\\dots,2\\\\ell\\\\} , for which z 0 : n − i y ∉ L z_{0:n-i}y\\\\notin L but z 0 : n − i y ∈ L ′ z_{0:n-i}y\\\\in L^{\\\\prime} (note that z 0 : n − i y z_{0:n-i}y is simply equal to z z ). This implies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Subcase 2b: z ∉ Pref ( L ) z\\\\notin\\\\mathrm{Pref}(L) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this case, let w ∈ Σ ∗ w\\\\in\\\\Sigma^{*} be the maximal prefix of z z that satisfies w ∈ Pref ( L ) w\\\\in\\\\mathrm{Pref}(L) , and let z = w a t z=wat , where a ∈ Σ a\\\\in\\\\Sigma , and t ∈ Σ ∗ t\\\\in\\\\Sigma^{*} . Here, by maximal, we mean that w a ∉ Pref ( L ) wa\\\\notin\\\\mathrm{Pref}(L) . We first claim that | w | ≥ 2 ℓ |w|\\\\geq 2\\\\ell . To see this, suppose that | w | < 2 ℓ |w|<2\\\\ell . In this case, | w a | ≤ 2 ℓ |wa|\\\\leq 2\\\\ell . But it also holds that w a ∉ Pref ( L ) wa\\\\notin\\\\mathrm{Pref}(L) , and w a ∈ Pref ( L ′ ) wa\\\\in\\\\mathrm{Pref}(L^{\\\\prime}) (since z = w a t ∈ L ′ z=wat\\\\in L^{\\\\prime} ). This contradicts ( 15 ). Thus, | w | ≥ 2 ℓ |w|\\\\geq 2\\\\ell .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"So, let w = w 0 w 1 , … , w n w=w_{0}w_{1},\\\\dots,w_{n} , where w 0 = ε w_{0}=\\\\varepsilon . We now argue that for every i ∈ { 0 , 1 , … , 2 ℓ } i\\\\in\\\\{0,1,\\\\dots,2\\\\ell\\\\} , NextLive L ( w 0 : n − i ) \\\\mathrm{NextLive}_{L}(w_{0:n-i}) is strictly smaller than NextLive L ′ ( w 0 : n − i ) \\\\mathrm{NextLive}_{L^{\\\\prime}}(w_{0:n-i}) . Again, this holds because there exists a suffix y = w n − i + 1 : n a y=w_{n-i+1:n}a having length | y | ∈ { 1 , … , 2 ℓ + 1 } |y|\\\\in\\\\{1,\\\\dots,2\\\\ell+1\\\\} , for which w 0 : n − i y ∉ Pref ( L ) w_{0:n-i}y\\\\notin\\\\mathrm{Pref}(L) but w 0 : n − i y ∉ Pref ( L ′ ) w_{0:n-i}y\\\\notin\\\\mathrm{Pref}(L^{\\\\prime}) (note that w 0 : n − i y w_{0:n-i}y is simply equal to w a wa ). This then implies that d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In both cases, we have shown the existence of an x ∈ L x\\\\in L satisfying d ( trace c L ( x ) , trace c L ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{L}}(x),{\\\\mathrm{trace}}_{c_{L^{\\\\prime}}}(x)\\\\right)>2\\\\ell , as required.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We give a formal definition of a DFA following the textbooks Sipser ( 1996 ); Hopcroft et al. ( 2001 ) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A DFA M M over a finite alphabet Σ \\\\Sigma is a 5-tuple M = ( Q ( M ) , Σ , δ M , q 0 ( M ) , F ( M ) ) M=(Q(M),\\\\Sigma,\\\\delta_{M},q_{0}(M),F(M)) , where Q ( M ) Q(M) is a finite set of states, δ M : Q ( M ) × Σ → Q ( M ) \\\\delta_{M}:Q(M)\\\\times\\\\Sigma\\\\to Q(M) is a state transition function, q 0 ( M ) ∈ Q ( M ) q_{0}(M)\\\\in Q(M) is the initial state, and F ( M ) ⊆ Q ( M ) F(M)\\\\subseteq Q(M) is the set of accepting states. The regular language L ( M ) L(M) that is accepted by the DFA M M is defined as the set L ( M ) = { x ∈ Σ ∗ : δ M ( q 0 ( M ) , x ) ∈ F ( M ) } L(M)=\\\\{x\\\\in\\\\Sigma^{*}:\\\\delta_{M}(q_{0}(M),x)\\\\in F(M)\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We can extend the domain of the transition function δ M \\\\delta_{M} from Σ \\\\Sigma to Σ ∗ \\\\Sigma^{*} by defining δ M ( q , ε ) = q \\\\delta_{M}(q,\\\\varepsilon)=q , and recursively defining δ M ( q , a y ) = δ M ( δ M ( q , a ) , y ) \\\\delta_{M}(q,ay)=\\\\delta_{M}(\\\\delta_{M}(q,a),y) , for every q ∈ Q ( M ) , a ∈ Σ q\\\\in Q(M),a\\\\in\\\\Sigma and y ∈ Σ ∗ y\\\\in\\\\Sigma^{*} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Corrupted State Traces\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As we did in \\\\Cref sec:state-machines, we describe how we can derive state coloring functions that are robust to corruptions in state traces from the trace coloring functions constructed in \\\\Cref lemma:corruptions-coloring-upper-bound above. Consider any DFA M M in a given collection 𝒞 = { M 1 , M 2 , … } \\\\mathcal{C}=\\\\{M_{1},M_{2},\\\\dots\\\\} , and fix ℓ > 0 \\\\ell>0 . For any q ∈ Q ( M ) q\\\\in Q(M) , define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that ShortAccept M \\\\mathrm{ShortAccept}_{M} and ShortPrefix M \\\\mathrm{ShortPrefix}_{M} are global properties of the DFA M M .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now fix the palette P = { 1 , 2 , … , B } P=\\\\{1,2,\\\\dots,B\\\\} , where B = max ( 8 ℓ + 4 , 8 k 2 ℓ + 1 ) B=\\\\max(8\\\\ell+4,8k^{2\\\\ell+1}) , and define the state coloring function c M : Q ( M ) → P c_{M}:Q(M)\\\\to P as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any x = x 0 x 1 … x n ∈ L ( M ) x=x_{0}x_{1}\\\\dots x_{n}\\\\in L(M) (where x 0 = ε x_{0}=\\\\varepsilon ), which traverses the states q 0 , q 1 , … , q n q_{0},q_{1},\\\\dots,q_{n} in M M , we have that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where c L ( M ) c_{L(M)} is exactly the trace coloring function defined in ( 13 ) in the proof of \\\\Cref lemma:corruptions-coloring-upper-bound. Thus, we have that for any M , M ′ ∈ 𝒞 M,M^{\\\\prime}\\\\in\\\\mathcal{C} satisfying L ( M ) ⊊ L ( M ′ ) L(M)\\\\subsetneq L(M^{\\\\prime}) , there exists an x ∈ L ( M ) x\\\\in L(M) for which d ( trace c M ( x ) , trace c M ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{M}}(x),{\\\\mathrm{trace}}_{c_{M^{\\\\prime}}}(x)\\\\right)>2\\\\ell , and we can identify 𝒞 \\\\mathcal{C} in the limit using state color traces having at most ℓ \\\\ell corruptions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analogous to \\\\Cref lemma:state-coloring-optimal-bound, we next show that the palette size used above is optimal in order for state coloring functions in general.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any k ≥ 1 k\\\\geq 1 and ℓ > 0 \\\\ell>0 , there exists a finite collection 𝒞 \\\\mathcal{C} of DFAs over the alphabet Σ = { 1 , 2 , … , k } \\\\Sigma=\\\\{1,2,\\\\dots,k\\\\} , such that any family of state coloring functions { c M } M ∈ 𝒞 \\\\{c_{M}\\\\}_{M\\\\in\\\\mathcal{C}} which satisfies: for every pair of DFAs M , M ′ ∈ 𝒞 M,M^{\\\\prime}\\\\in\\\\mathcal{C} satisfying that L ( M ) ⊊ L ( M ′ ) L(M)\\\\subsetneq L(M^{\\\\prime}) , there exists x ∈ L ( M ) x\\\\in L(M) for which d ( trace c M ( x ) , trace c M ′ ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{M}}(x),{\\\\mathrm{trace}}_{c_{M^{\\\\prime}}}(x)\\\\right)>2\\\\ell , necessarily requires a palette of size k 2 ℓ − 1 k^{2\\\\ell-1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let N = k 2 ℓ − 1 N=k^{2\\\\ell-1} , and S = { s 1 , … , s N } S=\\\\{s_{1},\\\\dots,s_{N}\\\\} be an ordering of all strings of size 2 ℓ − 1 2\\\\ell-1 in Σ 2 ℓ − 1 \\\\Sigma^{2\\\\ell-1} . Consider the collection 𝒞 = { M 1 , … , M N } \\\\mathcal{C}=\\\\{M_{1},\\\\dots,M_{N}\\\\} , where each M i M_{i} accepts the regular language\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Namely, L ( M i ) L(M_{i}) has strings of the form w y wy , where w w is any of the first i i strings in S S , and y y is any suffix in Σ ∗ \\\\Sigma^{*} . We have that L ( M 1 ) ⊊ L ( M 2 ) ⊊ ⋯ ⊊ L ( M N ) L(M_{1})\\\\subsetneq L(M_{2})\\\\subsetneq\\\\dots\\\\subsetneq L(M_{N}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now observe that each M i M_{i} can be specified by a DFA that first builds a prefix tree over the set { s 1 , … , s i } \\\\{s_{1},\\\\dots,s_{i}\\\\} , followed by an accepting sink state. Namely, for any x x , the prefix tree part checks whether the first 2 ℓ − 1 2\\\\ell-1 characters of x x belong to { s 1 , … , s i } \\\\{s_{1},\\\\dots,s_{i}\\\\} —if they do, then the string is accepted. Let q 0 ( M i ) q_{0}(M_{i}) be the initial state of M i M_{i} (i.e., the root of the prefix tree), and let q F ( M i ) q_{F}(M_{i}) be the single accepting sink state. Note then that any x ∈ L ( M i ) x\\\\in L(M_{i}) traverses a sequence of states of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, fix any state coloring functions c M 1 , … , c M N c_{M_{1}},\\\\dots,c_{M_{N}} that use a palette P P of size N − 1 N-1 , and satisfy that for every M i , M j M_{i},M_{j} with L ( M i ) ⊊ L ( M j ) L(M_{i})\\\\subsetneq L(M_{j}) , there exists x ∈ L ( M i ) x\\\\in L(M_{i}) for which d ( trace c M i ( x ) , trace c M j ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{M_{i}}}(x),{\\\\mathrm{trace}}_{c_{M_{j}}}(x)\\\\right)>2\\\\ell .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix any i , j ∈ { 1 , … , N } i,j\\\\in\\\\{1,\\\\dots,N\\\\} satisfying i < j i<j . For any x ∈ L ( M i ) x\\\\in L(M_{i}) , by the characterization ( 22 ) of the sequence of states traversed by x x in M i M_{i} , we get that trace c M i ( x ) {\\\\mathrm{trace}}_{c_{M_{i}}}(x) is some sequence in P 2 ℓ P^{2\\\\ell} , followed by the sequence ( c M i ( q F ( M i ) ) , … , c M i ( q F ( M i ) ) ) ⏟ | x | − 2 ℓ + 1 times \\\\underbrace{(c_{M_{i}}(q_{F}(M_{i})),\\\\dots,c_{M_{i}}(q_{F}(M_{i})))}_{|x|-2\\\\ell+1\\\\text{ times}} . Similarly, we have that trace c M j ( x ) {\\\\mathrm{trace}}_{c_{M_{j}}}(x) is some sequence in P 2 ℓ P^{2\\\\ell} , followed by the sequence c M j ( q F ( M j ) , … , c M j ( q F ( M j ) ) ) ⏟ | x | − 2 ℓ + 1 times \\\\underbrace{c_{M_{j}}(q_{F}(M_{j}),\\\\dots,c_{M_{j}}(q_{F}(M_{j})))}_{|x|-2\\\\ell+1\\\\text{ times}} . Thus, in order for d ( trace c M i ( x ) , trace c M j ( x ) ) > 2 ℓ d\\\\left({\\\\mathrm{trace}}_{c_{M_{i}}}(x),{\\\\mathrm{trace}}_{c_{M_{j}}}(x)\\\\right)>2\\\\ell to hold, we necessarily require that c M i ( q F ( M i ) ) ≠ c M j ( q F ( M j ) ) c_{M_{i}}(q_{F}(M_{i}))\\\\neq c_{M_{j}}(q_{F}(M_{j})) . But this is not possible for every i < j i<j , if the state coloring functions use only N − 1 N-1 colors.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Identifying Regular Languages with Two Colors\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let M M be a DFA satisfying | L ( M ) | = ∞ |L(M)|=\\\\infty , and let M M have n n states that are live and reachable. Then, for any k > 0 k>0 , there exists a DFA M ′ M^{\\\\prime} satisfying L ( M ′ ) = L ( M ) L(M^{\\\\prime})=L(M) , such that M ′ M^{\\\\prime} has n + k n+k states that are live and reachable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since L ( M ) L(M) is infinite, it must contain strings of arbitrarily large length. Let x = x 1 x 2 … x t x=x_{1}x_{2}\\\\dots x_{t} be any string in L ( M ) L(M) having length t ≥ n t\\\\geq n . Then, consider the sequence of states q 0 , q 1 , … , q t q_{0},q_{1},\\\\dots,q_{t} traversed by x x in M M : note that every state in this sequence is live and reachable. Then, since there are only n n live and reachable states in M M , and the number of states in this sequence is t + 1 > n t+1>n , by the pigeonhole principle, there exists at least one live and reachable state that appears twice in the sequence. Let s s be the smallest index for which there exists i < s i<s such that q i = q s q_{i}=q_{s} (i.e., this is the first time a state got repeated in the sequence), and consider the intermediate sequence of transitions q i → q i + 1 → … → q s − 1 → q s q_{i}\\\\to q_{i+1}\\\\to\\\\dots\\\\to q_{s-1}\\\\to q_{s} , together with the symbols x i + 1 , … , x s − 1 , x s x_{i+1},\\\\dots,x_{s-1},x_{s} that effected these transitions. Note that by the choice of s s , the states q 0 , … , q s − 1 q_{0},\\\\dots,q_{s-1} are all distinct.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Now, consider a DFA M ′ M^{\\\\prime} that operates on the same alphabet Σ \\\\Sigma . M ′ M^{\\\\prime} has a copy q ′ q^{\\\\prime} of every state q q in M M , plus an additional state q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} . The state transitions for all the states in M ′ M^{\\\\prime} are identical to their copies in M M , except for the states q s − 1 ′ q^{\\\\prime}_{s-1} and q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} , where q s − 1 q_{s-1} is the copy of the state q s − 1 q_{s-1} , and q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} is the additional new state. The state transitions for q s − 1 ′ q^{\\\\prime}_{s-1} and q new q_{{\\\\mathrm{new}}} are as follows. First, we make the transitions out of q new ′ q^{\\\\prime}_{\\\\mathrm{new}} to be identical to those out of q s q_{s} , i.e., set δ M ′ ( q new ′ , a ) \\\\delta_{M^{\\\\prime}}(q^{\\\\prime}_{{\\\\mathrm{new}}},a) to be the copy of δ M ( q s , a ) \\\\delta_{M}(q_{s},a) for every a ∈ Σ a\\\\in\\\\Sigma . Next, we redirect the single transition q s − 1 ′ → x s q s ′ q^{\\\\prime}_{s-1}\\\\xrightarrow{x_{s}}q^{\\\\prime}_{s} to be q s − 1 ′ → x s q new ′ q^{\\\\prime}_{s-1}\\\\xrightarrow{x_{s}}q^{\\\\prime}_{\\\\mathrm{new}} instead; namely, for any a ≠ x s a\\\\neq x_{s} , we set δ M ′ ( q s − 1 ′ , a ) \\\\delta_{M^{\\\\prime}}(q^{\\\\prime}_{s-1},a) to be the copy of the state δ M ( q s − 1 , a ) \\\\delta_{M}(q_{s-1},a) , and we make q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} reachable by setting δ M ′ ( q s − 1 ′ , x s ) = q new ′ \\\\delta_{M^{\\\\prime}}(q^{\\\\prime}_{s-1},x_{s})=q^{\\\\prime}_{{\\\\mathrm{new}}} . In essence, we have redirected the transition from q s − 1 ′ q^{\\\\prime}_{s-1} which was looping back into q s ′ q^{\\\\prime}_{s} to the new state q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} , whose outward transitions are then identical to q s q_{s} . The initial state in M ′ M^{\\\\prime} is set to the copy of the initial state in M M , and the accepting states in M ′ M^{\\\\prime} are set to the copies of the accepting states in M M , plus the state q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} if q s q_{s} was an accepting state. This completes the specification of M ′ M^{\\\\prime} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now argue that M ′ M^{\\\\prime} has n + 1 n+1 live and reachable states, and that L ( M ′ ) = L ( M ) L(M^{\\\\prime})=L(M) ; the lemma is then established by inducting this k k times. First, it is clear that for any state q q in M M that was either not live or reachable in M M , its copy q ′ q^{\\\\prime} has the same status in M ′ M^{\\\\prime} . Now, by construction, the newly added state q new ′ q^{\\\\prime}_{{\\\\mathrm{new}}} and the state q s ′ q^{\\\\prime}_{s} are both live and reachable, as witnessed by the traversal of x x through M ′ M^{\\\\prime} , which passes through both q s ′ q^{\\\\prime}_{s} and q new ′ q^{\\\\prime}_{\\\\mathrm{new}} on the way to an accepting state. Finally, for every state q ≠ q s q\\\\neq q_{s} that was live and reachable in M M , its copy q ′ q^{\\\\prime} remains live and reachable in M ′ M^{\\\\prime} . This can be seen by tracing out the sequence of states traversed by any y ∈ L ( M ) y\\\\in L(M) that passed through q q on the way to an accepting state in M M . In its traversal through M ′ M^{\\\\prime} , the only notable change is that a transition of the form q s − 1 → x s q s q_{s-1}\\\\xrightarrow{x_{s}}q_{s} gets replaced by q s − 1 ′ → x s q new ′ q^{\\\\prime}_{s-1}\\\\xrightarrow{x_{s}}q^{\\\\prime}_{{\\\\mathrm{new}}} . But since the transitions out of q new ′ q^{\\\\prime}_{\\\\mathrm{new}} are identical to q s q_{s} , y y continues on its path to an accepting state, and any occurrences of q q in the traversal through M M overlap with occurrences of q ′ q^{\\\\prime} in the traversal through M ′ M^{\\\\prime} ; thus, q ′ q^{\\\\prime} remains live and reachable in M ′ M^{\\\\prime} . Thus, we have argued that M ′ M^{\\\\prime} has n + 1 n+1 live and reachable states.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It remains to argue that L ( M ′ ) = L ( M ) L(M^{\\\\prime})=L(M) . Again, this is made clear by tracing out the traversal of any x x through M M and M ′ M^{\\\\prime} : if the traversal never uses the single redirected transition q s − 1 ′ → x s q new ′ q^{\\\\prime}_{s-1}\\\\xrightarrow{x_{s}}q^{\\\\prime}_{{\\\\mathrm{new}}} , then the end outcomes are clearly identical; if it does use this transition, then in M M , the transition leads into the state q s q_{s} , whereas in M ′ M^{\\\\prime} , the transition leads into the state q new ′ q^{\\\\prime}_{\\\\mathrm{new}} ; however, since the transitions out of q new ′ q^{\\\\prime}_{\\\\mathrm{new}} and q s q_{s} are identical, the end outcome is identical as well.\"\n }\n]","isConversational":false}}},{"rowIdx":299,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.14145"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"ToolAnchor: Anchoring Counterfactual Context to Boost Agentic Tool-use Capability"},"authors":{"kind":"list like","value":["Weiting Liu","Jieyi Bi","Wanqi Zhou","Jianfeng Feng","Yining Ma","Ai Han","Wenlian Lu"],"string":"[\n \"Weiting Liu\",\n \"Jieyi Bi\",\n \"Wanqi Zhou\",\n \"Jianfeng Feng\",\n \"Yining Ma\",\n \"Ai Han\",\n \"Wenlian Lu\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.14145v1"},"published":{"kind":"timestamp","value":"2026-07-14T00:00:00","string":"2026-07-14T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Recent advances in tool-augmented agents have significantly broadened the real-world utility of Large Language Models (LLMs) Singh et al. ( 2025b ); Jiang et al. ( 2025 ); Li et al. ( 2025b ); Fan et al. ( 2026 ); Li et al. ( 2026 ); Fang et al. ( 2026 ) . By integrating external tools into the reasoning loop, LLM-based agents have achieved strong performance across a wide spectrum of downstream tasks, spanning text-based question answering Wu et al. ( 2025b ); Wei et al. ( 2025 ); Zhou et al. ( 2025 ); Mialon et al. ( 2023 ) , visual question answering Yang et al. ( 2025 ); Huang et al. ( 2026 ); Zhang et al. ( 2026a ) , long-form text report generation Li et al. ( 2025e ); Du et al. ( 2025a ) , and domain-grounded tool use Yao et al. ( 2024 ); Barres et al. ( 2025 ) . Central to this progress are the ReAct paradigm Yao et al. ( 2022 ) , which interleaves reasoning with tool invocation, and agentic reinforcement learning (agentic RL) methods Shao et al. ( 2024 ); Yu et al. ( 2025 ); Zhang et al. ( 2025a ) , which further refine tool-use capabilities via post-training. However, these successes largely assume a fixed, predefined toolset available throughout training. In practice, task requirements evolve continuously, often necessitating new tools such as web search for enhanced retrieval Jin et al. ( 2025 ); Li et al. ( 2025c ) , python interpreters for scientific computation Feng et al. ( 2025 ); Xue et al. ( 2025 ) , file parsers for document understanding Team et al. ( 2025 ) , and visual tools for multimodal perception Huang et al. ( 2026 ); Fan et al. ( 2026 ) ."},{"type":"para","text":"A natural question then arises: how can an existing agent efficiently adapt to an expanded toolset? The most straightforward approach is to directly apply agentic RL, allowing the agent to discover effective reasoning strategies and action policies via exploration. However, this strategy faces a fundamental difficulty. Since the agent’s policy has already converged during prior agentic RL training, it tends to fall back on familiar behavioral patterns, underutilizing newly introduced tools and repeating suboptimal reasoning strategies. This behavioral inertia leads to sparse rewards and inefficient exploration, as the agent rarely ventures beyond its established routines. An alternative is to rely on the tool-use generalization capability of the agent Zou et al. ( 2025 ); Du et al. ( 2025b ) , hoping that it can adapt to new tools without additional training. Yet, as our preliminary analysis in Section 3.3 reveals, generalization alone is insufficient. To better understand this bottleneck, we conduct a motivation study on failed student trajectories. A teacher model first identifies a decision round where an alternative local thought–action pair could plausibly redirect the subsequent rollout; we call this decision point a counterfactual anchor round (Section 3.2 ). The teacher then proposes this alternative using only the history before that round, yielding a teacher-revised context from which the student resumes generation. Compared with resuming from the original context, these teacher-revised contexts improve task success in both textual and visual search settings. This suggests that failures arise not only from limited tool-use capability, but also from missing contextual anchors that redirect reasoning and tool selection at key decision points."},{"type":"para","text":"In this work, we propose ToolAnchor , a framework that bootstraps new tool-use capabilities through verified counterfactual context anchoring. ToolAnchor operates in three stages. In the Counterfactual Anchor hypothesis stage, diverse teacher models audit the student agent’s failed rollout trajectories and hypothesize anchor rounds where the agent’s decision deviated from a promising path. While the primary focus is on rounds where the agent failed to invoke a newly introduced tool, the teacher also considers suboptimal uses of existing tools and flawed reasoning steps, as these often compound the difficulty of adopting new tools effectively. For each identified anchor round, the teacher hypothesizes alternative reasoning and actions that would plausibly redirect the trajectory toward task completion. In the Counterfactual Anchor Verification stage, the student agent resumes generation from these teacher-anchored contexts under agentic rollout. Crucially, we retain for training only those anchor contexts that satisfy both verification criteria, namely task-level success and anchor-level usefulness, rather than accepting teacher hypotheses uncritically. In the Counterfactual Anchor Internalization stage, the verified contexts are used for agentic supervised fine-tuning (agentic SFT), allowing the agent to absorb the anchored behaviors into its own policy. This is followed by agentic RL, which realigns the internalized anchor behaviors with the agent’s previously learned agentic policy and supports robust, autonomous use of the expanded toolset."},{"type":"para","text":"Our contributions are summarized as follows: ( 1 ) We formalize the practical challenge of expanding a post-trained agent’s toolset. Through a motivation study, we identify behavioral inertia as the core bottleneck and reveal that counterfactual contextual anchoring at critical decision points can effectively redirect failed trajectories. ( 2 ) We propose ToolAnchor, a scalable, three-stage framework for capability expansion. It automates the discovery of counterfactual anchor rounds via diverse teacher models, strictly verifies their effectiveness through student rollouts, and internalizes these validated behaviors via agentic SFT and RL. ( 3 ) Extensive evaluations across general AI assistance (GAIA), textual search (BrowseComp), and visual search (VDR-Bench) demonstrate that ToolAnchor seamlessly integrates new visual tools. Furthermore, it simultaneously boosts the base agent’s reasoning and existing tool mastery, offering a robust paradigm for continuous agentic evolution."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"heading","level":4,"text":"Tool-augmented deep research agents."},{"type":"para","text":"Deep research agents tackle open-ended question answering through long-horizon tool use: iteratively planning, invoking external tools, verifying evidence, performing intermediate reasoning, and synthesizing final answers Li et al. ( 2025b ) . This paradigm naturally encompasses general AI assistance, textual search, and visual search. Textual search remains the most extensively explored direction, where agents retrieve external evidence to answer complex queries. For instance, Search-R1 Jin et al. ( 2025 ) , Search-o1 Li et al. ( 2025c ) , and WebThinker Li et al. ( 2025d ) directly integrate web search into reasoning, while subsequent work advances this via agentic SFT and RL Wu et al. ( 2025a ) , systematic uncertainty reduction Li et al. ( 2025a ) , scalable data synthesis Qiao et al. ( 2025 ) , dual-environment RL frameworks Li et al. ( 2025a ) , and continual pre-training Su et al. ( 2025 ) . Building on these techniques, the Tongyi DeepResearch Agent Team et al. ( 2025 ) achieves competitive performance, and OpenSeeker Du et al. ( 2026 ) provides high-quality training datasets. Visual search extends this paradigm to queries requiring visual evidence, demanding agents to not only retrieve web pages but also understand and ground visual content in external knowledge. Benchmarks like VDR-Bench Zeng et al. ( 2026 ) and VisBrowse Zhang et al. ( 2026a ) evaluate this capability, alongside VLM-driven agents such as T3-Agent Gao et al. ( 2024 ) and Vision-DeepResearch Huang et al. ( 2026 ) which explore multimodal tool integration. However, existing visual search approaches predominantly build agents from scratch with jointly designed toolsets, rather than extending well-trained text-oriented agents. This leaves a practical gap: given the immense cost of training capable deep research agents, incrementally augmenting them with visual tools is far more practical than starting over. We address this gap as a toolset expansion problem."},{"type":"heading","level":4,"text":"Agentic reinforcement learning."},{"type":"para","text":"Reinforcement learning has emerged as a powerful post-training paradigm for LLM-based agents Zhang et al. ( 2025a ); Yang and Thomason ( 2026 ); Zhang et al. ( 2026b ) . GRPO Shao et al. ( 2024 ) introduced group relative policy optimization for mathematical reasoning, and DAPO Yu et al. ( 2025 ) further enhanced its performance through a series of techniques. Although these methods can be applied to agentic RL settings, multi-turn tool-integrated reasoning introduces distinct challenges, such as distributional shift caused by tool feedback that deviates from the LLM’s training data. To mitigate these issues, several methods have been proposed: SimpleTIR Xue et al. ( 2025 ) filters trajectories containing void turns, GiGPO computes both episode-level and step-level relative advantages, and ARPO Dong et al. ( 2025b ) and AEPO Dong et al. ( 2025a ) use entropy-based adaptive beaming strategies to execute partial rollouts at high-entropy tokens. Despite these advances, existing agentic RL methods share a common assumption: the toolset is fixed throughout training. When new tools are introduced to a post-trained agent, its policy has already converged around the original toolsets, and standard agentic RL exploration mechanisms may be insufficient to overcome this behavioral inertia . Our work targets this setting through a teacher-guided anchoring mechanism that is then validated by student rollout. This design contrasts with standard knowledge distillation Hinton et al. ( 2015 ) and on-policy imitation methods such as DAgger Ross et al. ( 2011 ) , which treat teacher-generated reasoning and actions as ground-truth supervision. Our framework treats teacher guidance as a hypothesis to be verified through student rollouts rather than as a direct supervised target."},{"type":"heading","level":1,"text":"3 Preliminaries"},{"type":"heading","level":2,"text":"3.1 Tool-use agent setup"},{"type":"para","text":"Following Tongyi DeepResearch Agent Team et al. ( 2025 ) , we consider a post-trained tool-use agent equipped with an original toolset U old U_{\\mathrm{old}} . In our experiments, U old U_{\\mathrm{old}} consists of web search, web browsing, a Python interpreter, scholar search, and a file parser. To study toolset expansion, we introduce a set of new visual tools U new U_{\\mathrm{new}} , including an image descriptor, a bounding-box extractor, and crop-and-image search. The resulting expanded toolset is U = U old ∪ U new U=U_{\\mathrm{old}}\\cup U_{\\mathrm{new}} . Detailed tool descriptions are provided in Appendix C.1 . We follow the ReAct framework, which consists of thought, action, and observation in each round. At each round, the agent first generates a thought and an action, and then receives an observation from the environment. This process forms a trajectory, denoted as ℋ T \\mathcal{H}_{T} , which is a sequence of thought–action–observation triplets: ℋ T = ( τ 0 , a 0 , o 0 , … , τ i , a i , o i , … , τ T , a T ) \\mathcal{H}_{T}=(\\tau_{0},a_{0},o_{0},\\ldots,\\tau_{i},a_{i},o_{i},\\ldots,\\tau_{T},a_{T}) , where a T a_{T} represents the final answer to the given question. For each non-terminal round t < T t<T , the agent samples ( τ t , a t ) ∼ π θ ( ⋅ ∣ ℋ t − 1 ) (\\tau_{t},a_{t})\\sim\\pi_{\\theta}(\\cdot\\mid\\mathcal{H}_{t-1}) , after which the environment executes a t a_{t} and returns o t o_{t} . The terminal action a T a_{T} corresponds to the final answer, and no observation o T o_{T} is returned."},{"type":"heading","level":2,"text":"3.2 Counterfactual anchor contexts"},{"type":"para","text":"While existing diagnostics (e.g., AgentErrorBench Zhang et al. ( 2025b ) ) focus on categorizing root-cause errors in failed trajectories, we move beyond passive diagnosis to active intervention. Specifically, we investigate whether replacing a single local decision under a shared history can redirect the rollout and recover a failed task. Aligning with recent advances in agentic context engineering Zhang et al. ( 2025b ); Dou et al. ( 2026 ) , this motivates our formulation of the counterfactual anchor context. Formally, given a failed trajectory ℋ T \\mathcal{H}_{T} and a non-terminal round t < T t<T , the original local decision is the thought-action pair ( τ t , a t ) (\\tau_{t},a_{t}) generated under history ℋ t − 1 \\mathcal{H}_{t-1} . A counterfactual alternative retains ℋ t − 1 \\mathcal{H}_{t-1} unchanged but replaces this decision with a new pair ( τ t ′ , a t ′ ) (\\tau^{\\prime}_{t},a^{\\prime}_{t}) , conditioned solely on the available information in ℋ t − 1 \\mathcal{H}_{t-1} . The environment then executes a t ′ a^{\\prime}_{t} and returns the corresponding observation o t ′ o^{\\prime}_{t} , yielding the counterfactual anchor context c t = ( ℋ t − 1 , τ t ′ , a t ′ , o t ′ ) c_{t}=(\\mathcal{H}_{t-1},\\tau^{\\prime}_{t},a^{\\prime}_{t},o^{\\prime}_{t}) . Crucially, c t c_{t} isolates the divergent branch: it captures the precise local context arising from the alternative decision ( τ t ′ , a t ′ ) (\\tau^{\\prime}_{t},a^{\\prime}_{t}) without altering prior history. We term c t c_{t} an anchor because it serves as a revised contextual reset point, rather than a full trajectory, from which the student agent resumes generation"},{"type":"heading","level":2,"text":"3.3 Preliminary analysis"},{"type":"para","text":"To investigate tool-use adaptation, we benchmark the text-oriented Tongyi DeepResearch Agent, extended with three new visual tools (image descriptor, bbox extractor, crop-and-image search), against two general-purpose models (Gemini-2.5-Pro, Qwen3-30B-A3B-Instruct) on 100 random instances from VDR-Bench. As detailed in Table 1, the tool-call frequencies reveal a stark contrast: while general-purpose agents seamlessly adopt the visual tools, Tongyi remains heavily reliant on its original toolset. This discrepancy highlights a pronounced behavioral inertia in post-trained agents."},{"type":"para","text":"➤ Takeaway 1. The tool-call statistics support the behavioral inertia hypothesis: under an expanded toolset, the post-trained student agent still relies disproportionately on the original tools and uses the newly introduced visual tools less frequently than general-purpose agents."},{"type":"para","text":"We further conduct motivation experiments on the textual search dataset OpenSeeker Du et al. ( 2026 ) and the visual search dataset VDR-Bench Huang et al. ( 2026 ) . Specifically, we randomly sample 100 failed trajectories of the Tongyi DeepResearch Agent from OpenSeeker and VDR-Bench, respectively. For each failed trajectory, we use teacher models (Gemini-2.5-Pro and GPT-5) to hypothesize the counterfactual anchor round and generate an alternative thought-action pair at that round (see Appendix B.2 for the prompt). We then have the Tongyi DeepResearch Agent resume generation from the teacher-revised context. Table 2 compares the task success rate when the agent continues from the original context versus the teacher-revised context. The results show that resuming generation from the teacher-revised context improves task success accuracy."},{"type":"para","text":"➤ Takeaway 2. Teacher-hypothesized counterfactual anchor contexts can partially recover failed trajectories: compared with resuming from the original context, resuming from teacher-revised contexts yields higher task success in both textual and visual search settings."},{"type":"para","text":"These two findings motivate ToolAnchor, which systematically hypothesizes critical anchor rounds, verifies the counterfactual contexts, and internalizes the verified alternatives into the student agent."},{"type":"heading","level":1,"text":"4 Methodology"},{"type":"para","text":"In this section, we present ToolAnchor , a framework for adapting a post-trained tool-use agent to an expanded toolset through counterfactual anchor contexts. ToolAnchor consists of three stages: counterfactual anchor context hypothesis, counterfactual anchor context verification, and counterfactual anchor context internalization."},{"type":"heading","level":2,"text":"4.1 Problem formulation"},{"type":"para","text":"Building on the setup in Section 3.1 , we study toolset expansion for a post-trained tool-use agent π θ \\pi_{\\theta} . The agent has been trained with the original toolset U old U_{\\mathrm{old}} and is then exposed to the expanded toolset U = U old ∪ U new U=U_{\\mathrm{old}}\\cup U_{\\mathrm{new}} . The goal is to learn an updated policy π θ ′ \\pi_{\\theta^{\\prime}} that improves task success under U U while preserving the agent’s existing capability on tasks solvable with U old U_{\\mathrm{old}} ."},{"type":"para","text":"Our preliminary analysis in Section 3.3 reveals two key findings. First, the post-trained agent exhibits behavioral inertia , disproportionately relying on U old U_{\\mathrm{old}} even when U new U_{\\mathrm{new}} is available (Table 1 ). Second, failed trajectories can often be recovered when the agent resumes generation from a teacher-proposed counterfactual anchor context at a critical decision round (Table 2 ). These findings suggest that the bottleneck is not only missing tool-use capability, but also the lack of effective contextual anchors that redirect the agent away from its established behavioral patterns. This motivates ToolAnchor, which hypothesizes, verifies, and internalizes such contexts, as illustrated in Figure 2 ."},{"type":"heading","level":2,"text":"4.2 Counterfactual anchor context hypothesis"},{"type":"para","text":"Given a failed trajectory ℋ T \\mathcal{H}_{T} produced by the base policy π θ \\pi_{\\theta} under the expanded toolset U U , this stage uses teacher models to form a counterfactual anchor hypothesis. The hypothesis consists of a candidate anchor round t ∗ < T t^{*}<T and an alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) (\\tau^{\\prime}_{t^{*}},a^{\\prime}_{t^{*}}) , which together define a revised context that may redirect the subsequent student rollout toward a successful trajectory."},{"type":"heading","level":4,"text":"Failed trajectory collection."},{"type":"para","text":"We first roll out the base policy π θ \\pi_{\\theta} on the training datasets under the expanded toolset U U , collecting both successful and failed trajectories. We define the failed trajectory set as 𝒟 fail = { ℋ T ∣ Success ( ℋ T ) = 0 } \\mathcal{D}_{\\mathrm{fail}}=\\{\\mathcal{H}_{T}\\mid\\mathrm{Success}(\\mathcal{H}_{T})=0\\} , which serves as the input for the teacher-guided counterfactual anchor hypothesis."},{"type":"heading","level":4,"text":"Counterfactual round hypothesis."},{"type":"para","text":"We define an anchor round hypothesis as a candidate round at which an alternative local decision may induce a different subsequent rollout and potentially change the task outcome. Forming this hypothesis requires assessing how earlier decisions, observations, and subsequent reasoning steps interact across the trajectory, rather than relying on a fixed heuristic such as choosing the first or last suspicious step. We instantiate this step with a prompt-guided teacher audit: given the full failed trajectory, a teacher model sequentially reviews each round’s thought τ t \\tau_{t} , action a t a_{t} , and observation o t o_{t} , and selects a candidate anchor round t ∗ t^{*} where a revised context is likely to redirect the subsequent rollout. The prompt used in our implementation is provided in Appendix B.2 . In our implementation, the teacher is instructed to use the following criteria:"},{"type":"para","text":"Downstream impact. Prefer a round whose alternative local decision is likely to influence multiple subsequent steps in the trajectory."},{"type":"para","text":"Path redirection. Prefer a round that acts as a branching point, where a different local decision may redirect the subsequent rollout toward a different reasoning or tool-use path."},{"type":"para","text":"Strategic leverage. Prefer a round involving high-level agentic decisions, such as reasoning direction, tool selection, query formulation, evidence interpretation, information integration, or planning."},{"type":"para","text":"Counterfactual potential. Prefer a round where replacing the original local decision under the same history appears most likely to improve the final task outcome."},{"type":"para","text":"These criteria are intended to avoid trivial anchor-selection heuristics, such as always choosing the earliest suspicious step or the final decision step, both of which may fail to identify the round with the greatest potential to redirect the subsequent rollout."},{"type":"heading","level":4,"text":"Hypothesized context generation."},{"type":"para","text":"Once the anchor round t ∗ t^{*} is hypothesized, the teacher model generates the alternative thought-action pair in a separate step using only the history available before that round. Specifically, the teacher is given only ℋ t ∗ − 1 \\mathcal{H}_{t^{*}-1} , without access to subsequent rounds or the ground-truth answer. The teacher then proposes an alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) ∼ M ( ⋅ ∣ ℋ t ∗ − 1 ) (\\tau^{\\prime}_{t^{*}},a^{\\prime}_{t^{*}})\\sim M(\\cdot\\mid\\mathcal{H}_{t^{*}-1}) , where M M denotes the teacher model."},{"type":"para","text":"This separation is important: the anchor-round hypothesis can use the full failed trajectory for diagnosis, whereas the alternative thought-action pair must be generated under the same information constraint faced by the agent at round t ∗ t^{*} . This prevents the alternative context from encoding future observations or the ground-truth answer, making the resulting supervision compatible with the student’s inference-time information state. Although new-tool adoption is a central motivation, the teacher is not restricted to rounds involving U new U_{\\mathrm{new}} . It may also select rounds involving ineffective use of U old U_{\\mathrm{old}} , or reasoning steps that make the subsequent rollout less effective. The environment then executes a t ∗ ′ a^{\\prime}_{t^{*}} and returns the corresponding observation o t ∗ ′ o^{\\prime}_{t^{*}} . This produces the candidate anchor context c ∗ = ( ℋ t ∗ − 1 , τ t ∗ ′ , a t ∗ ′ , o t ∗ ′ ) c^{*}=(\\mathcal{H}_{t^{*}-1},\\tau^{\\prime}_{t^{*}},a^{\\prime}_{t^{*}},o^{\\prime}_{t^{*}}) ."},{"type":"heading","level":4,"text":"Teacher diversity."},{"type":"para","text":"A single teacher model may favor particular tool-use patterns or anchor-round choices when auditing failed trajectories. To improve coverage, we use a set of heterogeneous teacher models ℳ = { M j } j = 1 J \\mathcal{M}=\\{M_{j}\\}_{j=1}^{J} , each of which independently forms counterfactual anchor hypotheses for the same failed trajectory. Different teachers may select different anchor rounds or propose different alternative thought-action pairs, yielding a diverse set of candidate anchor contexts 𝒟 anchor = { c i ∗ } i = 1 m \\mathcal{D}_{\\mathrm{anchor}}=\\{c_{i}^{*}\\}_{i=1}^{m} . Each c i ∗ c_{i}^{*} is a teacher-hypothesized candidate counterfactual context and will be verified by student rollout in Section 4.3 ."},{"type":"heading","level":2,"text":"4.3 Counterfactual anchor context verification"},{"type":"para","text":"Teacher-generated anchor contexts are hypotheses rather than ground-truth demonstrations. A teacher may select an ineffective anchor round, propose a plausible but unhelpful alternative decision, or generate an observation that does not support the student’s subsequent reasoning. To ensure data quality without manual annotation or heuristic reward shaping, we introduce a student-side verification stage. This stage retains an anchor context only when the student successfully solves the task after resuming from it and a separate judge model, with access to the completed trajectory, verifies that the anchor context was materially useful for the final success."},{"type":"heading","level":4,"text":"Anchor-conditioned agentic rollout and dual-level verification."},{"type":"para","text":"For each candidate anchor context c ∗ ∈ 𝒟 anchor c^{*}\\in\\mathcal{D}_{\\mathrm{anchor}} , we let the base policy π θ \\pi_{\\theta} resume generation from c ∗ c^{*} under the standard agentic rollout procedure, producing a complete trajectory ℋ T ′ = Rollout ( π θ , c ∗ ) \\mathcal{H}^{\\prime}_{T}=\\mathrm{Rollout}(\\pi_{\\theta},c^{*}) . We retain c ∗ c^{*} only if it passes two verification checks. First, the final answer in ℋ T ′ \\mathcal{H}^{\\prime}_{T} must be judged correct by the task-level evaluator, denoted as J task ( ℋ T ′ ) ∈ { 0 , 1 } J_{\\mathrm{task}}(\\mathcal{H}^{\\prime}_{T})\\in\\{0,1\\} . Second, an anchor-level judge, denoted as J anchor ( c ∗ , ℋ T ′ ) ∈ { 0 , 1 } J_{\\mathrm{anchor}}(c^{*},\\mathcal{H}^{\\prime}_{T})\\in\\{0,1\\} , verifies from a global view that the proposed anchor context was materially useful in enabling the successful outcome. Formally, the verified effective-context dataset is"},{"type":"heading","level":4,"text":"Why verification matters."},{"type":"para","text":"Verification turns teacher hypotheses into training signals. The task-level success check filters out anchors from which the student cannot solve the task, and the anchor-level judge removes cases where success is not materially supported by the proposed anchor. Since the continuation after c ∗ c^{*} is generated by the student policy π θ \\pi_{\\theta} , the retained data avoid full teacher-trajectory imitation and remain closer to the student’s rollout distribution."},{"type":"heading","level":2,"text":"4.4 Counterfactual anchor context internalization"},{"type":"para","text":"The final stage internalizes verified counterfactual anchor contexts into the agent’s policy through a two-step pipeline, combining local policy internalization and policy alignment with exploration."},{"type":"heading","level":4,"text":"Local policy update (anchor-only agentic SFT)."},{"type":"para","text":"We perform anchor-only agentic SFT on the verified effective contexts, training the agent to reproduce the teacher-proposed thought-action pair at the verified counterfactual anchor round. For each verified counterfactual context ( c ∗ , ℋ T ′ ) ∈ 𝒟 eff (c^{*},\\mathcal{H}^{\\prime}_{T})\\in\\mathcal{D}_{\\mathrm{eff}} , where c ∗ = ( ℋ t ∗ − 1 , τ t ∗ ′ , a t ∗ ′ , o t ∗ ′ ) c^{*}=(\\mathcal{H}_{t^{*}-1},\\tau^{\\prime}_{t^{*}},a^{\\prime}_{t^{*}},o^{\\prime}_{t^{*}}) , the SFT loss is computed only on the alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) (\\tau^{\\prime}_{t^{*}},a^{\\prime}_{t^{*}}) , conditioned on the original history ℋ t ∗ − 1 \\mathcal{H}_{t^{*}-1} :"},{"type":"para","text":"This objective directly injects the verified anchor behaviors at high-leverage decision points, providing a precise behavioral guidance. While optimizing solely on 𝒟 eff \\mathcal{D}_{\\text{eff}} could risk localized overfitting, this anchor-only SFT serves as a targeted policy initialization. The subsequent post-SFT agentic RL stage then realigns these local updates with the agent’s prior broad capabilities, effectively mitigating catastrophic forgetting and preserving performance on tasks solvable with U old U_{\\text{old}} ."},{"type":"heading","level":4,"text":"Policy realignment and exploration (post-SFT agentic RL)."},{"type":"para","text":"Following agentic SFT, we apply agentic RL (GRPO) on the expanded toolset U U , using a binary task-level reward (1 for correct answers, 0 otherwise). This stage serves to integrate the internalized anchor behaviors from SFT with the agent’s prior RL policy, consolidating these local updates into a policy that is both aligned with prior knowledge and capable of effective inference under the expanded toolset. Simultaneously, it enables the agent to explore additional tool-use strategies beyond those explicitly provided by the teachers. Collectively, agentic SFT and agentic RL ensure that local counterfactual behaviors are internalized, properly aligned with the prior policy, and generalized, allowing the agent to leverage both newly introduced tools and previously learned capabilities."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"heading","level":2,"text":"5.1 Experimental setup"},{"type":"heading","level":4,"text":"Student and teacher models."},{"type":"para","text":"We use the Tongyi DeepResearch Agent Team et al. ( 2025 ) (Qwen3-30B-A3B) as our base model π θ \\pi_{\\theta} , which is trained via agentic RL on a fixed toolset U old U_{\\text{old}} consisting of web search, web browse, a Python interpreter, scholar search, and a file parser. We extend it with three visual tools U new U_{\\text{new}} : an image descriptor, a bbox extractor, and crop-and-image search. We choose the Tongyi DeepResearch Agent not only because its system prompts, toolsets, and model weights are open-source (allowing for training and customization), but also because it demonstrates strong performance on general AI assistant and textual search tasks. For the counterfactual anchor hypothesis, we employ Gemini-2.5-Pro and GPT-5 as diverse teacher models to identify anchor rounds and generate alternative reasoning and actions."},{"type":"heading","level":4,"text":"Training datasets and benchmarks."},{"type":"para","text":"We use two training datasets that span both textual and visual search tasks: (1) OpenSeeker-1.5k, which contains 1,500 instances sampled from OpenSeeker-v1-Data Du et al. ( 2026 ) ; and (2) VDR-1.5k Huang et al. ( 2026 ) , a visual search dataset that also contains 1,500 samples. We evaluate our method on three benchmarks corresponding to distinct tasks: GAIA Mialon et al. ( 2023 ) (General AI assistant, 165 samples), BrowseComp Wei et al. ( 2025 ) (Textual search, 200 samples subset following Du et al. ( 2026 ) ), and VDR-Bench Zeng et al. ( 2026 ) (Visual search, test-mini split with 500 samples, following Huang et al. ( 2026 ) )."},{"type":"heading","level":4,"text":"Evaluation metrics and baselines."},{"type":"para","text":"Following prior work Huang et al. ( 2026 ); Du et al. ( 2026 ) , we adopt an LLM-as-a-judge approach for evaluation, using Qwen3-235B-A22B as the judge model. Specifically, we use the official prompts provided by each benchmark to assess the correctness of the agent’s final answer against the ground truth. Our primary evaluation metric is task success rate under Pass@1, where each query is evaluated with a single agent rollout. All benchmark results are reported with Pass@1. Pass@3 is used only during counterfactual anchor verification, where a candidate context passes task-level verification if at least one of three student rollouts solves the task. Please refer to Appendix B.4 for the corresponding evaluation prompts. We compare ToolAnchor against both proprietary and open-source deep research agents, including GPT-5 Singh et al. ( 2025a ) , Gemini-2.5-Pro Comanici et al. ( 2025 ) , Claude-4-Sonnet, OpenAI DeepResearch, Qwen3-VL-30B-A3B-Thinking Bai et al. ( 2025 ) , Qwen3-VL-30B-A3B-Instruct, OpenSeeker Du et al. ( 2026 ) , Vision DeepResearch Huang et al. ( 2026 ) , and Tongyi DeepResearch Team et al. ( 2025 ) . Implementation details, including training hyperparameters, inference hyperparameters, and compute resources, are provided in Appendix C.2 ."},{"type":"heading","level":2,"text":"5.2 Main results"},{"type":"para","text":"As shown in Table 3 , we compare ToolAnchor-DeepResearch-30B-A3B against both proprietary and open-source deep research agents across three benchmarks."},{"type":"heading","level":4,"text":"Visual search."},{"type":"para","text":"On VDR-Bench, ToolAnchor achieves 28.8%, outperforming Gemini-2.5-Pro (18.8%), GPT-5 (20.4%), and Qwen3-VL-30B-A3B-Thinking (23.2%), despite using a smaller 30B LLM without prior visual-task training. This substantially narrows the gap with a vision-language model (VLM) agent trained from scratch for visual search, namely Vision-DeepResearch-30B-A3B (37.8%). This performance gap is a predictable trade-off: Vision-DeepResearch is a native vision-language model (VLM) agent jointly optimized from scratch on a 5 × \\times larger multimodal dataset (15K instances). In contrast, ToolAnchor operates on a frozen, text-only LLM backbone augmented purely via dynamic tool APIs. Achieving a competitive 28.8% without expensive multimodal alignment or massive data dependency underscores the remarkable parameter and compute efficiency of our toolset expansion paradigm."},{"type":"heading","level":4,"text":"General assistant and textual search."},{"type":"para","text":"On GAIA, ToolAnchor achieves 74.5%, improving over the base model (70.9%) by 3.6 percentage points and surpassing GPT-5 (67.4%), Claude-4-Sonnet (68.3%), and OpenAI DeepResearch (67.4%). On BrowseComp, ToolAnchor achieves 45.0%, a 1.6 percentage point improvement over the base model (43.4%). These results demonstrate that ToolAnchor enables the agent to acquire new visual tool-use capabilities while largely preserving its performance on the original textual tasks. The slight improvement observed on GAIA and BrowseComp suggests that internalizing effective counterfactual anchor contexts can also reinforce the agent’s existing tool usage and reasoning policies."},{"type":"heading","level":2,"text":"5.3 Ablation study"},{"type":"heading","level":4,"text":"Visual tools."},{"type":"para","text":"Table 4 compares the baseline Tongyi-DeepResearch-30B-A3B model with a variant equipped with our visual tools. The results demonstrate that while the integration of these visual tools significantly boosts performance on the VDR-Bench visual search (from 7.5% to 16.6%), it incurs a marginal performance degradation on textual search benchmarks (GAIA and BrowseComp)."},{"type":"heading","level":4,"text":"Post-training method."},{"type":"para","text":"Figure 3 ablates the post-training pipeline after visual-tool expansion. Agentic SFT on unverified counterfactual contexts performs poorly, especially on BrowseComp, indicating that teacher-hypothesized contexts can introduce noisy or ineffective supervision if applied without student-side verification. Using verified contexts improves over unverified agentic SFT on BrowseComp and VDR-Bench, supporting the utility of counterfactual anchor context verification. Agentic RL trained from scratch achieves moderate performance (70.3%, 41.5%, 21.6%), indicating that without access to counterfactual anchor contexts via SFT, RL alone exhibits limited exploration and fails to fully leverage the available toolset. However, agentic SFT with verified contexts alone still underperforms the ToolAnchor, suggesting that anchor-context internalization through SFT is insufficient to fully recover and improve the agent’s tool-use policy. By combining verified SFT with a subsequent agentic RL, ToolAnchor achieves the best performance across all three benchmarks. This indicates that verified counterfactual contexts provide effective local supervision, while the post-SFT RL stage aligns the internalized anchor behaviors with the agent’s previously learned policy."},{"type":"heading","level":2,"text":"5.4 Further analysis and discussions"},{"type":"heading","level":4,"text":"Does the choice of anchor round matter?"},{"type":"para","text":"To isolate the efficacy of our criterion-based anchor selection from the mere presence of teacher intervention, we benchmark it against three heuristic baselines: intervening at a random round, the first erroneous round, and the last erroneous round. Evaluated across 200 instances from both OpenSeeker and VDR-Train (using Gemini-3.1-Pro-Preview as the teacher), our criterion-based strategy consistently yields the highest recovery rates. We repeat teacher anchor selection 50 times on one failed trajectory. As illustrated in Figure 5 , effective anchors heavily cluster around strategic causal pivots rather than trivial first/last-error boundaries, confirming that a precise anchor location is as critical as the counterfactual context itself."},{"type":"heading","level":4,"text":"How many counterfactual contexts survive verification?"},{"type":"para","text":"For each teacher-generated counterfactual context, we sample three student rollouts and use Pass@3 task-level verification, where a context passes if any rollout solves the task. We then apply anchor-level verification to successful rollouts, retaining only contexts whose success is judged materially attributable to the anchor intervention. Overall, 33.67% of candidates pass task-level verification and 31.98% remain after anchor-level verification, showing that verification filters noisy teacher hypotheses while retaining contexts that both recover the task and validly intervene on the failure point."},{"type":"heading","level":2,"text":"5.5 Case study"},{"type":"para","text":"In Figure 1 , we present a case study to illustrate how ToolAnchor redirects a failed visual-search trajectory. In this example, the base Tongyi DeepResearch Agent first uses visual tools to locate the green-hooded character and invoke the crop-and-image search tool, but then dismisses the returned visual-search leads as generic and switches to broad text search, eventually producing an incorrect answer. The teacher hypothesizes that this decision is the counterfactual anchor round, since it is the key point where the agent abandons the visual-evidence path. The generated counterfactual context instead keeps the agent on the visual-search path by analyzing the crop-and-image search results and issuing a targeted follow-up query. This qualitative example illustrates how a local counterfactual context can redirect the subsequent rollout. Details are provided in Appendix A . Consistent with this qualitative example, Appendix C.3 shows that hypothesized counterfactual anchor rounds are broadly distributed across different trajectory positions, with most occurring within the first 10-20 rounds, and that each can potentially produce distinct effects on the agent’s subsequent behavior."},{"type":"heading","level":1,"text":"6 Conclusion and limitation"},{"type":"para","text":"In this paper, we propose ToolAnchor, a framework for extending post-trained tool-use agents with new tools through verified counterfactual context anchoring. Our main insight is that toolset expansion is not merely about adapting tool-selection and reasoning patterns to newly introduced tools. Post-trained agents may also need effective contextual anchors to break out of established reasoning and tool-use patterns. ToolAnchor turns teacher-hypothesized counterfactual contexts into training signals only after student-side verification, allowing the agent to internalize local high-leverage redirections rather than imitate full teacher trajectories. Experiments across general assistance, textual search, and visual search show that this mechanism improves new visual-tool use while also benefiting existing tool use and agentic reasoning, suggesting a practical path for capability expansion without retraining agents from scratch. In this work, the experiments rely on a limited set of teacher models and verified anchor contexts. Future research could investigate incorporating a more diverse set of teacher models and expanding the training dataset to enhance coverage, robustness, and generalization of internalized behaviors across a broader range of tasks and toolsets."},{"type":"para","text":"We first present a failed inference trajectory from the Tongyi DeepResearch Agent in Appendix A.1 , with the corresponding agent prompt provided in Appendix B.1 . We then show the teacher model’s counterfactual anchor-round hypothesis for this trajectory in Appendix A.2 , with the corresponding hypothesis prompt provided in Appendix B.2 . Finally, we provide the counterfactual anchor context generated for the hypothesized anchor round in Appendix A.3 , with the corresponding counterfactual context generation prompt provided in Appendix B.2 ."},{"type":"heading","level":2,"text":"A.1 Failed inference trajectory of the deep research agent"},{"type":"heading","level":2,"text":"A.2 Teacher counterfactual anchor round hypothesis"},{"type":"heading","level":2,"text":"A.3 Teacher counterfactual anchor context generation"},{"type":"heading","level":2,"text":"B.1 System prompt for deep research agent"},{"type":"para","text":"Following the Tongyi DeepResearch Agent, we provide the system prompt used by ToolAnchor, with the original tool-use instructions extended to include the new visual tools."},{"type":"heading","level":2,"text":"B.2 System Prompt for teacher model counterfactual anchor round hypothesis"},{"type":"para","text":"We provide the teacher prompts for counterfactual anchor round hypothesis and counterfactual anchor context generation."},{"type":"heading","level":2,"text":"B.3 Prompt for anchor-level LLM-As-Judge verification"},{"type":"para","text":"We provide the anchor-level LLM-as-judge verification prompt below."},{"type":"heading","level":2,"text":"B.4 Prompt for LLM-As-Judge evaluation"},{"type":"para","text":"For GAIA, BrowseComp, and VDR-Bench, we use the official evaluation prompts provided by each benchmark."},{"type":"heading","level":2,"text":"C.1 Tool descriptions"},{"type":"heading","level":4,"text":"Web search"},{"type":"para","text":"A web search tool that executes Google searches and retrieves the top 10 results. It accepts natural language queries and returns structured data, including page titles, URLs and snippets."},{"type":"heading","level":4,"text":"Web browse"},{"type":"para","text":"A web browse tool (visit tool) that visits a specified URL and returns a summary. It accepts a target URL and a specific goal, uses Jina to convert the page content into Markdown, and then utilizes the DeepSeek-V3.2 summarization model to extract information relevant to the goal."},{"type":"heading","level":4,"text":"Python interpreter"},{"type":"para","text":"A Python interpreter tool that executes Python code within a secure sandbox."},{"type":"heading","level":4,"text":"Scholar search"},{"type":"para","text":"A scholar search tool that executes Google Scholar searches and retrieves the top 10 results. It accepts natural language queries and returns structured data, including page titles, URLs and snippets."},{"type":"heading","level":4,"text":"File parser"},{"type":"para","text":"A file parser tool that analyzes various files and extracts their content as plain text. It accepts both local files and URLs, converting them into a readable text format for further use."},{"type":"heading","level":4,"text":"Image descriptor"},{"type":"para","text":"An image description tool powered by the multimodal model Qwen2.5-72B-Instruct. It supports both local image files and image URLs, and answers user queries about images."},{"type":"heading","level":4,"text":"Bbox extractor"},{"type":"para","text":"A bounding box (bbox) extractor that uses the multimodal model Qwen2.5-72B-Instruct to locate specific entities within an image. It accepts both local image files and URLs, as well as entity queries, and returns the bounding box coordinates of the requested entity."},{"type":"heading","level":4,"text":"Crop-and-image search"},{"type":"para","text":"An image cropping and search tool that executes Google Lens and retrieves the top 10 results. It accepts an image as input and returns structured data, including titles, URLs, and snippets. The snippets are generated by the multimodal model Qwen2.5-72B-Instruct, which compares the similarities and differences between the query image and the webpage images, a design motivated by VC-STaR Pan et al. ( 2026 ) ."},{"type":"para","text":"The Tongyi toolset includes a web search tool, a web browsing tool, a Python interpreter, a scholar search tool, and a file parser tool. The VDR toolset consists of a web search tool, a web browsing tool, a Python interpreter, and a visual search tool. Building on the Tongyi toolset, our toolset further incorporates an image description tool, a bounding-box extraction tool, and a crop-and-image-search tool."},{"type":"heading","level":2,"text":"C.2 Training and inference details"},{"type":"heading","level":4,"text":"Training hyperparameters"},{"type":"para","text":"All experiments were conducted on a cluster of eight NVIDIA B200 (192GB) GPUs using the Slime framework Zhu et al. ( 2025 ) . All training hyperparameters are listed in Table 5 , and Table 6 ."},{"type":"heading","level":4,"text":"Inference hyperparameters"},{"type":"para","text":"As shown in Table 7 , our hyperparameters are consistent with those of the Tongyi Deep Research Agent."},{"type":"heading","level":2,"text":"C.3 Statistical analysis of counterfactual anchor rounds"},{"type":"para","text":"Figure 6 shows the distribution of counterfactual anchor rounds hypothesized by Gemini-2.5-Pro on OpenSeeker-1.5k and VDR-1.5k. On both datasets, the hypothesized anchor rounds are heavily concentrated in the early stages of trajectories, with most anchors occurring within the first 10–20 rounds. This suggests that early counterfactual contexts exert greater influence on subsequent reasoning and tool-use decisions, while preserving sufficient rollout horizon for exploration and recovery. Similarly, on VDR-1.5k, the early-round concentration is even more pronounced, indicating that visual-search tasks particularly benefit from introducing visual-evidence-oriented contexts near the beginning of the trajectory. The long-tail of the distribution indicates that valuable counterfactual contexts also occur at later rounds in some trajectories. These observations support our use of teacher-guided anchor-round hypothesis generation rather than relying on fixed heuristics such as always selecting the first round."},{"type":"para","text":"To mitigate search-time data contamination Han et al. ( 2025 ) , we filter out specific domain names, such as github.com and huggingface.co, during the web search process."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent advances in tool-augmented agents have significantly broadened the real-world utility of Large Language Models (LLMs) Singh et al. ( 2025b ); Jiang et al. ( 2025 ); Li et al. ( 2025b ); Fan et al. ( 2026 ); Li et al. ( 2026 ); Fang et al. ( 2026 ) . By integrating external tools into the reasoning loop, LLM-based agents have achieved strong performance across a wide spectrum of downstream tasks, spanning text-based question answering Wu et al. ( 2025b ); Wei et al. ( 2025 ); Zhou et al. ( 2025 ); Mialon et al. ( 2023 ) , visual question answering Yang et al. ( 2025 ); Huang et al. ( 2026 ); Zhang et al. ( 2026a ) , long-form text report generation Li et al. ( 2025e ); Du et al. ( 2025a ) , and domain-grounded tool use Yao et al. ( 2024 ); Barres et al. ( 2025 ) . Central to this progress are the ReAct paradigm Yao et al. ( 2022 ) , which interleaves reasoning with tool invocation, and agentic reinforcement learning (agentic RL) methods Shao et al. ( 2024 ); Yu et al. ( 2025 ); Zhang et al. ( 2025a ) , which further refine tool-use capabilities via post-training. However, these successes largely assume a fixed, predefined toolset available throughout training. In practice, task requirements evolve continuously, often necessitating new tools such as web search for enhanced retrieval Jin et al. ( 2025 ); Li et al. ( 2025c ) , python interpreters for scientific computation Feng et al. ( 2025 ); Xue et al. ( 2025 ) , file parsers for document understanding Team et al. ( 2025 ) , and visual tools for multimodal perception Huang et al. ( 2026 ); Fan et al. ( 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A natural question then arises: how can an existing agent efficiently adapt to an expanded toolset? The most straightforward approach is to directly apply agentic RL, allowing the agent to discover effective reasoning strategies and action policies via exploration. However, this strategy faces a fundamental difficulty. Since the agent’s policy has already converged during prior agentic RL training, it tends to fall back on familiar behavioral patterns, underutilizing newly introduced tools and repeating suboptimal reasoning strategies. This behavioral inertia leads to sparse rewards and inefficient exploration, as the agent rarely ventures beyond its established routines. An alternative is to rely on the tool-use generalization capability of the agent Zou et al. ( 2025 ); Du et al. ( 2025b ) , hoping that it can adapt to new tools without additional training. Yet, as our preliminary analysis in Section 3.3 reveals, generalization alone is insufficient. To better understand this bottleneck, we conduct a motivation study on failed student trajectories. A teacher model first identifies a decision round where an alternative local thought–action pair could plausibly redirect the subsequent rollout; we call this decision point a counterfactual anchor round (Section 3.2 ). The teacher then proposes this alternative using only the history before that round, yielding a teacher-revised context from which the student resumes generation. Compared with resuming from the original context, these teacher-revised contexts improve task success in both textual and visual search settings. This suggests that failures arise not only from limited tool-use capability, but also from missing contextual anchors that redirect reasoning and tool selection at key decision points.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we propose ToolAnchor , a framework that bootstraps new tool-use capabilities through verified counterfactual context anchoring. ToolAnchor operates in three stages. In the Counterfactual Anchor hypothesis stage, diverse teacher models audit the student agent’s failed rollout trajectories and hypothesize anchor rounds where the agent’s decision deviated from a promising path. While the primary focus is on rounds where the agent failed to invoke a newly introduced tool, the teacher also considers suboptimal uses of existing tools and flawed reasoning steps, as these often compound the difficulty of adopting new tools effectively. For each identified anchor round, the teacher hypothesizes alternative reasoning and actions that would plausibly redirect the trajectory toward task completion. In the Counterfactual Anchor Verification stage, the student agent resumes generation from these teacher-anchored contexts under agentic rollout. Crucially, we retain for training only those anchor contexts that satisfy both verification criteria, namely task-level success and anchor-level usefulness, rather than accepting teacher hypotheses uncritically. In the Counterfactual Anchor Internalization stage, the verified contexts are used for agentic supervised fine-tuning (agentic SFT), allowing the agent to absorb the anchored behaviors into its own policy. This is followed by agentic RL, which realigns the internalized anchor behaviors with the agent’s previously learned agentic policy and supports robust, autonomous use of the expanded toolset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are summarized as follows: ( 1 ) We formalize the practical challenge of expanding a post-trained agent’s toolset. Through a motivation study, we identify behavioral inertia as the core bottleneck and reveal that counterfactual contextual anchoring at critical decision points can effectively redirect failed trajectories. ( 2 ) We propose ToolAnchor, a scalable, three-stage framework for capability expansion. It automates the discovery of counterfactual anchor rounds via diverse teacher models, strictly verifies their effectiveness through student rollouts, and internalizes these validated behaviors via agentic SFT and RL. ( 3 ) Extensive evaluations across general AI assistance (GAIA), textual search (BrowseComp), and visual search (VDR-Bench) demonstrate that ToolAnchor seamlessly integrates new visual tools. Furthermore, it simultaneously boosts the base agent’s reasoning and existing tool mastery, offering a robust paradigm for continuous agentic evolution.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Tool-augmented deep research agents.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Deep research agents tackle open-ended question answering through long-horizon tool use: iteratively planning, invoking external tools, verifying evidence, performing intermediate reasoning, and synthesizing final answers Li et al. ( 2025b ) . This paradigm naturally encompasses general AI assistance, textual search, and visual search. Textual search remains the most extensively explored direction, where agents retrieve external evidence to answer complex queries. For instance, Search-R1 Jin et al. ( 2025 ) , Search-o1 Li et al. ( 2025c ) , and WebThinker Li et al. ( 2025d ) directly integrate web search into reasoning, while subsequent work advances this via agentic SFT and RL Wu et al. ( 2025a ) , systematic uncertainty reduction Li et al. ( 2025a ) , scalable data synthesis Qiao et al. ( 2025 ) , dual-environment RL frameworks Li et al. ( 2025a ) , and continual pre-training Su et al. ( 2025 ) . Building on these techniques, the Tongyi DeepResearch Agent Team et al. ( 2025 ) achieves competitive performance, and OpenSeeker Du et al. ( 2026 ) provides high-quality training datasets. Visual search extends this paradigm to queries requiring visual evidence, demanding agents to not only retrieve web pages but also understand and ground visual content in external knowledge. Benchmarks like VDR-Bench Zeng et al. ( 2026 ) and VisBrowse Zhang et al. ( 2026a ) evaluate this capability, alongside VLM-driven agents such as T3-Agent Gao et al. ( 2024 ) and Vision-DeepResearch Huang et al. ( 2026 ) which explore multimodal tool integration. However, existing visual search approaches predominantly build agents from scratch with jointly designed toolsets, rather than extending well-trained text-oriented agents. This leaves a practical gap: given the immense cost of training capable deep research agents, incrementally augmenting them with visual tools is far more practical than starting over. We address this gap as a toolset expansion problem.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Agentic reinforcement learning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reinforcement learning has emerged as a powerful post-training paradigm for LLM-based agents Zhang et al. ( 2025a ); Yang and Thomason ( 2026 ); Zhang et al. ( 2026b ) . GRPO Shao et al. ( 2024 ) introduced group relative policy optimization for mathematical reasoning, and DAPO Yu et al. ( 2025 ) further enhanced its performance through a series of techniques. Although these methods can be applied to agentic RL settings, multi-turn tool-integrated reasoning introduces distinct challenges, such as distributional shift caused by tool feedback that deviates from the LLM’s training data. To mitigate these issues, several methods have been proposed: SimpleTIR Xue et al. ( 2025 ) filters trajectories containing void turns, GiGPO computes both episode-level and step-level relative advantages, and ARPO Dong et al. ( 2025b ) and AEPO Dong et al. ( 2025a ) use entropy-based adaptive beaming strategies to execute partial rollouts at high-entropy tokens. Despite these advances, existing agentic RL methods share a common assumption: the toolset is fixed throughout training. When new tools are introduced to a post-trained agent, its policy has already converged around the original toolsets, and standard agentic RL exploration mechanisms may be insufficient to overcome this behavioral inertia . Our work targets this setting through a teacher-guided anchoring mechanism that is then validated by student rollout. This design contrasts with standard knowledge distillation Hinton et al. ( 2015 ) and on-policy imitation methods such as DAgger Ross et al. ( 2011 ) , which treat teacher-generated reasoning and actions as ground-truth supervision. Our framework treats teacher guidance as a hypothesis to be verified through student rollouts rather than as a direct supervised target.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Preliminaries\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Tool-use agent setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following Tongyi DeepResearch Agent Team et al. ( 2025 ) , we consider a post-trained tool-use agent equipped with an original toolset U old U_{\\\\mathrm{old}} . In our experiments, U old U_{\\\\mathrm{old}} consists of web search, web browsing, a Python interpreter, scholar search, and a file parser. To study toolset expansion, we introduce a set of new visual tools U new U_{\\\\mathrm{new}} , including an image descriptor, a bounding-box extractor, and crop-and-image search. The resulting expanded toolset is U = U old ∪ U new U=U_{\\\\mathrm{old}}\\\\cup U_{\\\\mathrm{new}} . Detailed tool descriptions are provided in Appendix C.1 . We follow the ReAct framework, which consists of thought, action, and observation in each round. At each round, the agent first generates a thought and an action, and then receives an observation from the environment. This process forms a trajectory, denoted as ℋ T \\\\mathcal{H}_{T} , which is a sequence of thought–action–observation triplets: ℋ T = ( τ 0 , a 0 , o 0 , … , τ i , a i , o i , … , τ T , a T ) \\\\mathcal{H}_{T}=(\\\\tau_{0},a_{0},o_{0},\\\\ldots,\\\\tau_{i},a_{i},o_{i},\\\\ldots,\\\\tau_{T},a_{T}) , where a T a_{T} represents the final answer to the given question. For each non-terminal round t < T t<T , the agent samples ( τ t , a t ) ∼ π θ ( ⋅ ∣ ℋ t − 1 ) (\\\\tau_{t},a_{t})\\\\sim\\\\pi_{\\\\theta}(\\\\cdot\\\\mid\\\\mathcal{H}_{t-1}) , after which the environment executes a t a_{t} and returns o t o_{t} . The terminal action a T a_{T} corresponds to the final answer, and no observation o T o_{T} is returned.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Counterfactual anchor contexts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While existing diagnostics (e.g., AgentErrorBench Zhang et al. ( 2025b ) ) focus on categorizing root-cause errors in failed trajectories, we move beyond passive diagnosis to active intervention. Specifically, we investigate whether replacing a single local decision under a shared history can redirect the rollout and recover a failed task. Aligning with recent advances in agentic context engineering Zhang et al. ( 2025b ); Dou et al. ( 2026 ) , this motivates our formulation of the counterfactual anchor context. Formally, given a failed trajectory ℋ T \\\\mathcal{H}_{T} and a non-terminal round t < T t<T , the original local decision is the thought-action pair ( τ t , a t ) (\\\\tau_{t},a_{t}) generated under history ℋ t − 1 \\\\mathcal{H}_{t-1} . A counterfactual alternative retains ℋ t − 1 \\\\mathcal{H}_{t-1} unchanged but replaces this decision with a new pair ( τ t ′ , a t ′ ) (\\\\tau^{\\\\prime}_{t},a^{\\\\prime}_{t}) , conditioned solely on the available information in ℋ t − 1 \\\\mathcal{H}_{t-1} . The environment then executes a t ′ a^{\\\\prime}_{t} and returns the corresponding observation o t ′ o^{\\\\prime}_{t} , yielding the counterfactual anchor context c t = ( ℋ t − 1 , τ t ′ , a t ′ , o t ′ ) c_{t}=(\\\\mathcal{H}_{t-1},\\\\tau^{\\\\prime}_{t},a^{\\\\prime}_{t},o^{\\\\prime}_{t}) . Crucially, c t c_{t} isolates the divergent branch: it captures the precise local context arising from the alternative decision ( τ t ′ , a t ′ ) (\\\\tau^{\\\\prime}_{t},a^{\\\\prime}_{t}) without altering prior history. We term c t c_{t} an anchor because it serves as a revised contextual reset point, rather than a full trajectory, from which the student agent resumes generation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Preliminary analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To investigate tool-use adaptation, we benchmark the text-oriented Tongyi DeepResearch Agent, extended with three new visual tools (image descriptor, bbox extractor, crop-and-image search), against two general-purpose models (Gemini-2.5-Pro, Qwen3-30B-A3B-Instruct) on 100 random instances from VDR-Bench. As detailed in Table 1, the tool-call frequencies reveal a stark contrast: while general-purpose agents seamlessly adopt the visual tools, Tongyi remains heavily reliant on its original toolset. This discrepancy highlights a pronounced behavioral inertia in post-trained agents.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"➤ Takeaway 1. The tool-call statistics support the behavioral inertia hypothesis: under an expanded toolset, the post-trained student agent still relies disproportionately on the original tools and uses the newly introduced visual tools less frequently than general-purpose agents.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further conduct motivation experiments on the textual search dataset OpenSeeker Du et al. ( 2026 ) and the visual search dataset VDR-Bench Huang et al. ( 2026 ) . Specifically, we randomly sample 100 failed trajectories of the Tongyi DeepResearch Agent from OpenSeeker and VDR-Bench, respectively. For each failed trajectory, we use teacher models (Gemini-2.5-Pro and GPT-5) to hypothesize the counterfactual anchor round and generate an alternative thought-action pair at that round (see Appendix B.2 for the prompt). We then have the Tongyi DeepResearch Agent resume generation from the teacher-revised context. Table 2 compares the task success rate when the agent continues from the original context versus the teacher-revised context. The results show that resuming generation from the teacher-revised context improves task success accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"➤ Takeaway 2. Teacher-hypothesized counterfactual anchor contexts can partially recover failed trajectories: compared with resuming from the original context, resuming from teacher-revised contexts yields higher task success in both textual and visual search settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These two findings motivate ToolAnchor, which systematically hypothesizes critical anchor rounds, verifies the counterfactual contexts, and internalizes the verified alternatives into the student agent.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Methodology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we present ToolAnchor , a framework for adapting a post-trained tool-use agent to an expanded toolset through counterfactual anchor contexts. ToolAnchor consists of three stages: counterfactual anchor context hypothesis, counterfactual anchor context verification, and counterfactual anchor context internalization.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Problem formulation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on the setup in Section 3.1 , we study toolset expansion for a post-trained tool-use agent π θ \\\\pi_{\\\\theta} . The agent has been trained with the original toolset U old U_{\\\\mathrm{old}} and is then exposed to the expanded toolset U = U old ∪ U new U=U_{\\\\mathrm{old}}\\\\cup U_{\\\\mathrm{new}} . The goal is to learn an updated policy π θ ′ \\\\pi_{\\\\theta^{\\\\prime}} that improves task success under U U while preserving the agent’s existing capability on tasks solvable with U old U_{\\\\mathrm{old}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our preliminary analysis in Section 3.3 reveals two key findings. First, the post-trained agent exhibits behavioral inertia , disproportionately relying on U old U_{\\\\mathrm{old}} even when U new U_{\\\\mathrm{new}} is available (Table 1 ). Second, failed trajectories can often be recovered when the agent resumes generation from a teacher-proposed counterfactual anchor context at a critical decision round (Table 2 ). These findings suggest that the bottleneck is not only missing tool-use capability, but also the lack of effective contextual anchors that redirect the agent away from its established behavioral patterns. This motivates ToolAnchor, which hypothesizes, verifies, and internalizes such contexts, as illustrated in Figure 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Counterfactual anchor context hypothesis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a failed trajectory ℋ T \\\\mathcal{H}_{T} produced by the base policy π θ \\\\pi_{\\\\theta} under the expanded toolset U U , this stage uses teacher models to form a counterfactual anchor hypothesis. The hypothesis consists of a candidate anchor round t ∗ < T t^{*}<T and an alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) (\\\\tau^{\\\\prime}_{t^{*}},a^{\\\\prime}_{t^{*}}) , which together define a revised context that may redirect the subsequent student rollout toward a successful trajectory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Failed trajectory collection.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first roll out the base policy π θ \\\\pi_{\\\\theta} on the training datasets under the expanded toolset U U , collecting both successful and failed trajectories. We define the failed trajectory set as 𝒟 fail = { ℋ T ∣ Success ( ℋ T ) = 0 } \\\\mathcal{D}_{\\\\mathrm{fail}}=\\\\{\\\\mathcal{H}_{T}\\\\mid\\\\mathrm{Success}(\\\\mathcal{H}_{T})=0\\\\} , which serves as the input for the teacher-guided counterfactual anchor hypothesis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Counterfactual round hypothesis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We define an anchor round hypothesis as a candidate round at which an alternative local decision may induce a different subsequent rollout and potentially change the task outcome. Forming this hypothesis requires assessing how earlier decisions, observations, and subsequent reasoning steps interact across the trajectory, rather than relying on a fixed heuristic such as choosing the first or last suspicious step. We instantiate this step with a prompt-guided teacher audit: given the full failed trajectory, a teacher model sequentially reviews each round’s thought τ t \\\\tau_{t} , action a t a_{t} , and observation o t o_{t} , and selects a candidate anchor round t ∗ t^{*} where a revised context is likely to redirect the subsequent rollout. The prompt used in our implementation is provided in Appendix B.2 . In our implementation, the teacher is instructed to use the following criteria:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Downstream impact. Prefer a round whose alternative local decision is likely to influence multiple subsequent steps in the trajectory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Path redirection. Prefer a round that acts as a branching point, where a different local decision may redirect the subsequent rollout toward a different reasoning or tool-use path.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Strategic leverage. Prefer a round involving high-level agentic decisions, such as reasoning direction, tool selection, query formulation, evidence interpretation, information integration, or planning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Counterfactual potential. Prefer a round where replacing the original local decision under the same history appears most likely to improve the final task outcome.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These criteria are intended to avoid trivial anchor-selection heuristics, such as always choosing the earliest suspicious step or the final decision step, both of which may fail to identify the round with the greatest potential to redirect the subsequent rollout.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Hypothesized context generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once the anchor round t ∗ t^{*} is hypothesized, the teacher model generates the alternative thought-action pair in a separate step using only the history available before that round. Specifically, the teacher is given only ℋ t ∗ − 1 \\\\mathcal{H}_{t^{*}-1} , without access to subsequent rounds or the ground-truth answer. The teacher then proposes an alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) ∼ M ( ⋅ ∣ ℋ t ∗ − 1 ) (\\\\tau^{\\\\prime}_{t^{*}},a^{\\\\prime}_{t^{*}})\\\\sim M(\\\\cdot\\\\mid\\\\mathcal{H}_{t^{*}-1}) , where M M denotes the teacher model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This separation is important: the anchor-round hypothesis can use the full failed trajectory for diagnosis, whereas the alternative thought-action pair must be generated under the same information constraint faced by the agent at round t ∗ t^{*} . This prevents the alternative context from encoding future observations or the ground-truth answer, making the resulting supervision compatible with the student’s inference-time information state. Although new-tool adoption is a central motivation, the teacher is not restricted to rounds involving U new U_{\\\\mathrm{new}} . It may also select rounds involving ineffective use of U old U_{\\\\mathrm{old}} , or reasoning steps that make the subsequent rollout less effective. The environment then executes a t ∗ ′ a^{\\\\prime}_{t^{*}} and returns the corresponding observation o t ∗ ′ o^{\\\\prime}_{t^{*}} . This produces the candidate anchor context c ∗ = ( ℋ t ∗ − 1 , τ t ∗ ′ , a t ∗ ′ , o t ∗ ′ ) c^{*}=(\\\\mathcal{H}_{t^{*}-1},\\\\tau^{\\\\prime}_{t^{*}},a^{\\\\prime}_{t^{*}},o^{\\\\prime}_{t^{*}}) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Teacher diversity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single teacher model may favor particular tool-use patterns or anchor-round choices when auditing failed trajectories. To improve coverage, we use a set of heterogeneous teacher models ℳ = { M j } j = 1 J \\\\mathcal{M}=\\\\{M_{j}\\\\}_{j=1}^{J} , each of which independently forms counterfactual anchor hypotheses for the same failed trajectory. Different teachers may select different anchor rounds or propose different alternative thought-action pairs, yielding a diverse set of candidate anchor contexts 𝒟 anchor = { c i ∗ } i = 1 m \\\\mathcal{D}_{\\\\mathrm{anchor}}=\\\\{c_{i}^{*}\\\\}_{i=1}^{m} . Each c i ∗ c_{i}^{*} is a teacher-hypothesized candidate counterfactual context and will be verified by student rollout in Section 4.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Counterfactual anchor context verification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Teacher-generated anchor contexts are hypotheses rather than ground-truth demonstrations. A teacher may select an ineffective anchor round, propose a plausible but unhelpful alternative decision, or generate an observation that does not support the student’s subsequent reasoning. To ensure data quality without manual annotation or heuristic reward shaping, we introduce a student-side verification stage. This stage retains an anchor context only when the student successfully solves the task after resuming from it and a separate judge model, with access to the completed trajectory, verifies that the anchor context was materially useful for the final success.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Anchor-conditioned agentic rollout and dual-level verification.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each candidate anchor context c ∗ ∈ 𝒟 anchor c^{*}\\\\in\\\\mathcal{D}_{\\\\mathrm{anchor}} , we let the base policy π θ \\\\pi_{\\\\theta} resume generation from c ∗ c^{*} under the standard agentic rollout procedure, producing a complete trajectory ℋ T ′ = Rollout ( π θ , c ∗ ) \\\\mathcal{H}^{\\\\prime}_{T}=\\\\mathrm{Rollout}(\\\\pi_{\\\\theta},c^{*}) . We retain c ∗ c^{*} only if it passes two verification checks. First, the final answer in ℋ T ′ \\\\mathcal{H}^{\\\\prime}_{T} must be judged correct by the task-level evaluator, denoted as J task ( ℋ T ′ ) ∈ { 0 , 1 } J_{\\\\mathrm{task}}(\\\\mathcal{H}^{\\\\prime}_{T})\\\\in\\\\{0,1\\\\} . Second, an anchor-level judge, denoted as J anchor ( c ∗ , ℋ T ′ ) ∈ { 0 , 1 } J_{\\\\mathrm{anchor}}(c^{*},\\\\mathcal{H}^{\\\\prime}_{T})\\\\in\\\\{0,1\\\\} , verifies from a global view that the proposed anchor context was materially useful in enabling the successful outcome. Formally, the verified effective-context dataset is\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why verification matters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Verification turns teacher hypotheses into training signals. The task-level success check filters out anchors from which the student cannot solve the task, and the anchor-level judge removes cases where success is not materially supported by the proposed anchor. Since the continuation after c ∗ c^{*} is generated by the student policy π θ \\\\pi_{\\\\theta} , the retained data avoid full teacher-trajectory imitation and remain closer to the student’s rollout distribution.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Counterfactual anchor context internalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The final stage internalizes verified counterfactual anchor contexts into the agent’s policy through a two-step pipeline, combining local policy internalization and policy alignment with exploration.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Local policy update (anchor-only agentic SFT).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We perform anchor-only agentic SFT on the verified effective contexts, training the agent to reproduce the teacher-proposed thought-action pair at the verified counterfactual anchor round. For each verified counterfactual context ( c ∗ , ℋ T ′ ) ∈ 𝒟 eff (c^{*},\\\\mathcal{H}^{\\\\prime}_{T})\\\\in\\\\mathcal{D}_{\\\\mathrm{eff}} , where c ∗ = ( ℋ t ∗ − 1 , τ t ∗ ′ , a t ∗ ′ , o t ∗ ′ ) c^{*}=(\\\\mathcal{H}_{t^{*}-1},\\\\tau^{\\\\prime}_{t^{*}},a^{\\\\prime}_{t^{*}},o^{\\\\prime}_{t^{*}}) , the SFT loss is computed only on the alternative thought-action pair ( τ t ∗ ′ , a t ∗ ′ ) (\\\\tau^{\\\\prime}_{t^{*}},a^{\\\\prime}_{t^{*}}) , conditioned on the original history ℋ t ∗ − 1 \\\\mathcal{H}_{t^{*}-1} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This objective directly injects the verified anchor behaviors at high-leverage decision points, providing a precise behavioral guidance. While optimizing solely on 𝒟 eff \\\\mathcal{D}_{\\\\text{eff}} could risk localized overfitting, this anchor-only SFT serves as a targeted policy initialization. The subsequent post-SFT agentic RL stage then realigns these local updates with the agent’s prior broad capabilities, effectively mitigating catastrophic forgetting and preserving performance on tasks solvable with U old U_{\\\\text{old}} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Policy realignment and exploration (post-SFT agentic RL).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following agentic SFT, we apply agentic RL (GRPO) on the expanded toolset U U , using a binary task-level reward (1 for correct answers, 0 otherwise). This stage serves to integrate the internalized anchor behaviors from SFT with the agent’s prior RL policy, consolidating these local updates into a policy that is both aligned with prior knowledge and capable of effective inference under the expanded toolset. Simultaneously, it enables the agent to explore additional tool-use strategies beyond those explicitly provided by the teachers. Collectively, agentic SFT and agentic RL ensure that local counterfactual behaviors are internalized, properly aligned with the prior policy, and generalized, allowing the agent to leverage both newly introduced tools and previously learned capabilities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Experimental setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Student and teacher models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use the Tongyi DeepResearch Agent Team et al. ( 2025 ) (Qwen3-30B-A3B) as our base model π θ \\\\pi_{\\\\theta} , which is trained via agentic RL on a fixed toolset U old U_{\\\\text{old}} consisting of web search, web browse, a Python interpreter, scholar search, and a file parser. We extend it with three visual tools U new U_{\\\\text{new}} : an image descriptor, a bbox extractor, and crop-and-image search. We choose the Tongyi DeepResearch Agent not only because its system prompts, toolsets, and model weights are open-source (allowing for training and customization), but also because it demonstrates strong performance on general AI assistant and textual search tasks. For the counterfactual anchor hypothesis, we employ Gemini-2.5-Pro and GPT-5 as diverse teacher models to identify anchor rounds and generate alternative reasoning and actions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training datasets and benchmarks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use two training datasets that span both textual and visual search tasks: (1) OpenSeeker-1.5k, which contains 1,500 instances sampled from OpenSeeker-v1-Data Du et al. ( 2026 ) ; and (2) VDR-1.5k Huang et al. ( 2026 ) , a visual search dataset that also contains 1,500 samples. We evaluate our method on three benchmarks corresponding to distinct tasks: GAIA Mialon et al. ( 2023 ) (General AI assistant, 165 samples), BrowseComp Wei et al. ( 2025 ) (Textual search, 200 samples subset following Du et al. ( 2026 ) ), and VDR-Bench Zeng et al. ( 2026 ) (Visual search, test-mini split with 500 samples, following Huang et al. ( 2026 ) ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Evaluation metrics and baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following prior work Huang et al. ( 2026 ); Du et al. ( 2026 ) , we adopt an LLM-as-a-judge approach for evaluation, using Qwen3-235B-A22B as the judge model. Specifically, we use the official prompts provided by each benchmark to assess the correctness of the agent’s final answer against the ground truth. Our primary evaluation metric is task success rate under Pass@1, where each query is evaluated with a single agent rollout. All benchmark results are reported with Pass@1. Pass@3 is used only during counterfactual anchor verification, where a candidate context passes task-level verification if at least one of three student rollouts solves the task. Please refer to Appendix B.4 for the corresponding evaluation prompts. We compare ToolAnchor against both proprietary and open-source deep research agents, including GPT-5 Singh et al. ( 2025a ) , Gemini-2.5-Pro Comanici et al. ( 2025 ) , Claude-4-Sonnet, OpenAI DeepResearch, Qwen3-VL-30B-A3B-Thinking Bai et al. ( 2025 ) , Qwen3-VL-30B-A3B-Instruct, OpenSeeker Du et al. ( 2026 ) , Vision DeepResearch Huang et al. ( 2026 ) , and Tongyi DeepResearch Team et al. ( 2025 ) . Implementation details, including training hyperparameters, inference hyperparameters, and compute resources, are provided in Appendix C.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Main results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 3 , we compare ToolAnchor-DeepResearch-30B-A3B against both proprietary and open-source deep research agents across three benchmarks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Visual search.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On VDR-Bench, ToolAnchor achieves 28.8%, outperforming Gemini-2.5-Pro (18.8%), GPT-5 (20.4%), and Qwen3-VL-30B-A3B-Thinking (23.2%), despite using a smaller 30B LLM without prior visual-task training. This substantially narrows the gap with a vision-language model (VLM) agent trained from scratch for visual search, namely Vision-DeepResearch-30B-A3B (37.8%). This performance gap is a predictable trade-off: Vision-DeepResearch is a native vision-language model (VLM) agent jointly optimized from scratch on a 5 × \\\\times larger multimodal dataset (15K instances). In contrast, ToolAnchor operates on a frozen, text-only LLM backbone augmented purely via dynamic tool APIs. Achieving a competitive 28.8% without expensive multimodal alignment or massive data dependency underscores the remarkable parameter and compute efficiency of our toolset expansion paradigm.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"General assistant and textual search.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On GAIA, ToolAnchor achieves 74.5%, improving over the base model (70.9%) by 3.6 percentage points and surpassing GPT-5 (67.4%), Claude-4-Sonnet (68.3%), and OpenAI DeepResearch (67.4%). On BrowseComp, ToolAnchor achieves 45.0%, a 1.6 percentage point improvement over the base model (43.4%). These results demonstrate that ToolAnchor enables the agent to acquire new visual tool-use capabilities while largely preserving its performance on the original textual tasks. The slight improvement observed on GAIA and BrowseComp suggests that internalizing effective counterfactual anchor contexts can also reinforce the agent’s existing tool usage and reasoning policies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Ablation study\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Visual tools.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 compares the baseline Tongyi-DeepResearch-30B-A3B model with a variant equipped with our visual tools. The results demonstrate that while the integration of these visual tools significantly boosts performance on the VDR-Bench visual search (from 7.5% to 16.6%), it incurs a marginal performance degradation on textual search benchmarks (GAIA and BrowseComp).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Post-training method.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 ablates the post-training pipeline after visual-tool expansion. Agentic SFT on unverified counterfactual contexts performs poorly, especially on BrowseComp, indicating that teacher-hypothesized contexts can introduce noisy or ineffective supervision if applied without student-side verification. Using verified contexts improves over unverified agentic SFT on BrowseComp and VDR-Bench, supporting the utility of counterfactual anchor context verification. Agentic RL trained from scratch achieves moderate performance (70.3%, 41.5%, 21.6%), indicating that without access to counterfactual anchor contexts via SFT, RL alone exhibits limited exploration and fails to fully leverage the available toolset. However, agentic SFT with verified contexts alone still underperforms the ToolAnchor, suggesting that anchor-context internalization through SFT is insufficient to fully recover and improve the agent’s tool-use policy. By combining verified SFT with a subsequent agentic RL, ToolAnchor achieves the best performance across all three benchmarks. This indicates that verified counterfactual contexts provide effective local supervision, while the post-SFT RL stage aligns the internalized anchor behaviors with the agent’s previously learned policy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Further analysis and discussions\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Does the choice of anchor round matter?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To isolate the efficacy of our criterion-based anchor selection from the mere presence of teacher intervention, we benchmark it against three heuristic baselines: intervening at a random round, the first erroneous round, and the last erroneous round. Evaluated across 200 instances from both OpenSeeker and VDR-Train (using Gemini-3.1-Pro-Preview as the teacher), our criterion-based strategy consistently yields the highest recovery rates. We repeat teacher anchor selection 50 times on one failed trajectory. As illustrated in Figure 5 , effective anchors heavily cluster around strategic causal pivots rather than trivial first/last-error boundaries, confirming that a precise anchor location is as critical as the counterfactual context itself.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"How many counterfactual contexts survive verification?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each teacher-generated counterfactual context, we sample three student rollouts and use Pass@3 task-level verification, where a context passes if any rollout solves the task. We then apply anchor-level verification to successful rollouts, retaining only contexts whose success is judged materially attributable to the anchor intervention. Overall, 33.67% of candidates pass task-level verification and 31.98% remain after anchor-level verification, showing that verification filters noisy teacher hypotheses while retaining contexts that both recover the task and validly intervene on the failure point.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.5 Case study\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Figure 1 , we present a case study to illustrate how ToolAnchor redirects a failed visual-search trajectory. In this example, the base Tongyi DeepResearch Agent first uses visual tools to locate the green-hooded character and invoke the crop-and-image search tool, but then dismisses the returned visual-search leads as generic and switches to broad text search, eventually producing an incorrect answer. The teacher hypothesizes that this decision is the counterfactual anchor round, since it is the key point where the agent abandons the visual-evidence path. The generated counterfactual context instead keeps the agent on the visual-search path by analyzing the crop-and-image search results and issuing a targeted follow-up query. This qualitative example illustrates how a local counterfactual context can redirect the subsequent rollout. Details are provided in Appendix A . Consistent with this qualitative example, Appendix C.3 shows that hypothesized counterfactual anchor rounds are broadly distributed across different trajectory positions, with most occurring within the first 10-20 rounds, and that each can potentially produce distinct effects on the agent’s subsequent behavior.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion and limitation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we propose ToolAnchor, a framework for extending post-trained tool-use agents with new tools through verified counterfactual context anchoring. Our main insight is that toolset expansion is not merely about adapting tool-selection and reasoning patterns to newly introduced tools. Post-trained agents may also need effective contextual anchors to break out of established reasoning and tool-use patterns. ToolAnchor turns teacher-hypothesized counterfactual contexts into training signals only after student-side verification, allowing the agent to internalize local high-leverage redirections rather than imitate full teacher trajectories. Experiments across general assistance, textual search, and visual search show that this mechanism improves new visual-tool use while also benefiting existing tool use and agentic reasoning, suggesting a practical path for capability expansion without retraining agents from scratch. In this work, the experiments rely on a limited set of teacher models and verified anchor contexts. Future research could investigate incorporating a more diverse set of teacher models and expanding the training dataset to enhance coverage, robustness, and generalization of internalized behaviors across a broader range of tasks and toolsets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first present a failed inference trajectory from the Tongyi DeepResearch Agent in Appendix A.1 , with the corresponding agent prompt provided in Appendix B.1 . We then show the teacher model’s counterfactual anchor-round hypothesis for this trajectory in Appendix A.2 , with the corresponding hypothesis prompt provided in Appendix B.2 . Finally, we provide the counterfactual anchor context generated for the hypothesized anchor round in Appendix A.3 , with the corresponding counterfactual context generation prompt provided in Appendix B.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 Failed inference trajectory of the deep research agent\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 Teacher counterfactual anchor round hypothesis\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.3 Teacher counterfactual anchor context generation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 System prompt for deep research agent\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the Tongyi DeepResearch Agent, we provide the system prompt used by ToolAnchor, with the original tool-use instructions extended to include the new visual tools.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 System Prompt for teacher model counterfactual anchor round hypothesis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We provide the teacher prompts for counterfactual anchor round hypothesis and counterfactual anchor context generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 Prompt for anchor-level LLM-As-Judge verification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We provide the anchor-level LLM-as-judge verification prompt below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.4 Prompt for LLM-As-Judge evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For GAIA, BrowseComp, and VDR-Bench, we use the official evaluation prompts provided by each benchmark.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.1 Tool descriptions\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Web search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A web search tool that executes Google searches and retrieves the top 10 results. It accepts natural language queries and returns structured data, including page titles, URLs and snippets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Web browse\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A web browse tool (visit tool) that visits a specified URL and returns a summary. It accepts a target URL and a specific goal, uses Jina to convert the page content into Markdown, and then utilizes the DeepSeek-V3.2 summarization model to extract information relevant to the goal.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Python interpreter\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A Python interpreter tool that executes Python code within a secure sandbox.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scholar search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A scholar search tool that executes Google Scholar searches and retrieves the top 10 results. It accepts natural language queries and returns structured data, including page titles, URLs and snippets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"File parser\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A file parser tool that analyzes various files and extracts their content as plain text. It accepts both local files and URLs, converting them into a readable text format for further use.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Image descriptor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An image description tool powered by the multimodal model Qwen2.5-72B-Instruct. It supports both local image files and image URLs, and answers user queries about images.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Bbox extractor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A bounding box (bbox) extractor that uses the multimodal model Qwen2.5-72B-Instruct to locate specific entities within an image. It accepts both local image files and URLs, as well as entity queries, and returns the bounding box coordinates of the requested entity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Crop-and-image search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An image cropping and search tool that executes Google Lens and retrieves the top 10 results. It accepts an image as input and returns structured data, including titles, URLs, and snippets. The snippets are generated by the multimodal model Qwen2.5-72B-Instruct, which compares the similarities and differences between the query image and the webpage images, a design motivated by VC-STaR Pan et al. ( 2026 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Tongyi toolset includes a web search tool, a web browsing tool, a Python interpreter, a scholar search tool, and a file parser tool. The VDR toolset consists of a web search tool, a web browsing tool, a Python interpreter, and a visual search tool. Building on the Tongyi toolset, our toolset further incorporates an image description tool, a bounding-box extraction tool, and a crop-and-image-search tool.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.2 Training and inference details\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments were conducted on a cluster of eight NVIDIA B200 (192GB) GPUs using the Slime framework Zhu et al. ( 2025 ) . All training hyperparameters are listed in Table 5 , and Table 6 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 7 , our hyperparameters are consistent with those of the Tongyi Deep Research Agent.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.3 Statistical analysis of counterfactual anchor rounds\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 shows the distribution of counterfactual anchor rounds hypothesized by Gemini-2.5-Pro on OpenSeeker-1.5k and VDR-1.5k. On both datasets, the hypothesized anchor rounds are heavily concentrated in the early stages of trajectories, with most anchors occurring within the first 10–20 rounds. This suggests that early counterfactual contexts exert greater influence on subsequent reasoning and tool-use decisions, while preserving sufficient rollout horizon for exploration and recovery. Similarly, on VDR-1.5k, the early-round concentration is even more pronounced, indicating that visual-search tasks particularly benefit from introducing visual-evidence-oriented contexts near the beginning of the trajectory. The long-tail of the distribution indicates that valuable counterfactual contexts also occur at later rounds in some trajectories. These observations support our use of teacher-guided anchor-round hypothesis generation rather than relying on fixed heuristics such as always selecting the first round.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To mitigate search-time data contamination Han et al. ( 2025 ) , we filter out specific domain names, such as github.com and huggingface.co, during the web search process.\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":2,"numItemsPerPage":100,"numTotalItems":1000,"offset":200,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTkxNDI1OCwianRpIjoiZjE4OTkwZTAtMGRjMC00MDM2LWJhNjYtMTlhZDIzNmY1ZTI5Iiwic3ViIjoiL2RhdGFzZXRzL1ByaW1lSW50ZWxsZWN0L2Rlc2h1ZmZsZS1wYXBlcnMtdjEtY29ycHVzIiwiZXhwIjoxNzg1OTE3ODU4LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.fmhN1vyXkITX4jq418dE2FMRBGkEJ7GD17vM5wXW4IZH8Sff0b7-zeijLabHwrtP7mp8Hj5snrOmxKhzdf4kDw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1000,"numBytesParquet":49695546}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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|---|---|---|---|---|---|---|---|---|---|
arxiv | 2607.15275 | null | 1 | RoboTTT: Context Scaling for Robot Policies | [
"Yunfan Jiang",
"Yevgen Chebotar",
"Ruijie Zheng",
"Fengyuan Hu",
"Yunhao Ge",
"Jimmy Wu",
"Tianyuan Dai",
"Scott Reed",
"Li Fei-Fei",
"Yuke Zhu",
"Linxi \"Jim\" Fan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15275v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Figure 1: RoboTTT , a long-context visuomotor policy that integrates Test-Time Training (TTT) into robot foundation models, with context scaled to 8K timesteps. RoboTTT exhibits capabilities such as one-shot in-context imitation from human videos and on-the-fly policy improvement.... |
arxiv | 2607.15273 | null | 1 | MeanFlowNFT: Bringing Forward-Process RL to Average-Velocity Generators | [
"Yushi Huang",
"Xiangxin Zhou",
"Jun Zhang",
"Liefeng Bo",
"Tianyu Pang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15273v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "MeanFlowNFT: Bringing Forward-Process RL to Average-Velocity Generators"
},
{
"type": "para",
"text": "Yushi Huang 1,2,∗ Xiangxin Zhou 1,∗,‡ Jun Zhang 2 Liefeng Bo 1 Tianyu Pang 1,‡"
},
{
"type": "para",
"text": "1 Tencent Hunyuan 2 The Hong Kong University... |
arxiv | 2607.15271 | null | 1 | Online Neural Space Time Memory for Dynamic Novel View Synthesis | [
"Baback Elmieh",
"Lynn Tsai",
"Zeman Li",
"Srinivas Kaza",
"Tiancheng Sun",
"Gabor Csapo",
"Ali Behrouz",
"Yuan Deng",
"Stephen Lombardi",
"Steven M. Seitz",
"Xuan Luo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15271v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Online novel view synthesis (NVS) of dynamic scenes from multi-view streaming videos is a highly sought-after goal in computer vision, underpinning many applications such as 3D telepresence and live free-... |
arxiv | 2607.15217 | null | 1 | NeuronSoup: Evolving Asynchronous, Shared-Neuron Temporal Graphs without Backpropagation | [
"Subodh Kalia"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15217v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The deep learning revolution rests on a single algorithmic foundation: backpropagation through differentiable computation graphs. This one requirement dictates nearly every architectural decision we make.... |
arxiv | 2607.15207 | null | 1 | BadWAM: When World-Action Models Dream Right but Act Wrong | [
"Qi Li",
"Xingyi Yang",
"Xinchao Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15207v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Robotic foundation models are rapidly moving from action prediction toward world-action modeling [ kim2025openvla , black2024pi_0 , pi0.7 , zitkovich2023rt , yuan2026fastwam , shen2026world ] . A world-ac... |
arxiv | 2607.15200 | null | 1 | Mask-Aware Policy Gradients for Diffusion Language Models | [
"Haran Raajesh",
"Kulin Shah",
"Adam Klivans",
"Philipp Krähenbühl"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15200v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning has emerged as a powerful tool for improving large language models (Guo et al. , 2025 ; Jaech et al. , 2024 ) , with applications spanning instruction following (Ouyang et al. , 202... |
arxiv | 2607.15196 | null | 1 | Subjective Risk Decomposition: A New View for Uncertainty Quantification | [
"Raghad Alamri",
"Michele Caprio",
"Gavin Brown"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15196v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Keywords: epistemic, aleatoric, subjective risk, bias–variance, mutual information."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A popular framework for uncertainty quantification is the aleatoric/epistemic... |
arxiv | 2607.15180 | null | 1 | RTS Smoother-Guided Learning of Physics-Based Neural Differential Models | [
"Ahmet Demirkaya",
"Georgios Stratis",
"Tales Imbiriba",
"Zachary D. Danziger",
"Deniz Erdogmus"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15180v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Ordinary Differential Equations (ODEs) are a standard tool for describing how dynamical systems evolve over time in physics, biology, neuroscience, medicine, and engineering. In practice, however, the equ... |
arxiv | 2607.15161 | null | 1 | On-Policy Delta Distillation | [
"Byeongho Heo",
"Jaehui Hwang",
"Sangdoo Yun",
"Dongyoon Han"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15161v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "As the use of large language models (LLMs) continues to expand, the demands placed on them have become increasingly diverse, driving continuous advances in training methods. While next-token prediction Ra... |
arxiv | 2607.15094 | null | 1 | AlphaWiSE: Adaptive Weight Interpolation for Continual Multimodal Representation Learning | [
"Sarthak Jain",
"Qiran Hu",
"Zhen Zhu",
"Yaoyao Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15094v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Multimodal representation models have become the foundation of modern retrieval systems by learning a shared embedding space across modalities such as images, text, and audio. Models such as CLIP (Radford... |
arxiv | 2607.15065 | null | 1 | DriftWorld: Fast World Modeling through Drifting | [
"Susie Lu",
"Haonan Chen",
"Weirui Ye",
"Yilun Du"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15065v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Keywords: World Models, Action-Conditioned Video Generation"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Predictive world models have emerged as powerful tools for robot learning, letting robots predict the... |
arxiv | 2607.15047 | null | 1 | Parameter-efficient Prompt Tuning of Vision Foundation Model With Adaptive Focal Loss for Interpretable MCI Screening | [
"Javad Khoramdel",
"Farhad Hoseyni",
"Amirhossein Nikoofard"
] | CC-BY-4.0 | https://arxiv.org/html/2607.15047v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "K eywords Vision Foundation Models, Parameter-Efficient Fine-Tuning, Interpretable Deep Learning, Mild Cognitive Impairment, Neuropsychological Assessment, Medical Image Analysis"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "pa... |
arxiv | 2607.14962 | null | 1 | Multi-Axis Max@K Reinforcement Learning for Representative Diversity in Text-to-Image Generation | [
"Ku Onoda",
"Paavo Parmas",
"Hiroki Furuta",
"Soichiro Nishimori",
"Yuta Oshima",
"Shohei Taniguchi",
"Yutaka Matsuo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14962v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Text-to-image (T2I) models can synthesize many plausible images for a prompt [ 49 , 13 ] , but repeated samples for the same prompt often exhibit limited visual diversity. Existing diversity interventions... |
arxiv | 2607.14943 | null | 1 | Steering Robustness into World Action Models via Mechanistic Interpretability and Optimal Control | [
"Jihoon Hong",
"Julian Skifstad",
"Qiyue Dai",
"Alice Chan",
"Glen Chou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14943v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Keywords: mechanistic interpretability, world action models, optimal control"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Foundation models have advanced robot learning through policies that generalize acro... |
arxiv | 2607.14937 | null | 1 | A Minimal Interpretable Architecture for Zero-Shot Reconstruction of Dynamical Systems | [
"Christoph Jürgen Hemmer",
"Florian Plaswig",
"Daniel Durstewitz"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14937v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Many, if not most, natural and engineered systems, from chemical and molecular processes to climate, ecosystems, brain activity, or stock markets, are naturally described as dynamical systems (DS) [ 30 , ... |
arxiv | 2607.14921 | null | 1 | Random Logit Scaling: Defending Deep Neural Networks Against Black-Box Score-Based Adversarial Example Attacks | [
"Hamid Dashtbani",
"Mehdi Dousti Gandomani",
"AmirMahdi Sadeghzadeh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14921v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The remarkable performance of deep neural networks has led to a widespread increase in demand for their adoption in various domains ranging from image classification and (Deng et al. , 2009 ) , face recog... |
arxiv | 2607.14888 | null | 1 | Innocuous-Seeming Data, Latent Ideology: Ideological Generalisation in Finetuned LLMs | [
"Robert Graham",
"Edward Stevinson",
"Yariv Barsheshat"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14888v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "This paper contains text that might be offensive."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Practitioners often need to adapt language models to reflect specific beliefs, values, or domain expertise, yet... |
arxiv | 2607.14817 | null | 1 | Evaluating Epistemic Uncertainty: Beyond OOD Detection and Active Learning | [
"Jakub Paplhám",
"Willem Waegeman",
"Eyke Hüllermeier",
"Vojtěch Franc"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14817v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The primary goal of uncertainty quantification and disentanglement is to isolate epistemic uncertainty (model ignorance) from aleatoric uncertainty (irreducible data noise). Successfully isolating these c... |
arxiv | 2607.14770 | null | 1 | ChronoQG: Towards a Temporally Expressive and Hop-Bounded Benchmark for Temporal Knowledge Graph Question Generation | [
"Xuemeng Liu",
"Zhengpin Li",
"Wanpeng Tang",
"Haotong Xie",
"Wentao Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14770v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Knowledge graph question generation (KGQG) is a core task for converting structured graph evidence into natural-language question–answer pairs. Given a support subgraph and a target answer, KGQG aims to ... |
arxiv | 2607.14731 | null | 1 | What's in a Smoothness Constant? Tighter Rates for Local SGD with Bounded Second-order Heterogeneity | [
"Kumar Kshitij Patel",
"Rustem Islamov",
"Sebastian U Stich",
"Aurelien Lucchi",
"Eduard Gorbunov",
"Lingxiao Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14731v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Contents"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "In large-scale distributed training of machine learning algorithms, communication is often the primary bottleneck, outweighing comput... |
arxiv | 2607.14721 | null | 1 | Multimodality as Supervision: Self-Supervised Specialization to the Test Environment via Multimodality | [
"Kunal Pratap Singh",
"Ali Garjani",
"Rishubh Singh",
"Muhammad Uzair Khattak",
"Efe Tarhan",
"Jason Toskov",
"Andrei Atanov",
"Oğuzhan Fatih Kar",
"Amir Zamir"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14721v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction."
},
{
"type": "para",
"text": "Multimodality is a fundamental self-supervision mechanism . Consider this, you are a newly born agent with multiple sensory modalities, e.g., sight and sound, but no prior representation for what they re... |
arxiv | 2607.14706 | null | 1 | MESHA: Mechanism-Enforced Sequential Halving for Strategic Linear Bandits | [
"Xin Li",
"Zixin Zhong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14706v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I INTRODUCTION"
},
{
"type": "para",
"text": "Consider a hiring platform seeking to hire the optimal candidate from a pool of K K applicants within a limited budget of T T interview rounds. Each candidate i i is inherently characterized by a true bac... |
arxiv | 2607.14672 | null | 1 | Scalable Training of Continuous-Time Spiking Neural Networks with Differentiable Spike-Time Discretization | [
"Yusuke Sakemi",
"Tomoya Takeuchi",
"Takeo Hosomi",
"Kazuyuki Aihara"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14672v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Spiking neural networks provide a computational framework in which information is represented by discrete spike events and their timing [ 53 ] . This makes them relevant both for understanding temporal co... |
arxiv | 2607.14614 | null | 1 | Beyond Entropy: Correctness-Aware Advantage Shaping via Contrastive Policy Optimization | [
"Weiwen Xu",
"Jia Liu",
"Hou Pong Chan",
"Long Li",
"Deng Cai",
"Min Chen",
"Hao Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14614v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning with verifiable rewards (RLVR) has substantially advanced performance in domains like mathematics and programming by providing a simple yet effective recipe for improving reasoning ... |
arxiv | 2607.14571 | null | 1 | Gate-Zero Growth: A Geometric Framework for Function-Preserving Continual Learning | [
"Dante Lok"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14571v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The dominant paradigm for obtaining a more capable language model is to train a larger model from scratch. Model growth — expanding an existing trained model by adding parameters — offers a compelling alt... |
arxiv | 2607.14537 | null | 1 | MIDI-RAE-JEPA: Hierarchical Representation Learning and Generation for Symbolic Music | [
"Scott H. Hawley"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14537v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A long-standing goal in music technology is an intelligent AI assistant that can listen to a musical idea from a human and provide meaningful feedback to refine the composition, arrangement, or production... |
arxiv | 2607.14516 | null | 1 | Adaptive Runge-Kutta Step Control Buys Training Loss, Not Generalization: An Honest Compute-Matched Study of RK-Adam Optimizers | [
"Akhilesh Gogikar"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14516v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Since the observation that Nesterov’s method (Nesterov, 1983 ) is a discretization of a second-order ODE (Su et al. , 2016 ) , it has been tempting to run the logic in reverse: if optimizers are integrato... |
arxiv | 2607.14509 | null | 1 | Multi-Scale ViT Inference with Habitat-Fit Priors and kNN Retrieval for Multi-Species Plant Identification | [
"Alper Erten",
"Murilo Gustineli",
"Adrian Cheung"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14509v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "CLEF 2026: Conference and Labs of the Evaluation Forum, September 21-24, 2026, Jena, Germany"
},
{
"t... |
arxiv | 2607.14475 | null | 1 | One-Shot Generative Design for Disordered Metamaterials via Self-Organizing Neural Cellular Automata | [
"Yujie Xiang",
"Liwei Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14475v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Investigation, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing 1]organization=Department of Mechanical Engineering, Carnegie Mellon University, addressline=5000 Forbes Avenue, city=Pittsburgh, postcode=15213, state=PA, country=USA"
},... |
arxiv | 2607.14474 | null | 1 | Can Tokens Compete? Token Representations against Supervised CNN Backbones for BirdCLEF+ 2026 | [
"Anthony Miyaguchi",
"Murilo Gustineli",
"Adrian Cheung"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14474v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "CLEF 2026: Conference and Labs of the Evaluation Forum, September 21-24, 2026, Jena, Germany"
},
{
"t... |
arxiv | 2607.14466 | null | 1 | Interleaved Noise Injection Improves Clean, Corrupted, and OOD Performance | [
"Matt L. Wiemann",
"Peter Melchior",
"Andrew K. Saydjari"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14466v1 | 2026-07-16T00:00:00 | [
{
"type": "para",
"text": "black"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Stochastic gradient descent (SGD) and its variants are widely used as optimization algorithms for complex, non-convex functions, which often contain many u... |
arxiv | 2607.14439 | null | 1 | Active Real-World Factor-Based Evaluation for Generalist Robot Policies | [
"Andrew Liao",
"Hanchen Cui",
"Karthik Desingh",
"Aryan Deshwal"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14439v1 | 2026-07-16T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Recent advances in foundation models across vision, language, and multimodal domains [ 37 , 8 , 30 ] have inspired a parallel surge in generalist robot manipulation policies. These models use similar arch... |
arxiv | 2607.14424 | null | 1 | ConFlow: Constraints-Guided Learning with Flow Matching for Motion Generation | [
"Nutan Chen",
"Jianxiang Feng",
"Marvin Alles",
"Botond Cseke"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14424v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Recent advances in generative modeling, particularly diffusion models and flow matching, have driven interest in learning-based robot motion generation. Beyond their ability to model complex motion distri... |
arxiv | 2607.14396 | null | 1 | CatalogAgent: A Supervisor-mediated Self-Learning System Enabling Context Engineering for GenAI Models | [
"Zhu Cheng",
"Zhenming Wang",
" Yu",
" Tang",
"Dan Liu",
"Bryan Zhang",
"Athanasios N. Nikolakopoulos",
"Pranav Souri Itabada",
"Jing Zhang",
"Chih-Chi Chou",
"Peng Gao",
"Fatemeh Mansoori",
"Bharat Bojja",
"Sarath Chander",
"Sameer Thombare",
"Umit Batur",
"Tarik Arici"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14396v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Catalog enrichment plays a vital role in simplifying seller listing processes and enhancing customer shopping experiences in e-commerce. High-quality catalogs serve as crucial resources for customers and ... |
arxiv | 2607.14385 | null | 1 | MamaBench: Benchmarking LLM Robustness in Maternal and Child Health Diagnosis through Counterfactual Clinical Perturbation | [
"Thanni Adewuyi",
"Anuoluwa Sotome",
"Samuel Okoko",
"Angel Ezendu",
"Oluwafunke Akinbuwa",
"Oluwaseun Odunsi",
"Oluwasegun Oguntuase",
"Oluwadarasimi Oguntuase",
"Ifeoma Nwabueze",
"Abiodun Adereni"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14385v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "Keywords: counterfactual evaluation, clinical AI, maternal healthcare, retrieval-augmented generation, diagnostic robustness"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "LLMs now pass medical licensing exam... |
arxiv | 2607.14367 | null | 1 | Dysco: Dynamic Subspace Boosting to Mitigate LoRA Interference in Federated Learning | [
"Haobo Zhang",
"Jiankun Wang",
"Suraj Rajendran",
"Weishen Pan",
"Lam Tsoi",
"Yong Chen",
"Fei Wang",
"Jiayu Zhou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14367v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Federated learning (FL) [ 30 ] enables multiple clients to jointly train a model without centralizing raw data, making it attractive for privacy-sensitive applications and for the distributed adaptation o... |
arxiv | 2607.14338 | null | 1 | Beyond scalar losses: calibrating segmentation models via gradient vector field surgery | [
"Laurin Lux",
"Alexander H. Berger",
"Moritz Knolle",
"Daniel Rückert",
"Johannes C. Paetzold"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14338v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "315 \\firstpageno 3397 \\jmlryear 2026 \\jmlrworkshop Medical Imaging with Deep Learning \\midlauthor \\Name Laurin Lux \\nametag 1,2,3 \\Email laurin.lux@tum.de \\Name Alexander H. Berger \\nametag 1,3,4 \\Email a.berger@tum.de \\Name Moritz Knolle \\nametag 3 \\Email moritz.knol... |
arxiv | 2607.14272 | null | 1 | Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows | [
"Jingdong Zhang",
"Xinze Li",
"Yize Jiang",
"Luan Yang",
"Minkai Xu",
"Junhong Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14272v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Generative modeling has achieved substantial progress with the development of diffusion models [ 45 , 12 , 18 ] and their deterministic counterparts based on flow matching [ 27 , 29 , 49 ] . By learning a... |
arxiv | 2607.14246 | null | 1 | The Steering Budget: Examples beat Knobs | [
"Raj Kumar Rajendran"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14246v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction — Steering with knobs, and steering by example"
},
{
"type": "para",
"text": "Suppose you have a generative model and you want it to lean a certain way — images with the feel of nature photography, crystal structures with a wider band ga... |
arxiv | 2607.14205 | null | 1 | Privacy Leakage in Federated Learning in Radiology Reports: A Comparative Evaluation of Tokenizer-Driven Privacy Risks | [
"Santhosh Parampottupadam",
"Andres Martinez",
"Dimitrios Bounias",
"Sinem Sav",
"Klaus Maier-Hein",
"Ralf Floca"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14205v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Radiology reports serve as an indispensable tool in medical diagnostics, providing critical insights [1] that complement imaging data. These textual data contain rich clinical information, including patient... |
arxiv | 2607.14070 | null | 1 | Screening of Biosecurity Features in Metagenomic Data with Evo 2 Probes | [
"Jeremy Guntoro",
"Alexander Dack",
"Dylan Danno",
"Michaela Jančovičová",
"Križan Jurinović",
"Vanessa Smilansky"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14070v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Advances in genomic foundation models have enabled rich embedding representations of biological sequences, yet their application to biosecurity-relevant tasks remains limited. In this work, we investigate... |
arxiv | 2607.14037 | null | 2 | Early Adoption of Agentic Coding Tools by GitHub Projects | [
"Maliha Noushin Raida",
"Daqing Hou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14037v2 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Agentic coding tools are rapidly transitioning from completion-style assistants to semi-autonomous teammates that can independently generate, test, and submit pull requests (PRs) to large-scale software ... |
arxiv | 2607.14018 | null | 1 | Transforming Rank: How Architecture Navigates the Spectral Pathologies of Depth | [
"Katie Everett"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14018v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The modern Transformer feedforward block has a specific structure: each block normalizes the residual stream, expands the hidden dimension by a factor of roughly four through a pair of up- and down-projec... |
arxiv | 2607.14192 | null | 1 | Long-term User Engagement Optimization through Model-agnostic Downstream Rewards Learning | [
"Dingsu Wang",
"Filip Ryzner",
"Kelly He",
"Armando Ordorica",
"David Woo",
"Aditya Mantha",
"Liyao Lu",
"Usha Amrutha Nookala",
"Haoran Guo",
"Jiacong He",
"Olafur Gudmundsson",
"Matt Chun",
"Krystal Benitez",
"Dhruvil Deven Badani",
"Yijie Dylan Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14192v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Many modern recommendation systems have proven to be effective at optimizing short-term user actions. However, optimizing for immediate engagement alone could potentially hurt long-term user value. Prior... |
arxiv | 2607.14191 | null | 1 | TEDDY: A Pediatric Foundation Model for Risk Forewarning from ICD-Coded Diagnostic Histories | [
"Matthew Brady Neeley",
"Jorge Botas",
"Johnathan Jia",
"Lin Yao",
"Daniel Palacios",
"Benjamin Choi",
"Zhandong Liu",
"Hyun-Hwan Jeong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14191v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Pediatric patients differ from adults anatomically, physiologically, and in disease presentation, yet most clinical prediction tools are trained on adult data [ 1 ] . Doses, normal vital-sign ranges, and ... |
arxiv | 2607.13988 | null | 1 | TRACE: Turn-level Reward Assignment via Credit Estimation for Long-Horizon Agents | [
"Leitian Tao",
"Baolin Peng",
"Wenlin Yao",
"Tao Ge",
"Hao Cheng",
"Mike Hang Wang",
"Jianfeng Gao",
"Sharon Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13988v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language model (LLM) agents increasingly solve complex tasks by reasoning and acting through many interactions with external environments, including web navigation, software engineering, and general... |
arxiv | 2607.13874 | null | 1 | Relevance-Aware Rule: Structural Deletion of Irrelevant Conditions in Decision Trees | [
"Jung-Sik Hong",
"Jeongeon Lee",
"Min Kyu Sim",
"Sangheum Hwang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13874v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "Keywords: decision tree; interpretability; irrelevant condition; relevance-aware rule; pruning"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Decision trees translate model decisions into explicit root-to-le... |
arxiv | 2607.14180 | null | 1 | RENEW: Towards Learning World Models and Repairing Model Exploitation from Preferences | [
"Logan Mondal Bhamidipaty",
"Mykel Kochenderfer",
"Subramanian Ramamoorthy"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14180v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Model-based reinforcement learning offers a compelling path toward sample-efficient decision-making: by learning a dynamics model of the environment, an agent can plan and optimize behavior without costly... |
arxiv | 2607.13847 | null | 1 | Quantum Topological Data Encoding | [
"Adam Wesołowski",
"Dimitrios Thanos",
"Daniel Leykam",
"Lirandë Pira"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13847v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The ability to extract meaningful information from complex data lies at the heart of modern science and engineering. Across disciplines ranging from physics and chemistry to biology and the social science... |
arxiv | 2607.13837 | 10.1145/3690624.3709215 | 1 | NodeImport: Imbalanced Node Classification with Node Importance Assessment | [
"Nan Chen",
"Zemin Liu",
"Bryan Hooi",
"Bingsheng He",
"Jun Hu",
"Jia Chen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13837v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Graph data is prevalent across many domains, making graph analysis, particularly node classification, a significant research focus (Cai et al. , 2018 ; Wu et al. , 2020 ) . With the rise of deep learning... |
arxiv | 2607.13763 | null | 1 | MxGPS: Multiplex Graph Transformers for a Power Grid Foundation Model | [
"Charilaos Papaioannou",
"Ioannis Tsantilas",
"Dimitris Giannakakos",
"Vasilis Michalakopoulos",
"Sotiris Pelekis",
"Vangelis Marinakis",
"Arsam Aryandoust",
"Antonello Monti",
"Ricardo J. Bessa",
"Perdo P. Vergara",
"Jochen Cremer",
"Elissaios Sarmas"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13763v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The fundamental objective of power grid operation is the reliable, secure, and cost-effective delivery of energy. Traditionally, the core computational challenges of grid operation, namely AC power flow (... |
arxiv | 2607.13749 | null | 1 | Algebraic Representability as the Limiting Regime of Grokking: An Exactly Solvable Model with Holomorphic Activations | [
"Chon-Fai Kam",
"Xavier Cadet",
"Miloud Bessafi",
"Frederic Cadet"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13749v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "There is something genuinely puzzling about grokking. A network trained on a modular arithmetic task will, after memorising the training set, sit quietly for a long time—its training loss near zero, its t... |
arxiv | 2607.13735 | null | 1 | Constraint-Driven Model Optimization: An Industry Framework for Selecting Compression and Acceleration Techniques in Modern Machine Learning Systems | [
"Dhruv Shivkant",
"Saket Mohanty",
"Utkarsh Wadhwa"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13735v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "heading",
"level": 2,
"text": "1.1 The Deployment Imperative"
},
{
"type": "para",
"text": "Modern machine learning’s research interest, particularly in Large Language Models (LLMs), is shifting from furt... |
arxiv | 2607.13643 | null | 1 | Consensus as Privileged Context for Label-Free Self-Distillation | [
"John Gkountouras",
"Josip Jukić",
"Ivan Titov"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13643v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning with verifiable rewards has become the standard recipe for improving the reasoning abilities of large language models (Guo et al. , 2025 ; Shao et al. , 2024 ) , but it presupposes ... |
arxiv | 2607.13618 | null | 1 | STOCKTAKE: Measuring the Gap Between Perception and Action in LLM Agents with a Fair Oracle | [
"Sagar Deb",
"Ashwanth Krishnan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13618v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "LLM agents are increasingly evaluated on long-horizon decision tasks: running a vending machine for months [ 1 ] , managing a retail store [ 2 ] , operating supply chains [ 3 , 4 ] . These tasks share a s... |
arxiv | 2607.13612 | null | 1 | The SIGReg Objective as Variational Free Energy: A Theoretical Active-Inference Account of JEPA World Models | [
"Fabio Arnez",
"Alexandra Gomez-Villa"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13612v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Active Inference (AIF) and Joint-Embedding Predictive Architectures (JEPA) are two independently developed accounts of how an agent can learn and act from high-dimensional observations. AIF, rooted in the... |
arxiv | 2607.13609 | null | 1 | Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs | [
"Mohammad Forouhesh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13609v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Regularizing an ill-posed inverse problem on a graph requires two decisions: which penalty to add, and how strongly. They are usually treated separately, on the assumption that a reasonable penalty degrad... |
arxiv | 2607.14169 | null | 1 | When a Verified World Model Still Loses: Play-Adequacy vs Prediction-Accuracy in LLM-Synthesized Code World Models | [
"Javier Aguilar Martín"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14169v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "heading",
"level": 2,
"text": "1.1 The Code World Model paradigm"
},
{
"type": "para",
"text": "A central observation in recent work on LLMs for game playing is that a small language model plus a well-spe... |
arxiv | 2607.13568 | null | 1 | Graded Entity-Familiarity Readouts in Language Models: Polish Adaptation, Cross-Language Robustness, and Refusal Steering | [
"Grzegorz Brzezinka"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13568v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "Graded Entity-Familiarity Readouts in Language Models: Polish Adaptation, Cross-Language Robustness, and Refusal Steering † † thanks: Preprint. Under review."
},
{
"type": "para",
"text": "Grzegorz Brzezinka Prosit AS greg@prosit.no"
},
{
"type": "heading",
... |
arxiv | 2607.13566 | null | 1 | Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs | [
"Tianchi Yu",
"Ivan Oseledets"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13566v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Partial differential equations (PDEs) are the most widely used tools for solving physical and engineering problems (Karniadakis and Sherwin, 2005 ) . Many applications (e.g., in financial engineering, qua... |
arxiv | 2607.13479 | null | 1 | Topology-Agnostic Mesh Reconstruction of Deformable Objects from Sparse Touch | [
"Everest Yang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13479v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Manipulating a deformable object, whether cloth, rope, or cable, starts with a hard perception problem: its configuration is effectively infinite-dimensional, so a robot must estimate the object’s current... |
arxiv | 2607.13475 | null | 1 | Deformable State Estimation for Autonomous Surgical Tissue Retraction Under Partial Observability | [
"Everest Yang",
"Skye Thompson",
"George D. Konidaris"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13475v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I INTRODUCTION"
},
{
"type": "para",
"text": "Tissue retraction is a required step in many minimally invasive surgeries, where soft tissue must be manipulated to expose underlying anatomical targets. Autonomous robotic retraction remains challenging ... |
arxiv | 2607.13468 | null | 1 | HIVE-3D: Hierarchical Voxel Enhancement for High-Quality 3D Scene Generation | [
"Bin Zang",
"Wenting Zheng",
"Xiaoliang Luo",
"Zhiyuan Fang",
"Shi Li",
"Lvchun Wang",
"Wei Yu",
"Yi Zhao",
"Tian Xie",
"Yuchi Huo",
"Rengan Xie"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13468v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Rapid generation of a 3D scene from a single image is strongly desired across games, film, and industry. However, current scene generation methods (Gao et al. , 2024 ; Chen et al. , 2024a ; Nie et al. , 2... |
arxiv | 2607.13436 | null | 1 | Distributionally Robust and Safe Imitation Learning | [
"Ahmed Aboudonia",
"Naira Hovakimyan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13436v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I INTRODUCTION"
},
{
"type": "para",
"text": "Learning-from-demonstrations has emerged as a compelling paradigm for enabling autonomous systems to acquire control policies from expert trajectories [ 15 ] . In this context, imitation learning (IL) has... |
arxiv | 2607.13432 | null | 1 | Local Redundancy: An Information-Theoretic Measure of Plasticity from Synthetic Memorization | [
"Jiaxuan Cheng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13432v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Neural networks trained over long horizons or across multiple tasks often lose the ability to learn effectively, a phenomenon termed loss of plasticity (Lyle et al., 2023 ; Dohare et al., 2024 ) . Modern ... |
arxiv | 2607.13431 | null | 1 | Discrete Diffusion Models: A Unified Framework from Tokenization to Generation | [
"Ye Yuan",
"Weien Li",
"Rui Song",
"Zeyu Li",
"Haochen Liu",
"Xiangyu Kong",
"Zixuan Dong",
"Linfeng Du",
"Zipeng Sun",
"Weixu Zhang",
"Jiaxin Huang",
"Changjiang Han",
"Yonghan Yang",
"Zichen Zhao",
"Xiuyuan Hu",
"Haolun Wu",
"Yankai Chen",
"Fengran Mo",
"Jikun Kang",
"Bowei H... | CC-BY-4.0 | https://arxiv.org/html/2607.13431v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Autoregressive (AR) models have become the standard for generating discrete sequences. By factoring the joint distribution into a product of left-to-right conditionals, AR models enjoy a clean maximum-lik... |
arxiv | 2607.13416 | null | 1 | EXPLORE: Exploration with Guided Search for Analog Topology Generation using Language Models | [
"Guanglei Zhou",
"Chen-Chia Chang",
"Yikang Shen",
"Jonathan Ku",
"Isaac Jacobson",
"Jingyu Pan",
"Yiran Chen",
"Xin Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13416v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Analog circuit topology design sits at the heart of modern electronic systems, enabling everything from efficient power conversion to high-speed signal processing. As device requirements proliferate, var... |
arxiv | 2607.13413 | null | 1 | Is the Statistical Advantage Worth the Cost? An Empirical Comparison of KANs and MLPs for Structured Data Classification | [
"Matthew Steven P. Toledo",
"Justine Raphael H. Jacinto",
"Vivekjeet Singh Chambal",
"Rodolfo C. Camaclang",
"Jamlech Iram N. Gojo Cruz",
"Reginald Neil C. Recario"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13413v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Structured tabular data remains one of the most widely used input formats in production machine learning systems, spanning fields from healthcare to finance. In these settings, accuracy, reliability, and ... |
arxiv | 2607.13411 | null | 1 | Evaluating Frontier AI Agents as Autonomous Clinical Security Auditors | [
"Michael O. Eniolade"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13411v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "Evaluating Frontier AI Agents as Autonomous Clinical Security Auditors"
},
{
"type": "para",
"text": "Michael O. Eniolade University of the Cumberlands"
},
{
"type": "para",
"text": "meniolade20593@ucumberlands.edu"
},
{
"type": "heading",
"leve... |
arxiv | 2607.13399 | null | 1 | Demystifying On-Policy Distillation: Roles, Pathologies, and Regulations | [
"Rui Wang",
"Hongru Wang",
"Yi Chen",
"Boyang Xue",
"Tianqing Fang",
"Wenhao Yu",
"Kam-Fai Wong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13399v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "Demystifying On-Policy Distillation: Roles, Pathologies, and Regulations"
},
{
"type": "para",
"text": "Rui Wang † , Hongru Wang † , Yi Chen, Boyang Xue † Tianqing Fang ‡* , Wenhao Yu ‡ , Kam-Fai Wong †* † The Chinese University of Hong Kong ‡ Tencent AI Lab ruiwangnlp... |
arxiv | 2607.13395 | null | 1 | Self-Improving is Often Sudden: Enlightenment-style Finetuning for Large-Scale Models | [
"Jing-Xiao Liao",
"Tianwei Zhang",
"Yu-Hao Jiang",
"Feifei Zhang",
"Hang-Cheng Dong",
"Feng-Lei Fan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13395v1 | 2026-07-15T00:00:00 | [
{
"type": "para",
"text": "\\ul"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recently, enabling a large-scale model to improve itself has gained large traction in the AI community Wu et al. ( 2025 ); Qu et al. ( 2024 ); Huang et al. ... |
arxiv | 2607.13394 | null | 1 | GFlowRL: Scaling Distribution-Matching RL to Large Language Models | [
"Xiaodong Liu",
"Michael Xu",
"Jack W. Stokes",
"Paul Smolensky",
"Doug Burger",
"Jianfeng Gao"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13394v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning (RL) has become the defining post-training paradigm for the current generation of large language models. The most capable reasoning systems deployed today, including OpenAI o1 (Open... |
arxiv | 2607.14163 | null | 1 | A vision foundation model for single-cell biology via spatial gene cartography | [
"Ridvan Yesiloglu",
"Sakib Mostafa",
"James Zou",
"Ash Alizadeh",
"Jiajun Wu",
"Lei Xing",
"Ehsan Adeli",
"Md Tauhidul Islam"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14163v1 | 2026-07-15T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Single-cell RNA sequencing (scRNA-seq) has changed the scale at which biology can be studied. Instead of measuring average gene expression across mixed tissues, it now allows individual cells to be profiled... |
arxiv | 2607.13319 | null | 1 | Adapting Generalist Vehicle Models for High-Speed MPC Across Terrains | [
"Rwik Rana",
"Jesse Quattrociocchi",
"Christian Ellis",
"Nathan Tsoi",
"Garrett Warnell",
"Joydeep Biswas"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13319v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Keywords: Sampling-based MPC, FKD model learning, foundation model specialization, sim-to-real co-training, MPPI"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "High-speed driving over unstructured terrain und... |
arxiv | 2607.13246 | null | 1 | Reassessing Muon for Matrix Factorization | [
"Ali Parviz",
"Gal Mishne",
"Alex Cloninger"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13246v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": ": \\theoremsep \\jmlrvolume 334 \\jmlryear 2026 \\jmlrworkshop Topology, Algebra, and Geometry in Data Science"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recent advances in large-scale optimization have i... |
arxiv | 2607.13203 | null | 1 | BARS: Benign-Anchored Ranking and Selection for False Alarm Reduction in Network Intrusion Detection | [
"Abu Fuad Ahmad",
"Istiaque Ahmed"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13203v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Network intrusion detection systems (NIDS) are a core component of modern cybersecurity infrastructure, tasked with identifying malicious activity in high-volume network traffic. Although machine-learning... |
arxiv | 2607.13188 | null | 1 | Concurrent Image Understanding and Generation: Self-Correcting Coupled Markov Jump Processes | [
"Minh-Quan Le",
"Armand Comas",
"Alexandros Lattas",
"Stylianos Moschoglou",
"Pedro Vélez",
"Amit Raj",
"Aaron Germuth",
"Thabo Beeler",
"Dimitris Samaras",
"Di Qiu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13188v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "qqdd@google.com \\reportnumber 0001"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "When a teacher explains an idea at a whiteboard, language and drawing take place together : each utterance affects the sketch... |
arxiv | 2607.13164 | 10.1109/ACCESS.2026.3686260 | 1 | Text2Sign: A Single-GPU Diffusion Baseline for Text-to-Sign Language Video Generation | [
"Ruize Xia"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13164v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Keywords: Sign language generation, diffusion models, text-to-video synthesis, video generation, accessibility, spatio-temporal attention, transformer."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Accessibl... |
arxiv | 2607.13155 | null | 1 | HEDGEHOG: Hierarchical Evaluation of Drug Generators Through Rigorous Filtration | [
"Daria A. Ryabchenko",
"Pavel Gurevich",
"Shamil Kadyrov",
"Daria Frolova",
"Kseniia Fedisheva",
"Sergei A. Nikolenko",
"Alexander Shapeev",
"Marina A. Pak"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13155v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Generative models are widely used in molecular design for drug discovery, enabling the generation of large numbers of candidate compounds through efficient exploration and exploitation of chemical space [... |
arxiv | 2607.13003 | null | 1 | Watermark Forensics for Generative Models: An Information-Theoretic Perspective | [
"Xiaoyu Li",
"Zheng Gao",
"Xiaoyan Feng",
"Jiaojiao Jiang",
"Yulei Sui",
"Jiankun Hu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13003v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Watermark Forensics for Generative Models"
},
{
"type": "para",
"text": "An Information-Theoretic Perspective"
},
{
"type": "para",
"text": "1 University of New South Wales 2 Griffith University"
},
{
"type": "para",
"text": "{xiaoyu.li2, zheng.... |
arxiv | 2607.13120 | null | 1 | CoDiffGRN: Rethinking Gene Regulatory Network Inference via the BEELINE-KGC Benchmark and Co-evolutionary Discrete Diffusion | [
"Jiaze Song",
"Runhao Zhao",
"Minghao Xu",
"Bin Cui",
"Wentao Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13120v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Gene regulatory networks (GRNs) dictate cellular identity and physiological function by encoding the complex interactions between transcription factors (TFs) and target genes [ 37 , 8 , 14 , 39 ] . While ... |
arxiv | 2607.12868 | null | 1 | Deep4ge: DNN Training Trajectories for Fault Detection and Diagnosis | [
"Sigma Jahan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12868v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Deep neural network (DNN) components are now built into software systems, making training-time failures a major concern. Unlike traditional faults that produce explicit errors, DNN faults often appear as ... |
arxiv | 2607.12863 | null | 1 | Toward Localizing and Repairing Bias in Transformer Attention Heads | [
"Sigma Jahan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12863v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Real-world deployments of machine-learning software systems have produced documented unfair outcomes. Amazon withdrew an internal resume-screening system after engineers found that it ranked applicants lo... |
arxiv | 2607.12833 | null | 1 | ANGLE: Angular Neural Generative Learning via Engression | [
"Rajdeep Pathak",
"Archi Roy",
"Tanujit Chakraborty"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12833v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Ginwidth= \\Gin@nat@width ,height= \\Gin@nat@height ,keepaspectratio"
},
{
"type": "para",
"text": "Keywords: Circular data, distributional regression, Engression, extrapolation, pose estimation"
},
{
"type": "heading",
"level": 1,
"text": "1 Introducti... |
arxiv | 2607.12775 | null | 1 | Learning-enabled Acceleration of Scenario-based Model Predictive Control | [
"Trinh Tran",
"Binh Nguyen",
"Truong X. Nghiem"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12775v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Model Predictive Control (MPC) is a standard framework for constrained optimal control because it computes feedback actions by repeatedly solving a finite-horizon optimal control problem [ 7 ] . In many a... |
arxiv | 2607.13115 | null | 1 | Improving Molecular Property Prediction in Small Language Models Using Graph-based Tools | [
"Konstantinos Bougiatiotis",
"Dimitrios Kelesis",
"Georgios Paliouras"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13115v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "2nd Causal Neuro-symbolic Artificial Intelligence (Causal NeSy): Toward Agentic LLMs with Neuro-Symbolic and ... |
arxiv | 2607.14149 | null | 1 | Enhancing Small Language Models Reasoning through Knowledge Graph Grounding | [
"Dimitrios Kelesis",
"Konstantinos Bougiatiotis",
"Georgios Paliouras"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14149v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "2nd Causal Neuro-symbolic Artificial Intelligence (Causal NeSy): Toward Agentic LLMs with Neuro-Symbolic and ... |
arxiv | 2607.14148 | null | 1 | ITGPT: A Transformer Based Architecture for the Generation of Dance Dance Revolution and In the Groove Charts | [
"Miguel O'Malley"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14148v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Dance Dance Revolution (DDR), and its modern counterpart In the Groove (ITG), are rhythm games consisting of songs and charts. Players hit one of four arrows (up, down, left, right) as directed by a chart... |
arxiv | 2607.12726 | null | 1 | Physically Consistent Parameter Inference: Transparent Machine Learning Emulation in High Energy Physics and Cosmology | [
"Jorge Alda",
"Jacobo Asorey",
"Alejandro Mir",
"Siannah Peñaranda"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12726v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Physically Consistent Parameter Inference: Transparent Machine Learning Emulation in High Energy Physics and Cosmology"
},
{
"type": "para",
"text": "Jorge Alda a,b 1 1 1 jorge.aldagallo@unipd.it , Jacobo Asorey b,c 2 2 2 jasorey@unizar.es , Alejandro Mir b,c 3 3 3 ami... |
arxiv | 2607.12650 | null | 1 | Evidence-Grounded Verified Agentic Reasoning: A Path Toward Eliminating LLM Hallucination in Empirical Inference via Tool-Attested Kernel Proofs | [
"Junyu Ren"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12650v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "LLMs hallucinate on empirical facts when making inference, fabricating or reproducing outdated ones (Huang et al., 2025 ) . Tools and external memories were introduced as a mitigation, letting the LLM con... |
arxiv | 2607.12645 | null | 1 | AdaPCLA: Adaptive Prior-Calibrated Logit Adjustment for Long-Tailed Longitudinal EHR Generation | [
"Shuai Cui",
"Chen Wenxuan",
"Wenjie Du",
"Jian Lou",
"Dan Li",
"Wenjie Feng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12645v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Longitudinal Electronic Health Records (EHR) provide a rich view of patient trajectories, capturing diagnoses, treatments, medications, and clinical events over time. They are central to a wide range of c... |
arxiv | 2607.12626 | null | 1 | Gradient-free learning of a closed-loop wall controller for turbulent drag reduction | [
"Giorgio Maria Cavallazzi",
"Miguel Pérez Cuadrado",
"Alfredo Pinelli"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12626v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Skin-friction drag in wall-bounded turbulence is generated by quasi-streamwise vortices whose induced wall-normal velocities sustain the Reynolds shear stress above its laminar value (Orlandi & Jiménez, 1... |
arxiv | 2607.13107 | null | 1 | DeepCormack: Fermi surface tomography using model-based data-driven algorithms | [
"Georg F. B. Lovric",
"Bryn Drury",
"Carola-Bibiane Schönlieb",
"Stephen B. Dugdale",
"Ander Biguri"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13107v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The Fermi surface of a material is the surface in reciprocal space (or k -space) separating occupied from unoccupied electronic states at zero temperature ( T = 0 K T=0{\\rm K} ). Its shape is determine... |
arxiv | 2607.14147 | null | 1 | Breaking Refusal in the First Half: A Mechanistic Study of the Prefill Jailbreak | [
"Alex Kwon"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14147v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Aligned language models refuse harmful requests, and that refusal is easily removed. Prepend an affirmative prefix to the model’s own turn (“Sure, here is”) and many models continue straight into the harm... |
arxiv | 2607.12550 | null | 1 | A JoLT for the KV Cache: Near-Lossless KV Cache Compression via Joint Tucker and JL-Residual Allocation for LLMs | [
"Rahul Krishnan",
"Volker Schulz"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12550v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Every step of autoregressive transformer inference reuses the key and value projections of all previous tokens, so an implementation caches them at every layer rather than recomputing them [ 1 ] . This KV... |
arxiv | 2607.12488 | null | 1 | Sample Efficient Generative Optimization for Molecular Design | [
"Sarina Kopf",
"Cristina Nevado",
"Philippe Schwaller"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12488v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Molecular optimization is a central problem in chemistry, with applications ranging from drug discovery to catalyst design (Filella-Merce et al. , 2025 ; Strieth-Kalthoff et al. , 2024 ; Seumer et al. , 2... |
arxiv | 2607.13101 | null | 1 | TSSM: Triaxial State Space Model for Global Station Weather Forecasting with Temporal-Variable-Historical Modeling | [
"Songru Yang",
"Zili Liu",
"Tao Han",
"Ben Fei",
"Fenghua Ling",
"Lei Bai",
"Chang Liu",
"Xiangyang Ji",
"Zhenwei Shi",
"Zhengxia Zou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13101v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Global Station Weather Forecasting (GSWF) plays a vital role in delivering timely, localized weather predictions worldwide [ 56 , 59 , 18 ] . By improving average accuracy, capturing extreme events, and e... |
arxiv | 2607.12447 | null | 1 | The Computational Basis of Confidence in Large Language Models | [
"Dharshan Kumaran",
"Viorica Patraucean",
"Maks Ovsanikov",
"Petar Veličković",
"Nathaniel Daw"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12447v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Confidence is usually understood as a system’s estimate of whether its own answer is correct (Pouget et al. , 2016 , Fleming and Daw, 2017 , Kepecs and Mainen, 2012 , Mamassian, 2016 ) . For large language ... |
arxiv | 2607.12443 | null | 1 | Language Identification with Succinct Machine-Independent Traces | [
"Moses Charikar",
"Jon Kleinberg",
"Chirag Pabbaraju"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12443v1 | 2026-07-14T00:00:00 | [
{
"type": "para",
"text": "Moses Charikar \\Email moses@cs.stanford.edu \\addr Stanford University and \\Name Jon Kleinberg \\Email kleinberg@cornell.edu \\addr Cornell University and \\Name Chirag Pabbaraju \\Email cpabbara@cs.stanford.edu \\addr Stanford University"
},
{
"type": "heading",
"le... |
arxiv | 2607.14145 | null | 1 | ToolAnchor: Anchoring Counterfactual Context to Boost Agentic Tool-use Capability | [
"Weiting Liu",
"Jieyi Bi",
"Wanqi Zhou",
"Jianfeng Feng",
"Yining Ma",
"Ai Han",
"Wenlian Lu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.14145v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recent advances in tool-augmented agents have significantly broadened the real-world utility of Large Language Models (LLMs) Singh et al. ( 2025b ); Jiang et al. ( 2025 ); Li et al. ( 2025b ); Fan et al. ... |
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