| \chapter{Temporal evidence on a continuous trajectory} |
| \label{ch:temporal} |
| \section{The inherited pairing construction} |
| Let $\theta\in\{-1,+1\}$ be a source type. Let $F_t,G_t$ be mutually independent sign noises with means $\alpha,\beta\in[0,1]$, independent across events and independent of a sign-valued context process $Z_t$. Define |
| \begin{equation}\label{eq:channels} |
| A_t=\theta Z_tF_t,\qquad B_t=Z_tG_t,\qquad\kappa=\alpha\beta. |
| \end{equation} |
| The aligned products satisfy |
| \begin{equation} |
| U_t=A_tB_t=\theta F_tG_t,\qquad \E[U_t\mid\theta]=\theta\kappa. |
| \end{equation} |
| The context cancels exactly. Thus the products are independent even when the context process is correlated. This cancellation is a property of the declared sign model, not a universal theorem about arbitrary multimodal neural representations. |
|
|
| \begin{proposition}[Inherited balanced-summary erasure] |
| For an even-length record, suppose $Z$ has exactly equal numbers of positive and negative signs. The joint distribution of the two separate sums $S_A=\sum_tA_t$ and $S_B=\sum_tB_t$ is independent of $\theta$. |
| \end{proposition} |
| \begin{proof} |
| Condition on a particular balanced $Z$. Each channel sum is the difference of two sums of identically distributed independent signs of equal size, so its distribution is symmetric and does not depend on the arrangement of the signs in $Z$. The channels' noises are independent, giving conditional independence of the two sums. Changing $\theta$ only reverses the first symmetric sum. Since the conditional joint law does not depend on the arrangement, averaging over balanced $Z$ preserves the conclusion. |
| \end{proof} |
| Equal means alone would not prove this statement. Balanced occupancy and the complete joint-law argument are essential. A stationary Markov context does not generally have exact balanced occupancy in a finite record; the 50\% ceiling from this proposition is therefore not used as a baseline claim for the new continuous-trajectory study. |
|
|
| For the product likelihood with known $0<\kappa<1$, |
| \begin{equation} |
| P(U_t=u\mid\theta)=\frac{1+u\theta\kappa}{2}. |
| \end{equation} |
| The posterior log odds equal the prior log odds plus |
| \begin{equation} |
| 2\operatorname{atanh}(\kappa)\sum_tU_t. |
| \end{equation} |
| The signed count is sufficient for this product family. It need not be sufficient for the full sequence likelihood of $A,B$, where a model of context persistence can supply additional information. The new experiments retain full-HMM Bayes as a stronger reference. |
|
|
| \section{A synchronization law with dependence included} |
| Assume a stationary symmetric two-state Markov context. It can be written |
| \begin{equation} |
| Z_{t+1}=Z_tR_t, |
| \end{equation} |
| where $Z_0$ is fair and the increments $R_t$ are independent signs with mean $\rho\in[0,1]$. For a nonnegative pairing offset $\ell$, use |
| \begin{equation}\label{eq:lagged} |
| U_t^{(\ell)}=A_tB_{t+\ell}=\theta V_t\prod_{j=t}^{t+\ell-1}R_j, |
| \qquad V_t=F_tG_{t+\ell}. |
| \end{equation} |
| The $V_t$ are independent signs of mean $\kappa$. Empty increment products are one. This representation exposes the exact dependence that endpoint-only simulations omit. |
|
|
| \begin{theorem}[Continuous-record lag covariance]\label{thm:lag-cov} |
| For $n$ valid pairs from one trajectory, let $S_n^{(\ell)}=\sum_{t=1}^nU_t^{(\ell)}$. Then |
| \begin{align} |
| \E[U_t^{(\ell)}\mid\theta]&=\theta\kappa\rho^\ell,\\ |
| \Cov(U_t^{(\ell)},U_{t+h}^{(\ell)}\mid\theta) |
| &=\begin{cases} |
| \kappa^2(\rho^{2h}-\rho^{2\ell}),&1\le h<\ell,\\ |
| 0,&h\ge\ell, |
| \end{cases}\label{eq:lag-cov} |
| \end{align} |
| and |
| \begin{equation}\label{eq:lag-var} |
| \Var(S_n^{(\ell)}\mid\theta)=n(1-\kappa^2\rho^{2\ell})+ |
| 2\kappa^2\sum_{h=1}^{\min(n-1,\ell-1)}(n-h)(\rho^{2h}-\rho^{2\ell}). |
| \end{equation} |
| For $\ell=0$ the sum is empty and the products are independent with variance $n(1-\kappa^2)$. |
| \end{theorem} |
| \begin{proof} |
| Independence in \eqref{eq:lagged} gives the mean. For $0<h<\ell$, multiplying the two increment products cancels their overlapping increments because $R_j^2=1$. Exactly $2h$ nonoverlapping increments remain, so the product expectation is $\kappa^2\rho^{2h}$. Subtracting the product of means gives \eqref{eq:lag-cov}. If $h\ge\ell$, the increment intervals and all noise variables are disjoint, yielding independence. Sum the diagonal variances and all lagged covariance contributions; there are $n-h$ pairs at separation $h$. |
| \end{proof} |
| The cancellation of overlap explains an initially counterintuitive effect: a larger offset both weakens the identity signal and introduces correlations among neighboring evidence products. Counting each paired frame as independent can understate uncertainty. |
|
|
| \begin{theorem}[Dependence-aware sign-recovery bound]\label{thm:lag-tail} |
| Let $L=\min\{n,\max(1,\ell)\}$. Under the preceding assumptions, the sign-of-score estimator, with arbitrary tie handling, has error bounded by |
| \begin{equation}\label{eq:lag-tail} |
| \Prob(\widehat\theta\ne\theta)\le |
| \Prob(\theta S_n^{(\ell)}\le0) |
| \le\exp\left(-\frac{n\kappa^2\rho^{2\ell}}{2L}\right). |
| \end{equation} |
| For randomized ties the first inequality still holds. For $n\ge\ell\ge1$ and positive signal, a sufficient event count for this bound to be at most $\delta$ is |
| \begin{equation} |
| n\ge\frac{2\ell\log(1/\delta)}{\kappa^2\rho^{2\ell}}. |
| \end{equation} |
| \end{theorem} |
| \begin{proof} |
| For $\ell\ge1$, split event indices into nonempty residue classes modulo $\ell$. Within a class, the increment intervals in \eqref{eq:lagged} are disjoint, so the variables are independent. There are $L$ such classes. For $\ell=0$ use the single independent class. Put $X_t=\theta U_t^{(\ell)}$ and $\mu=\kappa\rho^\ell$. Hoeffding's lemma for a centered sign gives an exponential-moment bound $\exp(\lambda^2/2)$. Generalized H\"older applied across the $L$ class sums yields |
| \begin{equation} |
| \E\exp\left(-\lambda\sum_t(X_t-\mu)\right) |
| \le\exp(Ln\lambda^2/2). |
| \end{equation} |
| Exponential Markov inequality gives $\exp(-\lambda n\mu+Ln\lambda^2/2)$. Optimize at $\lambda=\mu/L$. The zero-signal case gives the valid but vacuous bound one. |
| \end{proof} |
| The proof is an instance of classical partly-dependent concentration \cite{janson,hoeffding}. The extra factor $\ell$ is a conservative bound cost, not an exact characterization of optimal statistical sample complexity. At large lags the bound becomes loose; the experiments report this instead of presenting the upper bound as the measured error rate. The v2 independent-episode result is recovered by setting the independent-class factor to one under its stronger sampling assumption. |
|
|
| \section{Clock offset is not the same as reaction delay} |
| \begin{proposition}[Clock--response observational equivalence]\label{prop:clock} |
| Suppose the likelihood of the observed stimulus-to-input delay depends on clock offset $\delta$ and actual response delay $\tau$ only through $d=\tau+\delta$. Every admissible pair $(\tau+c,\delta-c)$ has the same observation law. Neither parameter is separately identifiable from that likelihood alone. |
| \end{proposition} |
| \begin{proof} |
| The transformed sum equals $d$, so every observation probability is unchanged. A procedure distinguishing the transformed parameters would distinguish identical observation laws, which is impossible uniformly over those parameters. |
| \end{proof} |
| With an independently justified clock interval $[\delta_-,\delta_+]$, a noiseless observed delay $d$ restricts the response delay to |
| \begin{equation} |
| \tau\in[d-\delta_+,d-\delta_-]\cap[0,\infty). |
| \end{equation} |
| A correlation peak estimates an effective alignment under a model; it does not by itself establish a hardware clock calibration or a psychological reaction time. Clock anchors, input-device calibration, and uncertainty in event onset belong in the recording contract. This is a parameter-identifiability result, not a claim that practical timestamp estimation is impossible when additional anchors exist. |
|
|
| \chapter{Future queries determine the memory obligation} |
| \label{ch:memory} |
| \section{A small current identity can require a large archive} |
| The earlier cyclic lock made current marginal summaries uninformative about a relation that later calibration could unlock. The following construction goes further: the entropy of the current player identity does not bound the amount of source memory needed for every future calibration. |
|
|
| Let $\Theta$ be a fair sign, let $W_1,\ldots,W_n$ be independent fair signs independent of $\Theta$, and record |
| \begin{equation}\label{eq:many-locks} |
| D_i=\Theta W_i,\qquad i=1,\ldots,n. |
| \end{equation} |
| The vector $D$ is uniform on $2^n$ records and independent of $\Theta$. Thus every present identity posterior is exactly $(1/2,1/2)$. After consolidation, a future genuine probe selects an index $J$ and reveals the actual $W_J$. With the original record available, identity is recovered by $\Theta=D_JW_J$. |
|
|
| \begin{theorem}[One-bit identity, $n$-bit retrospective obligation]\label{thm:onebit} |
| In \eqref{eq:many-locks}, suppose the retained memory $M$ is the only accessible record-dependent information. To reconstruct $\Theta$ with zero error for every record and every future coordinate probe $(j,W_j)$, $M$ needs at least $2^n$ distinguishable states, or $n$ fixed-length bits. Storing $D$ attains the bound. |
| \end{theorem} |
| \begin{proof} |
| Take distinct records $d,d'$. They differ at some coordinate $j$. For any fixed $w\in\{-1,+1\}$, both $(D=d,W_j=w)$ and $(D=d',W_j=w)$ have positive probability in the model, but the required identities $d_jw$ and $d'_jw$ are opposite. If the two records share a memory state, that same state and same future transcript require two incompatible answers. Therefore their memory states must differ. All $2^n$ records need distinct states. For randomized zero-error encoding, the positive-probability memory supports of conflicting records must be disjoint; the same counting argument applies. The original record supplies a matching $n$-bit representation. |
| \end{proof} |
| The construction does not hide $n$ bits in a one-bit variable. It stores $n$ different \emph{relationships} to the one-bit variable. The future probe specifies which relationship matters. There is no contradiction with the fact that at most one bit of identity information can ultimately be learned. Information gained about an answer and information needed to compute it under every possible side observation are different quantities. |
|
|
| \begin{theorem}[Lossy retrospective lower bound]\label{thm:lossy} |
| Let $J$ be uniform on $\{1,\ldots,n\}$ and independent of the source variables. If a memory of at most $b$ bits permits identity recovery after $(J,W_J)$ with average error $\epsilon\le1/2$, then |
| \begin{equation}\label{eq:lossy} |
| b\ge n[1-h_2(\epsilon)], |
| \end{equation} |
| where $h_2$ is binary entropy. The claim counts all record-dependent readable storage and allows randomized encoding and decoding. |
| \end{theorem} |
| \begin{proof} |
| A predicted identity $\widehat\Theta(M,j,W_j)$ induces a predicted source bit $\widehat D_j=W_j\widehat\Theta$ with the same error $\epsilon_j$. The fair mask $W_j$ is independent of the entire uniform record $D$ and hence of $(D,M)$. Therefore |
| \begin{equation} |
| H(D_j\mid M)=H(D_j\mid M,W_j)\le h_2(\epsilon_j), |
| \end{equation} |
| by the binary form of Fano's inequality. Subadditivity and concavity give |
| \begin{equation} |
| H(D\mid M)\le\sum_jH(D_j\mid M)\le n h_2\left(\frac1n\sum_j\epsilon_j\right)=nh_2(\epsilon). |
| \end{equation} |
| Since $H(D)=n$ and $H(M)\le b$, $b\ge I(D;M)\ge n[1-h_2(\epsilon)]$. Independent public randomness can be conditioned on throughout, with the same averaged inequality. |
| \end{proof} |
| These are finite source-identity applications of established random-access and functional compression arguments \cite{doshi}. The underlying counting and entropy inequalities are not claimed as new information theory. The practical consequence is new to the integrated design: consolidating solely to a small current player posterior can discard precisely what future mechanics changes require. |
|
|
| \section{A one-shot contract through confusability} |
| Let $\mathcal D$ be a finite source-record space and $\mathcal Z$ a declared set of later transcripts. Let $F(d,z)$ be the deterministic requested answer on every supported pair $(d,z)$. An answer may be a discrete source choice, a predeclared certificate, or an exactly represented finite response object. The encoder sees $d$ before $z$ arrives. |
|
|
| \begin{definition}[Future-answer graph] |
| Construct a simple graph $G$ on $\mathcal D$. Join $d$ and $d'$ when there exists $z$ supported by both records such that $F(d,z)\ne F(d',z)$. |
| \end{definition} |
|
|
| \begin{theorem}[One-shot zero-error memory contract]\label{thm:graph} |
| The minimum number of deterministic memory states supporting every allowed future answer is $\chi(G)$, the chromatic number. The minimum fixed-length bit count is $\lceil\log_2\chi(G)\rceil$. Zero-error randomization does not reduce the number of states. |
| \end{theorem} |
| \begin{proof} |
| A memory state cannot be shared by adjacent records, because the defining transcript would require incompatible outputs. Any valid encoder therefore colors $G$. Conversely, take a proper coloring and encode the color. For a fixed color and later $z$, all compatible records in that color agree on their answer, since disagreement would create an edge. The decoder returns that common answer. For randomized zero-error encoders, adjacent records have disjoint positive-probability memory supports; selecting one supported state per record yields a proper coloring using no more states. |
| \end{proof} |
| This is the classical one-shot characteristic-graph perspective on functional coding, specialized to retrospective player reconstruction \cite{doshi}. It is not the asymptotic conditional graph-entropy rate theorem, and those distinct coding regimes are not conflated here. Computing chromatic number is hard in general; the implementation is an exact small-fixture solver, not a scalable compressor. |
|
|
| \begin{corollary}[Capability and query expansion cannot erase an obligation] |
| If a new capability expands the future transcript/answer contract while retaining all old requirements, the graph only acquires edges and the minimum zero-error memory-state count cannot decrease. |
| \end{corollary} |
| \begin{proof} |
| Every old conflict remains a conflict. Any coloring of the expanded graph is also a coloring of the old graph. |
| \end{proof} |
| For the one-bit construction, requiring all coordinates gives a complete graph on $2^n$ records. Requiring only $r$ known coordinates yields a complete multipartite graph with $2^r$ answer classes and requires exactly $r$ bits. The memory budget is therefore tied to what future questions are promised, not to a slogan about infinite context. |
|
|
| \section{How to use the lower bound constructively} |
| A practical system may choose a narrower future contract, retain a larger accessible archive, accept measured approximation error, or acquire the missing information again. It may not claim to answer unsupported questions with zero error from a smaller indistinguishable memory. A source pointer counts as additional usable memory only if it actually retrieves the source. A hash detects mismatch but does not reconstruct the missing record. |
|
|
| The positive task is to retain a compact sufficient object for a specified family, and to expose when a new model or question invalidates that object. The next chapter supplies such an object for fixed finite temporal models. Its full repairable form has a growing archive. Its root-only form has fixed dimension but a narrower query contract and nonconstant exact precision requirements. |
|
|
| \chapter{Revisable temporal evidence algebra} |
| \label{ch:operator} |
| \section{A fixed catalog and transient hidden state} |
| Fix a finite catalog $x\in\mathcal X$ with $H$ elements, such as source type and capture-mechanics hypotheses. A transient hidden state $z_t$ has $K$ values. Let $T_x$ be a transition matrix and $e_{x,t}(z)$ the conditional likelihood of the source event at time $t$ under the model. Set |
| \begin{equation} |
| M_{x,t}=\diag(e_{x,t})T_x. |
| \end{equation} |
| With initial vector $\nu_x$ and terminal vector $b_x$, define |
| \begin{equation}\label{eq:operator-likelihood} |
| L_x(D;b)=\nu_x^\top\left(\prod_{t=1}^n M_{x,t}\right)b_x. |
| \end{equation} |
| With $b_x=\1$, the harmless final transition sums to one. General nonnegative operators also represent boundary probes or other finite conditional factors when their order and model semantics are correctly specified. |
|
|
| This is ordinary finite-state likelihood computation. Compiled probabilistic inference and dynamic-network extensions are established prior art \cite{darwiche,brandherm,rabiner}. The particular design goal here is \emph{source-indexed repair}: later correct information may revise the interpretation of one old event without being counted as a new independent human event. |
|
|
| \section{Ordered composition and local replacement} |
| For a consecutive block $[a,b]$, retain the transfer operator |
| \begin{equation} |
| B_x[a,b]=M_{x,a}M_{x,a+1}\cdots M_{x,b}. |
| \end{equation} |
| Adjacent block composition is associative and order-sensitive. A balanced binary tree stores leaf factors and the product of each pair of children. A source index points to its existing leaf. Correcting that source replaces the leaf and recomputes its ancestors. |
|
|
| \begin{theorem}[Exact finite revision theorem]\label{thm:tree} |
| For a fixed catalog and fixed factor semantics, the root of the ordered tree yields exactly \eqref{eq:operator-likelihood}. Replacing one factor produces exactly the same likelihood as full ordered replay of the corrected factor sequence. With dense matrices, construction costs $O(nHK^3)$, a local correction costs $O(HK^3\log n)$, and retained operator payload is $O(nHK^2)$. Full vector-forward replay costs $O(nHK^2)$ for a complete sequence evaluation. |
| \end{theorem} |
| \begin{proof} |
| Induct on tree depth. Each leaf equals its source factor, and the product of two adjacent correct sub-blocks equals the transfer operator of their union. Associativity proves root correctness. A changed leaf affects only its ancestor path, and repeating the same induction proves correctness after replacement. A balanced tree has linear node count and logarithmic depth; dense multiplication gives the stated arithmetic costs. The vector-forward baseline multiplies a $K$-vector by one $K\times K$ matrix at each event. |
| \end{proof} |
| The tree is not uniformly faster: it uses more memory and more preprocessing than a single forward pass, and a batch replacing nearly every factor may be better handled by rebuilding. Its advantage is repeated sparse revision of an already compiled long archive. The experiment reports both this workload and the storage tradeoff. |
|
|
| \section{Root-only memory has a narrower promise} |
| The root $B_x[1,n]$ suffices for all subsequent likelihood readouts with the same internal factors and arbitrary specified boundary vectors $u_x,v_x$. Conversely, if all unnormalized bilinear readouts $u^\top Bv$ are promised, basis-vector queries recover every entry of $B$. This is a statement about the likelihood operator, not a general minimum-bit theorem for normalized identity posteriors. |
|
|
| \begin{example}[A scalar summary loses a boundary distinction] |
| With initial vector $(1,0)$, the operators |
| \begin{equation} |
| B_0=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad |
| B_1=\begin{pmatrix}0&1\\0&0\end{pmatrix} |
| \end{equation} |
| both have likelihood one under terminal vector $(1,1)$. A later terminal selector $(1,0)$ gives likelihood one under $B_0$ and zero under $B_1$. Retaining only the former scalar answer discards the distinction. |
| \end{example} |
| Retaining the root does not support arbitrary replacement of a forgotten internal leaf. Nor does it reconstruct raw frames for a newly introduced visual hypothesis. A sufficient operator for the old catalog is not automatically sufficient for a catalog with new states, new likelihoods, or a different action meaning. Such changes require a compatible transport certificate or a return to the source archive. |
|
|
| \section{Scaling, atomic revision, and source discipline} |
| The implementation normalizes each operator by its maximum entry and carries a per-hypothesis log scale. This reduces immediate numerical underflow in long products. Zero operators remain zero with log scale $-\infty$, and an all-hypothesis impossible observation is rejected rather than converted into a uniform posterior. |
|
|
| A proposed replacement is shape-checked, nonnegativity-checked, staged along the full ancestor path, and committed only after staging succeeds. Duplicate source IDs at construction and duplicate revision IDs are rejected. Replacing a source with its identical factor leaves the posterior unchanged. The library rejects a generated-rollout label as a source-factor update. |
|
|
| These checks are necessary interface discipline, not source authentication. A caller can lie about a label, distinct IDs do not prove independent observations, and the software does not validate the psychological or sensor likelihood. Actual person/engine authentication and capture provenance require an implementation outside this finite prototype. |
|
|
| \section{An approximation certificate that survives rare calibration} |
| Suppose every approximate source operator has exactly the same zero support as the corresponding exact operator and satisfies, entrywise, |
| \begin{equation}\label{eq:leaf-cert} |
| e^{-\epsilon_t}M_{x,t}\le\widetilde M_{x,t}\le e^{\epsilon_t}M_{x,t} |
| \quad\text{for every }x,t. |
| \end{equation} |
| Let $E=\sum_t\epsilon_t$. All compared systems use the same nonnegative boundaries and the same genuine additional likelihoods. Support deletion is not permitted by this certificate. |
|
|
| \begin{theorem}[Positive evidence-circuit certificate]\label{thm:circuit} |
| Under \eqref{eq:leaf-cert}, the likelihoods satisfy |
| \begin{equation} |
| e^{-E}L_x\le\widetilde L_x\le e^E L_x. |
| \end{equation} |
| For a common prior and any common nonnegative future calibration likelihood with a positive normalizer, the exact and approximate full-catalog posteriors obey |
| \begin{equation}\label{eq:posterior-bound} |
| \TV(P_X,Q_X)\le\tanh(E/2). |
| \end{equation} |
| The bound does not depend on how rare that particular valid calibration outcome is. |
| \end{theorem} |
| \begin{proof} |
| Expand an operator-product entry as a sum over hidden-state paths. Every approximate path product lies between $e^{-E}$ and $e^E$ times its exact product. Nonnegative sums and common boundaries preserve the same inequalities. Multiplying by a common prior and future likelihood does not alter the relative likelihood ratio range on the common surviving support. |
|
|
| For completeness, if unnormalized weight ratios lie in $[m,M]$ and the normalized comparison weights are $q_i$, let $\mu=\sum_iq_ir_i$. Then |
| \begin{equation} |
| \TV(p,q)=\frac{\E_q[(r-\mu)_+]}{\mu} |
| \le\frac{(M-\mu)(\mu-m)}{(M-m)\mu} |
| \le\frac{\sqrt M-\sqrt m}{\sqrt M+\sqrt m}. |
| \end{equation} |
| The first inequality bounds the convex positive-part function by its chord; the second maximizes at $\mu=\sqrt{mM}$. Here $M/m\le e^{2E}$, giving \eqref{eq:posterior-bound}. This is the finite form of the established sharp Hilbert-metric comparison credited to Cohen and Fausti \cite{cohen}, using half-$\ell_1$ TV. |
| \end{proof} |
| If newly multiplied approximate factors are added, their errors enter the sum $E$. Exact recomputation of old factors does not itself create new theoretical error debt. Floating-point arithmetic does introduce numerical error, however: the implementation does not automatically certify every rounding operation by \eqref{eq:leaf-cert}. The theorem is exact real-arithmetic reasoning; numerical tests check its behavior with tolerances and explicitly generated perturbation budgets. |
|
|
| \section{Why this is extensible context, not infinite physical storage} |
| For $H=4,K=2$, the root operator and its log scales have a 160-byte double-precision payload. This is fixed dimension, not arbitrary exact information in fixed bits. Exact long-history likelihoods can require growing precision, and the root supports only the stated query family. |
|
|
| The repairable tree in the largest executed workload has 16,384 leaves and a conceptual array payload of 5,242,880 bytes, excluding Python objects, metadata, source factors retained by the benchmark, and source archives. The two raw sign channels occupy 32,768 payload bytes before metadata. Thus the tree spends 160 times that narrow raw-data payload to obtain local revision. It is an update accelerator, not a compression win on this fixture. Rich video can have different byte economics, but those have not been measured here. |
|
|
| The combination with Chapter~\ref{ch:memory} is constructive: keep a small sufficient state where the future contract permits it; retain growing source-indexed structure where arbitrary future reinterpretation requires it; and make the unsupported boundary visible rather than claiming unrestricted infinite recall. |
|
|
