docs/RESEARCH_OVERVIEW.md

Research overview: AERÍN Continuation Algebra

Infinite Players, version 3.0.0 studies how an evolving game controller can improve execution while preserving an explicitly declared source behavior process. Its completed results concern finite mathematical models, exact small-instance audits, and reproducible synthetic experiments. The broader goal of reconstructing a real player from video and developing an agent beyond human expertise remains a research program.

The technical contribution is an integrated contract linking temporal source evidence, revisable memory, persistent latent identity, constrained control, and fresh improvement audits. Individual foundations—hidden Markov inference, characteristic graphs, constrained Markov decision processes, orthogonal projection, coupling, and alpha spending—are established methods credited in the manuscript. The release does not establish historical priority for those ingredients.

1. Why matching action preferences does not preserve a session

A controller may retain the same semantic action probabilities at every matched ground state while changing which situations occur next. The manuscript's binary example keeps the action label equal to the current semantic state but changes switching probability from 0.1 to 0.9. The conditional action rule is unchanged; the full path distributions become increasingly distinguishable. This separates local preference fidelity from trajectory fidelity.

For a known finite discounted MDP, write the ground state as $s$, motor action as $a$, transition model as $P(s'\mid s,a)$, and reward as $r(s,a)$. Declare semantic maps $q=\phi(s)$ and $c=\chi_s(a)$. A persistent source hypothesis $\theta$ specifies a target joint semantic kernel $K_\theta(c,q'\mid q)$.

The semantic transition signature has one column per motor action:

$$D_s[(c,q^{\mathrm{\prime }}),a]=\mathbf{1}\{{\chi }_s(a)=c\}\sum _{s^{\mathrm{\prime }}:\varphi (s^{\mathrm{\prime }})=q^{\mathrm{\prime }}}P(s^{\mathrm{\prime }}\mid s,a)\mathrm{.}D\_s[(c,q\text{'}),a]\; =\backslash mathbf\; 1\backslash \{\backslash chi\_s(a)=c\backslash \}\; \backslash sum\_\{s\text{'}:\backslash phi(s\text{'})=q\text{'}\}P(s\text{'}\backslash mid\; s,a).$$

Let $k_\theta(s)$ be the corresponding target row. The source-compatible policy fiber is

$${\mathcal{F}}_{\theta }(s)=\{p\ge 0:D_{s}p=k_{\theta }(s)\}\mathrm{.}\backslash mathcal\; F\_\backslash theta(s)=\backslash \{p\backslash ge0:D\_sp=k\_\backslash theta(s)\backslash \}.$$

Each signature column sums to one, so normalization follows from the equality. The fiber is nonempty exactly when the target belongs to the convex hull of the signature columns. An empty fiber diagnoses an unrealizable exact contract.

The semantic-trajectory theorem states that every possibly history-dependent, time-varying executor selecting distributions in these fibers generates the same joint law of $(q_0,c_0,q_1,\ldots,q_T)$ at every finite horizon. The assumptions include correct dynamics, fixed semantic maps and source target, and a shared initial semantic law. The equality concerns distributions, not identical sampled runs. Non-Markov source behavior needs richer semantic states or history-indexed targets.

Manuscript: semantic control, prop:feasible, thm:path. Implementation: fidelity.py.

2. Improving inside the protected policy fiber

For $0\le\gamma<1$, the constrained Bellman operator maximizes the ordinary discounted backup over $\mathcal F_\theta(s)$. It is a monotone $\gamma$-contraction. Its fixed point is attained by a stationary feasible policy and bounds every history-dependent controller obeying the same local fibers. The reference implementation uses LP policy iteration and reports primal, dual, and objective residuals.

An interior feasible policy also admits an explicit improving direction. Let $Q_s^\pi$ be exact action values and define

$${\mathrm{\Pi }}_s=I-D_s^{†}{D}_s,d_s={\mathrm{\Pi }}_s{Q}_s^{\pi }\mathrm{.}\backslash Pi\_s=I-D\_s^\backslash dagger\; D\_s,\; \backslash qquad\; d\_s=\backslash Pi\_sQ\_s^\backslash pi.$$

The projector removes changes that would alter the protected signature. Because its directions sum to zero, a sufficiently small update $\pi^+=\pi+\eta_sd_s$ remains normalized and feasible. The manuscript chooses a strict fraction of the largest step allowed by decreasing action probabilities. Its advantage at the old value is $g(s)=\eta_s|d_s|_2^2$, giving

$$V^{{\pi }^{+}}-V^{\pi }=(I-\gamma P_{{\pi }^{+}}{)}^{-1}g\ge 0.V^\{\backslash pi^+\}-V^\backslash pi\; =(I-\backslash gamma\; P\_\{\backslash pi^+\})^\{-1\}g\backslash ge0.$$

For an interior feasible policy, zero projection at every state is equivalent to optimality under the contract. Boundary policies require nonnegative feasible directions or LP dual certificates; a nonzero nullspace dimension alone does not prove improvement. Likewise, calibration can remove false-world constraints and enlarge a fiber without creating a useful reward direction. The release includes both a positive calibration fixture and a negative control.

If reward is entirely determined by the protected semantic trajectory, preserving that law also fixes expected reward. Demonstrated gains therefore use execution dimensions excluded from the declared semantics.

Manuscript: control geometry, thm:ceiling, thm:nullspace, thm:noheadroom, prop:rank-release.

3. Temporal evidence must account for overlapping observations

The temporal construction uses binary source type $\theta$, context $Z_t$, and independent sign noises:

$$A_t=\theta Z_t{F}_t,B_t=Z_t{G}_t,\kappa =\mathbb{E}[F_t]\mathbb{E}[G_t]\mathrm{.}A\_t=\backslash theta\; Z\_tF\_t,\backslash qquad\; B\_t=Z\_tG\_t,\; \backslash qquad\; \backslash kappa=\backslash mathbb\; E[F\_t]\backslash mathbb\; E[G\_t].$$

Aligned products cancel context exactly. With Markov increments of mean $\rho$, lagged products have mean $\theta\kappa\rho^\ell$. Overlapping increment intervals introduce covariance $\kappa^2(\rho^{2h}-\rho^{2\ell})$ for $1\le h<\ell$; covariance is zero for $h\ge\ell$. The resulting score variance includes all overlapping-pair terms.

A conservative sign-recovery bound is

$$\mathrm{Pr}(\widehat{\theta }\mathrm{\ne }\theta )\le \exp \text{\hspace{0.17em}⁣}[-\frac{n{\kappa }^2{\rho }^{2\mathrm{\ell }}}{2L}],L=\min \{n,\max (1,\mathrm{\ell })\}\mathrm{.}\backslash Pr(\backslash widehat\backslash theta\backslash ne\backslash theta)\; \backslash le\backslash exp\backslash !\backslash left[-\backslash frac\{n\backslash kappa^2\backslash rho^\{2\backslash ell\}\}\{2L\}\backslash right],\; \backslash qquad\; L=\backslash min\backslash \{n,\backslash max(1,\backslash ell)\backslash \}.$$

This specializes classical partly dependent concentration to the declared sign model. It is not a generic multimodal representation theorem or an inferred biological clock. The separate clock-response result also shows that a likelihood depending only on observed delay $d=\tau+\delta$ cannot identify reaction delay $\tau$ separately from clock offset $\delta$ without additional anchors.

Manuscript: temporal evidence, thm:lag-cov, thm:lag-tail, prop:clock. Implementation: temporal.py.

4. A small current identity can require a large archive

Let $\Theta$ be one fair sign and $W_1,\ldots,W_n$ independent fair masks. Record $D_i=\Theta W_i$. The current identity posterior is fair for every full record, yet a later genuine coordinate probe revealing $(j,W_j)$ makes $\Theta=D_jW_j$ recoverable.

If retained memory must answer every such future probe with zero error, distinct source records must have distinguishable memory states. Exactly $n$ fixed-length bits are necessary and sufficient. With a uniform random queried coordinate and average error $\epsilon\le1/2$, the lower bound becomes

$$b\ge n[1-h_2(ϵ)]\mathrm{.}b\backslash ge\; n[1-h\_2(\backslash epsilon)].$$

The general one-shot formulation connects records whenever a common supported future transcript demands different answers. Its chromatic number $\chi(G)$ gives the minimum number of memory states. This is a characteristic-graph specialization, not a new universal coding theorem or scalable graph-coloring algorithm.

The practical lesson is contractual: current posterior entropy is not a sufficient memory budget for every future reinterpretation. A system must restrict promised future queries, retain an accessible archive, accept measured error, or reacquire missing evidence.

Manuscript: memory contracts, thm:onebit, thm:lossy, thm:graph. Implementation: memory_contract.py.

5. Source-indexed revision with ordered evidence operators

For a fixed catalog of $H$ source/world hypotheses and $K$ hidden states, finite sequence evidence is represented by nonnegative operators:

$$M_{x,t}=\mathrm{diag}(e_{x,t})T_x,L_x={\nu }_x^{\mathrm{\top }}\left(\prod _t{M}_{x,t}\right)b_x\mathrm{.}M\_\{x,t\}=\backslash operatorname\{diag\}(e\_\{x,t\})T\_x,\; \backslash qquad\; L\_x=\backslash nu\_x^\backslash top\backslash left(\backslash prod\_tM\_\{x,t\}\backslash right)b\_x.$$

A balanced ordered product tree retains consecutive block operators. Revising an existing source interpretation replaces its leaf and recomputes ancestors, rather than counting the correction as another independent observation. Associativity gives the same likelihood as corrected full replay.

Dense construction costs $O(nHK^3)$, sparse local revision costs $O(HK^3\log n)$, and operator payload costs $O(nHK^2)$. A full vector-forward evaluation costs $O(nHK^2)$. The different matrix and vector operation units matter. The tree benefits repeated sparse repairs after preprocessing; it is larger than the raw sign archive in the supplied fixture.

The root alone supports fixed-factor boundary readouts, not arbitrary forgotten-leaf repair. A new hypothesis catalog or likelihood semantics requires recompilation or justified transport. Source labels and digests detect declared inconsistencies; they do not authenticate that evidence came from a human.

Manuscript: revisable evidence, thm:tree. Implementation: evidence_tree.py.

6. Persistent branches and an end-to-end error bound

Suppose exact and stored evidence operators share zero support and each factor has entrywise multiplicative error at most $e^{\pm\epsilon_t}$. With $E=\sum_t\epsilon_t$, nonnegative composition preserves the likelihood bound and yields posterior TV at most $\tanh(E/2)$, including after common valid rare calibration outcomes. Missing support is excluded. The implementation tests constructed perturbations; it does not automatically certify every floating-point operation or neural encoder.

For full source/world latent $X$, let posterior mismatch be at most $\epsilon$, and let the executor's conditional next-semantic-output discrepancies be uniformly bounded by $\zeta_t$. Persistent latent coupling gives

Exact controller revisions keep all $\zeta_t=0$, leaving the posterior error independent of horizon. The branch is selected persistently, rather than resampled each action. Preserving only the identity marginal can lose unresolved world dependence needed for continuation.

Manuscript: approximation certificate, thm:circuit; composition, thm:composition.

7. Promotion is a separate statistical contract

Candidate $j$ receives fresh fixed-size paired outcomes in $[-1,1]$. Allocating $\delta_j=\delta/[j(j+1)]$ and promoting only when the sample mean exceeds the minimum gain plus $\sqrt{2\log(1/\delta_j)/n_j}$ bounds the probability of ever falsely promoting by $\delta$. This is classical alpha spending with bounded concentration. It assumes fresh conditional independence, fixed audit size, and correctly allocated metrics; reused holdouts and optional stopping are outside the guarantee.

The theorem controls false acceptance. It does not ensure useful proposals, sustained growth, or modification of the learning algorithm itself.

Manuscript: promotion, thm:lifetime. Implementation: promotion.py.

8. Evidence and the next implementation boundary

The synthetic six-state arena's path-constrained LP raises mean normalized value from 0.511568 to 0.642583 over 16 seeds while six-step semantic TV is 4.30e-16. Weaker constraints obtain higher reward with substantial path drift. Tree inference matches exact HMM replay within 5.33e-15 over 3,000 short records; both reach 999/1000 classification at 32 events after simulated calibration. These are finite reference findings, not real-player or commercial-game benchmarks.

Related work includes VPT, ToMnet, Voyager, SIMA 2, Robo-Saber, style consistency, nullspace imitation, probabilistic circuits, abstraction, functional compression, and shielding. The foundation chapter and references distinguish their roles. Neuroscience motivates time-resolved evidence handling; no biological data were reanalyzed.

The next empirical program requires person-disjoint capture splits, independently calibrated clocks, held-out source sessions, learned-model error bounds, explicit privileged-information boundaries, and matched engine baselines. Native engine adapters, local-model proposers, GPU scaling, human recognizability, expert comparisons, and sustained recursive self-improvement remain open. The integration contract defines interfaces for that work; it does not claim those integrations already exist.