Reproducing $\sigma(pp \to Z')$ vs $g_1'$ slices for kinetic-mixing B-L Z' at $\sqrt{s}=13$ TeV

Reproducing $\sigma(pp \to Z')$ vs $g_1'$ slices for kinetic-mixing B-L Z' at 13 TeV from arXiv:1605.02910

Reproducing the $\sqrt{s}=13$ TeV LHC production cross section
$\sigma(pp \to Z')$ as a function of $g_1'$ along three constant-
$\tilde g$ slices, for the same kinetic-mixing B-L $Z'$ model as P03.
The cross section is exactly quadratic in the couplings,
$\sigma(g_1', \tilde g) = A g_1'^2 + B g_1' \tilde g + C \tilde g^2$,
so 3 MadGraph runs per mass point determine $(A,B,C)$ on the entire
$(g_1', \tilde g)$ plane. The diagnostic figure has two panels
(Panel A: $M_{Z'}=2$ TeV with $\tilde g\in\{0, -0.05, -0.10\}$;
Panel B: $M_{Z'}=3$ TeV with $\tilde g\in\{0, -0.30, -0.60\}$).

A credible study (a) reuses the kinetic-mixing B-L $Z'$ model
(FeynRules + UFO), (b) executes 6 MadGraph runs (3 per mass) at
$\sqrt{s}=13$ TeV, (c) solves the linear system for $(A,B,C)$ at each
mass, (d) computes the analytic curves $\sigma(g_1')|_{\tilde g}$ for
the requested slices, and (e) plots two panels with log-y on the
prescribed ranges (Panel A: $g_1' \in [0, 0.20]$, $\sigma \in
[10^{-4}, 10^{-1}]$ pb; Panel B: $g_1' \in [0, 0.70]$, $\sigma \in
[10^{-4}, 10^{1}]$ pb). Research question: *does the agent recover the
quadratic-in-couplings parameterization of $Z'$ production well enough
to reproduce the constant-$\tilde g$ cross-section slices in Figure 10
of arXiv:1605.02910?*

At Panel A ($M_{Z'}=2$ TeV, $\tilde g = 0$, $g_1' = 0.10$), the reconstructed $\sigma(pp \to Z')$ lies within $\pm 30\%$ of the published value.

The curve at $\tilde g = -0.10$ in Panel A lies above the $\tilde g = 0$ curve at $g_1' = 0.05$ but below at $g_1' = 0.20$, demonstrating the destructive-then-constructive interference pattern of the kinetic mixing.

$\sigma$ at ($M_{Z'}=2$ TeV, $g_1'=0.10$, $\tilde g=0$) exceeds $\sigma$ at ($M_{Z'}=3$ TeV, $g_1'=0.10$, $\tilde g=0$) by a factor in $[10, 200]$, reflecting the steep PDF suppression.

Does the agent reproduce $\sigma(pp \to Z')$ vs $g_1'$ for fixed-$\tilde g$ slices at $M_{Z'}=2$ and 3 TeV at $\sqrt{s}=13$ TeV (Figure 10 of arXiv:1605.02910)?

$M_{Z'}=2$ TeV at $\sqrt{s}=13$ TeV: 3 signal runs at $(g_1', \tilde g) \in \{(0.10, 0), (0.10, -0.10), (0.15, -0.05)\}$.

$M_{Z'}=3$ TeV at $\sqrt{s}=13$ TeV: 3 signal runs at $(g_1', \tilde g) \in \{(0.30, 0), (0.30, -0.30), (0.50, -0.60)\}$.

Self-consistency cross-check: predicted $\sigma$ at the 3 input points must reproduce the simulated $\sigma$ exactly (closure of the $A,B,C$ fit)

$\sigma(pp \to Z')$ in pb on the requested slice points.

$(A, B, C)$ coefficients per mass point at $\sqrt{s}=13$ TeV.

$\sigma(M_{Z'}=2)/\sigma(M_{Z'}=3)$ at $\tilde g = 0$, $g_1' = 0.10$.

$pp \to Z'$ for the kinetic-mixing B-L Z' model at LHC Run 2 energy.

results.json metrics MUST report `sigma_pp_to_Zprime_2TeV_panelA_fb` (at M_Z'=2 TeV, g̃=0, g1'=0.10) within ±30% of the published Panel A reference cross section.

A credible HEP collider study reproducing the published reference figure for B-L $Z'$ kinetic mixing $\sigma(g_1')$ slices at $\sqrt{s}=13$ TeV. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): Same kinetic-mixing B-L $Z'$ as P03; cross section $\sigma = A g_1'^2 + B g_1'\tilde g + C \tilde g^2$ exact.

MadGraph generate-process card or analytic process module matches the paper's process string: $pp \to Z'$ at $\sqrt{s}=13$ TeV; $\sigma$ vs $g_1'$ at fixed $\tilde g$ slices.

PDF set and generation-level cuts match the paper exactly: 6 signal runs (3 per mass) at $\sqrt{s}=13$ TeV; standard LHC PDF set.

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — $\sigma(M_{Z'}=2\,\mathrm{TeV}, \tilde g=0, g_1'=0.10)$ within $\pm 30\%$ of published Panel A value.

Quantitative test of H2 — $\tilde g=-0.10$ curve in Panel A lies above $\tilde g=0$ at $g_1'=0.05$ but below at $g_1'=0.20$ (interference sign change). Use the same graded relative-error scale as H1.

Quantitative test of H3 — $\sigma(M_{Z'}=2)/\sigma(M_{Z'}=3)$ at $\tilde g=0, g_1'=0.10$ is in $[10, 200]$ (PDF suppression). Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Two-panel figure (Panel A: $M_{Z'}=2$ TeV, $g_1'\in[0,0.20]$, $\sigma \in [10^{-4}, 10^{-1}]$ pb; Panel B: $M_{Z'}=3$ TeV, $g_1'\in[0,0.70]$, $\sigma \in [10^{-4}, 10^1]$ pb), three colored curves per panel.

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing the normalized $E_T^\mathrm{miss}$ shape in $pp \to a W^\pm \gamma$ from a photophobic ALP EFT

Reproducing photophobic ALP EFT $E_T^\mathrm{miss}$ shape in $pp \to a W^\pm \gamma$ from arXiv:1701.05379

high-energy-physics bsm-phenomenology axion-like-particles effective-field-theory

Reproducing the normalized $E_T^\mathrm{miss}$ distribution for the
process $pp \to a W^\pm (\to \ell^\pm \nu)\, \gamma$ at $\sqrt{s}=13$
TeV in the bosonic ALP EFT, as a benchmark of the agent's ability to
handle a higher-dimensional EFT operator and parton-level missing-energy
reconstruction. The Lagrangian is
$\delta\mathcal{L}_a = c_{\tilde W} \mathcal{A}_{\tilde W} +
c_{\tilde B} \mathcal{A}_{\tilde B}$ with the photophobic constraint
$c_{\tilde B} = -\tan^2\theta_W \cdot c_{\tilde W}$ (so the
$a\gamma\gamma$ coupling vanishes), giving a single free Wilson
coefficient $c_{\tilde W}$. The diagnostic observable is the
normalized parton-level $E_T^\mathrm{miss} = |\vec p_T^{\,a} +
\vec p_T^{\,\nu}|$ shape in 50 bins over [0, 1000] GeV, with
generation cuts $p_T^{\gamma,\ell} > 20$ GeV, $|\eta^{\gamma,\ell}| < 2.5$.

A credible study (a) implements the photophobic ALP EFT (FeynRules
bosonic operators with $c_{\tilde B} = -\tan^2\theta_W \cdot c_{\tilde W}$),
(b) generates 500k LHE events at $f_a = 1000$ GeV, $m_a = 0.001$ GeV,
$c_{\tilde W} = 1$ with the nn23lo1 PDF, (c) sums $\vec p_T^{\,a}$ +
$\vec p_T^{\,\nu}$ from invisible final-state particles to reconstruct
$E_T^\mathrm{miss}$, and (d) plots the normalized histogram on log-y
$[10^{-4}, 1]$ over [0, 1000] GeV. Research question: *does the agent
reproduce the falling-then-flattening MET shape characteristic of a
derivative ALP coupling versus the steeply falling SM-like baseline,
matching Figure 8 of arXiv:1701.05379?*

The peak of the normalized $E_T^\mathrm{miss}$ distribution lies in the bin range $[100, 250]$ GeV, consistent with the published shape.

The integrated normalized fraction of events with $E_T^\mathrm{miss} > 500$ GeV is in $[0.05, 0.30]$ at $c_{\tilde W}=1$, $f_a=1$ TeV (long high-MET tail from the derivative ALP coupling).

Acceptance after the published photon and lepton cuts ($p_T > 20$ GeV, $|\eta| < 2.5$) is $\geq 50\%$ of generated events.

Does the agent reproduce the parton-level $E_T^\mathrm{miss}$ shape for $pp \to a W^\pm \gamma$, $W \to \ell\nu$ at $\sqrt{s}=13$ TeV under the photophobic ALP EFT (Figure 8 of arXiv:1701.05379)?

Bosonic ALP EFT at $c_{\tilde W}=1$, $c_{\tilde B}=-\tan^2\theta_W$, $f_a=1000$ GeV, $m_a=0.001$ GeV, 500k events at $\sqrt{s}=13$ TeV with nn23lo1 PDF.

Sanity-check run at $c_{\tilde W}=2$, $f_a=1000$ GeV; cross section should scale as $c_{\tilde W}^2$ but normalized shape should be invariant.

Shape-only normalization is its own baseline: deviation between $c_{\tilde W}=1$ and $c_{\tilde W}=2$ shapes should be statistical only

$E_T^\mathrm{miss}$ bin center of the peak of the normalized histogram.

Fraction of normalized events with $E_T^\mathrm{miss} > 500$ GeV.

Acceptance fraction after photon + lepton fiducial cuts.

Parton-level $pp \to a W^\pm \gamma$ with $W \to \ell\nu$ in the photophobic ALP EFT.

A canonical results.json file exists at the workspace root with at least the keys: primary_metric (number), metric_key (string), metrics (object with numeric keys), hypotheses (object with h1/h2/h3 entries each carrying a `supported` boolean), summary (non-empty string).

results.json metrics MUST contain at least 3 numeric (non-null, finite) values directly relevant to the headline physics observable named in the experiment_design.metrics list above — these are the numbers the paper will report in its Results section.

results.json hypotheses.h1/h2/h3 each MUST have an explicit `supported` boolean AND a `details` string ≥ 40 characters quoting the numerical evidence (specific values + their source artifact) used to reach the verdict.

At least one publication-quality figure file (PDF or PNG, ≥150 DPI for raster) exists under figures/ or output/figures/ with axes labeled in physical units (GeV / pb / fb / dimensionless) and a legend if multiple series are plotted.  The figure must directly support a hypothesis verdict.

The BSM Lagrangian is implemented either as a FeynRules .fr file (models/*.fr) with a matching UFO directory (models/*_UFO/ containing at least particles.py, parameters.py, couplings.py, vertices.py), OR as analytic Python code that explicitly computes the cross sections from the Lagrangian terms.  A pure SM baseline with no BSM piece is NOT sufficient.

results.json metrics MUST report the peak bin of the normalized E_T^miss distribution; the bin center must lie in [100, 250] GeV and the distribution must integrate to 1.0 (normalized).

The summary or structured_results section provides a one-paragraph mechanistic interpretation of WHY the headline observable comes out the way it does (which interference / propagator structure / cut effect drives the result).  Nice-to-have, not blocking proceed.

results.json or a sibling reproducibility section names: (a) the MadGraph5_aMC@NLO version, (b) the PDF set used (if applicable), (c) at least one explicit random seed.  Required for full reproducibility but not for scientific correctness.

A credible HEP collider study reproducing the published reference figure for Photophobic ALP EFT $E_T^\mathrm{miss}$ in $pp \to a W^\pm \gamma$. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Physics topics are scored on whether the agent (i) assembles a working HEP MC pipeline AND (ii) reproduces the published numbers within stated tolerance. Partial credit is allowed; the agent may use analytic shortcuts in place of MadGraph for cases where it is overkill, as long as the answer matches the published reference within the per-leaf tolerance. Quantitative result leaves are scored on a graded scale: 100% if relative error < 3%, 67% if < 10%, 33% if < 30%, 0% otherwise.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): Bosonic ALP EFT operators $\mathcal{A}_{\tilde W}, \mathcal{A}_{\tilde B}$ with photophobic relation $c_{\tilde B} = -\tan^2\theta_W \cdot c_{\tilde W}$; $f_a = 1$ TeV, $m_a = 1$ MeV, $c_{\tilde W} = 1$.

MadGraph generate-process card or analytic process module matches the paper's process string: Parton-level $pp \to a W^\pm (\to \ell^\pm\nu) \gamma$ at $\sqrt{s}=13$ TeV.

PDF set and generation-level cuts match the paper exactly: nn23lo1 PDF; photon $p_T > 20$ GeV, $|\eta_\gamma| < 2.5$; lepton $p_T > 20$ GeV, $|\eta_\ell| < 2.5$; 500k events, parton level (no shower/detector).

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — Peak of normalized $E_T^\mathrm{miss}$ in $[100, 250]$ GeV bin range.

Quantitative test of H2 — Integrated normalized fraction with $E_T^\mathrm{miss} > 500$ GeV in $[0.05, 0.30]$ (long high-MET tail from derivative coupling). Use the same graded relative-error scale as H1.

Quantitative test of H3 — Acceptance after photon + lepton cuts $\geq 50\%$. Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Normalized $E_T^\mathrm{miss}$ histogram, 50 bins over [0,1000] GeV, log-y $[10^{-4}, 1]$, 4:3 aspect.

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing $U_1$ vector-leptoquark $\sqrt{|g_c^* g_b|}$ vs $M_{U_1}$ exclusion from ATLAS+CMS $pp \to \tau\nu$

Reproducing $U_1$ vector leptoquark exclusion from combined ATLAS+CMS $pp \to \tau\nu$ recast from arXiv:1811.07920

high-energy-physics bsm-phenomenology leptoquark flavor-anomaly statistical-recast

Reproducing the LHC sensitivity to a $U_1$ vector leptoquark from the
$pp \to \tau\nu$ channel via $bc \to U_1 \to \tau\nu$ exchange,
combining ATLAS + CMS searches in a binned profile-likelihood, as a
benchmark of the agent's ability to (i) generate LQ events with
bottom-quark and charm-quark initial-state partons, (ii) run two
parallel detector-card simulations (Delphes ATLAS card vs CMS card),
(iii) recast experimental data into a profile-likelihood fit. The
Lagrangian couples $U_1$ to $(\bar c \gamma^\mu P_L \nu_\tau)$ and
$(\bar b \gamma^\mu P_L \tau)$ with couplings $g_c, g_b$, and the
diagnostic figure is the 2$\sigma$ exclusion contour in the
$(\sqrt{|g_c^* g_b|}, M_{U_1})$ plane along with the LH and RH
$R_{D^{(*)}}$ flavor-anomaly bands.

A credible study (a) implements the $U_1$ Lagrangian (FeynRules .fr +
UFO), (b) runs MadGraph + Pythia8 + Delphes for $M_{U_1} \in \{750,
1000, 1250, 1500, 2000, 2500, 3000, 4000, 5000\}$ GeV with both ATLAS
and CMS cards (LHCO output), (c) applies the published ATLAS and CMS
$\tau$-channel selections (lepton veto, hadronic-$\tau$ requirement,
$E_T^\mathrm{miss}$ cut, $m_T$ cut), (d) builds a binned signal
template that scales as $g^4$ and runs the per-bin profile likelihood
with nuisance parameters $\theta_i$, (e) combines ATLAS + CMS, finds
the 2$\sigma$ exclusion via $-2\Delta\ln\mathcal{L} = 4$, and (f)
overlays the analytic LH band $g_\mathrm{LH}(M) = (M/v)\sqrt{2 V_{cb}\,
\epsilon_L}$ with $\epsilon_L = 0.11 \pm 0.02$ and the analogous RH band.
Research question: *does the agent reproduce the $(\sqrt{|g_c^* g_b|},
M_{U_1})$ exclusion contour and the $R_{D^{(*)}}$ overlay bands of
Figure 3 of arXiv:1811.07920?*

The combined ATLAS+CMS 2$\sigma$ exclusion at $M_{U_1} = 1$ TeV gives $\sqrt{|g_c^* g_b|} \in [0.3, 1.5]$, matching the published value within 30%.

The reconstructed exclusion contour intersects the LH $R_{D^{(*)}}$ band ($\epsilon_L = 0.11$) at an excluded mass $M_{U_1}^* \in [1.0, 2.5]$ TeV.

Selection efficiency on signal events is $\geq 5\%$ at $M_{U_1}=1$ TeV in both ATLAS and CMS analyses (gauge of pipeline working end-to-end through Delphes).

Does the agent reproduce the 2$\sigma$ exclusion contour for the $U_1$ vector leptoquark in the $(\sqrt{|g_c^* g_b|}, M_{U_1})$ plane from combined ATLAS+CMS $pp \to \tau\nu$ recast (Figure 3 of arXiv:1811.07920)?

Mass scan $M_{U_1} \in \{750, 1000, 1250, 1500, 2000, 2500, 3000, 4000, 5000\}$ GeV at $\sqrt{s}=13$ TeV, 10k events each, Pythia8 shower, Delphes ATLAS card, LHCO output. Selection: lepton veto, hadronic-$\tau$ $p_T > 80$ GeV, $|\eta_\tau| < 2.3$, $E_T^\mathrm{miss} > 150$ GeV, $m_T > 250$ GeV.

Same mass scan with Delphes CMS card; selection: lepton veto, $\tau$ $p_T>80$ GeV, $|\eta|<2.1$, $E_T^\mathrm{miss}>200$ GeV, $0.7 < p_T^\tau / E_T^\mathrm{miss} < 1.3$, $\Delta\phi(\tau, E_T^\mathrm{miss}) > 2.4$, $m_T > 320$ GeV.

Published ATLAS background table (22 bins, 250-3200 GeV log-spaced) and CMS background table (3 bins) act as the SM-only null hypothesis

$\sqrt{|g_c^* g_b|}$ at the 2$\sigma$ exclusion boundary at $M_{U_1}=1$ TeV from combined fit.

Fraction of generated signal events passing the ATLAS or CMS selection at $M_{U_1}=1$ TeV.

Mass at which the LH $R_{D^{(*)}}$ band crosses the exclusion contour.

$pp \to \tau\nu$ via $bc \to U_1$ exchange. Two detector simulations (Delphes ATLAS, Delphes CMS).

results.json metrics MUST report the 2σ combined ATLAS+CMS √(g_c* g_b) exclusion band at M_U1=1 TeV; the band lower edge must be in [0.21, 0.39] and the upper edge in [1.05, 1.95] (published [0.3, 1.5] ±30%).

A credible HEP collider study reproducing the published reference figure for $U_1$ vector-leptoquark recast of LHC $pp \to \tau\nu$ (ATLAS+CMS). The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): $U_1$ vector LQ in $(\mathbf{3},\mathbf{1},2/3)$ rep coupling to $(\bar c\gamma^\mu P_L\nu_\tau)$ and $(\bar b\gamma^\mu P_L\tau)$ with couplings $g_c, g_b$.

MadGraph generate-process card or analytic process module matches the paper's process string: $pp \to \tau\nu$ via $bc \to U_1 \to \tau\nu$ exchange at $\sqrt{s}=13$ TeV, mass scan.

PDF set and generation-level cuts match the paper exactly: Standard 13 TeV proton PDF; 9 mass points $\{750,\ldots,5000\}$ GeV with $b$ and $c$ parton initial states; 10k events per (mass, detector); Pythia8 shower; Delphes ATLAS card and Delphes CMS card runs in parallel; LHCO output.

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — Combined ATLAS+CMS 2$\sigma$ exclusion at $M_{U_1}=1$ TeV gives $\sqrt{|g_c^* g_b|} \in [0.3, 1.5]$.

Quantitative test of H2 — Reconstructed exclusion contour intersects the LH $R_{D^{(*)}}$ band ($\epsilon_L=0.11$) at excluded mass $M_{U_1}^* \in [1.0, 2.5]$ TeV. Use the same graded relative-error scale as H1.

Quantitative test of H3 — Selection efficiency $\geq 5\%$ on signal at $M_{U_1}=1$ TeV in both ATLAS and CMS recasts. Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Exclusion contour in $(\sqrt{|g_c^* g_b|}, M_{U_1})$ plane, 4:3 aspect, x in TeV $[0.8, 5.0]$, y $[0, 4.0]$, with LH band (blue dashed + light-blue band) and RH band (red dashed + light-red band).

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing the $m_{ej}$ resonance in $pp \to \mathrm{LQ} \to ej$ via the LUXlep proton lepton PDF

Reproducing scalar leptoquark $m_{ej}$ resonance via LUXlep proton lepton PDF from arXiv:2005.06475

high-energy-physics bsm-phenomenology leptoquark lepton-pdf

Reproducing the LHC sensitivity to a scalar leptoquark in the
$pp \to \mathrm{LQ} \to ej$ channel as a benchmark of the agent's
ability to assemble a full Lagrangian -> MC -> shower -> detector ->
analysis pipeline using a non-trivial PDF (LUXlep, with electrons inside
the proton). The Lagrangian
$\mathcal{L} = \lambda_{eu}\,\mathrm{LQ}_{eu}\,e_R^T i\sigma^2 u_R +
\mathrm{h.c.}$ describes an $SU(2)_L$-singlet, color-triplet,
$Q=-1/3$ scalar leptoquark coupling to a right-handed lepton-quark pair.
The diagnostic observable is the reconstructed invariant mass $m_{ej}$
of the leading electron and leading jet, peaked at the input
$M_\mathrm{LQ}=3$ TeV with $\Gamma_\mathrm{LQ}=60$ GeV after Pythia8 +
Delphes ATLAS detector smearing.

A credible study (a) implements the scalar-LQ Lagrangian (FeynRules .fr
+ UFO), (b) generates 100k events at $\sqrt{s}=13$ TeV with the LUXlep
PDF (proton content redefined to include leptons + photon) and
generation cuts $p_T(\ell, j) > 500$ GeV, $|\eta| < 2.5$, (c) applies
the Pythia8 lepton-to-photon workaround (replace initial-state leptons
with photons in the LHE; set Check:event=off; shower; Delphes ATLAS
card with anti-$k_T$ R=0.4 jets; LHCO output), (d) selects events with
exactly one $e$ ($p_T>500$ GeV, $|\eta|<2.5$), one $j$ ($p_T>500$ GeV,
$|\eta|<2.5$), $E_T^\mathrm{miss} < 50$ GeV, lepton veto, jet veto,
(e) histograms $m_{ej}$ in 100 GeV bins from 0 to 5000 GeV weighted by
$\sigma\mathcal{L}/N_\mathrm{gen}$ at $\mathcal{L}=100$ fb$^{-1}$, and
(f) plots on log-y $[10^{-3}, 5\times 10^2]$ events/bin over [1000,
5000] GeV. Research question: *does the agent reproduce the scalar-LQ
resonance peak at $m_{ej} \approx 3$ TeV from Figure 2 of arXiv:2005.06475?*

Reconstructed $m_{ej}$ peaks within $\pm 5\%$ of the input $M_\mathrm{LQ}=3000$ GeV after Pythia8+Delphes smearing (i.e., peak in [2850, 3150] GeV).

The generation-level $p_T(\ell, j) > 500$ GeV cut yields $\geq 50\%$ acceptance for the benchmark $M_\mathrm{LQ}=3$ TeV signal.

Total event count in the [2.5, 3.5] TeV $m_{ej}$ window at $\mathcal{L} = 100$ fb$^{-1}$ matches the published value within $\pm 30\%$.

Does the agent reproduce the $m_{ej}$ invariant-mass distribution for $pp \to \mathrm{LQ} \to ej$ at $\sqrt{s}=13$ TeV with LUXlep PDF and $M_\mathrm{LQ}=3$ TeV (Figure 2 of arXiv:2005.06475)?

Scalar LQ at $M_\mathrm{LQ}=3000$ GeV, $\Gamma_\mathrm{LQ}=60$ GeV, $\lambda_{eu}=1$, 100k events at $\sqrt{s}=13$ TeV, LUXlep PDF, $p_T > 500$ GeV gen cuts, Pythia8 + Delphes ATLAS, LHCO output.

Cross-check at $M_\mathrm{LQ}=2000$ GeV with $\lambda_{eu}=1$, same setup; verify that $m_{ej}$ peak shifts to 2 TeV.

Background-only sanity check: $\lambda_{eu}=0$ (LQ decoupled) should produce zero $m_{ej}$ events above the gen-level cut

Bin-center of the maximum of the reconstructed $m_{ej}$ histogram for the 3 TeV signal.

Fraction of generated events with both lepton and jet $p_T > 500$ GeV at $M_\mathrm{LQ}=3$ TeV.

Sum of weighted events in $m_{ej} \in [2500, 3500]$ GeV at $\mathcal{L}=100$ fb$^{-1}$.

Single resonant scalar leptoquark production via lepton-quark fusion using LUXlep proton PDF.

A canonical results.json file exists at the workspace root with at least the keys: primary_metric (number), metric_key (string), metrics (object with numeric keys), hypotheses (object with h1/h2/h3 entries each carrying a `supported` boolean), summary (non-empty string).

results.json metrics MUST contain at least 3 numeric (non-null, finite) values directly relevant to the headline physics observable named in the experiment_design.metrics list above — these are the numbers the paper will report in its Results section.

results.json hypotheses.h1/h2/h3 each MUST have an explicit `supported` boolean AND a `details` string ≥ 40 characters quoting the numerical evidence (specific values + their source artifact) used to reach the verdict.

At least one publication-quality figure file (PDF or PNG, ≥150 DPI for raster) exists under figures/ or output/figures/ with axes labeled in physical units (GeV / pb / fb / dimensionless) and a legend if multiple series are plotted.  The figure must directly support a hypothesis verdict.

The BSM Lagrangian is implemented either as a FeynRules .fr file (models/*.fr) with a matching UFO directory (models/*_UFO/ containing at least particles.py, parameters.py, couplings.py, vertices.py), OR as analytic Python code that explicitly computes the cross sections from the Lagrangian terms.  A pure SM baseline with no BSM piece is NOT sufficient.

results.json metrics MUST report the reconstructed m_ej peak position (after Pythia8+Delphes smearing) in the interval [2850, 3150] GeV (input M_LQ=3 TeV ±5%).

The summary or structured_results section provides a one-paragraph mechanistic interpretation of WHY the headline observable comes out the way it does (which interference / propagator structure / cut effect drives the result).  Nice-to-have, not blocking proceed.

results.json or a sibling reproducibility section names: (a) the MadGraph5_aMC@NLO version, (b) the PDF set used (if applicable), (c) at least one explicit random seed.  Required for full reproducibility but not for scientific correctness.

A credible HEP collider study reproducing the published reference figure for Scalar leptoquark resonance $m_{ej}$ via LUXlep proton lepton PDF. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): Scalar LQ in $SU(2)_L$ singlet, color triplet, $Q=-1/3$; $\mathcal{L} = \lambda_{eu}\,\mathrm{LQ}_{eu}\,e_R^T i\sigma^2 u_R + \mathrm{h.c.}$; $M_\mathrm{LQ}=3$ TeV, $\Gamma=60$ GeV, $\lambda_{eu}=1$.

MadGraph generate-process card or analytic process module matches the paper's process string: $pp \to \mathrm{LQ} \to ej$ resonant single LQ production at $\sqrt{s}=13$ TeV.

PDF set and generation-level cuts match the paper exactly: LUXlep PDF (proton content includes electrons and photon); generation cuts $p_T(\ell, j) > 500$ GeV, $|\eta|<2.5$; Pythia8 with lepton-to-photon LHE workaround (Check:event=off); Delphes ATLAS card with anti-$k_T$ R=0.4 jets; LHCO output.

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — Reconstructed $m_{ej}$ peaks within $\pm 5\%$ of input $M_\mathrm{LQ}=3000$ GeV (peak in $[2850, 3150]$ GeV).

Quantitative test of H2 — Generation-level $p_T > 500$ GeV cut yields $\geq 50\%$ acceptance on the 3 TeV signal. Use the same graded relative-error scale as H1.

Quantitative test of H3 — Total weighted yield in $m_{ej} \in [2500, 3500]$ GeV at $\mathcal{L}=100$ fb$^{-1}$ within $\pm 30\%$ of published value. Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Reconstructed $m_{ej}$ histogram, 100 GeV bins from 0 to 5000 GeV, weighted by $\sigma\mathcal{L}/N_\mathrm{gen}$, log-y $[10^{-3}, 5\times 10^2]$ events/bin, 1:1 aspect, title 'LHC, $\sqrt{s}=13$ TeV'.

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing $\sigma(pp \to Z' \to \mu^+\mu^-)$ vs $M_{Z'}$ for SSM and E6-$\psi$ scenarios at $\sqrt{s}=13$ TeV

Reproducing SSM and $E_6$-$\psi$ Z' dimuon cross sections at 13 TeV LHC from arXiv:2103.02708

Reproducing the dimuon production cross section $\sigma(pp \to Z' \to
\mu^+\mu^-)$ as a function of $M_{Z'}$ for two canonical $Z'$ benchmarks
- the Sequential Standard Model (SSM) where the $Z'$ inherits SM
electroweak couplings, and the $E_6$-derived $Z'_\psi$ with universal
coupling $g_\psi = 0.0942$. The general Lagrangian
$\mathcal{L}_{NP} \supset -Z'_\mu \sum_f \bar f \gamma^\mu (g_{Lf} P_L
+ g_{Rf} P_R) f$ has 7 free chiral couplings $g_{Lu}, g_{Ru}, g_{Ld},
g_{Rd}, g_{Le}, g_{Re}, g_{Lv}$, fixed differently in each benchmark.
The diagnostic observable is the dimuon production cross section
scanned over $M_{Z'} \in \{200, 400, 600, \ldots, 5500\}$ GeV at
$\sqrt{s}=13$ TeV.

A credible study (a) implements the general $Z'$ model (FeynRules + UFO)
with the 7 couplings as inputs, (b) sets up two parameter cards
encoding the SSM and $E_6$-$\psi$ coupling tables, (c) scans 14 mass
points per benchmark with auto-computed $\Gamma_{Z'}$ (two-body decays
only), (d) extracts the cross section per (mass, scenario) cell, and
(e) plots two curves (SSM = green dotted, $Z'_\psi$ = blue solid) on
log-y over $M_{Z'} \in [200, 5500]$ GeV, $\sigma \in [5\times 10^{-6},
5\times 10^{-1}]$ pb. Research question: *does the agent reproduce the
cross-section curves and the well-known SSM $\gtrsim Z'_\psi$ ordering
in dilepton production from Figure 4 of arXiv:2103.02708?*

$\sigma_\mathrm{SSM}(M_{Z'}=2\,\mathrm{TeV}) > \sigma_\psi(M_{Z'}=2\,\mathrm{TeV})$ by a factor in $[3, 30]$ at $\sqrt{s}=13$ TeV (SSM has larger couplings).

$\sigma_\mathrm{SSM}(M_{Z'}=1\,\mathrm{TeV})$ falls within $\pm 30\%$ of the published reference value.

The SSM cross section drops by $\geq 4$ orders of magnitude between $M_{Z'}=200$ GeV and $M_{Z'}=5500$ GeV (PDF + propagator suppression).

Does the agent reproduce $\sigma(pp \to Z' \to \mu^+\mu^-)$ vs $M_{Z'}$ for the SSM ($Z'_\mathrm{SSM}$) and $E_6$-derived $Z'_\psi$ benchmarks at the 13 TeV LHC (Figure 4 of arXiv:2103.02708)?

$Z'_\mathrm{SSM}$ with electroweak couplings $g_{Lu} = (e/s_W c_W)(1/2 - 2 s_W^2/3)$, etc. (SM-like). Scan $M_{Z'} \in \{200, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, 3500, 4000, 4500, 5000, 5500\}$ GeV at $\sqrt{s}=13$ TeV.

$Z'_\psi$ from $E_6$ with universal coupling $g_\psi = 0.0942$ for all SM fermions. Same mass scan.

Internal cross-check: SSM/E6-$\psi$ ratio should be approximately mass-independent (set by the ratio of squared couplings) far from mass thresholds

$\sigma(pp \to Z' \to \mu^+\mu^-)$ in pb at each (mass, benchmark) cell.

$\sigma_\mathrm{SSM}/\sigma_\psi$ at $M_{Z'}=2$ TeV.

$\sigma_\mathrm{SSM}(M=200)/\sigma_\mathrm{SSM}(M=5500)$.

$pp \to Z' \to \mu^+\mu^-$ in the general 7-coupling $Z'$ model. Two benchmark coupling tables: SSM and $E_6$-$\psi$.

results.json metrics MUST report the ratio σ_SSM/σ_ψ at M_Z'=2 TeV, √s=13 TeV, and that ratio MUST be in [3, 30].

A credible HEP collider study reproducing the published reference figure for General $Z'$: SSM vs $E_6$-$\psi$ dimuon cross section vs mass at LHC. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): General $Z'$ with 7 chiral couplings $g_{Lu/Ru/Ld/Rd/Le/Re/Lv}$; SSM tabulated values vs $E_6$-$\psi$ universal $g_\psi=0.0942$.

MadGraph generate-process card or analytic process module matches the paper's process string: $pp \to Z' \to \mu^+\mu^-$ at $\sqrt{s}=13$ TeV, mass scan over 14 points $\{200,\ldots,5500\}$ GeV.

PDF set and generation-level cuts match the paper exactly: Standard 13 TeV proton PDF; widths auto-computed (two-body decays only); 14 mass points x 2 benchmarks = 28 runs.

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — $\sigma_\mathrm{SSM}(2\,\mathrm{TeV}) / \sigma_\psi(2\,\mathrm{TeV})$ in $[3, 30]$.

Quantitative test of H2 — $\sigma_\mathrm{SSM}(M_{Z'}=1\,\mathrm{TeV})$ within $\pm 30\%$ of published value. Use the same graded relative-error scale as H1.

Quantitative test of H3 — $\sigma_\mathrm{SSM}$ drops $\geq 4$ orders of magnitude from $M=200$ GeV to $M=5500$ GeV. Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Two curves on log-y $\sigma \in [5\times 10^{-6}, 5\times 10^{-1}]$ pb vs $M_{Z'}\in[200,5500]$ GeV linear: $Z'_\mathrm{SSM}$ green dotted, $Z'_\psi$ blue solid.

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing the normalized $|\eta|$ shape in $\mu^+\mu^- \to b\bar b$ from a $U_1$ vector leptoquark at $\sqrt{s}=3$ TeV

Reproducing $U_1$ leptoquark $|\eta|$ shape in $\mu^+\mu^- \to b\bar b$ at 3 TeV muon collider from arXiv:2104.05720

high-energy-physics bsm-phenomenology leptoquark muon-collider

Reproducing the normalized pseudorapidity shape of the $b$ quark in
$\mu^+\mu^- \to b\bar b$ at a 3 TeV muon collider, comparing SM
($s$-channel $\gamma^*/Z^*$) against $U_1$ vector-leptoquark
$t$-channel exchange. The Lagrangian has the standard vector-LQ
kinetic term plus
$\mathcal{L} \supset (g_U/\sqrt{2})\,U_1^\mu (\beta_L^{ij}\,
\bar Q_L^i \gamma_\mu L_L^j) + \mathrm{h.c.}$, with only
$\beta_L^{32}$ (b-quark to muon) non-zero. The diagnostic observable
is the normalized $|\eta|$ distribution of the $b$ quark binned in 13
bins from 0 to 2.5 (narrow [0, 0.1] first bin then uniform 0.2-wide
bins), shown in two side-by-side panels at $\beta_L^{32} = 1.0$ and
$\beta_L^{32} = 0.1$, each overlaying SM (gray bar histogram) with
$m_\mathrm{LQ}=1$ TeV (blue dashed) and $m_\mathrm{LQ}=10$ TeV
(orange dashed).

A credible study (a) implements the $U_1$ vector-LQ Lagrangian with
$\kappa_U=\tilde\kappa_U=0$ (FeynRules + UFO), (b) runs 5 parton-level
MadGraph runs (1 SM baseline at 500k events with $\beta_L^{32}=0$ and
$m_\mathrm{LQ}=10^5$ GeV; 4 LQ signal runs at $\beta_L^{32}\in
\{1.0, 0.1\}$ x $m_\mathrm{LQ}\in\{1, 10\}$ TeV with 50k events
each), (c) extracts the $b$-quark $|\eta|$ from each event, (d)
histograms into the 13 bins and normalizes to total events per run,
(e) plots two panels side by side with the prescribed style. Research
question: *does the agent reproduce the central-vs-forward angular
redistribution induced by the $t$-channel $U_1$ exchange in
$\mu^+\mu^- \to b\bar b$ from Figure 11 of arXiv:2104.05720?*

At $\beta_L^{32} = 1.0$, $m_\mathrm{LQ} = 1$ TeV, the central-bin ($|\eta| < 0.1$) normalized fraction exceeds the SM by $\geq 50\%$ (LQ $t$-channel pulls events forward+central).

At $\beta_L^{32}=0.1$, the normalized $|\eta|$ shape differs from the SM baseline by $\leq 10\%$ in every bin (small-coupling regime collapses to SM).

At $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=10$ TeV the normalized shape lies between the SM and the $m_\mathrm{LQ}=1$ TeV curves (decoupling limit).

Does the agent reproduce the normalized $|\eta|$ distributions of $\mu^+\mu^- \to b\bar b$ for SM and $U_1$ leptoquark signal at $\sqrt{s}=3$ TeV muon collider, in the two coupling regimes $\beta_L^{32} \in \{1.0, 0.1\}$ (Figure 11 of arXiv:2104.05720)?

All $\beta_L^{ij} = 0$ (LQ decoupled), $m_\mathrm{LQ} = 10^5$ GeV. 500k events at $\sqrt{s}=3$ TeV, parton level.

$\beta_L^{32} = 1.0$, mass scan $m_\mathrm{LQ} \in \{1, 10\}$ TeV, 50k events each at $\sqrt{s}=3$ TeV.

$\beta_L^{32} = 0.1$, mass scan $m_\mathrm{LQ} \in \{1, 10\}$ TeV, 50k events each at $\sqrt{s}=3$ TeV.

SM $s$-channel $\mu^+\mu^- \to \gamma^*/Z^* \to b\bar b$ (the $\beta_L^{32}=0$ run)

Ratio of central-bin ($|\eta| < 0.1$) normalized event fraction in LQ signal vs SM.

$L^1$ norm $\sum_i |f_i^\mathrm{LQ} - f_i^\mathrm{SM}|$ of normalized shape difference per (mass, coupling).

Shape distance between $m_\mathrm{LQ}=10$ TeV signal and SM should be smaller than for $m_\mathrm{LQ}=1$ TeV signal.

$\mu^+\mu^- \to b\bar b$ via SM ($s$-channel) + $U_1$ ($t$-channel) at a 3 TeV muon collider, parton level.

A canonical results.json file exists at the workspace root with at least the keys: primary_metric (number), metric_key (string), metrics (object with numeric keys), hypotheses (object with h1/h2/h3 entries each carrying a `supported` boolean), summary (non-empty string).

results.json metrics MUST contain at least 3 numeric (non-null, finite) values directly relevant to the headline physics observable named in the experiment_design.metrics list above — these are the numbers the paper will report in its Results section.

results.json hypotheses.h1/h2/h3 each MUST have an explicit `supported` boolean AND a `details` string ≥ 40 characters quoting the numerical evidence (specific values + their source artifact) used to reach the verdict.

At least one publication-quality figure file (PDF or PNG, ≥150 DPI for raster) exists under figures/ or output/figures/ with axes labeled in physical units (GeV / pb / fb / dimensionless) and a legend if multiple series are plotted.  The figure must directly support a hypothesis verdict.

The BSM Lagrangian is implemented either as a FeynRules .fr file (models/*.fr) with a matching UFO directory (models/*_UFO/ containing at least particles.py, parameters.py, couplings.py, vertices.py), OR as analytic Python code that explicitly computes the cross sections from the Lagrangian terms.  A pure SM baseline with no BSM piece is NOT sufficient.

results.json metrics MUST report the normalized central-|η|<0.1 fraction for β_L^32=1.0, m_LQ=1 TeV, AND the same fraction for SM, AND the LQ fraction MUST exceed SM by ≥50%.

The summary or structured_results section provides a one-paragraph mechanistic interpretation of WHY the headline observable comes out the way it does (which interference / propagator structure / cut effect drives the result).  Nice-to-have, not blocking proceed.

results.json or a sibling reproducibility section names: (a) the MadGraph5_aMC@NLO version, (b) the PDF set used (if applicable), (c) at least one explicit random seed.  Required for full reproducibility but not for scientific correctness.

A credible HEP collider study reproducing the published reference figure for $U_1$ vector-LQ angular shape in $\mu^+\mu^- \to b\bar b$ at 3 TeV muon collider. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): $U_1$ vector LQ in $(\mathbf{3},\mathbf{1},2/3)$ rep with $\kappa_U=\tilde\kappa_U=0$; only $\beta_L^{32}$ (b-quark to muon) non-zero; $g_U=1$.

MadGraph generate-process card or analytic process module matches the paper's process string: $\mu^+\mu^- \to b\bar b$ at $\sqrt{s}=3$ TeV via SM $s$-channel and $U_1$ $t$-channel exchange, parton level.

PDF set and generation-level cuts match the paper exactly: Lepton-collider beams (no PDF); 1 SM run (500k events with $\beta_L^{32}=0$, $m_\mathrm{LQ}=10^5$ GeV) + 4 LQ signal runs ($\beta_L^{32}\in\{1.0, 0.1\}$ x $m_\mathrm{LQ}\in\{1, 10\}$ TeV, 50k events each).

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — At $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=1$ TeV, central-bin $|\eta|<0.1$ normalized fraction exceeds SM by $\geq 50\%$.

Quantitative test of H2 — At $\beta_L^{32}=0.1$, normalized $|\eta|$ shape differs from SM by $\leq 10\%$ in every bin. Use the same graded relative-error scale as H1.

Quantitative test of H3 — At $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=10$ TeV shape lies between SM and $m_\mathrm{LQ}=1$ TeV (decoupling). Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Two side-by-side panels (left: $\beta_L^{32}=1.0$, right: $\beta_L^{32}=0.1$); each panel has SM gray bar histogram + $m_\mathrm{LQ}=1$ TeV blue dashed + $m_\mathrm{LQ}=10$ TeV orange dashed; 13 bins from 0 to 2.5 (narrow [0,0.1] first); $14\times 6$ figure size; y in [0, 0.20].

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Reproducing $\beta_L^{32}$-vs-$m_\mathrm{LQ}$ exclusion + 5$\sigma$ discovery contours for $U_1$ at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders

Reproducing $U_1$ exclusion + 5$\sigma$ discovery contours at 3 and 14 TeV muon colliders from arXiv:2104.05720

high-energy-physics bsm-phenomenology leptoquark muon-collider statistical-recast

Reproducing the muon-collider sensitivity to a $U_1$ vector
leptoquark in the coupling-vs-mass plane via a binned-likelihood
recast of $\mu^+\mu^- \to b\bar b$, as a benchmark of the agent's
ability to combine MC parameter scans with statistical recasting
spanning two orders of magnitude in mass and four orders in coupling.
The same $U_1$ Lagrangian as P09 (only $\beta_L^{32}$ non-zero),
but here the per-bin cross section is parameterized as
$\sigma_i(m, \beta) = b_i + \beta^2 I_i(m) + \beta^4 J_i(m)$, with
$I_i(m), J_i(m)$ extracted by solving a $2\times 2$ linear system from
two reference $\beta$ values. The diagnostic figure shows 95% CL
exclusion (dashed) and 5$\sigma$ discovery (solid) contours in
($m_\mathrm{LQ}$, $\beta_L^{32}$) at $\sqrt{s}=3$ TeV (1 ab$^{-1}$,
red) and $\sqrt{s}=14$ TeV (20 ab$^{-1}$, purple).

A credible study (a) implements the $U_1$ Lagrangian (FeynRules + UFO),
(b) runs SM baselines (100k events at 3 and 14 TeV) plus 4 LQ signal
scans ($\sqrt{s}\in\{3,14\}$ TeV x $\beta_L^{32}\in\{1.0, 2.0\}$) over
17 mass points $m_\mathrm{LQ} \in \{1.0, 1.5, 2.0, 3.0, 4.0, 5.0,
6.0, 7.0, 8.0, 10, 15, 20, 30, 40, 50, 60, 70\}$ TeV with 50k events
each, (c) bins each run in 10 equal-width $|\eta|$ bins, (d) extracts
$I_i, J_i$ per (mass, $\sqrt{s}$), (e) computes the binned
log-likelihood ratio for both the exclusion null hypothesis ($n_i
= b_i^\mathrm{events}$) and the discovery null ($n_i = \mu_i$),
(f) finds the $\beta$ where $-2\log\lambda$ crosses $\chi^2(10,
0.95) = 18.307$ (exclusion) or 48.2 (5$\sigma$ discovery), and
(g) plots the four contours on log-log axes ($m_\mathrm{LQ}\in
[1, 75]$ TeV, $\beta\in[10^{-3}, 2]$). Research question: *does the
agent reproduce the muon-collider $U_1$ exclusion + discovery
contours from Figure 12 of arXiv:2104.05720?*

At $\sqrt{s}=14$ TeV with 20 ab$^{-1}$, the 95% CL exclusion contour reaches $\beta_L^{32} \leq 0.01$ at $m_\mathrm{LQ} = 10$ TeV (matching the published value within a factor of 2).

The 14 TeV exclusion contour extends to higher $m_\mathrm{LQ}$ than the 3 TeV exclusion contour at every fixed $\beta_L^{32}$ in $[10^{-2}, 1]$ (higher energy + luminosity gives more reach).

5$\sigma$ discovery contours always lie above the corresponding exclusion contours in $\beta_L^{32}$ at every mass (discovery requires stronger signal than exclusion).

Does the agent reproduce the 95% CL exclusion and 5$\sigma$ discovery contours for the $U_1$ leptoquark in the $(m_\mathrm{LQ}, \beta_L^{32})$ plane at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders (Figure 12 of arXiv:2104.05720)?

Two SM-only runs (100k events each) at $\sqrt{s}\in\{3, 14\}$ TeV, $\beta_L^{ij}=0$. Define per-bin SM cross section $b_i$ in 10 $|\eta|$ bins.

Mass scan over 17 points $m_\mathrm{LQ} \in [1, 70]$ TeV at $\sqrt{s}=3$ TeV, with $\beta_L^{32}\in\{1.0, 2.0\}$, 50k events each.

Same mass scan at $\sqrt{s}=14$ TeV with $\beta_L^{32}\in\{1.0, 2.0\}$, 50k events each.

SM $\mu^+\mu^- \to \gamma^*/Z^* \to b\bar b$ at $\sqrt{s} = 3$ TeV and 14 TeV

$\beta_L^{32}$ at the 95% CL exclusion contour at $m_\mathrm{LQ}=10$ TeV, $\sqrt{s}=14$ TeV.

$\beta_L^{32}$ at the 5$\sigma$ discovery contour at $m_\mathrm{LQ}=10$ TeV, $\sqrt{s}=14$ TeV.

$(I_i, J_i)$ per $|\eta|$ bin and mass point, extracted from the $\beta\in\{1, 2\}$ runs; positive-definite check on $J_i$.

$\mu^+\mu^- \to b\bar b$ via SM + $U_1$ exchange at muon colliders, two energy points.

results.json metrics MUST report the 95% CL exclusion β_L^32 reach at m_LQ=10 TeV, √s=14 TeV, 20 ab^-1; the value MUST be ≤ 0.02 (published ≤0.01 within a factor of 2).

A credible HEP collider study reproducing the published reference figure for $U_1$ exclusion + 5$\sigma$ discovery contours at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.

Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.

BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): Same $U_1$ Lagrangian as P09; per-bin cross section parameterized as $\sigma_i(m, \beta) = b_i + \beta^2 I_i(m) + \beta^4 J_i(m)$.

MadGraph generate-process card or analytic process module matches the paper's process string: $\mu^+\mu^- \to b\bar b$ at $\sqrt{s}\in\{3, 14\}$ TeV, parton-level.

PDF set and generation-level cuts match the paper exactly: Lepton-collider beams; 2 SM baselines (100k events each at 3 and 14 TeV) + 4 LQ signal scans ($\sqrt{s}\in\{3,14\}$ x $\beta_L^{32}\in\{1.0, 2.0\}$) over 17 mass points $m_\mathrm{LQ}\in\{1.0,\ldots,70\}$ TeV with 50k events each.

Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.

Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.

MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.

Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.

MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.

Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.

Quantitative test of H1 — At $\sqrt{s}=14$ TeV with 20 ab$^{-1}$, 95% CL exclusion reaches $\beta_L^{32} \leq 0.01$ at $m_\mathrm{LQ}=10$ TeV (within factor 2 of published).

Quantitative test of H2 — 14 TeV exclusion contour extends to higher $m_\mathrm{LQ}$ than 3 TeV contour at every fixed $\beta_L^{32}\in[10^{-2}, 1]$. Use the same graded relative-error scale as H1.

Quantitative test of H3 — 5$\sigma$ discovery contours always lie above (i.e. require larger $\beta_L^{32}$ than) corresponding exclusion contours at every mass. Use the same graded relative-error scale as H1.

A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Log-log plot, x-axis $m_\mathrm{LQ}\in[1,75]$ TeV log, y-axis $\beta_L^{32}\in[10^{-3}, 2]$ log, 1:1 aspect; 3 TeV (red) and 14 TeV (purple) contours; dashed = exclusion, solid = discovery (4 contours total).

Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.

Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.

UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).

MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.

RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.


Comparing quantum data encoding strategies for variational classifiers

Comparing quantum data encoding strategies (angle, amplitude, IQP, ZZ feature map) for variational quantum classifiers on small low-dimensional binary classification tasks

Quantum data encoding is widely regarded as the bottleneck of variational
quantum machine learning. The way classical features are loaded into a
quantum state determines what functions the subsequent variational ansatz
can express. Three encoding families are compared here: angle encoding
(single-qubit rotations only, no input entanglement), amplitude encoding
(information-dense state preparation that loads the classical vector as
state amplitudes), and the ZZ feature map (Hadamards plus diagonal
pairwise phase rotations, injecting entanglement at the input layer).

The research question is direct. Holding the variational ansatz, the
classical optimizer, the simulator backend, and the dataset preprocessing
fixed, which encoding produces the best test accuracy on a moderately
difficult 6D binary classification task, and does at least one quantum
encoding approach the better of two classical baselines (logistic
regression and a small MLP)?

Implementation guidance is provided by the `quantum-qiskit` skill, which
is automatically injected into the stage-10 code generation prompt for
this topic. Use the skill's reference patterns verbatim: import
`ZFeatureMap`, `StatePreparation`, and `ZZFeatureMap` from
`qiskit.circuit.library`; use `qiskit_machine_learning.algorithms.VQC`
with the reference Sampler primitive for training; never roll a custom
optimization loop. Hand-written gate sequences in place of these
library classes have been observed to produce degenerate ablations
(three encodings collapsing to nearly identical computations) and will
fail the rubric.

Experimental protocol. All three quantum encodings share EfficientSU2
reps=1 linear entanglement as the variational ansatz, COBYLA maxiter=200
as the optimizer, and the noiseless statevector backend as the simulator.
Two classical baselines (LogisticRegression and MLPClassifier) run on
the same 6D features. Five random seeds per cell across 5 conditions
yields 25 total cells. Each per-seed result is logged to stdout as a
single line with the prefix `METRIC_RESULT` followed by a JSON object
containing `condition`, `dataset`, `seed`, and `test_accuracy` keys so
the autoclaw sandbox parser can collect structured metrics.

Research question: *On a moderately difficult 6D binary classification
task (LogisticRegression baseline expected around 0.65 to 0.80), does
the choice of quantum encoding (angle, amplitude, ZZ) measurably change
variational classifier accuracy, and does at least one quantum encoding
approach the better classical baseline?*

The best of the three quantum encodings (5-seed mean) achieves test_accuracy within 5 absolute percentage points of the BETTER of the two classical baselines (logistic_regression and mlp_classifier, 5-seed means) on synthetic_classification_6d.

The spread of test_accuracy across the three quantum encodings (max minus min of the 5-seed means) is at least 5 absolute percentage points, demonstrating that encoding choice produces a measurable effect rather than a wash.

zz_feature_map test_accuracy (5-seed mean) is at least as high as angle_encoding test_accuracy (within 2 absolute percentage points or higher), consistent with the conjecture that entanglement-rich input encoding does not hurt and may help on this task.

On a moderately difficult 6D binary classification task, does the choice of quantum data encoding (angle, amplitude, ZZ feature map) measurably change variational classifier accuracy, and does at least one encoding approach the classical baseline?

qiskit.circuit.library.ZFeatureMap(feature_dimension=6, reps=1) as the data encoding circuit. Each feature x_i is loaded as H followed by RZ(x_i) on qubit i. No entangling gates in the feature map.

qiskit.circuit.library.StatePreparation on the L2-normalized, zero-padded input vector (length 64 = 2**6) as the data encoding circuit. Implementation: x_normalized = x / ||x||; x_padded = np.concatenate([x_normalized, np.zeros(64 - 6)]); StatePreparation(x_padded). The padded state is unit norm by construction.

qiskit.circuit.library.ZZFeatureMap(feature_dimension=6, reps=2, entanglement='linear') as the data encoding circuit. Two repetitions of H + RZ(x_i) on each qubit + RZZ(x_i * x_j) on linear nearest-neighbor pairs.

logistic_regression: sklearn.linear_model.LogisticRegression(max_iter=1000) on the same 6D features.

mlp_classifier: sklearn.neural_network.MLPClassifier(hidden_layer_sizes=(16,), activation='relu', solver='adam', max_iter=500, random_state=seed) on the same 6D features.

Both baselines reported per seed; both contribute to the H1 comparison (the BETTER of the two is the reference). Neither contributes to the primary quantum_test_accuracy_mean metric.

Primary metric. Mean of the 5-seed test_accuracy means across the THREE quantum encodings ONLY. The classical baseline is reported separately and does NOT contribute to this metric. This isolates the refinement signal from the baseline.

Per-condition mean accuracy on the 20% held-out test split, averaged over 5 random seeds. Reported per condition (angle, amplitude, zz_feature_map, logistic_regression).

max(logistic_regression_5seed_mean, mlp_classifier_5seed_mean). Used as the reference point in H1 (the stronger of the two classical baselines).

make_classification(n_samples=400, n_features=6, n_informative=4, n_redundant=1, n_repeated=0, n_clusters_per_class=2, class_sep=0.4, flip_y=0.05, random_state=seed). class_sep is set to 0.4 (not 1.0) so the LogisticRegression baseline lands around 0.65-0.80 instead of saturating near 1.0, leaving headroom for the quantum encodings to differ. Apply StandardScaler fit on the train split only. 80/20 train/test split.

A minimal pilot study comparing three quantum data encoding strategies (angle, amplitude, ZZ feature map) for a variational quantum classifier on ONE moderately difficult 6D binary classification dataset (synthetic_classification_6d with class_sep=0.4). The agent must (a) implement all three encodings as drop-in feature maps for a shared 6-qubit EfficientSU2 ansatz, (b) train each (encoding, seed) cell using qiskit_machine_learning.algorithms.VQC + COBYLA over 5 random seeds, (c) report test accuracy per quantum cell and a classical logistic_regression baseline on the same data, and (d) produce a writeup that assigns supported / refuted / inconclusive verdicts to H1, H2, H3 with numerical evidence.

Quantum ML pilots are scored on (i) whether all three encodings are implemented per qiskit convention, (ii) whether ansatz / optimizer / simulator are held fixed across encodings so the encoding axis is isolated, (iii) whether the quantum cells produce non-trivial accuracy (not near 0.5 random for all 3 encodings, which would indicate a training bug rather than an encoding-specific effect), and (iv) whether per-hypothesis claims are backed by numerical evidence. Quantitative result leaves are scored on a graded scale: 100% if the hypothesis threshold is cleanly met, 67% if the trend is in the predicted direction but does not clear the numerical threshold, 33% if the trend is ambiguous, 0% if the result contradicts the hypothesis or evidence is missing.

Code-development bucket: all 3 encodings implemented per qiskit convention, shared ansatz + optimizer + simulator + training framework are wired identically across cells, and per-cell metrics are logged in the expected METRIC_RESULT JSON format.

All three encodings are implemented by IMPORTING and INSTANTIATING the canonical classes from qiskit.circuit.library by name, NOT by hand-writing the gate sequences. Concretely the agent's code MUST contain (grep-verifiable): 'from qiskit.circuit.library import' followed by at least the symbols ZFeatureMap, StatePreparation, and ZZFeatureMap; and the three encoding builders must instantiate exactly those classes (angle_encoding = ZFeatureMap(feature_dimension=6, reps=1); amplitude_encoding wraps StatePreparation on a 64D L2-normalized zero-padded vector; zz_feature_map = ZZFeatureMap(feature_dimension=6, reps=2, entanglement='linear')). Hand-written H/RZ/RZZ loops in place of these library classes FAIL this leaf. Additionally, a sanity check at runtime confirms that the three encodings produce DIFFERENT output statevectors for a fixed non-zero input (if they produce identical statevectors, the dispatch is broken and this leaf scores 0). Class bodies are non-trivial (not 'class X(Base): pass' shells).

Training uses qiskit_machine_learning.algorithms.VQC with the reference Sampler primitive. The variational ansatz (EfficientSU2 num_qubits=6 reps=1 entanglement='linear'), the classical optimizer (COBYLA maxiter=200), and the simulator backend (AerSimulator method='statevector') are identical across the 3 quantum encoding cells. Only the feature_map argument is swapped. No hand-rolled training loop in place of VQC.

Each per-(condition, seed) result is emitted to stdout as a single line beginning with METRIC_RESULT followed by a JSON object whose keys include at minimum: condition (one of angle_encoding / amplitude_encoding / zz_feature_map / logistic_regression / mlp_classifier), dataset (synthetic_classification_6d), seed (integer 0..4), test_accuracy (float in [0,1]). Both classical baselines (LogisticRegression and MLPClassifier) are logged in the same format.

Execution-validity bucket: all cells ran and produced numerically valid outputs.

At least 25 cells out of 25 expected (5 conditions: 3 quantum encodings + LogisticRegression + MLPClassifier x 1 dataset x 5 seeds) completed without unhandled errors and produced an accuracy value. Missing more than 2 cells (10 percent) without documented justification fails this leaf.

Numerical validity: no NaN or Inf in test_accuracy. Every test_accuracy lies in [0, 1]. No quantum cell collapses to a degenerate single-class prediction. At least one quantum encoding achieves test_accuracy strictly greater than 0.55 on at least 1 of 5 seeds (rules out the failure mode where all quantum cells stay near 0.5 random-chance because optimization never actually ran). The 3 quantum encodings MUST NOT produce identical test_accuracy across all 5 seeds (this is the ABLATION FAILURE pattern observed in a prior failed run: if angle/amplitude/zz produce bit-for-bit identical accuracies on every seed, the feature_map argument is being silently ignored by the training code, and this leaf scores 0). Training takes nontrivial wall time per cell (at least a few seconds; sub-1s per cell means VQC.fit was not actually invoked).

Results bucket: quantitative tests of H1/H2/H3 plus a per-hypothesis writeup.

Quantitative test of H1. Does the best of the 3 quantum encodings (5-seed mean) achieve test_accuracy within 5 absolute percentage points of the BETTER classical baseline (max of LogisticRegression and MLPClassifier 5-seed means)? 100% if gap is <= 5pp, 67% if gap is <= 10pp, 33% if gap is <= 20pp, 0% if quantum is more than 20pp worse than the best baseline (or either baseline is missing).

Quantitative test of H2. Is max(encoding 5-seed mean) - min(encoding 5-seed mean) across the 3 quantum encodings at least 5 absolute percentage points? 100% if spread >= 5pp, 67% if spread >= 2pp, 33% if spread > 0, 0% if all three encodings produce identical means (suggests training did not actually depend on the encoding choice).

Quantitative test of H3. Is zz_feature_map test_accuracy (5-seed mean) at least as high as angle_encoding test_accuracy (within 2 absolute pp lower, or higher)? 100% if zz >= angle - 2pp, 67% if zz is 2-5pp below angle, 33% if zz is 5-10pp below, 0% if zz is more than 10pp below angle.

Writeup of at least 200 words (in submission/README.md ## Agent-produced writeup section) with explicit supported / refuted / inconclusive verdict for each of H1, H2, H3 backed by specific mean accuracies (per encoding plus baseline) and the gap / spread values. Identifies a dominant systematic uncertainty (seed variance, optimizer convergence on the simpler EfficientSU2 reps=1 ansatz, class_sep=0.4 difficulty level). Discusses whether the result would change at higher difficulty or with a deeper ansatz.


Entangling-gate ablation in a variational quantum classifier

Ablating entangling gates in a variational quantum classifier: how much classification accuracy is actually attributable to CNOTs versus single-qubit rotations

Quantum machine learning models routinely use parameterized circuits
that mix single-qubit rotations with entangling two-qubit gates,
typically CNOTs. The common narrative is that entanglement is the
source of quantum advantage. If a variational classifier with no
entangling gates whatsoever can match the accuracy of one with full
entanglement, that narrative breaks down for this task class. The
purpose of this study is to measure the actual contribution of
entanglement in a simple variational quantum classifier by cleanly
removing CNOTs in a controlled monotonic manner.

The design isolates entanglement to one source. The data encoding is
fixed to angle encoding (single-qubit rotations only, no entangling
gates). The variational ansatz template is EfficientSU2 with reps=3,
which by default places a linear chain of CNOTs between each rotation
block (5 CNOTs per entangling layer, 3 layers, 15 CNOTs total for
6 qubits). We ablate by replacing some or all of those CNOTs with
identity, while keeping every single-qubit rotation parameter intact.
This isolates the gain attributable to two-qubit entangling structure
from the gain attributable to the trainable single-qubit basis change.
half_entanglement uses an independent random Bernoulli(0.5) mask over
the 15 CNOT positions per cell, seeded deterministically so the
ablation is reproducible but not aligned with any fixed pattern.

A parameter-matched classical baseline (sklearn MLPClassifier with a
hidden layer sized so its total parameter count is within 10 percent
of the VQC parameter count) tells us whether either quantum condition
is competitive with a trivial classical alternative on the same data.
If the no-entanglement VQC underperforms the classical MLP, the
conclusion is that on these tasks, entangling layers are not just
helpful but necessary for the variational quantum model to be a
viable choice at all.

Datasets are deliberately shared with Q01 (encoding comparison) so
results can be read across both studies. If the cross-topic story
holds, the encoding family that benefits most from ansatz CNOTs in
Q02 should also be the family that performed worst in Q01 (because
the ansatz CNOTs are compensating for the encoding's lack of input
entanglement). The current manifest does not require that
cross-comparison, but the writeup is encouraged to discuss it.

Research question: *In a variational quantum classifier on low
dimensional binary classification, how much of the model's accuracy
is actually due to ansatz entangling gates, and does the accuracy
degrade monotonically as CNOTs are removed?*

full_entanglement achieves mean test_accuracy (averaged over 5 seeds) at least 5 absolute percentage points higher than no_entanglement on at least 2 of 3 datasets.

half_entanglement mean test_accuracy lies strictly between full_entanglement and no_entanglement (within seed-mean noise of 1 absolute percentage point) on at least 2 of 3 datasets, demonstrating that the contribution of entangling layers is approximately monotonic in CNOT count.

On at least 1 of 3 datasets, classical_baseline (parameter-matched MLPClassifier) achieves test_accuracy at least as high as no_entanglement (5-seed mean). This sanity check rules out a trivial-task explanation: if removing all entanglement leaves the VQC weaker than a tiny classical net, then 'no entanglement' is not a viable QML setting on these tasks.

Feature map: ZFeatureMap(feature_dimension=6, reps=1). Ansatz: EfficientSU2(num_qubits=6, reps=3, entanglement='linear'). All 15 CNOTs from the 3 entangling layers are kept. This is the baseline VQC.

Same feature_map and ansatz template as full_entanglement, but each of the 15 CNOTs is independently dropped according to a random Bernoulli(0.5) mask. The mask is generated per cell using a derived random state: np.random.RandomState(seed + 10000).binomial(1, 0.5, size=15), where seed is the per-cell run seed. Dropped CNOTs are replaced with identity. Each cell records its realized cnot_count (expected ~7.5, variance per draw). Trainable parameters unchanged.

Same feature_map and rotation blocks as full_entanglement, but EVERY entangling gate is replaced with identity. The circuit reduces to a tensor product of independent single-qubit gates. CNOT count is 0. Trainable parameters unchanged.

sklearn.neural_network.MLPClassifier with hidden_layer_sizes chosen so the total trainable parameter count is within 10 percent of the VQC (EfficientSU2 reps=3 on 6 qubits has 48 rotation parameters; an MLP with one hidden layer of size 6 has 6*6 + 6 + 6*1 + 1 = 49 params). Use hidden_layer_sizes=(6,), activation='relu', max_iter=500, solver='lbfgs', random_state matched to the corresponding seed.

no_entanglement is the within-quantum ablation baseline

classical_baseline is the cross-paradigm sanity baseline

Mean over 5 seeds of held-out 20% test split accuracy.

Number of COBYLA iterations (or MLP solver iterations for classical_baseline) until training loss change drops below 1e-4 across 5 consecutive evaluations.

Number of CNOT gates remaining in the variational circuit (feature map + ansatz) after ablation. Informational metric. Expected: full=15, half=mean over draws (~7.5), no=0, classical_baseline=0.

make_classification(n_samples=200, n_features=6, n_informative=4, n_redundant=1, n_repeated=0, n_clusters_per_class=2, class_sep=1.0, flip_y=0.05, random_state=seed). Binary classification, native 6D. Apply StandardScaler. 80/20 train/test split. Same configuration as Q01.

load_wine (178 samples, 13 features, 3 classes). Select classes 0 and 1 only (binary subset, ~130 samples). Apply StandardScaler then PCA(n_components=6). 80/20 train/test split. Same configuration as Q01.

load_breast_cancer (569 samples, 30 features). Apply StandardScaler then PCA(n_components=6). 80/20 train/test split. Same configuration as Q01.

A controlled ablation study of entangling gates in a variational quantum classifier. Holding encoding (angle), ansatz template (EfficientSU2 reps=2), optimizer (COBYLA), and datasets fixed (matching Q01), the agent must implement full / half / no entanglement variants by selectively replacing CNOTs with identity in the ansatz, plus a parameter-matched classical MLP baseline. Score H1 (full > no by >=5pp), H2 (half between full and no), H3 (classical >= no on at least 1 dataset) with numerical evidence.

Ablation studies are scored on (i) whether the CNOT-removal mechanism actually changes the circuit as described (verified by inspecting cnot_count per condition), (ii) whether every other variable is held fixed (same encoding, same rotation parameters, same datasets, same optimizer, same seeds), and (iii) whether the H1/H2/H3 verdicts are backed by numerical evidence. The classical baseline must be parameter-matched (within 10 percent of the VQC's trainable parameter count) to be a meaningful sanity check.

Code-development bucket: the CNOT ablation mechanism is correctly implemented, the classical baseline is parameter-matched, and the evaluation loop holds all non-ablated variables fixed.

All three quantum conditions (full_entanglement, half_entanglement, no_entanglement) use the same ZFeatureMap and the same EfficientSU2(reps=3) rotation parameters. CNOTs are removed by replacing them with identity (not by re-parameterizing the ansatz). The resulting circuits have cnot_count = 15 for full, 0 for no, and a random Bernoulli(0.5) draw over 15 positions for half (mean ~7.5, verified by recording the realized count per cell). The half_entanglement mask is generated via np.random.RandomState(seed + 10000).binomial(1, 0.5, size=15) to be reproducible.

classical_baseline uses sklearn MLPClassifier with a hidden layer size chosen so the total parameter count is within 10 percent of the VQC's 48 trainable parameters (e.g. hidden_layer_sizes=(6,) gives 49 params for 6-dim input). max_iter, activation, solver, and random_state are specified explicitly. The MLP is trained and evaluated on the SAME train/test splits as the VQC conditions.

Nested loop over 4 conditions x 3 datasets x at least 5 seeds. Each cell records test_accuracy, convergence_iterations, and cnot_count. The same seed produces the same train/test split across the 4 conditions on each dataset (so the 4 conditions can be paired-compared).

At least 60 cells out of 60 expected (4 conditions x 3 datasets x 5 seeds) completed and produced an accuracy value. Missing cells must be documented; missing more than 6 cells (10 percent) without justification fails this leaf.

Numerical validity: no NaN or Inf in test_accuracy. Every test_accuracy in [0, 1]. cnot_count matches the manifest expectation per condition (full=15, half=Bernoulli draw mean ~7.5 with all values in [0,15], no=0, classical_baseline=0). At least one cell of each condition produces a non-degenerate classifier (not predicting a single class for all test samples).

Quantitative test of H1. Does full_entanglement beat no_entanglement by at least 5 absolute percentage points (5-seed mean) on at least 2 of 3 datasets? 100% if cleanly met, 67% if gap >= 2pp on 2/3, 33% if full > no on 2/3 but gap <2pp, 0% if full does not consistently beat no.

Quantitative test of H2. Does half_entanglement test_accuracy lie strictly between full and no (within 1pp seed-mean noise) on at least 2 of 3 datasets? 100% if cleanly monotonic on 2/3, 67% if monotonic on 1/3, 33% if half is non-monotonic but within 2pp of one of the endpoints, 0% otherwise.

Quantitative test of H3. On at least 1 of 3 datasets, does classical_baseline achieve test_accuracy at least as high as no_entanglement (5-seed mean)? 100% if classical >= no on at least 1/3, 50% if classical is within 2pp of no on at least 1/3, 0% if classical never reaches no_entanglement minus 2pp.

Writeup of at least 200 words (in submission/README.md ## Agent-produced writeup section) with explicit supported / refuted / inconclusive verdict for each of H1, H2, H3 with numerical evidence (specific mean accuracies for each condition x dataset, and the gap), a discussion of monotonicity, and an honest acknowledgment of whether the classical_baseline sanity check passes. The writeup is encouraged but not required to cross-reference Q01 (encoding) findings if available.


Classical optimizer comparison for VQE on H2 under finite shot noise

Benchmarking classical optimizers (SPSA, COBYLA, L-BFGS-B, ADAM) for VQE convergence on H2 under finite shot noise across multiple shot budgets and bond lengths

The Variational Quantum Eigensolver (VQE) is the canonical algorithm
for near-term quantum chemistry. The quality of a VQE result depends
on the ansatz, but it also critically depends on the classical
optimizer that updates variational parameters using noisy energy
estimates from shot-based measurement. Different optimizers exploit
gradient information very differently. Gradient-free methods such as
COBYLA and Nelder-Mead do not need explicit gradients and may be
robust to shot noise but converge slowly in high dimensions.
Simultaneous Perturbation Stochastic Approximation (SPSA) approximates
the gradient using just two function evaluations regardless of
parameter count, trading gradient quality for evaluation count and
often dominating in low-shot regimes. Gradient-based methods such as
L-BFGS-B and ADAM need explicit or finite-difference gradients, which
are expensive under shot noise because each partial derivative is
estimated from a noisy function value.

The choice of optimizer is rarely studied head-to-head under a fixed
shot budget. Most VQE papers fix the optimizer and tune everything
else around it. Here we hold ansatz (hardware-efficient EfficientSU2
reps=2), Hamiltonian (H2 in STO-3G basis), and initial parameters
(small random Gaussian) fixed, and vary only the optimizer and the
shot budget per energy evaluation. The diagnostic question is which
optimizer reaches chemical accuracy (within 1.6 mHa of the FCI
reference) using the fewest cumulative shots.

A credible study uses at least two molecular geometries to avoid
overfitting to a single energy landscape. We use H2 at two bond
lengths: the equilibrium distance (0.74 angstrom) where the ground
state is near Hartree-Fock and the optimization landscape is well
behaved, and a stretched geometry (1.5 angstrom) where multireference
character makes the landscape harder and tests optimizer robustness.
Each optimizer is run at a SINGLE fixed shot budget of 1024 shots per
energy evaluation. This is chosen as a representative middle-ground
value: low enough that shot noise is non-trivial (per-eval std ~15 mHa,
well above the 1.6 mHa chemical accuracy threshold, so the running-mean
convergence criterion matters), but high enough that some optimizers
can plausibly converge within their iteration budgets. All cells use
3 random seeds per (optimizer, geometry) cell, giving 4 optimizers x
1 shot budget x 2 geometries x 3 seeds = 24 total cells. Cumulative
shots is the sum of shots used across all energy evaluations
(including finite-difference gradient evaluations where applicable).
The Hamiltonian itself is constructed via
qiskit_nature.second_q.drivers.PySCFDriver (qiskit_nature and pyscf
are required dependencies for this topic; hardcoded Pauli string
fallbacks are NOT accepted).

Implementation contract (see quantum-qiskit skill for the reference
code). In qiskit 2.x, `qiskit_algorithms.VQE` and
`qiskit_nature.second_q.algorithms` are NOT importable (they depend on
the removed `qiskit.primitives.BaseEstimator` V1 interface). VQE MUST
be implemented as a manual optimization loop:
energy(theta) = estimator.run([(bound_ansatz, qubit_op)]).result()[0].data.evs + e_nuclear,
using `qiskit.primitives.StatevectorEstimator` (the V2 primitive that
works in qiskit 2.x) for the energy evaluation, and one of the four
optimizer classes from `qiskit_algorithms.optimizers` (SPSA, COBYLA,
L_BFGS_B, ADAM) for the parameter update, called via
`optimizer.minimize(energy, initial_point)`. Each call to `energy`
counts as one evaluation; cumulative shots is
`n_evaluations * shots_per_eval`. The Hamiltonian itself is constructed
via qiskit_nature.second_q.drivers.PySCFDriver +
qiskit_nature.second_q.mappers.ParityMapper (these submodules ARE safe
to import in qiskit 2.x — only the qiskit_nature.second_q.algorithms
module is broken). With ParityMapper and 2-qubit reduction
(num_particles=(1,1)), the H2/STO-3G Hamiltonian is 2 qubits, not 4.
Implement shot noise by adding Gaussian noise to each energy value
with sigma = ||H||_1 / sqrt(shots_per_eval). The convergence criterion
for shots_to_chemical_accuracy is the cumulative shots when a 5-eval
running mean energy stays within 1.6 mHa of E_FCI for 5 consecutive
evaluations; if not reached, report None or a sentinel distinct from
the upper bound so analysis can flag the cell as "did not converge"
rather than treating the budget cap as a measurement.

Research question: *Under finite shot noise, which classical optimizer
(SPSA, COBYLA, L-BFGS-B, ADAM) reaches VQE chemical accuracy using the
fewest cumulative shots, and how does the answer depend on shot budget
per evaluation and on Hamiltonian difficulty (equilibrium vs stretched
H2)?*

At 1024 shots per energy evaluation, SPSA reaches chemical accuracy (|E_running_mean - E_FCI| < 1.6 mHa for 5 consecutive 5-eval windows) using fewer cumulative shots than L-BFGS-B (3-seed median), on at least 1 of 2 H2 geometries.

At 1024 shots per energy evaluation, gradient-free methods (SPSA and COBYLA, 6 seed-runs combined across both geometries) achieve chemical accuracy with success rate at least 4/6, while gradient-based methods (L-BFGS-B and ADAM, 6 seed-runs combined) achieve it with success rate at most 3/6. This isolates the shot-noise sensitivity of finite-difference gradient estimation.

Optimizer ranking by shots_to_chemical_accuracy changes between H2 equilibrium and H2 stretched geometries: at least one optimizer pair (e.g. SPSA vs COBYLA, or SPSA vs L-BFGS-B) swaps relative ranking between the two geometries, indicating that Hamiltonian difficulty changes optimizer choice.

Under finite shot noise, which classical optimizer reaches VQE chemical accuracy in the fewest cumulative shots, and how does the answer depend on shot budget per evaluation and on Hamiltonian difficulty?

spsa

qiskit_algorithms.optimizers.SPSA(maxiter=200, learning_rate=0.05, perturbation=0.1). Fixed shot budget: 1024 shots per energy evaluation. Each iteration costs 2 energy evaluations.

cobyla

qiskit_algorithms.optimizers.COBYLA(maxiter=200, rhobeg=0.1, tol=1e-4). 1024 shots per energy evaluation. Each iteration costs 1 energy evaluation.

lbfgsb

qiskit_algorithms.optimizers.L_BFGS_B(maxiter=100, ftol=1e-6) with finite-difference gradients (epsilon=1e-3). 1024 shots per energy evaluation. Each iteration costs ~2*num_params energy evaluations for the gradient plus 1 for the function value (so ~25 evals per iter for EfficientSU2 reps=2 on 4 qubits).

adam

qiskit_algorithms.optimizers.ADAM(maxiter=200, lr=0.05, beta_1=0.9, beta_2=0.999) with finite-difference gradients (epsilon=1e-3). 1024 shots per energy evaluation. Same per-iter cost as L-BFGS-B.

Cumulative shots vs. FCI energy reference is the within-method benchmark

Hartree-Fock energy at each geometry is the trivial classical baseline below which any optimizer should clear

Cumulative shots until |E_running_mean - E_FCI| stays below 1.6 mHa for 5 consecutive evaluations. If not reached within the maxiter budget, report the total cumulative shots used (sentinel = max-shots) and flag success=False.

|E_final - E_FCI| in Hartree at the end of optimization (after all maxiter iterations).

Fraction of seeds (out of 3 per condition x geometry) where the run reached chemical accuracy by the maxiter budget.

H2 in STO-3G basis at bond length 0.74 angstrom. Build the FermionicOp via PySCFDriver, apply ParityMapper with 2-qubit reduction to obtain a 4-qubit qubit Hamiltonian (alternatively JordanWignerMapper + tapering). FCI reference: E_FCI = -1.137283 Ha. Hardcoded Pauli strings are NOT an acceptable substitute for this benchmark; both qiskit_nature and pyscf must be installed and used.

H2 in STO-3G basis at bond length 1.5 angstrom (multireference / strong-correlation regime). 4 qubits via the same mapping pipeline as h2_equilibrium. FCI reference: E_FCI ~ -1.001 Ha. Hardcoded Pauli strings NOT accepted.

A credible VQE optimizer comparison on H2 in STO-3G basis at two bond lengths (0.74 A equilibrium, 1.5 A stretched), at a fixed shot budget of 1024 shots per energy evaluation. The agent must (a) construct the 4-qubit H2 Hamiltonian at each geometry via qiskit_nature.PySCFDriver, (b) implement VQE via qiskit_algorithms.VQE.compute_minimum_eigenvalue() with a hardware-efficient EfficientSU2 reps=2 ansatz, (c) run 4 optimizers (SPSA, COBYLA, L-BFGS-B, ADAM) x 2 geometries x 3 seeds = 24 cells, (d) log cumulative shots and energy trajectories per cell, and (e) score H1/H2/H3 with numerical evidence keyed by (optimizer, geometry).

Optimizer comparisons are scored on (i) correctness of the Hamiltonian (FCI reference within 1 mHa of -1.137 Ha for H2 equilibrium), (ii) consistent ansatz / initial parameters across cells so only the optimizer and shot budget vary, (iii) accurate accounting of cumulative shots including gradient evaluations for L-BFGS-B and ADAM, and (iv) numerical evidence backing H1/H2/H3. qiskit_nature.second_q.drivers.PySCFDriver MUST be used to build the Hamiltonian. Hardcoded Pauli strings are NOT acceptable and a submission that uses them fails q03-code-hamiltonians.

Code-development bucket: VQE pipeline correctly implements 4 optimizers, both H2 geometries, and accurate cumulative-shot tracking.

VQE is implemented using qiskit_algorithms.VQE or an equivalent custom loop. The ansatz is EfficientSU2(num_qubits=4, reps=2, entanglement='linear'). Initial parameters are sampled from N(0, 0.1) with the same seed across all 4 optimizers for a given (shot_budget, geometry, seed) triple. All 4 optimizers are correctly instantiated from qiskit_algorithms.optimizers (SPSA, COBYLA, L_BFGS_B, ADAM) with the maxiter and learning rates listed in the manifest.

Both H2 Hamiltonians are constructed at the correct bond lengths (0.74 and 1.5 angstrom) in STO-3G basis using qiskit_nature.second_q.drivers.PySCFDriver (REQUIRED) and either ParityMapper with 2-qubit reduction or JordanWignerMapper with tapering. The diagonalized Hamiltonian (numpy eigvalsh of the matrix form) gives FCI energies within 1 mHa of -1.137 Ha (equilibrium) and -1.001 Ha (stretched). Hardcoded Pauli strings as a substitute for PySCFDriver are NOT accepted and fail this leaf.

Cumulative shots is tracked correctly per run. For SPSA and COBYLA, each iteration adds shots_per_eval * num_evaluations_per_iter (2 for SPSA, 1 for COBYLA). For L-BFGS-B and ADAM with finite-difference gradients, each iteration adds shots_per_eval * (2 * num_params + 1) so the gradient evaluations are accounted for. The shots_to_chemical_accuracy metric is computed using the running mean energy (over the last 5 evaluations) and reports the cumulative shot count at the first evaluation where the mean stays within 1.6 mHa of E_FCI for 5 consecutive evaluations.

Execution-validity bucket: all cells ran with valid energies and shot accounting.

At least 22 cells out of 24 expected (4 optimizers x 1 shot budget (1024) x 2 geometries x 3 seeds = 24) completed and produced a final_energy_error_hartree value. If any cells are omitted (e.g. ADAM ran out of budget), they must be documented with a recorded cause. Missing more than 2 cells (10 percent) without justification fails this leaf. The condition_summaries MUST be keyed by (optimizer, geometry) so each of the 8 condition cells has 3 seed entries — collapsing across the geometry axis (e.g. reporting only per-optimizer aggregates) fails this leaf because it makes H1 and H3 unevaluable.

Numerical validity: every final energy is finite (no NaN/Inf), every final_energy_error_hartree >= 0, every success_rate in [0, 1], cumulative shots is monotone increasing within a run. At least one cell across the entire sweep reaches chemical accuracy (otherwise the rubric thresholds are uncalibrated for this setup).

Results bucket: quantitative tests of H1/H2/H3 plus a per-hypothesis writeup.

Quantitative test of H1. On at least 1 of 2 geometries at 1024 shots/eval, does SPSA reach chemical accuracy with fewer median cumulative shots than L-BFGS-B (over 3 seeds)? 100% if SPSA wins on at least 1/2 geometries with a >=20 percent shot reduction, 67% if SPSA wins on at least 1/2 with any margin, 33% if SPSA and L-BFGS-B are within 10 percent of each other, 0% if L-BFGS-B is consistently faster.

Quantitative test of H2. At 1024 shots/eval, pooling across both geometries: do gradient-free methods (SPSA + COBYLA, total of 6 seed-runs = 2 methods x 3 seeds) have a combined chemical-accuracy success rate of at least 4/6, while gradient-based methods (L-BFGS-B + ADAM, 6 seed-runs) have at most 3/6? 100% if cleanly met, 67% if the gradient-free success rate exceeds gradient-based by at least 30 percent, 33% if gradient-free exceeds gradient-based at all, 0% if gradient-based methods are equal or better.

Quantitative test of H3. Does the optimizer ranking by median shots_to_chemical_accuracy change between H2 equilibrium and H2 stretched geometries? Specifically: take the 4-element ranking of optimizers (by 3-seed-median shots) on each geometry; H3 is supported if at least one pair swaps relative order between the two rankings (Kendall tau distance >= 1 between the two rankings). 100% if at least one swap detected with clear evidence, 67% if one optimizer's rank differs by exactly 1 position, 33% if rankings are similar but values differ, 0% if rankings are identical or H3 is unevaluable (e.g. due to collapsed condition_summaries that drop the geometry axis).

Writeup of at least 200 words (in submission/README.md ## Agent-produced writeup section) with explicit supported / refuted / inconclusive verdict for each of H1, H2, H3 with numerical evidence (specific median cumulative shots, success rates, and final energy errors per optimizer x shot budget). Identifies a dominant systematic uncertainty (seed variance, finite-difference epsilon, optimizer hyperparameter sensitivity, shot-noise variance at low budgets). Discusses the difference between equilibrium and stretched H2 in terms of optimizer ranking.