\appendix
\chapter{Additional memory and correctness arguments}
\section{The inherited conditional-memory certificate}
The positive-circuit certificate is useful when complete unnormalized likelihood ratios can be controlled. A different v1 guarantee applies when a representation separately approximates world-conditioned identity laws and world weights. It is retained here because the contracts are not interchangeable.

Let $\alpha_w$ be exact identity conditionals and $\beta_w$ their approximations, with $\TV(\alpha_w,\beta_w)\le\delta$ for every world in a shared positive world support. Suppose positive unnormalized world weights $r_w,\widetilde r_w$ satisfy $e^{-\eta}r_w\le\widetilde r_w\le e^{\eta}r_w$ up to one common scaling convention. Let $\Delta=\max_{u,v}\TV(\alpha_u,\alpha_v)$.

\begin{proposition}[Inherited diameter-sensitive certificate]
After any shared nonnegative world likelihood with a positive normalizer, the identity posterior error is at most
\begin{equation}
\min\{1,\delta+\Delta\tanh(\eta/2)\}.
\end{equation}
The corresponding full joint identity/world TV is bounded by $\min\{1,\delta+\tanh(\eta/2)\}$.
\end{proposition}
\begin{proof}
Let $a_w,b_w$ be the two updated world mixing weights. The likelihood does not change their unnormalized relative weight ratios, so the sharp ratio argument in Theorem~\ref{thm:circuit} gives $\TV(a,b)\le\tanh(\eta/2)$. Replacing each exact conditional with its approximation at fixed mixing weights contributes at most $\delta$. For the remaining difference, maximally couple the world weights: with probability $1-\TV(a,b)$ they choose the same world and hence the same exact conditional; on a mismatch, the conditional laws differ by at most $\Delta$. This gives the identity bound. For the joint variable, a world mismatch itself is distinguishable, so replace $\Delta$ by one.
\end{proof}
This is a source-derived result from v1, with the classical sharp ratio inequality credited separately \cite{ip1,cohen}. It assumes uniform conditional accuracy, not merely an average representation loss. It can tolerate small conditional identity errors that do not satisfy a multiplicative entrywise certificate, but it is limited to world-only additional likelihoods under the stated conditional-independence model.

\section{Why small current TV does not protect every rare query}
Take binary identity and world variables with exact joint mass $1-\varepsilon$ at $(0,0)$ and $\varepsilon$ at $(1,1)$. Let the approximate joint law put the same masses at $(0,0)$ and $(0,1)$. The current joint TV is only $\varepsilon$, and the world marginals agree. Conditioning on the rare but supported measurement $W=1$ makes the identity posteriors opposite point masses, with TV one.

There is no contradiction with any uniform theorem in this manuscript. The multiplicative common-support certificate fails at the omitted exact identity/world atom, and the conditional identity approximation error in world one is one. Nor is there a contradiction with an average conditional error bound: the failing world occurs with probability $\varepsilon$. This example is why a plausible current summary cannot substitute for an explicit future-probe contract.

\section{Exact preservation under arbitrary internal controller revisions}
Theorem~\ref{thm:path} permits a new policy distribution after any full history, including revisions chosen by a learning algorithm. The only requirement for the semantic proof is that the executed distribution remains in the same state-dependent fiber. The proof does not depend on a particular optimizer, number of descendants, or equality of internal weights. Thus exact functional preservation can survive arbitrarily many internal changes even though the algorithm that chooses those changes has not been proved to improve.

This observation is useful for engineering modularity, but it must not be confused with a guarantee about actual implementation bugs or model changes. A changed sensor decoder can alter the interpreted state, a changed feature map can change the meaning of the signature, and a game update can change transition probabilities. Each such change lies outside the fixed-model theorem until separately revalidated.

\section{Scope of complete trajectory enumeration}
The reference path audit expands every word of $(c,q')$ outputs up to a chosen finite horizon. Each word carries a vector of ground-state probability mass; extension by a symbol multiplies that vector by the corresponding ground transition operator. Summing the vector gives the word's probability. This is independently compared across source and descendant policies.

Its work grows exponentially with horizon, which is acceptable only for a small audit. Deployment uses the local signature equality and its inductive theorem, not exhaustive path enumeration. The enumeration therefore tests the implementation on tractable cases without being proposed as an infinite-horizon runtime algorithm.

\chapter{Interface and interpretation checklist}
\begin{longtable}{@{}p{.28\textwidth}p{.66\textwidth}@{}}
\toprule
Question & Required interpretation\\\midrule\endhead
What counts as a source? & One authentic human event or a declared synthetic fixture; multiple labels of it do not create independent source samples.\\
What counts as calibration? & An actual observation from the appropriate world/build, with an explicit conditional likelihood; generated alternatives remain proposals.\\
What is memory preserving? & A declared future-answer or likelihood contract, including player--world dependence where needed.\\
What is fixed in the operator tree? & The catalog, state dimension, event likelihood semantics, source order, and boundary-query interpretation.\\
What changes at a local revision? & The old source factor is replaced atomically; its interpretation history is retained separately.\\
What does the fidelity shield preserve? & The declared joint semantic action/state law, conditional on the specified source target and correct mechanics.\\
What can still change? & All excluded observables, including the implemented arena's motor choice, microstate distribution, reward, and unmodeled timing.\\
When is gain impossible? & If the chosen contract fixes the reward law, the policy fiber is a singleton, or the value/dual no-headroom condition holds.\\
What does a rank drop prove? & Additional linear directions; an interior feasible point makes them locally available, but reward gain and boundary feasibility remain separate.\\
What does the lifetime gate prove? & A false-promotion probability budget under fresh independent fixed-size audits, not the existence of good candidates.\\
What do the numerical results represent? & Constructed finite sequences and original small MDPs with known generative models, not real humans or game backbones.\\
What does complete mean here? & A complete research handoff with proofs, code, results, audit trail, and an empirical program; no production or real-player claim.\\\bottomrule
\end{longtable}

\chapter{Notation and evidence ledger}
\begin{longtable}{@{}p{.20\textwidth}p{.73\textwidth}@{}}
\toprule Symbol & Meaning\\\midrule\endhead
$\Theta$ & Persistent source type within a declared epoch.\\
$W$ & Explanatory mechanics/capture variables, not automatically the player's belief.\\
$D$ & Recorded source evidence.\\
$Z_t$ & Transient sign context in the temporal model; distinct from later transcript $Z$.\\
$F_t,G_t$ & Independent observation noise signs with means $\alpha,\beta$.\\
$\kappa,\rho$ & Product reliability $\alpha\beta$ and Markov-increment mean.\\
$\ell$ & Pairing offset in event units; not a physical frequency.\\
$M_{x,t}$ & Ordered nonnegative source likelihood/transition operator.\\
$H,K$ & Number of static catalog hypotheses and transient hidden states.\\
$E$ & Sum of explicitly assumed log-factor approximation budgets.\\
$\phi,\chi$ & Declared ground-to-semantic state and action maps.\\
$D_s,k_\theta(s)$ & Ground action signature matrix and target semantic output law.\\
$\mathcal F_\theta(s)$ & Feasible policy distributions preserving the target signature.\\
$\Pi_s$ & Orthogonal projector onto $\ker D_s$.\\
$V_{\mathcal F}$ & Best discounted value compatible with the path contract.\\
$\zeta_t$ & Assumed/validated conditional semantic output discrepancy.\\
$\delta_j$ & Per-version false-promotion budget.\\
$\TV$ & Half-$\ell_1$ total variation for finite probability vectors.\\\bottomrule
\end{longtable}

Numerical results have machine-readable counterparts. Proof-to-code and study mappings are in \texttt{CLAIMS.json}; the supplied studies remain separately attributed.