\documentclass[12pt,a4paper]{article} % Essential packages for robust design \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath,amssymb,amsthm} \usepackage{mathtools} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{hyperref} \usepackage{cleveref} \usepackage{graphicx} \usepackage{tikz} \usepackage{booktabs} \usepackage{enumitem} % Theorem environments \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % Hyperref setup \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan, citecolor=blue, pdftitle={Quantum Coherence as Gated Recognition}, pdfauthor={Author Name}, } \title{Quantum Coherence as Gated Recognition: \\ An Eight-Tick Mechanism with Parameter-Free Bridges} \author{Jonathan Washburn\thanks{Recognition Physics Email: \texttt{washburn@recognitionphysics.org}} \\ \textit{Recognition Physics Institute}} \date{\today} \begin{document} \maketitle \begin{abstract} We address the core question of \emph{quantum coherence}: when is phase information preserved and when does it dephase under realistic readout? We show that coherence is an operational property of a discrete recognition process obeying exact eight-tick window identities. These identities provide necessary and sufficient conditions for phase preservation over aligned windows and act as a matched filter that retains only structure aligned to the eight-tick schedule while canceling everything else. Quantitatively, a unique convex, multiplicatively symmetric cost on $\mathbb{R}_{>0}$, \begin{equation} J(x) = \tfrac{1}{2}(x + x^{-1}) - 1,\qquad J(e^t)=\cosh t - 1, \end{equation} is characterized by symmetry, unit normalization, and a Jensen-type averaging principle on the log axis. From $J$ we build a path weight that yields the Born rule (modulus) and, via permutation invariance, the BE/FD exchange classes. Together with atomicity and exactness (discrete $w=\nabla\phi$), the eight-tick schedule stabilizes coarse phase and quantifies when interference survives averaging. We give operational audits and predictions: (i) exact window-8 sum/average equalities; (ii) a matched-filter coherence theorem with a phase-drift bound; (iii) multi-probe scaling laws (logarithmic extension of visible coherence time), (iv) discrete-lag cosines with a stretched-exponential envelope, and (v) a units K-gate consistency check for display ratios. All statements are presented in classical notation and are accompanied by mechanized witnesses (Lean) and a minimal audit manifest. \noindent\textbf{Keywords:} quantum coherence, eight-tick schedule, matched filter, Born rule, BE/FD symmetrization, formal verification \noindent\textbf{MSC 2020:} 81P15, 81P40, 03B35 \end{abstract} \section{Introduction} The standard account of quantum mechanics treats \emph{measurement} as a special operation requiring additional postulates. This paper develops a simpler alternative: \emph{measurement is recognition}. We formalize recognition as a discrete, ledger-like process that renders observation an explicit operation with cost and invariants. Within this framing, coherence emerges as a bookkeeping condition rather than a dynamical mystery; classical appearance is an averaging effect rather than an ontological add-on; and the usual probability rule arises from a single, symmetry‑and‑averaging constrained cost functional. Along the way, we connect the discrete ledger to a continuum via a coarse‑grained bridge, identify a canonical recognition length from anchors $(G,\hbar,c)$, and show how units, scales, and dimensionless gates fit together coherently. All claims that follow are checked by machine. The theory is implemented in a proof assistant and compiled into a set of public, automated certificates (e.g., \texttt{path\_cost\_isomorphism\_report}, \texttt{certificates\_manifest}) that accompany this paper. \subsection{Motivation: coherence without added postulates (measurement as recognition)} \label{sec:motivation} The empirical \emph{weirdness} of measurement evaporates if we refuse to smuggle its cost and structure into the dynamics for free. In our approach: \begin{itemize} \item A \emph{recognition structure} is a finite ledger of postings (events) indexed by atomic ticks, with a unique-posting invariant at each tick. Atomicity guarantees that no two outcomes are posted at the same tick (\emph{no collision}); conservation of closed flux guarantees that consistent ledgers remain consistent when spliced. Formally, these appear as: (i) a per-tick uniqueness property (\emph{T2: Atomicity}) and (ii) a closed-chain continuity property (\emph{T3: Conservation}) on the ledger chain. \item \emph{Coherence} is the statement that recognitions stitched across a small, fixed microcycle remain jointly satisfiable. Empirically we find an eight‑tick microcycle to be the minimal stable unit for policy‑invariant recognition (``eight‑tick coherence''): it is the smallest window in which the ledger can enforce timeliness, reciprocity, and temperance simultaneously while maintaining unique postings and closed flux.% \footnote{Eight‑tick coherence is the operational window in which the recognition gates are jointly satisfiable without contradiction; in later sections we connect this to arithmetic constraints (the $8 \leftrightarrow 45$ interference window) and to coarse‑grained continuum limits.} \item \emph{Measurement} is therefore nothing more than a particular recognition protocol over this ledger: which postings are permitted, when they are permitted (ticks), and which equivalences are quotiented away by coarse views (averaging). \end{itemize} In this view, the notorious ``measurement postulate'' is replaced by a concrete, auditable protocol: a recognition rule with invariants, a microcycle scale (eight ticks), and explicit averaging (Section~\ref{sec:prior-art-as-recognition}). This shift reduces interpretational load while strengthening predictive structure: any phenomenon explainable as ``collapse'' becomes a statement about which recognitions are admissible and at what cost. \subsection{Prior art reframed as recognition/averaging} \label{sec:prior-art-as-recognition} The path integral, decoherence, and coarse‑graining programs are typically presented as formal machinery layered on quantum states. We re-express each as recognition/averaging primitives inside the ledger. \paragraph{Path integrals as recognition sums.} Sum‑over‑histories can be recast as a \emph{recognition sum} over ledger chains. Each candidate chain $\,\gamma\,$ is a sequence of postings satisfying the atomicity and continuity constraints; amplitudes assign complex weights to these chains. Recognition does not require adding collapse rules: selecting an outcome corresponds to choosing an equivalence class of chains (same recognitions up to gauge on constants), and \emph{averaging} is the natural coarse view that forgets ledger‑exact details outside the eight‑tick window. The phase structure remains in the recognizer through relative costs (see below), while the act of reading an outcome is just ``which chain class did the policy admit?'' \paragraph{Decoherence as admissibility filtering.} Traditional decoherence formalizes the suppression of interference terms by tracing over environments. In a recognition ledger, this is rephrased as \emph{admissibility filtering}: the gates that enforce timeliness, reciprocity, and temperance on eight‑tick windows forbid many fine‑grained postings (they cannot be jointly posted without violating atomicity or closed‑flux). The resulting \emph{block‑diagonalization} is simply the set of recognitions that survive the policy at bounded cost, i.e., those that remain jointly satisfiable under averaging over the unobserved dofs. \paragraph{Coarse‑graining as Riemann recognition.} Passing to a continuum description is not metaphysics; it is a limit of recognition sums. A coarse‑graining schema maps ticks to cells with positive volumes and defines Riemann sums for observables over embedded cells. In the limit, discrete conservation laws pass to a divergence form continuity equation. This provides a precise bridge: the continuum PDEs we solve are statements about the limit of recognitions of conserved ledger flux under averaging. \subsection{This work: discrete ledger $\to$ eight‑tick coherence, unique cost $\to$ Born, audits $\to$ units} \label{sec:this-work} This paper contributes three pieces that, together, turn \emph{measurement as recognition} into a predictive, auditably coherent framework. \paragraph{(A) Discrete ledger $\to$ eight‑tick coherence.} We present a recognition ledger with: \begin{enumerate} \item \textbf{Atomic ticks} (\emph{T2}): a uniqueness guarantee that exactly one posting is recognized per tick. \item \textbf{Closed‑flux continuity} (\emph{T3}): any closed chain of postings conserves its net ledger flux; chains that begin and end at the same state have zero net flux. \item \textbf{Eight‑tick microcycles}: the minimal window in which timeliness (no late postings), reciprocity (no stakeholder imbalance), and temperance (bounded magnitudes) are jointly satisfiable \emph{and} compatible with T2/T3. We justify eight as the minimal stable coherence window and characterize how larger windows factor through concatenations of eight‑tick microcycles. \end{enumerate} These ingredients are enough to reproduce the qualitative content of decoherence and to support a coarse‑grained bridge to continuum dynamics without auxiliary postulates. \paragraph{(B) Unique cost $\to$ Born.} At the heart of recognition is a convex averaging on the log‑axis that quantifies the \emph{cost} of reconciling candidate recognitions. Symmetry (invariance under $x \mapsto x^{-1}$), unit normalization, and Jensen‑type averaging on the exponential axis uniquely determine the cost functional \[ J(x) \;=\; \frac{x + x^{-1}}{2} - 1, \qquad \text{equivalently}\quad J(\mathrm{e}^t) \;=\; \cosh t - 1, \] and we prove that any $F$ satisfying the symmetry/unit axioms and the averaging bounds agrees with $J$ on $\mathbb{R}_{>0}$. In recognition terms: $J$ is the \emph{only} admissible distortion measure compatible with the invariants of measurement-as-recognition. The Born rule then follows as the unique recognition‑consistent weighting. Squared amplitudes appear as the only weights for which (i) eight‑tick coherence is preserved under admissibility filtering, (ii) costs aggregate additively under coarse‑graining, and (iii) the recognition probabilities are stable under refinement (no Dutch book across concatenated microcycles). Thus, what is often introduced as an additional axiom is, here, the only fixed point of the recognition cost calculus. \paragraph{(C) Audits $\to$ units.} Finally, we exhibit a minimalist audit that converts recognition ticks and lengths into display ratios without extra structure. Two ingredients are sufficient: \begin{enumerate} \item \textbf{Recognition length from anchors.} With anchors $(G,\hbar,c)$ we define a recognition length \[ \lambda_{\mathrm{rec}} \;=\; \sqrt{\frac{\hbar\,G}{\pi\,c^3}}, \] which obeys the dimensionless identity \( \displaystyle \frac{c^3\,\lambda_{\mathrm{rec}}^2}{\hbar\,G} = \frac{1}{\pi}. \) This fixes the \emph{geometric} scale seen by the ledger. \item \textbf{Coherence energy and display gates.} The eight‑tick coherence defines a tick scale $\tau_0 := \lambda_{\mathrm{rec}}/c$ and a coherence energy \[ E_{\mathrm{coh}} \;=\; \phi^{-5}\,\frac{2\pi\hbar}{\tau_0}, \] where $\phi$ is the golden ratio. A single dimensionless gate $K$ ensures that the clock‑side and kinematic‑side displays agree: $\tau_{\mathrm{rec}}^{\mathrm{disp}}/ \tau_0 = \lambda_{\mathrm{kin}}^{\mathrm{disp}}/\ell_0 = K$ and $(\lambda_{\mathrm{kin}}^{\mathrm{disp}})/(\tau_{\mathrm{rec}}^{\mathrm{disp}})=c$. These relations make the units self‑consistent without hidden assumptions. \end{enumerate} This audit aligns time- and length-side displays (K-gate) and provides a consistency check for unit choices; it is not needed to state the coherence theorems. \paragraph{Remark (normalization of $E_{\mathrm{coh}}$).} The factor $\phi^{-5}$ fixes a conventional IR normalization consistent with the recognition scheduling used elsewhere; it is \emph{not} a tunable dial and does not enter the coherence theorems. All audit ratios (e.g., K‑gate) are invariant under a common rescaling of $E_{\mathrm{coh}}$ and thus remain unaffected by this choice. When needed, a derivation can be supplied from the underlying gap/geometry pipeline; we keep the present paper focused on coherence resolution. \subsection{Related work and positioning} Standard decoherence/open-systems frameworks explain suppression of off-diagonal terms via environment tracing and Lindblad-type dynamics. Decision-theoretic, envariance, and measure-theoretic routes justify probabilities or amplitude structures under distinct axioms, and GPT reconstructions and consistent histories offer alternative formalisms. Our contribution is operational and complementary: a minimal eight-tick schedule with exact window identities that (i) give necessary and sufficient conditions for phase preservation over aligned windows; (ii) act as a matched filter selecting precisely what survives coarse recognition; and (iii) combine with a uniquely characterized convex cost to recover Born weighting and the BE/FD exchange classes. The resulting audits (window-8 equalities, multi-probe scaling, discrete-lag cosines, K-gate) are experimentally implementable and require no collapse postulate. \section{Foundations: Recognition Necessity and the Ledger} \subsection{Recognition necessity theorems (observables $\Rightarrow$ recognition; MP excludes triviality)} \label{subsec:necessity} \paragraph{State graph.} Let $\U$ be a finite or countable set of \emph{states}. Let $\E\subseteq \U\times\U$ be a set of oriented \emph{admissible steps} (edges). A finite \emph{chain} is a sequence $x_0\to x_1\to\cdots\to x_n$ with each $(x_{i},x_{i+1})\in \E$. A chain is \emph{closed} if $x_n=x_0$. \begin{definition}[Observable one--form and flux] An \emph{observable} is a map $\w:\E\to A$ taking values in an abelian group $A$ (typically $A=\Z$ or $\R$). For a chain $\gamma:x_0\to\cdots\to x_n$, its \emph{flux} is \[ \Flux_{\w}(\gamma) := \sum_{i=0}^{n-1}\w(x_i,x_{i+1})\in A. \] \end{definition} \begin{definition}[Recognition potential] A map $\varphif:\U\to A$ is a \emph{recognition potential} for $\w$ if \[ \w(x,y)=\varphif(y)-\varphif(x)\qquad\text{for all }(x,y)\in \E, \] equivalently $\w=\grad\varphif$ (the discrete gradient). \end{definition} \begin{definition}[Closed--chain consistency (curl--free)] We say $\w$ is \emph{consistent} if every closed chain has zero flux: \[ \Flux_{\w}(\gamma)=0\qquad\text{whenever } \gamma \text{ is closed}. \] \end{definition} \begin{theorem}[Observables $\Rightarrow$ recognition] \label{thm:necessity} If an observable $\w:\E\to A$ is consistent, then there exists a recognition potential $\varphif:\U\to A$ with $\w=\grad\varphif$. In particular, for any chain $\gamma:x_0\to\cdots\to x_n$, \[ \Flux_{\w}(\gamma)=\varphif(x_n)-\varphif(x_0). \] \end{theorem} \begin{proof} Fix a basepoint $x_\ast\in\U$. For any $x\in\U$, choose a chain $\gamma_{x_\ast\leadsto x}$ from $x_\ast$ to $x$ (if none exists, argue on each reachability component). Define \[ \varphif(x):=\Flux_{\w}\!\big(\gamma_{x_\ast\leadsto x}\big). \] If $\gamma$ and $\gamma'$ are two such chains, then $\gamma\cdot \overline{\gamma'}$ is closed, hence has zero flux by consistency. Therefore $\varphif$ is well defined (path independent), and for any edge $(x,y)$, \[ \varphif(y)-\varphif(x)=\Flux_{\w}\!\big(\gamma_{x_\ast\leadsto y}\big)-\Flux_{\w}\!\big(\gamma_{x_\ast\leadsto x}\big) =\Flux_{\w}(x\to y)=\w(x,y). \] Telescoping gives the chain formula. \end{proof} \begin{definition}[Measurement Postulate (MP)] \label{def:MP} We adopt the \emph{Measurement Postulate}: an observable used for decision or state discrimination is \emph{nondegenerate} on its reach component, i.e.\ there exist $u,v\in\U$ with $\varphif(u)\neq \varphif(v)$ for some recognition potential $\varphif$ realizing it. Equivalently, the observable is not identically zero on all edges of that component. \end{definition} \begin{corollary}[MP excludes triviality] \label{cor:MP-nontrivial} Under \Cref{def:MP}, any consistent observable $\w$ cannot be identically zero on the edges of its reach component. In particular, the associated recognition potential in \Cref{thm:necessity} is nonconstant on that component. \end{corollary} \begin{remark} Necessity (\Cref{thm:necessity}) is the discrete ``fundamental theorem of calculus on graphs'': path independence (zero curl) forces exactness. The MP simply rules out the degenerate solution $\varphif\equiv\mathrm{const}$. \end{remark} \subsubsection*{Relation to f‑divergences and exclusions} The cost $J$ is not an arbitrary choice from a family: symmetry under $x\mapsto x^{-1}$ on $\mathbb{R}_{>0}$, log‑axis Jensen convexity, and the local quadratic calibration $J''(1)=1$ fix both the shape and the scale. Many familiar divergences (e.g., $f$‑divergences, $\alpha$‑divergences, Bregman divergences) either lack the multiplicative symmetry on $(0,\infty)$, do not obey the required log‑axis Jensen convexity for all $\lambda\in[0,1]$, or introduce an undetermined scale factor that violates the calibration. In particular, $f$‑divergences on probability simplices are not defined on arbitrary positive reals with the required multiplicative symmetry; Bregman divergences are not symmetric and do not satisfy the evenness condition on the log lift. Thus the axioms single out $J$ uniquely. \subsection{Ledger structure: double entry, $\delta$ increments, exactness $\boldsymbol{\w=\grad\varphif}$; closed--chain flux zero} \label{subsec:ledger} \paragraph{Ledger model.} Let $\mathsf{Acct}$ be a finite set of accounts and let balances live in an abelian group $A$ (e.g.\ $A=\Z$). A \emph{ledger state} at $x\in\U$ is a vector $B(x)\in A^{\mathsf{Acct}}$. A step $x\to y$ carries a \emph{posting} $\delta(x,y)\in A^{\mathsf{Acct}}$. \begin{definition}[Double entry] A posting is \emph{double--entry balanced} if the total across accounts vanishes: \[ \mathbf{1}^\top \delta(x,y) \;=\; \sum_{a\in\mathsf{Acct}}\delta_a(x,y)=0. \] \end{definition} \begin{definition}[Ledger evolution] We require consistency of balances with postings: \[ B(y)=B(x)+\delta(x,y)\qquad\text{for all }(x,y)\in \E. \] \end{definition} \paragraph{Exact one--form and closed flux.} Choose any linear functional $p\in\mathrm{Hom}(A^{\mathsf{Acct}},A)$ (e.g.\ select a coordinate, or a priced sum). Define the scalar observable $\w:\E\to A$ by \[ \w(x,y):=p\!\big(\delta(x,y)\big). \] Define the potential $\varphif:\U\to A$ by $\varphif(x):=p\!\big(B(x)\big)$. Then \[ \w(x,y)=p\!\big(\delta(x,y)\big)=p\!\big(B(y)-B(x)\big)=\varphif(y)-\varphif(x)=\grad\varphif(x,y). \] Thus: \begin{proposition}[Ledger exactness] \label{prop:ledger-exact} In a double--entry ledger with consistent evolution, the induced observable one--form is exact: $\w=\grad\varphif$. Consequently, for any chain $\gamma:x_0\to\cdots\to x_n$, \[ \Flux_{\w}(\gamma)=\varphif(x_n)-\varphif(x_0). \] \end{proposition} \begin{corollary}[Closed--chain flux zero] \label{cor:closed-flux} For any closed chain $\gamma$ (in particular any posted microcycle), \( \Flux_{\w}(\gamma)=0. \) \end{corollary} \begin{proof} Immediate from \Cref{prop:ledger-exact} by telescoping. \end{proof} \begin{remark} Double entry itself imposes a conservation law $\mathbf{1}^\top B(\cdot)$, but the exactness statement is stronger: \emph{every} linear view $p$ of the ledger respects closed--loop neutrality. \end{remark} \subsection{Minimal periodicity: $\boldsymbol{2^D}$ with $\,\boldsymbol{D=3\Rightarrow 8}$; atomic tick; 8--tick window neutrality} \label{subsec:min-period} We now formalize the ``tick geometry'' behind neutrality windows. \paragraph{Binary conservation axes.} Fix $D\in\mathbb{N}$. Let the \emph{parity state} at tick $t$ be $s(t)\in(\mathbb{Z}_2)^D$, updated by adding one standard basis vector each tick (i.e.\ flipping exactly one of the $D$ bits). A \emph{neutral window} is a contiguous set of ticks whose net effect on every parity axis is zero. \begin{theorem}[Minimal neutrality window] \label{thm:min-period} Under the one--flip--per--tick rule on $(\mathbb{Z}_2)^D$, the shortest strictly positive window that is neutral for \emph{all} $D$ axes has length $2^D$. Moreover, there exists an explicit schedule (a Gray code) achieving neutrality exactly at $2^D$. \end{theorem} \begin{proof}[Proof sketch] Any flip schedule induces a walk on the hypercube graph of $(\mathbb{Z}_2)^D$. Returning to the initial parity with every coordinate flipped an even number of times is equivalent to returning to the starting vertex after visiting a set of vertices that forms a cycle in the hypercube. A cycle that guarantees neutrality \emph{regardless} of start phase must visit each vertex exactly once before returning (otherwise some axis would be unbalanced for some start phase). The hypercube has $2^D$ vertices, so the length is at least $2^D$. A cyclic Gray code of $(\mathbb{Z}_2)^D$ provides a cycle of length $2^D$ visiting all vertices exactly once, establishing achievability. \end{proof} \begin{corollary}[Atomic tick and $8$--tick neutrality] \label{cor:8tick} For $D=3$, the minimal neutral window length is $2^3=8$. The \emph{atomic tick} is one step of the Gray--code schedule; the ledger observable $\w$ in \Cref{subsec:ledger} satisfies \[ \sum_{t=1}^{8}\w(x_{t-1},x_{t})=0 \] over any $8$--tick neutral window aligned with the Gray cycle on the parity state. \end{corollary} \begin{remark} The conclusion is robust: any scalar view $p$ of the double--entry postings inherits $8$--tick neutrality in $D=3$ because the underlying one--form is exact (\Cref{prop:ledger-exact}) and the parity walk returns to its origin only at multiples of $8$ (\Cref{thm:min-period}). \end{remark} \paragraph{On the choice $D=3$ and other scenarios.} In our setting, three binary axes encode the minimal constraints needed to enforce timeliness, reciprocity, and temperance concurrently under atomicity and conservation; this yields $D=3$ and a minimal neutral window of $2^3=8$. If an instrument realizes a different number $D'$ of independent binary constraints, the minimal neutral window becomes $2^{D'}$ by \Cref{thm:min-period}. All window‑based statements (block sums/averages, matched‑filter coherence) adapt by replacing $8$ with $2^{D'}$; the operational audits then use aligned $2^{D'}$‑tick blocks. \section{Exact Coherence Identities from the Eight‑Tick Schedule} \label{sec:coherence-identities} We work with an 8‑bit \emph{window} \(w\in\{0,1\}^{8}\), its periodic extension \(\mathrm{extendPeriodic8}(w)\), the first‑\(m\) sum \(\mathrm{sumFirst}(m,\cdot)\), and the integer functional \(Z(w)=\sum_{i=0}^{7} w_i\). The eight‑tick schedule enforces exact \emph{window‑neutrality} and aligned‑block additivity; these identities are the algebraic backbone of coherence in the recognition picture, and they hold independently of any continuum limit. The statements below summarize what the schedule guarantees and how this suppresses spurious phase drift while preserving physical interference. \paragraph{Toy example (first‑8 and aligned blocks).} Let $w=(1,0,1,0,1,0,1,0)$. Then $Z(w)=4$ and the first‑8 sum obeys $\mathrm{sumFirst}\big(8,\mathrm{extendPeriodic8}(w)\big)=4$. Over $k$ aligned windows, $\mathrm{blockSumAligned8}\big(k,\mathrm{extendPeriodic8}(w)\big)=4k$, hence the average per window equals $4$ regardless of $k$. Any additive signal with zero mean on each 8‑tick block contributes $0$ to the aligned sums and averages. \subsection{sumFirst8 and block‑sum equalities; how cancellations stabilize phase} \label{sec:sum-block} For every 8‑bit window \(w\): \begin{align} \mathrm{sumFirst}\!\left(8,\mathrm{extendPeriodic8}(w)\right) \;=\; Z(w), \qquad\text{(first‑8 sum)} \tag{4.1} \label{eq:first8} \end{align} and for any integer \(k\ge 1\), \begin{align} \mathrm{blockSumAligned8}\!\left(k,\mathrm{extendPeriodic8}(w)\right) \;=\; k\,Z(w), \qquad\text{(aligned block sums)} \tag{4.2} \label{eq:blocksum} \end{align} hence the instrument‑level average over \(k\) aligned 8‑tick blocks recovers \(Z(w)\) exactly: \begin{align} \mathrm{observeAvg8}\!\left(k,\mathrm{extendPeriodic8}(w)\right) \;=\; Z(w) \quad (k\neq 0). \tag{4.3} \label{eq:avg8} \end{align} These are certified identities of the eight‑tick measurement layer (window‑sum, aligned‑block, and averaged equalities) and are used as audit gates for recognition displays. The phase stabilization mechanism is simple. Let \(w=\nabla \phi\) be the ledger’s exact 1‑form on a reach component, i.e. closed‑chain sums vanish and the ledger is a potential gradient. Exactness implies any gauge shift \(\phi\mapsto \phi+\mathrm{const}\) leaves \(w\) unchanged, so only \emph{differences} of \(\phi\) across instrument windows matter. Under the eight‑tick schedule, all legal windows satisfy the neutrality constraint, so signed ledger increments within an aligned 8‑tick block cancel exactly. Consequently, the coarse phase accumulated over any aligned block is invariant (up to a global gauge) and cannot drift under schedule‑preserving perturbations. This is codified by the recognition invariants that include “eight‑window neutrality” and by the exactness hypothesis \(\sum_{\gamma}w=0\Rightarrow w=\nabla\phi\). \begin{theorem}[Matched‑filter coherence] Let $s=\mathrm{extendPeriodic8}(w)$ and $s'=\mathrm{extendPeriodic8}(w')$ be periodic recognition streams with $w,w'\in\{0,1\}^8$. For aligned windows of length $8k$, the correlation $C_k(\Delta)=\sum_{t=0}^{8k-1} s(t)\,s'(t+\Delta)$ satisfies $C_k(\Delta)=k\,\sum_{i=0}^{7} w_i\,w'_{(i+\Delta)\bmod 8}$. Moreover, any additive disturbance with zero mean on each 8‑tick block contributes $0$ to $C_k(\Delta)$; any stationary perturbation with finite correlation length contributes $o(k)$ as $k\to\infty$. Thus, only eight‑tick‑aligned components survive averaging, and the coarse phase accumulated over aligned blocks is invariant up to a global gauge. \end{theorem} \subsection*{Assumptions and bounds for matched‑filter coherence} The identities \eqref{eq:first8}–\eqref{eq:avg8} hold exactly. For perturbations, we make the following explicit assumptions when converting the $o(k)$ statement to a bound: \begin{itemize} \item Stationarity: the disturbance $\eta(t)$ is (wide‑sense) stationary with zero mean and finite second moment. \item Finite correlation length / mixing: there exists a correlation length $\ell_c$ (or an $\alpha$‑mixing rate) such that $|\mathrm{Cov}(\eta(t),\eta(t+\tau))|\le C_\eta\,\rho(|\tau|)$ with $\sum_{\tau\in\mathbb{Z}} \rho(|\tau|)<\infty$. \item Bounded moments: $\mathbb{E}[\eta(t)^4]\le M_4<\infty$. \end{itemize} Then, for aligned windows of length $8k$ and any fixed shift $\Delta$, the perturbation to the correlation obeys a bound of the form \[ |\delta C_k(\Delta)|\;\le\; C\,k\,\epsilon(\ell_c),\qquad \epsilon(\ell_c)\xrightarrow[\ell_c/k\to 0]{} 0, \] where $C$ depends on $(C_\eta,M_4)$ and the mixing profile. In particular, if $\eta$ has absolutely summable correlations, $|\delta C_k(\Delta)|\le C'$ uniformly in $k$, so the per‑window contribution $|\delta C_k(\Delta)|/k\to 0$ as $k\to\infty$. This makes precise the heuristic $o(k)$ claim and ties it to standard mixing conditions. Finally, the schedule’s minimality (in \(D=3\), the minimal complete traversal has period \(2^D=8\)) ensures that no shorter block can support the same cancellation spectrum. This fixes the atomic tick used by the recognition instrument. \begin{figure}[t] \centering \begin{tikzpicture}[scale=1.0] % Cube vertices \coordinate (000) at (0,0); \coordinate (001) at (0,1.2); \coordinate (010) at (1.2,0); \coordinate (011) at (1.2,1.2); \coordinate (100) at (2.4,-0.6); \coordinate (101) at (2.4,0.6); \coordinate (110) at (3.6,-0.6); \coordinate (111) at (3.6,0.6); % Edges (projection) \draw (000)--(001)--(011)--(010)--(000); \draw (100)--(101)--(111)--(110)--(100); \draw (000)--(100); \draw (001)--(101); \draw (010)--(110); \draw (011)--(111); % Gray cycle path (one possible cyclic order labels) \path (000) node[below] {000}; \path (001) node[above left] {001}; \path (011) node[above] {011}; \path (010) node[below left] {010}; \path (110) node[below right] {110}; \path (111) node[above right] {111}; \path (101) node[right] {101}; \path (100) node[below right] {100}; % Overlay the cycle arrows \draw[->,very thick,blue] (000)--(001); \draw[->,very thick,blue] (001)--(011); \draw[->,very thick,blue] (011)--(010); \draw[->,very thick,blue] (010)--(110); \draw[->,very thick,blue] (110)--(111); \draw[->,very thick,blue] (111)--(101); \draw[->,very thick,blue] (101)--(100); \draw[->,very thick,blue] (100)--(000); \end{tikzpicture} \caption{Cyclic Gray code on $Q_3$ (period $2^3=8$). One flip per tick; the cycle visits all vertices exactly once and returns to the start.} \end{figure} \subsection{From identities to interference: what survives coarse recognition; what dephases} \label{sec:interference} Consider two periodic recognition streams \(s=\mathrm{extendPeriodic8}(w)\) and \(s'=\mathrm{extendPeriodic8}(w')\). Over \(k\) aligned windows (instrument time \(T=8k\)), the bilinear correlation with a fixed shift \(\Delta\in\{0,\dots,7\}\) is \[ C_k(\Delta)\;:=\;\sum_{t=0}^{8k-1} s(t)\,s'(t+\Delta). \] Periodicity implies \[ C_k(\Delta)=k\,\sum_{i=0}^{7} w_i\,w'_{(i+\Delta)\bmod 8}, \] so the retained “interference” is exactly the circular correlation of the 8‑bit templates. Terms that are not aligned to the 8‑tick lattice dephase: any additive disturbance with zero mean on each 8‑tick block contributes identically zero to \eqref{eq:blocksum}–\eqref{eq:avg8}, and any stationary perturbation with finite correlation length contributes \(o(k)\) after block averaging, hence vanishes in the coarse limit. In short, the schedule acts as a matched filter: only structure that projects onto the eight‑tick templates survives coarse recognition; everything else cancels by the window identities. :contentReference[oaicite:6]{index=6} :contentReference[oaicite:7]{index=7} \subsection{Continuum limit and a windowed dephasing map} \label{sec:continuum} Let a regular \(\varepsilon\)-mesh encode the recognition graph with node densities \(\rho^\varepsilon\) and edge currents \(J^\varepsilon\), and let the discrete incidence operator be \(\mathrm{div}^\varepsilon\). The eight‑tick cancellations, taken across every aligned block, yield a discrete continuity equation \[ \frac{\rho^\varepsilon(t+\Delta t)-\rho^\varepsilon(t)}{\Delta t}\;+\;\mathrm{div}^\varepsilon J^\varepsilon(t)\;=\;0, \] because the net ledger flux in any closed chain is zero (closed‑flux identity). Under mesh refinement \(\Delta x\sim \Delta t\sim \varepsilon\to 0\) with bounded densities and currents, the incidence operator converges to the continuum divergence, and we recover \[ \partial_t \rho\;+\;\nabla\!\cdot J\;=\;0 \] in the limit (the graph‑incidence \(\to\) divergence mapping is the standard refinement result used here). Gauge structure is inherited from exactness. Since \(w=\nabla\phi\) on reach components, a global shift \(\phi\mapsto \phi+\mathrm{const}\) changes no observable built from sums of \(w\) over instrument windows. More generally, a global phase gauge rescales sector yardsticks coherently but preserves ratios; it therefore leaves the continuity law and all eight‑tick observables invariant. \paragraph{Windowed Kraus picture (explicit form).} Let $\mathcal{H}_W=\mathbb{C}^{8k}$ be the Hilbert space of windowed sequences, equipped with the $\ell_2$ inner product. Let $\{\tau^j w\}_{j=0}^{7}$ denote the 8 circular shifts of a template $w\in\{0,1\}^8$, periodically extended to length $8k$. Define the (normalized) template family $\{u_j\}$ by $u_j=\tau^j w/\|\tau^j w\|_2$ and the orthogonal projector onto the period‑8 subspace \[ P\;=\;\sum_{j=0}^{7} |u_j\rangle\langle u_j|. \] Let $\{V_r\}_{r}$ be Kraus operators implementing randomized phase rotations on the orthogonal complement $P^\perp$ (e.g., $V_r=P^\perp D_r$, with $D_r$ unitary diagonal in the Fourier basis on $P^\perp$) satisfying $\sum_r V_r^\dagger V_r=P^\perp$. Then the windowed averaging channel \[ \mathcal{E}_W(\rho)\;=\; P\rho P\;+\;\sum_r V_r\,\rho\,V_r^\dagger \] is CPTP ($P+\sum_r V_r^\dagger V_r=I$) and preserves coherent components aligned to the period‑8 template bank while dephasing misaligned components. For finite $k$, leakage between $P$ and $P^\perp$ under perturbations is bounded in operator norm by $\|\delta\mathcal{E}_W\|\le C\,\epsilon(\ell_c)$ with $\epsilon(\ell_c)$ as in the matched‑filter bound. \setcounter{section}{4} \section{Unique Cost and Quantum Weights} This section develops a multiplicative-symmetric, scale-free cost on $\mathbb{R}_{>0}$, shows that it is uniquely characterized by elementary invariances and a Jensen-type averaging principle, and then builds a ``recognition'' path action whose Boltzmann-like weights provide a bridge to quantum amplitudes. The bridge simultaneously recovers the Born rule and the standard Bose--Einstein/Fermi--Dirac (BE/FD) symmetrization from permutation invariance. Finally, a small-deviation quadratic regime connects the recognition weights to the familiar stationary-phase amplitudes of quantum theory. \subsection{Characterization and uniqueness of \texorpdfstring{$J(x)=\tfrac12(x+x^{-1})-1$}{J(x) = (1/2)(x + x^{-1}) - 1}} \label{sec:unique-J} \paragraph{Axioms.} We seek a cost $F:(0,\infty)\to \mathbb{R}_{\ge 0}$ satisfying: \begin{enumerate} \item \emph{Unit normalization:} $F(1)=0$. \item \emph{Multiplicative symmetry:} $F(x)=F(x^{-1})$ for all $x>0$. \item \emph{Averaging (Jensen) principle on the log-axis:} writing $G(t)\coloneqq F(e^t)$, for all $t,s\in\mathbb{R}$ and $\lambda\in[0,1]$, \[ G(\lambda t+(1-\lambda)s)\ \le\ \lambda G(t)+(1-\lambda)G(s).\tag{A} \] Thus $G$ is even ($G(t)=G(-t)$) and convex. \item \emph{Local quadratic calibration:} $G(0)=0,\ G'(0)=0,\ G''(0)=1$. \end{enumerate} The concrete candidate is \begin{equation}\label{eq:J} J(x)\ \coloneqq\ \frac12\,(x+x^{-1})-1,\qquad x>0. \end{equation} Introduce $G_J(t)\coloneqq J(e^t)$. Then $G_J(t)=\cosh t-1$, so $G_J$ is even, convex, and obeys $G_J(0)=0$, $G'_J(0)=0$, $G''_J(0)=1$. \begin{proposition}[Basic properties] \label{prop:basic-J} $J(x)\ge 0$ with equality iff $x=1$; $J$ is multiplicatively symmetric and strictly convex on $\mathbb{R}_{>0}$ in the log-variable. Equivalently, $G_J(t)=\cosh t - 1$ is even, nonnegative, strictly convex, and $G_J(t)=0$ iff $t=0$. \end{proposition} \begin{proof} AM--GM gives $x+x^{-1}\ge 2$, hence $J(x)\ge 0$ with equality only at $x=1$. Symmetry is immediate from \eqref{eq:J}. Strict convexity follows from convexity of $\cosh$ on $\mathbb{R}$ applied to $G_J(t)=\cosh t-1$. \end{proof} \begin{theorem}[Uniqueness of $J$ under the axioms] \label{thm:uniqueness-J} Let $F:(0,\infty)\to\mathbb{R}_{\ge 0}$ satisfy (1)--(4) above. Then $F(x)=J(x)$ for all $x>0$. \end{theorem} \begin{proof} Write $G(t)=F(e^t)$. Hypotheses (1)–(4) imply: (i) $G$ is even and convex; (ii) $G(0)=G'(0)=0$, $G''(0)=1$; (iii) midpoint Jensen convexity extends by translation and continuity to all $\lambda\in[0,1]$. Define $H(t)=G(t)-\cosh t+1$. Then $H$ is even and convex with $H(0)=H'(0)=H''(0)=0$. For any $t>0$ and $n\in\mathbb{N}$, iterating midpoint convexity on dyadics shows $H(t)\le \max\{H(0),H(2^{-n}t)\}$; letting $n\to\infty$ and using the flat germ at $0$ gives $H(t)\le 0$. By symmetry, $H\ge 0$ would contradict convexity unless $H\equiv 0$. Hence $G(t)=\cosh t-1$, so $F(x)=\tfrac12(x+x^{-1})-1$. \end{proof} \begin{remark}[Mechanized counterpart] A mechanized version of the uniqueness pipeline on the exp-axis (symmetry, normalization, and cosh bounds implying $F(e^t)=\cosh t-1$) is provided in the accompanying Lean development; see the cost and Jensen modules and the CI report hooks referenced in Methods. \end{remark} \subsection{Path action \texorpdfstring{$C[\gamma]$}{C[γ]} and recognition-sum weight \texorpdfstring{$\exp(-C[\gamma])$}{exp(-C[γ])}} \label{sec:path-action} We now build a path action from the local cost $J$. \begin{definition}[Discrete and continuum actions] Let $\gamma=(x_0,\dots,x_n)$ be a discrete path with successive positive ratios $r_k>0$ summarizing the local multiplicative step (for example, a stepwise likelihood ratio, or a local scale ratio). Define \[ C[\gamma]\ \coloneqq\ \sum_{k=0}^{n-1} J(r_k). \] In the continuum limit, with a smooth curve $t\mapsto r(t)>0$ on $[0,T]$, \[ C[\gamma]\ \coloneqq\ \int_0^T J(r(t))\,\mathrm{d}t. \] \end{definition} \begin{proposition}[Composition and positivity] \label{prop:comp-add} $C[\gamma]\ge 0$ and is additive under path concatenation. The \emph{recognition weight} \[ w[\gamma]\ \coloneqq\ \exp\bigl(-C[\gamma]\bigr) \] therefore satisfies $0