%===================================================================== % En02-A % Finite-Resource Geometry for the Enter Calculus % Cut Monotones, Certificate Costs, and Conditional Born Rigidity % % Self-contained mathematical revision of: % 085WvIp1v2_DiscQuantSci.tex % % Compile: % pdflatex En02-A.tex % pdflatex En02-A.tex % % The Python appendix is a standalone listing, not a compilation % dependency. No Python execution or local LaTeX compilation is % asserted by this source file. %===================================================================== \documentclass[11pt]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[margin=1.02in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable} \usepackage{enumitem} \usepackage{xcolor} \usepackage{fvextra} \usepackage[hidelinks]{hyperref} \setcounter{secnumdepth}{3} \setcounter{tocdepth}{2} \numberwithin{equation}{section} \allowdisplaybreaks \setlength{\emergencystretch}{2em} %--------------------------------------------------------------------- % Environments %--------------------------------------------------------------------- \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \newtheorem{openproblem}[theorem]{Open Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{convention}[theorem]{Convention} \DefineVerbatimEnvironment{CodeBlock}{Verbatim}{ fontsize=\scriptsize, breaklines=true, breakanywhere=true, frame=single, framesep=3mm, numbers=left, numbersep=6pt, tabsize=4 } \DefineVerbatimEnvironment{ShellBlock}{Verbatim}{ fontsize=\small, breaklines=true, breakanywhere=true, frame=single, framesep=3mm } %--------------------------------------------------------------------- % Notation %--------------------------------------------------------------------- \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \newcommand{\floor}[1]{\left\lfloor #1\right\rfloor} \newcommand{\code}[1]{\texttt{\detokenize{#1}}} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Choi}{Choi} \DeclareMathOperator{\Born}{Born} \DeclareMathOperator{\Reach}{Reach} \DeclareMathOperator{\Obs}{Obs} \newcommand{\Q}{\mathsf Q} \newcommand{\Bit}{\mathsf{Bit}} \newcommand{\Unit}{\mathbf 1} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\Activate}{\mathsf{Activate}} \newcommand{\Send}{\mathsf{Send}} \newcommand{\Recv}{\mathsf{Recv}} \newcommand{\Strategy}{\mathsf S} \newcommand{\Prob}{\mathsf P} \newcommand{\Update}{\mathsf U} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\Eact}{E_{\mathrm{act}}} \newcommand{\Llat}{L_{\mathrm{lat}}} \newcommand{\FG}{\mathsf{FG}} \newcommand{\Rfg}{\mathcal R_{\mathrm{fg}}} \newcommand{\Rcert}{\mathcal R_{\mathrm{cert}}} \newcommand{\Aclass}{\mathcal A} \newcommand{\Ray}{\mathbb P} \newcommand{\DFh}{\mathrm{DF}_{0\). \end{enumerate} These data depend only on the current ray and measured names. They are equivariant under simultaneous relabeling of names, tensor factors, and outcome coordinates. All finite-register unitaries and fresh \(\ket0\) allocation have their stipulated linear-algebraic meanings. Allocation is allowed only when the resulting live occupancy is at most \(N\). No hidden memory, silent quantum deletion, or apparatus-dependent measurement parameter is included. A finite experimental program uses allocation, gates, destructive readout, finite classical control, and return. Every quantum name is consumed or returned. Local programs exchange neither quantum names nor classical messages. Single programs use their prescribed evaluation order; bipartite experiments permit interleavings of their local orders. For \(S=\varnothing\), readout has its unique unit outcome and leaves the state unchanged. \end{assumption} This is the finite restriction of the relevant structural interfaces of manuscript I. It is not obtained from the operational requirements. \subsection{The five capacity-restricted requirements} \begin{definition}[Operational admissibility] \label{def:admissible} A strategy in Assumption~\ref{ass:class} is \emph{operationally admissible at capacity \(N\)} if it satisfies the following requirements on every specified experiment whose entire execution remains within \(N\) live qubits. Write \(\Strategy\in\Aclass_N\). \paragraph{\(\textnormal{(C)}_N\): composability.} For disjoint \(S,T\subseteq L\), joint readout and both sequential orders have the same ordered joint law. Thus \begin{equation} \Prob^{S\cup T}_\psi(\kappa,\lambda) = \Prob^S_\psi(\kappa)\, \Prob^T_{\Update^S_\psi(\kappa)}(\lambda) \label{eq:C} \end{equation} on positive first branches, with value zero on a zero first branch; the corresponding equation holds in the reverse order. \paragraph{\(\textnormal{(NS)}_N\): internal no-signaling.} For every bipartition, the terminal classical law of one local program is independent of the other local program and of the interleaving. Locally allocated qubits and returned quantum spectators are included. There is no exchange of names or messages. \paragraph{\(\textnormal{(N)}_N\): nullity.} \begin{equation} M^S_\kappa\psi=0 \quad\Longrightarrow\quad \Prob^S_\psi(\kappa)=0. \label{eq:nullity} \end{equation} \paragraph{\(\textnormal{(K)}_N\): continuity.} For each fixed \(L,S\), the map \[ [\psi]\longmapsto\Prob^S_\psi \] is continuous in total variation. No continuity of the unknown update is presumed. \paragraph{\(\textnormal{(R)}_N\): typed records.} Let \(L=S\sqcup B\), and let \(u\) consume all of \(B\), returning classical output and not using the first reported outcome. Compare: \begin{enumerate}[label=\textup{(\alph*)}] \item read \(S\), then run \(u\) on the residual \(B\), and report the ordered classical outputs; \item allocate a fresh register \(R\) of \(\abs S\) zero qubits, copy \(S\) computationally to \(R\) by bitwise CNOT, run the same \(u\) on \(B\), then read \(R\), report the same ordered classical outputs, and return the original \(S\). \end{enumerate} Their classical observation laws must agree whenever both complete executions fit the budget. \end{definition} The second program retains the original \(S\) as quantum output. When \(B\ne\varnothing\), a continuation that omits \(B\) is not a legal record instance. This is the typed interface of I's \code{def:record}, restated rather than replaced by silent discard \cite{I}. \begin{lemma}[Born inhabitation] \label{lem:born-inhabitation} The strategy \[ \Prob^S_\psi(\kappa)=b^S_\psi(\kappa), \qquad \Update^S_\psi(\kappa)=[\psi^S_{\mathrm L,\kappa}] \] belongs to \(\Aclass_N\) for every \(N\). \end{lemma} \begin{proof} Normalization follows from \(\sum_\kappa(M^S_\kappa)^\dagger M^S_\kappa=I\). Disjoint destructive components commute after identifying surviving factors, proving composability. Nullity and continuity follow directly from squared norms. For a finite local program, compose allocation isometries, unitaries, and destructive measurement bras along each history. Backward induction on the outcome tree gives a complete family of path maps: \(\sum_hK_h^\dagger K_h=I\). For bipartite histories the maps factor as \(K_a\otimes L_b\). Summing over the remote histories gives \[ \sum_b \bra\psi K_a^\dagger K_a\otimes L_b^\dagger L_b\ket\psi = \bra\psi K_a^\dagger K_a\otimes I\ket\psi. \] This proves the specified no-signaling requirement. For records write \(\psi=\sum_\kappa\ket\kappa_S\ket{\xi_\kappa}_B\). A continuation history \(h\), with path map \(K_h\), has joint weight \(\norm{K_h\xi_\kappa}^2\) in the direct program. The copied program has pre-readout vector \[ \sum_\kappa \ket\kappa_S K_h\ket{\xi_\kappa}_B\ket\kappa_R, \] and record readout gives the same weight. Summing histories proves record equality. All compared paths obey the capacity restriction. \end{proof} \subsection{Locality and record marginalization} \begin{lemma}[Marginals and remote invariance] \label{lem:remote} For \(\Strategy\in\Aclass_N\): \begin{enumerate}[label=\textup{(\roman*)}] \item composability implies \[ \Prob^S_\psi = \operatorname{marg}_S\Prob^{S\cup T}_\psi; \] \item applying a unitary outside \(S\) does not change \(\Prob^S_\psi\); \item adjoining fresh zero-state spectators outside \(S\) does not change \(\Prob^S_\psi\), provided the enlarged configuration fits. \end{enumerate} \end{lemma} \begin{proof} For (i), sum \eqref{eq:C} over \(\lambda\) and use normalization. For (ii), compare the local readout of \(S\) against a remote gate followed by return, and against remote return alone. For (iii), replace that remote gate experiment by allocation followed by return. These are capacity-expressible no-signaling comparisons. \end{proof} \begin{lemma}[Budgeted purification locality] \label{lem:locality} Let two pure configurations have the same reduced state on a register \(S\). If padding both environments to a common size fits capacity \(N\), their \(S\)-laws agree. \end{lemma} \begin{proof} Pad the smaller environment by zero spectators. Two purifications of the same reduced state with equal environment dimension are related by an environment unitary: choose matching Schmidt vectors on \(S\), map one orthonormal environment family to the other, and extend that isometry to a unitary. Apply Lemma~\ref{lem:remote}. \end{proof} \begin{lemma}[Record marginal identity] \label{lem:record-marginal} Suppose \(n=\abs L\), \(s=\abs S\), and \(n+s\le N\). Define the copied state \begin{equation} \mathcal C_S\psi = \sum_\kappa \ket\kappa_S\ket{\xi_\kappa}_B\ket\kappa_R, \qquad \psi=\sum_\kappa\ket\kappa_S\ket{\xi_\kappa}_B. \label{eq:copy} \end{equation} Then \begin{equation} \Prob^S_\psi=\Prob^R_{\mathcal C_S\psi}, \qquad \rho_R = \sum_\kappa\norm{\xi_\kappa}^2\ketbra{\kappa}{\kappa}. \label{eq:record-marginal} \end{equation} \end{lemma} \begin{proof} Use the legal continuation \(u=\Enter(B)\), retaining all its classical outputs. If \(B\) is empty, use the unit continuation. The direct joint law has \(S\)-marginal \(\Prob^S_\psi\) by composability. In the deferred program, the marginal law of \(R\) equals its law before the remote readout of \(B\), by no-signaling. Record equality identifies these marginals. The density-operator formula follows by tracing the copied vector: the original computational labels in \(S\) are orthogonal. \end{proof} \begin{remark}[What this improves, and what it does not] \label{rem:record-budget} Lemma~\ref{lem:record-marginal} uses \(s\) fresh record qubits and requires \(n+s\le N\). I's budgeted stability theorem uses whole-register records, including further records of intermediate deferred states; its displayed conditions include \(2n\le N\) and \(2(n+s)\le N\) \cite[label \texttt{thm:stability}]{I}. The budgets price different comparisons. The present exact marginal argument does not remove any register from that approximate continuation theorem, nor does it supply an approximate update estimate. \end{remark} \subsection{Uniform correlations without an extra outcome-covariance axiom} \begin{lemma}[Uniform correlations inside the budget] \label{lem:uniformity} Suppose \[ \psi= \frac1{\sqrt d} \sum_{k\in K}e^{i\theta_k}\ket k_S\ket{\eta_k}_B \] fits capacity \(N\), where the strings \(k\) are distinct and the computational strings \(\eta_k\) are distinct. Then \(\Prob^S_\psi\) is uniform on \(K\). \end{lemma} \begin{proof} A diagonal unitary on \(B\) removes the phases without changing the \(S\)-law. Write \(p=\Prob^S_\psi\), \(q=\Prob^B_\psi\), and \(\nu=\Prob^{S\cup B}_\psi\). Nullity restricts \(\nu\) to \((k,\eta_k)\); composability gives \[ p(k)=\nu(k,\eta_k)=q(\eta_k). \] For a permutation \(\pi\) of \(K\), apply the basis permutation on \(S\) sending \(k\) to \(\pi(k)\). The resulting vector can also be obtained by a unitary on \(B\) sending \(\eta_k\) to \(\eta_{\pi^{-1}(k)}\). Remote invariance therefore leaves the \(S\)-law equal to \(p\) and the \(B\)-law equal to \(q\). Nullity for the new pairing gives \[ p(m)=q(\eta_{\pi^{-1}(m)})=p(\pi^{-1}(m)). \] Thus every permutation preserves \(p\), and normalization makes it uniform. No extra qubits are allocated. \end{proof} This is the finite version of I's \code{thm:uniformity}. It derives the required symmetry; it does not assume covariance of unknown probabilities under arbitrary computational-basis permutations \cite{I}. %===================================================================== \section{Resource geometry, ownership, and legal updates} \label{sec:geometry} %===================================================================== \subsection{Identified links and capacity} \begin{definition}[Resource geometry] \label{def:geometry} A geometry is \[ g=(V,\mathrm{cap},\Eact,\Llat), \] where: \begin{enumerate}[label=\textup{(\roman*)}] \item \(V\) is a fixed finite set of regions; \item \(\mathrm{cap}:V\to\mathbb N\) is a capacity function; \item \(\Llat\) is a finite family of identified latent links \((\ell,\{u,v\})\), with distinct endpoints in \(V\); \item \(\Eact\) is a finite family of identified directed edges \((\ell,u,v)\). \end{enumerate} Identifiers are unique and no identifier occurs in both components. The directed graph underlying \(\Eact\) is acyclic. Distinct identified edges may have the same endpoints. Set \[ \mu(g)=\sum_{v\in V}\mathrm{cap}(v) +\abs{\Llat}+\abs{\Eact}. \] \end{definition} Identified parallel edges are retained because each activation consumes one identified token and creates one identified edge. If active edges were instead an unlabelled set of endpoint pairs, activating a second parallel token would not preserve the scalar count. A configuration carries: \[ [g,\psi,t,\lambda], \] where \(\lambda:L\to V\) labels live names and \[ o(v)=\#\set{q\in L:\lambda(q)=v} \le\mathrm{cap}(v). \] Measurement removes names and their tensor factors. It does not remove the region containing them. Permanent capacity retirement is a separate update. \subsection{Ownership of cells, not merely endpoints} \begin{definition}[Geometric ownership and footprints] \label{def:footprint} Mutable geometric cells comprise capacity cells and identified link cells. Their permissions are linearly owned. An endpoint region is not, by itself, a permission to modify every incident link. A transition has a write footprint \(W\), a read footprint \(D\), and a guard. Its output on \(W\) depends only on data in \(D\cup W\), and it fixes all cells outside \(W\). Two transitions are \emph{independent} if \[ W_1\cap(D_2\cup W_2)= W_2\cap(D_1\cup W_1)=\varnothing \] and each guard remains true after the other transition. \end{definition} An acyclicity check can read nonlocal graph data. Consequently, disjoint endpoint sets or disjoint quantum names do not automatically establish geometric independence. \begin{definition}[Activation authority] \label{def:authority} Each latent link has one linear activation authority \(\Send_\ell@u\), with partner receive capability \(\Recv_\ell@v\). Only the authority can activate the link, from \(u\) to \(v\). The orientation and authority are preparation data. The receive capability is not a second activation authority. Activation consumes \(\Send_\ell\), replaces the latent link cell by the active edge with the same identifier, and is legal only if the resulting active graph is acyclic. \end{definition} This avoids a two-party race in which two independently owned ``halves'' both attempt to consume the same global latent object. A different two-sided activation protocol would require its own synchronization and progress proof. \subsection{The update grammar} \begin{definition}[Legal primitive moves] \label{def:moves} The only geometric updates are finite compositions of: \begin{enumerate}[label=\textnormal{(m\arabic*)}] \item retire one capacity unit at an owned region, with \(\mathrm{cap}(v)>o(v)\); \item delete one owned identified active edge; \item delete one owned identified latent link; \item activate one owned latent link as in Definition~\ref{def:authority}. \end{enumerate} No move creates a new identifier, changes endpoints, reverses an existing edge, or creates an edge without consuming its latent link. \end{definition} The strict inequality in (m1) protects occupied slots. If measurement is followed by permanent retirement, name deletion is performed first and the resulting free unit is then retired. \begin{proposition}[Preservation and the geometric diamond] \label{prop:diamond} Legal primitive moves preserve geometric well-formedness and the occupancy bound. Independent geometric transitions commute. For the specified Born quantum semantics, a pair of name-disjoint quantum redexes equipped with independent geometric transitions has the concrete probabilistic diamond. \end{proposition} \begin{proof} Capacity retirement leaves capacity at least occupancy. Deletion preserves acyclicity. Activation preserves it by its guard and preserves identifier uniqueness by moving, rather than copying, the link identifier. For independent transitions, neither writes data used by the other, and neither invalidates the other's guard. Each therefore produces the same update on its own write footprint in either order. Outside their union both act identically. The Born quantum diamond follows from the commuting destructive components, disjoint gates, and fresh-name identification. Its joint branch weights are the same in both orders. Combine that equality with the geometric commutation just proved. \end{proof} \begin{remark}[Inherited and new parts] Manuscript I's \code{lem:diamond} concerns quantum-name disjointness in the specified concrete calculus. The geometric part of Proposition~\ref{prop:diamond} is new and needs the footprint and guard conditions above. It is not a theorem about arbitrary unknown updates merely because their names are disjoint. \end{remark} \begin{example}[Local writes need not commute] \label{ex:read-conflict} Let \(x,y\in\{0,1\}\) initially both be zero. One map writes \(x:=1-y\) and leaves \(y\) fixed; the other writes \(y:=1-x\) and leaves \(x\) fixed. Their write footprints are disjoint, but both read the other's cell. The two orders give \((1,0)\) and \((0,1)\). Similarly, a collection of individually acyclic edge insertions may form a cycle when combined. A global acyclicity guard is therefore not automatically stable under disjoint writes. These are new counterexamples to insufficient geometric hypotheses, not inherited quantum counterexamples. \end{example} %===================================================================== \section{Cut monotones and decreasing potential reachability} \label{sec:cuts} %===================================================================== For a cut \(X\mid Y\), with \(Y=V\setminus X\), define \[ \mu_{X\mid Y}(g) = \sum_{v\in X}\mathrm{cap}(v) +\#\set{\text{latent links crossing }X\mid Y} +\#\set{\text{active edges directed }X\to Y}, \] and \[ \sigma_{X\mid Y}(g) = \#\set{\text{latent links crossing }X\mid Y} +\#\set{\text{active edges crossing in either direction}}. \] All counts include identifiers and hence multiplicity. \begin{example}[Scalar accounting is insufficient] Deleting a crossing token \(\{A,B\}\) and an internal token \(\{B,B'\}\), then creating edges \(B\to A\) and \(B'\to A\), preserves the scalar count of links. It increases the number of crossing links from one to two. The grammar excludes the second edge because no consumed token had endpoints \(\{B',A\}\). \end{example} \begin{theorem}[E: cut monotonicity and exact conversion] \label{thm:E} Under Definition~\ref{def:moves}: \begin{enumerate}[label=\textup{(\roman*)}] \item \(\mu\) decreases by one under (m1)--(m3) and is preserved by (m4); \item for every cut, \(\mu_{X\mid Y}\) and \(\sigma_{X\mid Y}\) are non-increasing; \item activation preserves \(\sigma_{X\mid Y}\) for every cut: one crossing token becomes one crossing identified edge; \item along a run, the number of cross-cut activations is at most the initial number of crossing latent links, and therefore at most \(\sigma_{X\mid Y}(g_0)\). \end{enumerate} \end{theorem} \begin{proof} Use \(\mathbf 1_A\) for the indicator of a condition \(A\). For a deleted or activated link with endpoints \(u,v\), let \(c=\mathbf 1_{\text{endpoints cross}}\). The changes are: \[ \begin{array}{c|c|c|c} \text{move}&\Delta\mu&\Delta\mu_{X\mid Y}& \Delta\sigma_{X\mid Y}\\ \hline \text{retire capacity at }v &-1&-\mathbf 1_{v\in X}&0\\ \text{delete }u\to v &-1&-\mathbf 1_{u\in X,v\in Y}&-c\\ \text{delete latent link} &-1&-c&-c\\ \text{activate }u\to v &0&-c+\mathbf 1_{u\in X,v\in Y}&0 \end{array} \] Every entry in the cut columns is non-positive. The activation row counts an identified edge, so it remains valid for parallel endpoint pairs. Finite composition proves (i)--(iii). No move adds a latent link. Each crossing latent identifier can be activated at most once. This proves (iv). \end{proof} The word ``conservation'' is appropriate for the latent-to-active conversion in (iii). The complete quantities are monotones, not constants along arbitrary runs. \begin{definition}[Potential reachability] \[ R(g)= \Reach\!\left( \Eact\ \cup\ \set{\text{both orientations of every latent link}} \right), \] including reflexive pairs and forgetting identifiers only when forming reachability. \end{definition} \begin{theorem}[F: potential coarsening and one-shot activation] \label{thm:F} For every legal run, \[ R(g_{t+1})\subseteq R(g_t), \qquad \Eact(g_t)\subseteq R(g_0) \] at the level of endpoint pairs. Each latent identifier activates at most once. If initially \(\sigma_{X\mid Y}=0\), no crossing active edge or latent link is ever reachable. \end{theorem} \begin{proof} Retiring capacity changes no reachability generator. Deleting structure removes generators. Activation replaces the two potential orientations of its latent link by one of those orientations. Thus its generator set only shrinks, and so does its reachability closure. An activated identifier never returns to the latent component. The final assertion follows either from Theorem~\ref{thm:E} or directly from the absence of crossing generators. \end{proof} \subsection{Observation boundaries and no-signaling} \begin{proposition}[Private observations before communication] \label{prop:private-ns} Consider two local quantum programs with disjoint quantum ownership. Suppose their geometric actions leave the other party's private readable cells fixed, and neither program inspects shared link status, uses a cross-party communication primitive, or has its local enablement changed by the other's geometric actions. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item the specified Born semantics has program-independent unconditioned remote classical laws; \item an operationally admissible unknown strategy has the same independence by \(\textnormal{(NS)}_N\); \item the remote private geometric observation is unchanged. \end{enumerate} \end{proposition} \begin{proof} Part (i) is the path-map marginalization in Lemma~\ref{lem:born-inhabitation}. Part (ii) is the operational requirement, applied to the stated local experiments. Part (iii) follows because no transition writes a remotely readable private cell or changes its local enablement. \end{proof} \begin{remark}[Activation is not literally invisible in every view] An agent's view that includes the actual latent/active status of an incident link changes when that link activates. Literal equality of such views is false. Only an observation interface that cannot inspect or exploit that shared status before the authorized communication step supports an invisibility claim. The cut theorem itself does not prove physical no-signaling, nor does edge counting determine a channel's information capacity. \end{remark} %===================================================================== \section{Capacity restriction and its monotonicity} \label{sec:capacity} %===================================================================== \begin{lemma}[Restriction in a fixed experimental library] \label{lem:restriction} If \(N\le N'\) and \(\Strategy\in\Aclass_{N'}\), restricting its laws and residuals to inputs and experiments of peak occupancy at most \(N\) gives a member of \(\Aclass_N\). \end{lemma} \begin{proof} Every capacity-\(N\) experiment is a capacity-\(N'\) experiment. Its state spaces, fixed gates, and observation interface are unchanged. Every required equation is therefore inherited. \end{proof} For multiple regions one can make the same statement with pointwise occupancy and effect bounds, provided the experimental library is fixed independently of the current active graph. \begin{remark}[No automatic closure for changing access libraries] Activation can enable a previously unavailable communication or gate interface. In such a graph-dependent language, pointwise capacity decrease does not imply that the new set of experiments is a subset of the old set. Run-invariant admissibility then needs a separate simulation or library-inclusion theorem. The present downward-closure claim is only the fixed-library claim of Lemma~\ref{lem:restriction}. \end{remark} A total capacity sum is also not sufficient for local allocation. A register needed in one region must fit its local capacity, unless an explicitly permitted distributed placement implements the construction. All forcing results below concern a single laboratory of capacity \(N\), with the coherent gates of Assumption~\ref{ass:class}. %===================================================================== \section{Exact fine-graining certificate cost and cheaper symmetry} \label{sec:forcing} %===================================================================== \subsection{Primitive multiplicities and the certificate family} \begin{definition}[Reduced rational law] \label{def:primitive} Let a probability law on \(s\) qubits have positive support \(K\subseteq\{0,1\}^s\). Write \[ p_k=\frac{m_k}{M}, \qquad m_k\in\mathbb N_{>0}, \quad M=\sum_{k\in K}m_k, \quad \gcd\set{m_k:k\in K}=1. \] Zero weights are omitted. This is the primitive integer representation. \end{definition} \begin{definition}[Decohered fine-graining certificates] \label{def:fg} An \(\FG\)-certificate for this law uses disjoint registers \(S,E',E''\), with \(\abs S=s\), and a state \[ \Phi = \frac1{\sqrt{M'}} \sum_{\alpha=1}^{M'} \ket{k(\alpha)}_S \ket{r(\alpha)}_{E'} \ket{e(\alpha)}_{E''}, \] such that: \begin{enumerate}[label=\textup{(\roman*)}] \item the labels \(r(\alpha)\) are pairwise distinct; \item the companions \((k(\alpha),e(\alpha))\) are pairwise distinct; \item the diagonal marginal on \(S\) is the target law. \end{enumerate} Its certificate cost is the number of qubits simultaneously hosting \(S,E',E''\). Let \(C_{\FG}(s,\vec m)\) be the minimum within this specified family. \end{definition} The family is intentionally narrower than all protocols capable of forcing a law. It is the register pattern of I's decohered fine-graining proof \cite[label \texttt{thm:fine-anchor}]{I}. \begin{theorem}[H: exact certificate cost] \label{thm:H} For primitive multiplicities, \begin{equation} C_{\FG}(s,\vec m) = s+\ceil{\log_2 M} +\ceil{\log_2\max_{k\in K}m_k}. \label{eq:fgcost} \end{equation} \end{theorem} \begin{proof} For the upper bound, choose \(a=\ceil{\log_2 M}\) and \(b=\ceil{\log_2\max_km_k}\). Partition \(M\) distinct \(a\)-bit labels into blocks of sizes \(m_k\). Within block \(k\), assign distinct \(b\)-bit labels. Use the resulting state with \(M'=M\). For the lower bound, the number of terms in block \(k\) is \[ c_k=\frac{m_kM'}M\in\mathbb N. \] Thus \(M\) divides \(m_kM'\) for every \(k\). Since the multiplicities have greatest common divisor one, an integer linear combination of them equals one; hence \(M\) divides \(M'\), and \(M'\ge M\). Distinct refined labels require \(2^{\abs{E'}}\ge M'\ge M\). Within block \(k\), distinct companions require \[ 2^{\abs{E''}}\ge c_k\ge m_k. \] These two bounds are attained by the upper-bound construction. \end{proof} Without reduction to primitive multiplicities the formula is not representation-invariant: \((2,2)/4=(1,1)/2\). The revised code reduces multiplicities before computing the cost. \begin{proposition}[Fine-graining anchoring] \label{prop:fg-anchor} If \(C_{\FG}(s,\vec m)\le N\), every \(\Strategy\in\Aclass_N\) has the target law on any capacity-\(N\) purification whose measured reduction is \(\diag(\vec m/M)\). \end{proposition} \begin{proof} On the certificate, Lemma~\ref{lem:uniformity} makes the \(E'\)-law uniform on its \(M\) labels. Nullity restricts joint \(S,E'\) outcomes to the prescribed pairs \((k(r),r)\). Composability therefore gives weight \(1/M\) to each pair and marginal \(m_k/M\) on \(S\). For another purification, pad the smaller environment to the larger of its size and the certificate size. Both fit \(N\). Lemma~\ref{lem:locality} transfers the law. \end{proof} No record register is included in \eqref{eq:fgcost}. A separate coherent-to-record comparison on an \(n\)-qubit input adds \(\abs S\) live qubits. The two costs are not to be added when the corresponding witness configurations are used in separate experiments, but neither may be silently omitted from its own experiment. \subsection{Symmetry certificates} \begin{proposition}[Permutation-symmetric certificates] \label{prop:symmetry} For the same rational law, capacity \(sM\) suffices to anchor its diagonal reduction, using composability, nullity, and the structural tensor-permutation equivariance, followed by purification locality. \end{proposition} \begin{proof} Take \(M\) blocks of \(s\) qubits. Let \(\mathcal W\) contain all sequences \((k_1,\ldots,k_M)\) with exactly \(m_k\) occurrences of label \(k\), and set \[ \Omega_{\vec m} = \frac1{\sqrt{\abs{\mathcal W}}} \sum_{(k_1,\ldots,k_M)\in\mathcal W} \ket{k_1}\cdots\ket{k_M}. \] Block permutations are tensor-factor permutations. They preserve the vector and act transitively on its computational support. Equivariance and nullity make its full outcome law uniform on \(\mathcal W\). Composability gives the first-block marginal \[ \frac{\#\set{w\in\mathcal W:w_1=k}}{\abs{\mathcal W}} = \frac{m_k}{M}. \] The first-block reduction is diagonal: completions corresponding to different first labels have different remaining multisets and hence disjoint support. Purification locality gives the stated transfer. \end{proof} \begin{corollary}[The three-qubit \((1/3,2/3)\) anchor] \label{cor:third} For every \(\Strategy\in\Aclass_3\), measurement of the first qubit of any two-qubit purification of \(\diag(1/3,2/3)\) has law \((1/3,2/3)\). \end{corollary} \begin{proof} The state \[ \Omega= \frac{\ket{011}+\ket{101}+\ket{110}}{\sqrt3} \] is invariant under tensor-factor permutations, which act transitively on its three support strings. Its full law is uniform by equivariance and nullity. Its first-qubit marginal is \((1/3,2/3)\), and its reduced state is diagonal with those weights. Pad the two-qubit purification by one zero spectator and apply Lemma~\ref{lem:locality}. \end{proof} This is I's \code{prop:three}, included with its finite proof. In contrast, \[ C_{\FG}(1,(1,2))=4. \] Both statements are true; they optimize different certificate families. \subsection{Forcing thresholds} \begin{definition}[A fixed witness threshold] For \(a\in(0,1)\), put \[ \chi_a=\sqrt a\,\ket{00}+\sqrt{1-a}\,\ket{11}. \] Let \(N_{\mathrm{force}}(a)\) be the least \(N\ge2\) for which every \(\Strategy\in\Aclass_N\) satisfies \(\Prob^{\{0\}}_{\chi_a}(0)=a\), if such an \(N\) exists. \end{definition} Lemma~\ref{lem:restriction} makes this forcing property upward monotone. Born inhabitation ensures that the admissible class is not empty. \begin{corollary}[Corrected status of the third-weight threshold] \label{cor:L} \[ N_{\mathrm{force}}(1/3)\le3. \] In particular, a capacity-three strategy deviating at \(\chi_{1/3}\) cannot satisfy the retained structural and operational requirements. The exact lower threshold is not determined here. \end{corollary} A proof that uses three qubits does not, by itself, exclude forcing at two. The earlier claimed equivalence between a four-qubit unrestricted threshold and a capacity-three deviant at this witness is therefore not retained. %===================================================================== \section{Certified rational nets and a budgeted mesh theorem} \label{sec:mesh} %===================================================================== \begin{definition}[Certified nets] For \(N\ge s\), define \[ \Rfg(N,s) = \set{\vec m/M:C_{\FG}(s,\vec m)\le N}, \] including zero coordinates at unsupported outcomes. Also set \[ \Rcert(N,s) = \Rfg(N,s)\cup \set{\vec m/M:sM\le N}. \] The second term uses primitive multiplicities and the symmetry certificate of Proposition~\ref{prop:symmetry}. Both nets are finite. They are sufficient anchoring sets, not a classification of all forceable laws. For either net \(\mathcal R\), define its covering radius \[ \delta_{\mathcal R}(N,s) = \max_{p\in\Delta_{2^s}} \min_{q\in\mathcal R(N,s)}\TV(p,q). \] \end{definition} For binary laws the parameter \(p(0)\) identifies the simplex with \([0,1]\), and total variation is ordinary absolute distance. Thus the covering radius is half the largest gap between successive net points. \begin{proposition}[Binary mesh values] \label{prop:mesh-values} The fine-graining covering radii at capacities \(4,5,6\) are \[ \delta_{\mathrm{fg}}(4,1)=\frac16,\qquad \delta_{\mathrm{fg}}(5,1)=\frac18,\qquad \delta_{\mathrm{fg}}(6,1)=\frac1{10}. \] Including symmetry certificates gives \[ \delta_{\mathrm{cert}}(3,1)=\frac16,\quad \delta_{\mathrm{cert}}(4,1)=\frac18,\quad \delta_{\mathrm{cert}}(5,1)=\frac1{10},\quad \delta_{\mathrm{cert}}(6,1)=\frac1{12}. \] \end{proposition} \begin{proof} The binary fine-graining nets at capacities \(4\) and \(5\) are, respectively, \[ \set{0,\tfrac13,\tfrac12,\tfrac23,1}, \qquad \set{0,\tfrac14,\tfrac13,\tfrac12,\tfrac23,\tfrac34,1}. \] At capacity \(6\), the net is \[ \set{ 0,\tfrac15,\tfrac14,\tfrac13,\tfrac25,\tfrac37, \tfrac12,\tfrac47,\tfrac35,\tfrac23,\tfrac34,\tfrac45,1}. \] These lists follow directly from \eqref{eq:fgcost}; their largest gaps are \(1/3,1/4,1/5\). For capacities \(3\le N\le6\), symmetry includes every grid point \(j/N\), so all gaps in the union are at most \(1/N\). The cost formula shows that fine-graining introduces no positive point below \(1/N\) in these four cases. The initial gap is therefore exactly \(1/N\). \end{proof} \begin{lemma}[Common probability modulus] \label{lem:modulus} For a fixed admissible strategy at capacity \(N\), there is a nondecreasing function \(\omega_N\), tending to zero at zero, such that \[ \TV(\Prob^S_\psi,\Prob^S_\phi) \le \omega_N(d_{\mathrm{ray}}([\psi],[\phi])) \] for every same-size pair with at most \(N\) qubits. \end{lemma} \begin{proof} Each fixed ray space is compact, so continuity implies uniform continuity. Up to name relabeling there are finitely many register sizes and measured subsets at capacity \(N\). Take the maximum of their uniform moduli. \end{proof} \begin{theorem}[C\(^\star\): anchoring and mesh control] \label{thm:Cstar} Let \(\Strategy\in\Aclass_N\), let \(n=\abs L\), and let \(s=\abs S\ge1\). \begin{enumerate}[label=\textup{(\roman*)}] \item Every diagonal measured reduction whose weights belong to \(\Rcert(N,s)\) is exactly Born. \item If \(n+s\le N\), then \begin{equation} \TV(\Prob^S_\psi,\Born^S_\psi) \le \omega_N\!\left(\sqrt{2\delta_{\mathrm{cert}}(N,s)}\right) +\delta_{\mathrm{cert}}(N,s). \label{eq:meshbound} \end{equation} The same bound holds with the fine-graining net and its radius. \end{enumerate} \end{theorem} \begin{proof} Part (i) is Propositions~\ref{prop:fg-anchor} and~\ref{prop:symmetry}, including purification locality. For (ii), use Lemma~\ref{lem:record-marginal}. Write the copied vector with the record first as \[ \Psi_p=\sum_\kappa\sqrt{p_\kappa}\ket\kappa_R\ket{v_\kappa}_E, \] where the \(v_\kappa\) are orthonormal on positive weights. The environment has \(n\ge s\) qubits, so extend this family orthonormally to every record label. Choose \(q\) in the certified net with \(\TV(p,q)\le\delta_{\mathrm{cert}}\), and define \[ \Psi_q=\sum_\kappa\sqrt{q_\kappa}\ket\kappa_R\ket{v_\kappa}_E. \] Then \[ \norm{\Psi_p-\Psi_q}^2 = \sum_\kappa(\sqrt{p_\kappa}-\sqrt{q_\kappa})^2 \le \sum_\kappa\abs{p_\kappa-q_\kappa} = 2\TV(p,q). \] The \(R\)-law at \(\Psi_q\) is \(q\) by (i). Uniform continuity and the triangle inequality give \[ \TV(\Prob^R_{\Psi_p},p) \le \omega_N(\sqrt{2\TV(p,q)})+\TV(p,q). \] The record identity transfers this to direct \(S\)-readout. \end{proof} \begin{remark}[No boundary cocycle estimate without residual control] A residual law close to Born on its own state need not be close to the Born conditional law of the original input: the unknown update may change that residual state. Consequently, \eqref{eq:meshbound} alone does not justify replacing sequential conditionals by L\"uders conditionals at full capacity. No support-graph classification or state-uniform boundary \(O(\varepsilon)\) estimate is claimed here. \end{remark} For fixed \(s\), the certified radii tend to zero as \(N\) grows. For example, taking denominator \(2^h\) with \(h=\floor{(N-s)/2}\) gives a fine-graining grid whose rounding radius is at most \[ (2^s-1)2^{-h}. \] The argument of \(\omega_N\) is the square root of this probability radius. There is no strategy-uniform rate without a common modulus assumption. %===================================================================== \section{Diagonal-faithful rigidity} \label{sec:rigidity} %===================================================================== \begin{definition}[Sub-capacity and global diagonal faithfulness] \label{def:df} A strategy satisfies \(\DFh\) if, for every input on \(n\le N-1\) qubits and every positive-probability measurement branch, \[ \diag\!\left( \ketbra{\Update^S_\psi(\kappa)} {\Update^S_\psi(\kappa)} \right) = \diag\!\left( \ketbra{\psi^S_{\mathrm L,\kappa}} {\psi^S_{\mathrm L,\kappa}} \right). \] Nullity ensures that the L\"uders residual exists on such a branch. The diagonals are taken on the surviving full computational basis. The condition \(\DFall\) imposes the same equality on inputs with \(n\le N\). Neither condition identifies residual phases. \end{definition} \begin{remark}[Relationship to the registered record condition] Diagonal faithfulness isolates the projective-diagonal property needed in the cocycle proof. A continuation record argument that has already identified Born future-test laws can imply this property by taking computational diagonals. The registered record equality itself, however, is an equality of classical observations of typed programs, not this residual equation. We therefore retain Definition~\ref{def:df} as an explicit conditional hypothesis. \end{remark} \subsection{The scalar functional equation} \begin{lemma}[Open-domain cocycle rigidity] \label{lem:functional} Let \(\varphi:[0,1]\to[0,1]\) be continuous, with \[ \varphi(0)=0,\qquad \varphi(1)=1,\qquad \varphi(1/2)=1/2. \] Suppose \begin{equation} \varphi(a)\varphi(u) = \varphi(m)\varphi(au/m) \label{eq:cocycle} \end{equation} for all \[ a,u\in(0,1), \qquad au0. \] If it vanishes on \((0,\lambda)\) with \(\lambda\le1/2\), then for \(x\in(\lambda/2,\lambda)\) we have \(x<1/2\), hence \(U(x)=1\); \eqref{eq:doubling} makes it vanish on \((0,2x)\). Taking the union gives vanishing on \((0,2\lambda)\). Finite repetition produces an interval containing \(1/2\), contradicting the anchor. Thus \(\varphi>0\) on \((0,1)\). Put \(H=\log\varphi\). For \(x,y\in(0,1)\), set \(p=xy\). The admissible intervals for the factorizations \((x,y)\) and \((\sqrt p,\sqrt p)\) are \[ (p,p+1-x), \qquad (p,p+1-\sqrt p). \] Their intersection is nonempty. Choosing \(m\) in that intersection and applying \eqref{eq:cocycle} twice gives \[ H(x)+H(y)=2H(\sqrt{xy}). \] Hence, with \(f(t)=2H(\sqrt t)\), \[ H(x)+H(y)=f(xy). \] Write \(G(s)=H(e^{-s})\) and \(F(t)=f(e^{-t})\). Then \[ G(s)+G(t)=F(s+t),\qquad s,t>0. \] For every \(\tau>0\), \[ G(s+\tau)-G(s)=G(t+\tau)-G(t). \] The common increment \(\Delta(\tau)\) is continuous and additive on positive arguments. Additivity gives \(\Delta(q)=q\Delta(1)\) for positive rational \(q\); continuity gives the same formula for every positive real argument. Thus \[ G(s)=-cs+d, \qquad H(x)=c\log x+d, \qquad \varphi(x)=e^dx^c. \] Continuity at one and \(\varphi(1)=1\) give \(d=0\). The half anchor gives \(2^{-c}=1/2\), so \(c=1\). The endpoint anchors complete the result. \end{proof} \subsection{Operational derivation of the cocycle} \begin{theorem}[I: sub-capacity diagonal-faithful rigidity] \label{thm:I} Let \(N\ge3\) and \(\Strategy\in\Aclass_N\). If \(\Strategy\) satisfies \(\DFh\), then on every input with \(n\le N-1\) and every measured register \(S\), \[ \Prob^S_\psi(\kappa)=b^S_\psi(\kappa). \] \end{theorem} \begin{proof} \paragraph{Step 1: one marginal function.} Define \[ \varphi(a)=\Prob^{\{0\}}_{\chi_a}(0), \qquad \chi_a=\sqrt a\,\ket{00}+\sqrt{1-a}\,\ket{11}. \] For any single-qubit readout on an \(n\le N-1\) input, let \(a=\bra0\rho_S\ket0\). One record qubit fits. By Lemma~\ref{lem:record-marginal}, its law equals the direct law, and its reduction is \(\diag(a,1-a)\). Pad \(\chi_a\) with \(n-1\) zero spectators. The padded reference and copied input are equal-size purifications of that same record reduction, both on \(n+1\le N\) qubits. Purification locality and spectator invariance therefore give \[ \Prob^S_\psi=(\varphi(a),1-\varphi(a)). \] Continuity of \(a\mapsto\chi_a\) and \(\textnormal{(K)}_N\) make \(\varphi\) continuous. Only the reference and cocycle witnesses below have a fixed three-qubit maximum. The general marginal comparison uses \(n+1\le N\), not a universal three-qubit bound. \paragraph{Step 2: anchors.} Nullity at \(\chi_0,\chi_1\) gives \(\varphi(0)=0\), \(\varphi(1)=1\). Uniform correlations at \((\ket{00}+\ket{11})/\sqrt2\) give \(\varphi(1/2)=1/2\). \paragraph{Step 3: the exact witness domain.} Fix \(a,u\in(0,1)\) and \(m\in(au,au+1-a)\). Define \[ \psi_{a,u,m} = \sqrt{au}\,\ket{00} +\sqrt{a(1-u)}\,\ket{01} +\sqrt{m-au}\,\ket{10} +\sqrt{1-a-m+au}\,\ket{11}. \] All four squared amplitudes are strictly positive and sum to one. The computational diagonals of its one-qubit reductions are \[ \diag(\rho_{q_0})=(a,1-a), \qquad \diag(\rho_{q_1})=(m,1-m). \] The reductions themselves generally have nonzero off-diagonal entries; no diagonality claim is needed. The projective residual after \(q_0=0\) has zero-outcome weight \(u\); after \(q_1=0\) it has zero-outcome weight \(au/m\). Both quantities lie in \((0,1)\). The input occupies two qubits and hence is sub-capacity at every \(N\ge3\). Its one-qubit residuals are sub-capacity as well. If the first probability is positive, \(\DFh\) makes the strategy's residual diagonal equal to the corresponding projective diagonal, so Step 1 evaluates the second law by \(\varphi\). If the first probability is zero, the joint probability is zero and the same product expression remains valid. Composability in the two orders therefore gives \[ \varphi(a)\varphi(u) = \varphi(m)\varphi(au/m) \] on precisely the stated open domain. \paragraph{Step 4: solve the scalar equation.} Lemma~\ref{lem:functional} gives \(\varphi=\mathrm{id}\). Thus every single-qubit readout on a sub-capacity input is Born. \paragraph{Step 5: arbitrary registers.} Induct on \(s=\abs S\). For \(s\ge2\), write \(S=\{i\}\sqcup S'\). Composability gives \[ \Prob^S_\psi(\kappa_i,\kappa') = \Prob^{\{i\}}_\psi(\kappa_i) \Prob^{S'}_{\Update^{\{i\}}_\psi(\kappa_i)}(\kappa'). \] A positive first branch has its projective residual diagonal by \(\DFh\). The residual occupies \(n-1\le N-2\) qubits. The induction hypothesis makes its \(S'\)-law Born, and that Born law depends only on the residual computational diagonal. The Born chain rule then gives the joint Born law. Zero first branches contribute zero to both sides. The empty register has its unique outcome. \end{proof} \begin{corollary}[K: a precise update-slack separation] \label{cor:K} If an admissible capacity-\(N\) strategy, \(N\ge3\), deviates from Born on a sub-capacity input, it violates \(\DFh\). Equivalently, under \(\DFh\), any outcome-law deviation must be at full capacity. \end{corollary} This is a statement about the strategy's update conditions, not a claim that the observed deviating branch itself must be the branch on which diagonal faithfulness fails. \subsection{Global diagonal faithfulness removes the boundary} \begin{theorem}[Full-capacity outcome rigidity under global faithfulness] \label{thm:boundary} Let \(N\ge3\), \(\Strategy\in\Aclass_N\), and assume \(\DFall\). Then every outcome law on every input with \(n\le N\) is Born. \end{theorem} \begin{proof} Theorem~\ref{thm:I} handles \(n0\), \[ b_\kappa \abs{\bra\lambda V\Update^S_\psi(\kappa)\rangle}^2 = \abs{\bra\lambda V\xi_\kappa\rangle}^2. \] Choose \(V\) sending \(\xi_\kappa/\sqrt{b_\kappa}\) to a fixed basis vector. The equality makes the strategy residual have unit overlap with that ray. Thus the rays agree. The chosen continuation is fixed within each experiment and does not depend on the first reported outcome. \end{proof} %===================================================================== \section{The remaining finite-capacity question and partial slack} \label{sec:slack} %===================================================================== \begin{openproblem}[G: unrestricted finite-capacity uniqueness] \label{op:G} For a fixed \(N\ge3\), do the requirements defining \(\Aclass_N\), without diagonal faithfulness, force every capacity-\(N\) outcome law and every positive residual ray to be Born and destructive projective, respectively? A witness-specific version asks whether an admissible strategy can deviate at a specified input not already anchored by a certificate. For capacity three, one possible diagnostic is \[ D_3(\Strategy) = \abs{\Prob^{\{0\}}_{\chi_{1/5}}(0)-1/5}. \] No claim is made that this diagnostic captures every possible deviation. \end{openproblem} The diagnostic at \(\chi_{1/3}\) is identically zero on \(\Aclass_3\) by Corollary~\ref{cor:third}. It is therefore unsuitable as an open capacity-three witness. \begin{remark}[Static slack is a different problem] I's static appendix assigns marginal probabilities without the full operational record equation and without a finite-capacity protocol model. It is not a capacity-three admissible strategy. The operational question here additionally constrains conditional residuals, sequential experiments, null branches, and records. Neither a static deformation nor a finite partial assignment settles Open Problem~\ref{op:G}. \end{remark} \subsection{An exact finite partial compensation} The following example replaces an unsupported claim of a globally admissible compensated strategy by an explicitly partial construction. \begin{example}[A non-Born scalar candidate and a compensated table] \label{ex:partial} For \(0<\eta\le1/2\), set \[ \varphi_\eta(x) = x+\eta x(1-x)(1-2x)(x-1/3)(x-2/3). \] It has anchors at \(0,1/3,1/2,2/3,1\) and obeys \(\varphi_\eta(1-x)=1-\varphi_\eta(x)\). The derivative of the added polynomial has absolute value at most one on \([0,1]\): the product rule bounds its terms by \(4/9\), \(2/9\), and \(1/3\). Thus \(\varphi_\eta'\ge1-\eta>0\), and it maps \([0,1]\) increasingly onto itself. At \(\eta=1/2\), \[ \varphi_\eta(1/5)-1/5=\frac{28}{9375}>0. \] However, the cocycle instance \[ a=\frac13,\qquad u=\frac12,\qquad m=\frac14, \qquad \frac{au}{m}=\frac23 \] has defect \begin{equation} \varphi_\eta(a)\varphi_\eta(u) - \varphi_\eta(m)\varphi_\eta(au/m) = -\frac5{4608} \quad(\eta=1/2). \label{eq:partial-defect} \end{equation} The associated Born table is \[ B= \begin{pmatrix} 1/6&1/6\\ 1/12&7/12 \end{pmatrix}. \] One can nevertheless satisfy both sequential orders on this one strictly supported input by setting the proposed joint table to \[ J_{ij}=r_i c_j, \quad r=(\varphi_\eta(a),1-\varphi_\eta(a)), \quad c=(\varphi_\eta(m),1-\varphi_\eta(m)). \] After the first-qubit readout, assign a one-qubit residual with computational zero-weight \(m\); after the second-qubit readout, assign one with zero-weight \(a\). If one-qubit readout uses \(\varphi_\eta\), both sequential products equal \(J_{ij}\). The first assigned residual has weight \(m=1/4\), whereas the projective residual after first outcome zero has weight \(u=1/2\). The compensation is therefore diagonally unfaithful. This is an exact partial assignment at the displayed state and continuations. It is not a strategy on all rays. It does not prove continuity across support boundaries, all record equations, or no-signaling for arbitrary local programs. \end{example} A record-law equality can be assigned on a finite copied family as well, but extending all these assignments simultaneously is precisely an additional feasibility obligation. A passing selected record test is not that extension. %===================================================================== \section{Conditional compactness without an LP claim} \label{sec:compactness} %===================================================================== Continuity of outcome laws alone does not compactify unknown update maps. A correct compactness statement must control selective outputs. \begin{definition}[Selective pure outputs] \label{def:selective} Represent a strategy branch by \[ \tau^S_\kappa([\psi]) = \Prob^S_\psi(\kappa) \ketbra{\Update^S_\psi(\kappa)} {\Update^S_\psi(\kappa)}. \] At zero probability set \(\tau^S_\kappa=0\). Then \(\tau\) is positive semidefinite, has rank at most one, and \[ \tr\tau^S_\kappa=\Prob^S_\psi(\kappa). \] \end{definition} \begin{assumption}[Additional selective-output regularity] \label{ass:selective-modulus} Fix a nondecreasing modulus \(\nu\) with \(\nu(t)\to0\) as \(t\downarrow0\). For every branch at capacity \(N\), require \[ \norm{ \tau^S_\kappa([\psi])-\tau^S_\kappa([\phi]) }_1 \le \nu(d_{\mathrm{ray}}([\psi],[\phi])). \] This is additional to \(\textnormal{(K)}_N\). \end{assumption} Let \(\mathcal K_{N,\nu}\) be the family of all finite tuples of continuous branch maps satisfying this modulus, positivity, rank at most one, normalization of their traces, the empty-readout convention, and relabeling equivariance. \begin{lemma}[Compactness of the selective-output family] \label{lem:compact} The space \(\mathcal K_{N,\nu}\), with the uniform trace-norm topology on all branch maps, is compact. \end{lemma} \begin{proof} There are finitely many canonical register sizes, subsets, and outcomes at capacity \(N\). Each ray space is compact, and each finite-dimensional output set is closed and bounded. The common modulus gives equicontinuity. For completeness, choose a countable dense subset of each input ray space. Repeated subsequence extraction makes a sequence converge on every selected point. The common modulus then makes that subsequence uniformly Cauchy: cover each input space by finitely many sufficiently small balls centered at selected points. The resulting uniform limits satisfy the same modulus. Positivity, rank at most one, trace normalization, and equivariance are closed under uniform limits. Thus every sequence has a convergent subsequence in \(\mathcal K_{N,\nu}\). \end{proof} \begin{lemma}[Closed finite-experiment constraints] \label{lem:closed} The law of any fixed finite experiment is a continuous functional of \(\tau\in\mathcal K_{N,\nu}\). Consequently, each fixed instance of composability, nullity, no-signaling, and record equality is closed. \end{lemma} \begin{proof} At a positive branch of weight \(p\), the rank-one operator \(\tau/p\) determines the residual ray continuously. A gate or allocation is a continuous map on that ray. Evaluation of a subsequent branch map is continuous by the common modulus. At a branch whose weight tends to zero, every downstream terminal contribution is bounded by that first weight. Such a contribution therefore tends to zero independently of the normalized residual. Induction on the finite history tree proves continuity of every reported law. Equality of continuous laws and exact zero conditions are closed. \end{proof} \begin{theorem}[J: compact constraint exhaustion] \label{thm:J} Choose a countable collection of experiment constraints dense in the parameter spaces of all required instances, including dense samples within each nullity subspace. Let \(K_k\subseteq\mathcal K_{N,\nu}\) impose the first \(k\) constraints, with \(K_{k+1}\subseteq K_k\). Assume \[ K_\infty=\bigcap_{k=1}^\infty K_k \] is nonempty and equals the admissible family with the additional modulus assumption. For a continuous witness functional \(D\), define \[ \varepsilon_k=\max_{\tau\in K_k}D(\tau), \qquad \varepsilon_\infty=\max_{\tau\in K_\infty}D(\tau). \] Then \[ \varepsilon_{k+1}\le\varepsilon_k, \qquad \lim_{k\to\infty}\varepsilon_k=\varepsilon_\infty, \] and the limiting maximum is attained. \end{theorem} \begin{proof} The sets \(K_k\) are nonempty compact sets by Lemmas~\ref{lem:compact} and~\ref{lem:closed}. The maximum exists, and nesting gives monotonicity. Choose maximizers \(\tau_k\). A convergent subsequence has limit \(\tau_\infty\). For each fixed \(j\), its tail lies in \(K_j\), so the limit lies in every \(K_j\) and hence in \(K_\infty\). Continuity gives \[ D(\tau_\infty)=\lim_k\varepsilon_k. \] The reverse inequality follows because \(K_\infty\subseteq K_k\) for every \(k\). \end{proof} \begin{remark}[Exactly what J does not establish] \label{rem:not-lp} Theorem~\ref{thm:J} is a compactness theorem with finitely many constraints at each stage. Its unknowns are still functions on ray spaces. It is not a finite-dimensional LP. Sequential probabilities contain products and evaluation at unknown residual states. A grid discretization must prove an interpolation or extension theorem, error bounds, and preservation of all required constraints before its optimum can be identified with \(\varepsilon_k\). Neither effective computability nor a convergence rate follows from compactness alone. If \(\varepsilon_\infty=0\), every positive target is eventually excluded by a sufficiently small upper bound. This supports rigidity; it does not ``refute rigidity.'' Finite tests of one candidate do not optimize over \(K_k\). Without Assumption~\ref{ass:selective-modulus}, this theorem does not settle Open Problem~\ref{op:G}. \end{remark} %===================================================================== \section{A one-shot instrument ledger and the adaptive separation} \label{sec:instruments} %===================================================================== This section concerns the specified Born execution interface. An unknown admissible finite-capacity strategy is not assumed to produce linear completely positive maps. Let \(\Phi=\{\Phi_x\}_{x\in X}\) be a finite instrument from \(s\) input qubits to \(s'\) output qubits: each \(\Phi_x\) is completely positive and \(\sum_x\Phi_x\) is trace preserving. Discard zero outcome maps and write \(X_+\) for the remaining labels. For a Kraus family, \[ \Choi(\Phi_x) = \sum_j \operatorname{vec}(K_{xj}) \operatorname{vec}(K_{xj})^\dagger. \] Set \[ r_x=\rank\Choi(\Phi_x), \qquad R=\sum_{x\in X_+}r_x. \] The finite-dimensional Choi, Kraus, and Stinespring conventions are those of \cite[Chapter 2]{Watrous}. \begin{definition}[One-shot realization model] \label{def:oneshot} A one-shot realization consists of: \begin{enumerate}[label=\textup{(\roman*)}] \item a pure initialized environment; \item an isometry from the input into output tensored with environment; \item a projective environment readout whose orthogonal blocks are labelled by \(x\); \item optional classical coarse-graining of a fully read fine index. \end{enumerate} Arbitrary coherent unitaries are available. A classical outcome register, if explicitly retained quantumly, is counted separately; it is not a free output qubit. \end{definition} \begin{theorem}[D\('\): exact one-shot environment ledger] \label{thm:Dprime} In Definition~\ref{def:oneshot}: \begin{enumerate}[label=\textup{(\roman*)}] \item the minimum environment dimension is \(R\); \item full computational readout followed by classical coarse-graining needs \[ e=\ceil{\log_2 R} \] environment qubits; \item within a rectangular outcome/index register split, the exact register cost is \[ e_{\mathrm{split}} = \ceil{\log_2\abs{X_+}} +\ceil{\log_2\max_xr_x}; \] \item the full-readout one-shot peak is \(s'+e\). For an input already occupying \(s\) slots in a region with \(o_{\mathrm{other}}\) unrelated occupied slots, realization is possible precisely when \[ o_{\mathrm{other}}+s'+e\le\mathrm{cap}(v). \] Equivalently, the additional free slots must cover \(s'+e-s\). \end{enumerate} \end{theorem} \begin{proof} A positive Choi matrix of rank \(r_x\) has a spectral decomposition with exactly \(r_x\) nonzero vectors. Unvectorizing them gives a Kraus family of size \(r_x\). Conversely, a family of \(d_x\) Kraus operators gives a Choi sum of \(d_x\) rank-one matrices, so \(r_x\le d_x\). If an environment block for outcome \(x\) has dimension \(d_x\), expanding the dilation in a basis of that block gives at most \(d_x\) Kraus operators for \(\Phi_x\). Thus \[ \dim E=\sum_xd_x\ge\sum_xr_x=R. \] For the upper bound, choose minimum Kraus families and define \[ V\ket\psi = \sum_{x,j}K_{xj}\ket\psi\otimes\ket{x,j}. \] Completeness gives \(V^\dagger V=I\). The environment has \(R\) basis states, and its block readout realizes the instrument. Encode those states in \(e\) qubits and measure the full fine index. On reporting only \(x\), the conditional output density operator is \[ \frac{\sum_jK_{xj}\rho K_{xj}^\dagger} {\tr\Phi_x(\rho)}. \] This is a classical mixture of fine pure branches, not a single pure primitive residual. No unmeasured quantum environment needs an affine discard operation; classical forgetting is explicit. For a prescribed product split, the outcome register must encode \(\abs{X_+}\) labels and the common index register must encode at least \(\max_xr_x\) indices. The displayed cost is both necessary within that model and sufficient. The isometry embeds a \(2^s\)-dimensional space in a \(2^{s'}R\)-dimensional one, so \(s'+e\ge s\). Allocate \(s'+e-s\) zero ancillas, embed the isometry in the \(s'+e\)-qubit space, and extend it to a unitary. The peak and local-capacity condition follow. \end{proof} The classically correlated channel \[ \rho\longmapsto\sum_x\ketbra{x}{x}\otimes\Phi_x(\rho) \] has Choi rank \(R\). Its explicit classical output is not to be counted twice as both a retained quantum register and a destructively read environment. \begin{example}[Adaptive peak need not obey a prefix Choi bound] \label{ex:adaptive} A qubit depolarizing channel is the uniform mixture of the four Pauli conjugations. Its four Pauli Kraus vectors are orthogonal, so its Choi rank is four. Its one-shot environment ledger requires two environment qubits and peak three including the input. An adaptive implementation needs only one coin qubit at a time. Prepare and destructively read a Hadamard coin, apply \(X\) or \(I\) to the input according to that bit, reuse the freed slot for a second coin, and apply \(Z\) or \(I\). Classically forgetting the two bits gives the depolarizing channel. The peak is two live qubits. Thus the Choi rank of the accumulated channel bounds a coherent environment that retains its whole dilation, not the instantaneous quantum memory of every adaptive implementation. \end{example} \begin{remark}[Capacity retirement is not a physical decoherence rate] Allocation occupancy, recyclable ancillas, permanent capacity retirement, and classical forgetting are distinct operations. One may define a bookkeeping flux as the number of permanently retired capacity units per step. No theorem here identifies that quantity with a physical decoherence rate, entropy production rate, or coupling constant. \end{remark} %===================================================================== \section{Status, evidence, and retained open boundaries} \label{sec:status} %===================================================================== \begin{center} \fbox{\begin{minipage}{0.92\linewidth} \textbf{Inherited layer: registered Enter characterization.} Manuscript I characterizes Born probabilities and destructive projective residual rays in its stated unbounded finite structural class under its five starred requirements. Its concrete metatheory belongs first to the specified Born semantics. \end{minipage}} \medskip \fbox{\begin{minipage}{0.92\linewidth} \textbf{New layer: explicitly resource-relative statements.} Identified-link grammar gives all-cut monotones and decreasing potential reachability. Restricted certificate families have exact register costs. Budget-expressible records give certified-net bounds. Sub-capacity diagonal faithfulness gives exact Born laws; global diagonal faithfulness also fixes boundary outcome laws. \end{minipage}} \medskip \setlength{\unitlength}{1pt} \begin{picture}(400,66) \dashbox{4}(400,66){ \parbox{370pt}{ \textbf{Open frontier, not completed.} Unrestricted finite-capacity uniqueness, including capacity three, remains undecided here. The \((1/3,2/3)\) diagonal witness is already closed by the three-qubit symmetry certificate. Partial slack assignments and sampled checks do not provide an admissible counterexample. }} \end{picture} \end{center} \begin{remark}[Package evidence boundary] \label{rem:package} A passing script, regression, or manifest is package evidence, not a proof of a universally quantified theorem. This is the same evidential separation emphasized in the package remarks of Em02-B and XVI-O \cite{Em02B,XVIO}. Those remarks supply a scope convention, not a mathematical premise for the present results. Theorem~\ref{thm:E} quantifies over every cut and legal run. The appendix can enumerate all cuts only for its finite examples. Theorem~\ref{thm:I} quantifies over every admissible input and the full open cocycle domain. The numerical diagnostic samples finitely many points. Theorem~\ref{thm:J} is conditional compactness, not a solved optimization hierarchy. No formal proof-checker certificate, compilation certificate, field preparation, or physical admission is claimed. \end{remark} \begin{center} \small \begin{tabular}{@{}p{0.26\linewidth}p{0.34\linewidth}p{0.30\linewidth}@{}} \toprule Result & Mathematical status & Appendix evidence\\ \midrule E and F & Proved for the identified-link grammar & Finite all-cut and reachability checks\\ Geometric diamond & Proved for independent, guard-stable transitions & Illegal-cycle and frame regressions\\ H & Exact within \(\FG\), primitive multiplicities & Exact integer cost arithmetic\\ Three-qubit anchor & Proved under retained tensor equivariance & Explicit symmetric support\\ C\(^\star\) & Proved where records fit & Exact binary net radii\\ I and boundary extension & Conditional on the stated diagonal faithfulness & Sampled Born identities, not proof execution\\ Partial compensation & Exact only on the specified finite assignments & Rational defects and joint-table identities\\ J & Conditional compactness, not an LP theorem & No optimizer supplied\\ D\('\) & Exact in the stated one-shot model & Ledger arithmetic with supplied ranks\\ \bottomrule \end{tabular} \end{center} %===================================================================== \appendix \section{Standalone computational specification} \label{app:code} %===================================================================== The following single-file Python listing uses only the standard library. It has no dependency on an unavailable \code{enter-calculus} package. Copy it to \code{en02_a_checks.py} and run: \begin{ShellBlock} python en02_a_checks.py \end{ShellBlock} The exact arithmetic portions use integers and \code{fractions.Fraction}. Floating-point portions are explicitly sampled diagnostics. This listing has been statically revised to separate rejected legal guards from invariant failures, retain parallel link identifiers, reject cycles, reduce multiplicities, bound denominator enumeration from capacity, and check every cut of each finite test geometry. It is supplied without an execution log; no particular machine epsilon or generated session is asserted. \begin{CodeBlock} """En02-A: finite regressions and sampled diagnostics. This file is package evidence, not a proof certificate. No finite test establishes global operational admissibility. """ from __future__ import annotations from dataclasses import dataclass, replace from fractions import Fraction as F from itertools import combinations, product from math import gcd, sqrt from functools import reduce from random import Random class GuardFailure(Exception): """Expected refusal of an illegal operation.""" class InvariantError(Exception): """A bug or malformed geometry; never silently swallowed.""" def require(condition, message): if not condition: raise InvariantError(message) @dataclass(frozen=True) class Geometry: # cap: ((region, nonnegative integer), ...) # active: ((identifier, source, target), ...) # latent: ((identifier, frozenset({u, v})), ...) cap: tuple active: tuple = () latent: tuple = () def __post_init__(self): regions = [v for v, _ in self.cap] require(len(regions) == len(set(regions)), "duplicate capacity region") require(all(isinstance(c, int) and c >= 0 for _, c in self.cap), "invalid capacity") V = set(regions) ids_a = [tid for tid, _, _ in self.active] ids_l = [tid for tid, _ in self.latent] require(len(ids_a) == len(set(ids_a)), "duplicate active identifier") require(len(ids_l) == len(set(ids_l)), "duplicate latent identifier") require(not (set(ids_a) & set(ids_l)), "identifier is both active and latent") require(all(u in V and v in V and u != v for _, u, v in self.active), "invalid active endpoint") require(all(len(ends) == 2 and set(ends) <= V for _, ends in self.latent), "invalid latent endpoint") require(self.acyclic(), "active graph contains a cycle") def regions(self): return frozenset(v for v, _ in self.cap) def capacity(self): return dict(self.cap) def adjacency(self, include_latent=False): adj = {v: set() for v in self.regions()} for _, u, v in self.active: adj[u].add(v) if include_latent: for _, ends in self.latent: u, v = tuple(ends) adj[u].add(v) adj[v].add(u) return adj def acyclic(self): adj = self.adjacency() colour = {v: 0 for v in adj} def visit(v): if colour[v] == 1: return False if colour[v] == 2: return True colour[v] = 1 for w in adj[v]: if not visit(w): return False colour[v] = 2 return True return all(visit(v) for v in adj) def has_path(self, source, target): seen = set() todo = [source] adj = self.adjacency() while todo: v = todo.pop() if v == target: return True if v not in seen: seen.add(v) todo.extend(adj[v] - seen) return False def potential_relation(self): adj = self.adjacency(include_latent=True) out = set() for source in adj: seen = set() todo = [source] while todo: v = todo.pop() if v not in seen: seen.add(v) out.add((source, v)) todo.extend(adj[v] - seen) return frozenset(out) def mu(self): return (sum(c for _, c in self.cap) + len(self.active) + len(self.latent)) def cut_mu(self, X): X = frozenset(X) require(X <= self.regions(), "cut contains unknown region") Y = self.regions() - X cap_x = sum(c for v, c in self.cap if v in X) latent_x = sum(len(ends & X) == 1 for _, ends in self.latent) active_xy = sum(u in X and v in Y for _, u, v in self.active) return cap_x + latent_x + active_xy def sigma(self, X): X = frozenset(X) require(X <= self.regions(), "cut contains unknown region") return ( sum(len(ends & X) == 1 for _, ends in self.latent) + sum((u in X) != (v in X) for _, u, v in self.active) ) def all_cuts(g): V = sorted(g.regions()) for k in range(len(V) + 1): for members in combinations(V, k): yield frozenset(members) def owned(cell, frame): if cell not in set(frame): raise GuardFailure("missing ownership permission") def finish(g, g2, expected_mu_delta): require(g2.mu() - g.mu() == expected_mu_delta, "incorrect scalar ledger") for X in all_cuts(g): require(g2.cut_mu(X) <= g.cut_mu(X), "cut_mu increased") require(g2.sigma(X) <= g.sigma(X), "sigma increased") require(g2.potential_relation() <= g.potential_relation(), "potential reachability increased") return g2 def retire_capacity(g, region, frame, occupancy=None): owned(("cap", region), frame) occ = dict(occupancy or {}) caps = g.capacity() if region not in caps or caps[region] <= occ.get(region, 0): raise GuardFailure("no unoccupied capacity unit") caps[region] -= 1 g2 = replace(g, cap=tuple(sorted(caps.items()))) return finish(g, g2, -1) def delete_active(g, tid, frame): owned(("link", tid), frame) found = [edge for edge in g.active if edge[0] == tid] if not found: raise GuardFailure("active identifier absent") g2 = replace(g, active=tuple(edge for edge in g.active if edge[0] != tid)) return finish(g, g2, -1) def delete_latent(g, tid, frame): owned(("link", tid), frame) found = [link for link in g.latent if link[0] == tid] if not found: raise GuardFailure("latent identifier absent") g2 = replace(g, latent=tuple(link for link in g.latent if link[0] != tid)) return finish(g, g2, -1) def activate(g, tid, source, frame): # The frame represents the single activation authority. # Production term typing must issue that authority linearly. owned(("link", tid), frame) found = [link for link in g.latent if link[0] == tid] if not found: raise GuardFailure("latent identifier absent or spent") _, ends = found[0] if source not in ends: raise GuardFailure("source is not an endpoint") target = next(iter(ends - {source})) if g.has_path(target, source): raise GuardFailure("activation would create a cycle") g2 = replace( g, latent=tuple(link for link in g.latent if link[0] != tid), active=g.active + ((tid, source, target),), ) return finish(g, g2, 0) def expect_guard_failure(fn): try: fn() except GuardFailure: return raise InvariantError("expected guard refusal did not occur") def ceil_log2(n): require(isinstance(n, int) and n >= 1, "ceil_log2 requires a positive integer") return (n - 1).bit_length() def primitive_multiplicities(m): m = tuple(m) require(bool(m), "empty multiplicity list") require(all(isinstance(x, int) and x >= 0 for x in m), "multiplicities must be nonnegative integers") positive = [x for x in m if x] require(bool(positive), "all multiplicities vanish") d = reduce(gcd, positive) return tuple(x // d for x in m) def finegrain_cost(m, s=1): require(isinstance(s, int) and s >= 0, "invalid register size") m = primitive_multiplicities(m) require(len(m) <= 2 ** s, "too many outcome labels") M = sum(m) return s + ceil_log2(M) + ceil_log2(max(m)) def finegrain_words(m, s=1): """Return certificate basis labels; amplitudes are 1/sqrt(M).""" m = primitive_multiplicities(m) require(len(m) <= 2 ** s, "too many outcome labels") words = [] refined = 0 for label, count in enumerate(m): for tie in range(count): words.append((label, refined, tie)) refined += 1 require(len({r for _, r, _ in words}) == len(words), "refined labels repeat") require(len({(k, e) for k, _, e in words}) == len(words), "companions repeat") return tuple(words) def binary_finegrain_net(N): require(isinstance(N, int) and N >= 1, "invalid capacity") out = {F(0), F(1)} # Every certificate has ceil(log2(M)) <= N - 1. # This is a complete capacity-derived bound, not a fixed cutoff. max_M = 1 << (N - 1) for M in range(2, max_M + 1): for m0 in range(1, M): if finegrain_cost((m0, M - m0), 1) <= N: out.add(F(m0, M)) return tuple(sorted(out)) def binary_certified_net(N): out = set(binary_finegrain_net(N)) for M in range(2, N + 1): for m0 in range(1, M): out.add(F(m0, M)) return tuple(sorted(out)) def binary_mesh(points): points = tuple(sorted(points)) require(points[0] == 0 and points[-1] == 1, "net must include endpoints") return max((b - a) / 2 for a, b in zip(points, points[1:])) def phi_candidate(x, eta=F(1, 2)): return (x + eta * x * (1 - x) * (1 - 2 * x) * (x - F(1, 3)) * (x - F(2, 3))) def exact_partial_compensation(): a, u, m = F(1, 3), F(1, 2), F(1, 4) pa, pm = phi_candidate(a), phi_candidate(m) row = (pa, 1 - pa) col = (pm, 1 - pm) J = {(i, j): row[i] * col[j] for i in (0, 1) for j in (0, 1)} for i in (0, 1): require(sum(J[i, j] for j in (0, 1)) == row[i], "wrong joint row") for j in (0, 1): require(sum(J[i, j] for i in (0, 1)) == col[j], "wrong joint column") # Assigned residual zero-weights are m and a respectively. for i in (0, 1): for j in (0, 1): require(J[i, j] == row[i] * col[j], "forward sequential mismatch") require(J[i, j] == col[j] * row[i], "reverse sequential mismatch") defect = (phi_candidate(a) * phi_candidate(u) - phi_candidate(m) * phi_candidate(a * u / m)) deviation = phi_candidate(F(1, 5)) - F(1, 5) require(defect == -F(5, 4608), "unexpected cocycle defect") require(deviation == F(28, 9375), "unexpected witness deviation") require(m != u, "compensation accidentally faithful") return { "partial_joint": J, "cocycle_defect": defect, "candidate_deviation_at_1/5": deviation, "assigned_minus_projective_residual_weight": m - u, "status": "finite partial assignment, not admissibility", } def chi(a): require(0.0 <= a <= 1.0, "a outside [0,1]") return (sqrt(a), 0j, 0j, sqrt(1.0 - a)) def witness(a, u, m): if not (0.0 < a < 1.0 and 0.0 < u < 1.0): raise ValueError("a and u must be in (0,1)") if not (a * u < m < a * u + 1.0 - a): raise ValueError("m outside exact open cocycle domain") w = (a * u, a * (1 - u), m - a * u, 1 - a - m + a * u) require(min(w) > 0.0, "nonpositive witness amplitude weight") require(abs(sum(w) - 1.0) < 1e-12, "witness is not normalized") return tuple(complex(sqrt(x)) for x in w) def qubit_count(psi): length = len(psi) require(length >= 1 and length & (length - 1) == 0, "state length is not a power of two") return length.bit_length() - 1 def project_delete(psi, S, outcome): n = qubit_count(psi) S = tuple(S) require(len(S) == len(set(S)), "duplicate measured coordinate") require(all(0 <= i < n for i in S), "invalid measured coordinate") require(len(outcome) == len(S), "wrong outcome arity") survivors = tuple(i for i in range(n) if i not in S) out = [0j] * (1 << len(survivors)) for index, amp in enumerate(psi): bits = tuple((index >> (n - 1 - i)) & 1 for i in range(n)) if tuple(bits[i] for i in S) == tuple(outcome): target = 0 for i in survivors: target = 2 * target + bits[i] out[target] = complex(amp) p = sum(abs(z) ** 2 for z in out) if p == 0.0: return p, None return p, tuple(z / sqrt(p) for z in out) def born_law(psi, S): return { outcome: project_delete(psi, S, outcome)[0] for outcome in product((0, 1), repeat=len(S)) } class BornStrategy: def P(self, psi, S): return born_law(psi, S) def U(self, psi, S, outcome): p, residual = project_delete(psi, S, outcome) if p == 0.0: raise ValueError("zero branch has no residual") return residual def diagonal_residual_defect(strategy, psi, S, outcome): p = strategy.P(psi, S)[tuple(outcome)] if p <= 0.0: return 0.0 b, projective = project_delete(psi, S, outcome) require(b > 0.0, "positive strategy branch violates nullity") residual = strategy.U(psi, S, outcome) require(len(residual) == len(projective), "residual has wrong dimension") return max(abs(abs(x) ** 2 - abs(y) ** 2) for x, y in zip(residual, projective)) def sample_rigidity_identities(strategy, samples=200, seed=0): """Sample necessary identities; this is not proof execution.""" rng = Random(seed) def phi(x): return strategy.P(chi(x), (0,))[(0,)] worst_df = 0.0 worst_cocycle = 0.0 worst_product = 0.0 worst_identity = 0.0 for x in (0.0, 0.5, 1.0): require(abs(phi(x) - x) < 1e-9, "anchor diagnostic failed") for _ in range(samples): a = rng.uniform(0.05, 0.95) u = rng.uniform(0.05, 0.95) lo, hi = a * u, a * u + 1.0 - a # Fractional positioning avoids a fixed endpoint margin. m = lo + rng.uniform(0.1, 0.9) * (hi - lo) psi = witness(a, u, m) for S in ((0,), (1,)): worst_df = max( worst_df, diagonal_residual_defect(strategy, psi, S, (0,)) ) worst_cocycle = max( worst_cocycle, abs(phi(a) * phi(u) - phi(m) * phi(a * u / m)) ) x = rng.uniform(0.05, 0.95) y = rng.uniform(0.05, 0.95) worst_product = max( worst_product, abs(phi(x) * phi(y) - phi(sqrt(x * y)) ** 2) ) worst_identity = max(worst_identity, abs(phi(x) - x)) return { "sampled_diagonal_defect": worst_df, "sampled_cocycle_defect": worst_cocycle, "sampled_product_defect": worst_product, "sampled_phi_identity_defect": worst_identity, "status": "sampled diagnostics only", } def one_shot_ledger_from_ranks(ranks, s, s_out): ranks = tuple(ranks) require(bool(ranks) and all(isinstance(r, int) and r > 0 for r in ranks), "supply positive Choi ranks after removing zero maps") R = sum(ranks) e = ceil_log2(R) e_split = ceil_log2(len(ranks)) + ceil_log2(max(ranks)) require(s_out + e >= s, "ranks/dimensions cannot support the required isometry") return { "environment_dimension": R, "environment_qubits": e, "aligned_environment_qubits": e_split, "one_shot_peak": s_out + e, "additional_slots_beyond_input": s_out + e - s, "status": "arithmetic with supplied ranks", } def run_exact_regressions(): g = Geometry( cap=(("A", 2), ("B", 2), ("C", 1)), latent=( ("ab1", frozenset(("A", "B"))), ("ab2", frozenset(("A", "B"))), ("bc", frozenset(("B", "C"))), ("ca", frozenset(("C", "A"))), ), ) g1 = activate(g, "ab1", "A", {("link", "ab1")}) g2 = activate(g1, "ab2", "A", {("link", "ab2")}) require(len(g2.active) == 2, "parallel identifiers collapsed") require(g2.mu() == g.mu(), "activation did not conserve mu") expect_guard_failure( lambda: activate(g2, "ab1", "A", {("link", "ab1")}) ) g3 = activate(g2, "bc", "B", {("link", "bc")}) expect_guard_failure( lambda: activate(g3, "ca", "C", {("link", "ca")}) ) expect_guard_failure( lambda: retire_capacity( g3, "A", {("cap", "A")}, occupancy={"A": 2} ) ) g4 = retire_capacity(g3, "A", {("cap", "A")}, occupancy={"A": 1}) g5 = delete_active(g4, "ab2", {("link", "ab2")}) g6 = delete_latent(g5, "ca", {("link", "ca")}) require(g6.mu() == g3.mu() - 3, "retirement ledger wrong") require(finegrain_cost((1, 2)) == 4, "wrong FG cost") require(finegrain_cost((1, 1)) == 2, "wrong uniform cost") require(finegrain_cost((2, 2)) == 2, "multiplicities were not reduced") require(finegrain_cost((3, 5)) == 7, "wrong FG cost") require(len(finegrain_words((1, 2))) == 3, "wrong certificate support") expected_fg = {4: F(1, 6), 5: F(1, 8), 6: F(1, 10)} for N, value in expected_fg.items(): require(binary_mesh(binary_finegrain_net(N)) == value, "wrong fine-graining mesh") for N in range(3, 7): require(binary_mesh(binary_certified_net(N)) == F(1, 2 * N), "wrong combined certificate mesh") exact_partial_compensation() # Supplied symbolic ranks: projective instrument and depolarization. require(one_shot_ledger_from_ranks((1, 1), 1, 1) ["one_shot_peak"] == 2, "wrong projective-instrument ledger") require(one_shot_ledger_from_ranks((4,), 1, 1) ["one_shot_peak"] == 3, "wrong depolarizing one-shot ledger") def main(): run_exact_regressions() print("Exact finite regressions: PASS") print("This pass is package evidence, not a quantified certificate.") for N in range(3, 7): print("capacity", N, "FG radius", binary_mesh(binary_finegrain_net(N)), "combined radius", binary_mesh(binary_certified_net(N))) print(exact_partial_compensation()) print(sample_rigidity_identities(BornStrategy())) if __name__ == "__main__": main() \end{CodeBlock} \section{Revision and proof-obligation ledger} \label{app:repairs} The changes from the preceding draft are substantive. \begin{enumerate}[leftmargin=*,label=\textup{(\arabic*)}] \item \textbf{Stable inheritance.} Registered I labels replace drifting internal numbers as the primary source locators. The characterization remains restricted to its declared strategy class. \item \textbf{Three-qubit symmetry retained.} The four-qubit fine-graining optimum is no longer identified with unrestricted forcing of \((1/3,2/3)\). The registered three-qubit certificate is included and proved. \item \textbf{Primitive rational representation.} The register formula and code reduce multiplicities by their greatest common divisor. Zero outcomes are handled separately. \item \textbf{Correct reduced-state calculation.} The cocycle witness has specified reduced-state diagonals, not generally diagonal reduced states. \item \textbf{Correct capacity bookkeeping.} General one-qubit record comparisons use \(n+1\le N\). Only the scalar witnesses have the fixed three-qubit bound. \item \textbf{Faithfulness scope separated.} Sub-capacity faithfulness gives the headroom theorem. Global faithfulness additionally gives full-capacity outcome rigidity. Neither condition is silently equated with the full record requirement. \item \textbf{Geometric identity counts.} Active edges retain identifiers, including parallel edges. Activation then really preserves the scalar and crossing-link counts. \item \textbf{Ownership and read dependencies.} Quantum-name disjointness no longer implies disjoint geometric frames. Read footprints and guard stability are explicit. \item \textbf{Acyclicity enforced.} An activation that creates a directed cycle is refused before constructing the new geometry. \item \textbf{Occupied slots protected.} Permanent capacity retirement requires a free unit. Measurement removes names, not their regions. \item \textbf{No-signaling observations delimited.} Incident link status is not declared literally invisible. Private observations, shared status, enablement, and communication are distinguished. \item \textbf{Fixed-library restriction.} Downward closure is proved for a fixed experimental library. No unproved closure under newly enabled graph-dependent operations is claimed. \item \textbf{Boundary approximation withdrawn.} The mesh bound is not used to identify an unknown residual with the original state's L\"uders conditional. Unsupported support-graph classification and state-uniform boundary constants are removed. \item \textbf{Compactness repaired.} Selective-output regularity is added explicitly to the compactness theorem. Finite constraints are not called a finite LP, and compactness is not called effective computability. \item \textbf{Partial slack honestly labelled.} An exact partial compensation replaces the asserted all-state level-one witness. Its extension to all required experiments remains an open obligation. \item \textbf{One-shot and adaptive instruments separated.} The one-shot environment lower bound is proved in its own model. An adaptive counterexample excludes the former prefix Choi-rank claim about instantaneous quantum memory. \item \textbf{Physical flux identification removed.} Permanent capacity retirement is a bookkeeping operation, not a derived physical decoherence rate. \item \textbf{Artefact scope corrected.} Invariant failures are not caught as ordinary guard refusals. The denominator bound follows from capacity. Numerical fitting and sampling are not described as execution of a universal mathematical proof. \end{enumerate} %===================================================================== \section*{Declarations} %===================================================================== \paragraph{Competing interests.} The author declares no competing interests. \paragraph{Funding.} No specific grant supported this work. \paragraph{Author contribution.} N.C.\ developed the mathematical framework and prepared the manuscript and computational specification. \paragraph{Data and code availability.} The exact finite constructions and standalone computational source are included in this manuscript. No external package execution log or formal verification certificate is supplied. \paragraph{Ethics and clinical studies.} Not applicable. %===================================================================== \begin{thebibliography}{99} \bibitem{I} N.~Cheung, \emph{Born--L\"uders as the unique measurement strategy of a strictly linear quantum \(\lambda\)-calculus: The Enter Calculus}. Registered manuscript I, version 1, \code{I.tex}. Stable locators used here include \code{ass:strategy-class}, \code{def:requirements}, \code{def:record}, \code{lem:diamond}, \code{thm:confluence}, \code{thm:uniformity}, \code{prop:three}, \code{thm:fine-anchor}, \code{thm:born}, \code{thm:update}, \code{thm:stability}, and \code{thm:main}. Source interface: \url{https://enter.grok.me/browse/I}. \bibitem{XEnter} N.~Cheung, \emph{X-Enter}. Registered manuscript, version 1, \code{X-Enter.tex}. Finite readout interface and complete selective-record formulas, including \code{eq:instrument} and \code{eq:recordchannel}. Source interface: \url{https://enter.grok.me}. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Chapter 2: completely positive maps, Kraus representations, Stinespring representations, and instruments. Author's reference interface: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \bibitem{Em02B} N.~Cheung, \emph{Em02-B}. Manuscript, version 1. Package-boundary remark \code{conv:package}. Cited only for the distinction between package evidence and proof. \bibitem{XVIO} N.~Cheung, \emph{XVI-O}. Manuscript, version \code{o70-sync}. Package-boundary remark \code{conv:package}. Cited only for the distinction between an archive or pass flag and a mathematical certificate. \end{thebibliography} \end{document}