\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={X-K-ST: Certified composition of destructive clocks and finite instruments on a prescribed conical Enter spacetime}, pdfsubject={Finite adaptive instruments, fresh-clock typing, complete-channel errors, resources, and prescribed causal transport} } \setlength{\emergencystretch}{3em} \setlist{itemsep=3pt,topsep=5pt} \setcounter{tocdepth}{1} \numberwithin{equation}{section} \allowdisplaybreaks \urlstyle{same} \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} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\NN}{\mathbb N} \newcommand{\HH}{\mathcal H} \newcommand{\BB}{\mathcal B} \newcommand{\Alg}{\mathcal A} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\out}{\operatorname{out}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddia}{d_\diamond} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\Mclock}{\mathcal M} \newcommand{\Qlab}{Q_{\mathrm{lab}}} \newcommand{\Qext}{Q_{\mathrm{ext}}} \newcommand{\Qres}{Q_{\mathrm{res}}} \newcommand{\Bres}{B_{\mathrm{res}}} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\proj}[1]{\kb{#1}{#1}} \newcommand{\loc}[1]{\texttt{\detokenize{#1}}} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\rank}{rank} \newcommand{\statementbox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{X-K-ST: Certified composition of destructive clocks\\ and finite instruments}\\[4pt] \large Complete histories, causal record delivery, and resource bounds\\ on the unchanged prescribed conical Enter spacetime} \author{} \date{} \begin{document} \maketitle \begin{abstract} We give a single finite certificate for successive destructive clock readouts, conditional instruments, regular transport, and declared releases on the conical host retained from I-B and X-A-ST. The certificate records the live quantum names at every cut, all accessible classical controls, the provenance of each new clock encoding, and every communication needed before a record can control a remote operation. Its semantics is one complete instrument on arbitrary reference-entangled inputs, with all failure histories and declared surviving outputs retained. The main assembly theorem supplies the complete history law, Kraus completeness, scheduled causal containment, and separate laboratory, returned-storage, and external-reference ledgers. For matched adaptive trees, complete-channel errors admit a worst-path bound rather than a sum over mutually exclusive branches. A two-readout, two-route example gives an explicit nonempty realization, a failure probability of \(1/4\), and a complete referenced output formula, subject to an explicit fit inside the existing register and time reserve. Several necessary corrections are part of the result. Fresh quantum names do not imply statistically independent clocks. A trace-distance estimate at one tested input is not a diamond-norm estimate. Releasing a register does not erase its storage cost, and a joint output formula does not deliver a record to a distant controller. The novelty claimed is the audited assembly of these requirements, not a new general instrument or channel-composition theorem. Its prospective use is to make finite multi-instrument continuations checkable without concealing consumed inputs, memory, failures, or external timing. The host, supported chart, signature ledger, regulator, and fixed positive width are unchanged. No continuum acquisition, convolution repair, interacting quantum BRST differential, characteristic quantum coefficient, ultraviolet completion, or spacetime emergence is constructed. In particular, the composed instrument does not provide quantized relational time without an external clock or external scheduling time. \end{abstract} \statementbox{Exact release}{% A finite, well-typed, resource-fitting instruction certificate defines one complete finite instrument. This is a theorem about the declared finite interface. It is not a theorem that every finite tree fits an already fixed laboratory, that clock inputs can be reused after consumption, or that a finite channel determines a continuum evolution. External preparation, calibration, locations, durations, and causal routing remain inputs.} \tableofcontents %===================================================================== \section{Purpose, provenance, and the Enter discipline} \label{sec:introduction} %===================================================================== \subsection{The gap addressed} The predecessor records already contain destructive clock instruments, reference-preserving routes, and finite adaptive composition patterns. In particular, X-I-ST explicitly permits a finite adaptive tree with fresh or separately returned clock encodings \cite{XIST}. X-J-ST combines selective coefficient reports with complete instruments and error margins \cite{XJST}. It would therefore be incorrect to advertise X-K-ST as the first proof that finite instruments compose. The narrower contribution is an assembled certificate with all of the following data present at once: \begin{enumerate} \item the types before and after every destructive readout; \item the provenance of every subsequent clock input; \item the complete public history and the distinction between public outcomes and unobserved Kraus indices; \item the actual memory and retained residuals; \item the causal delivery of every remotely used classical code; \item a branchwise occupancy and storage ledger; \item a complete-channel comparison on matching interfaces; \item an explicit fit inside the unchanged host reserve. \end{enumerate} The source comparisons are I-B, En09-A, X-Enter-ST, X-H-ST, X-I-ST, and the supplied X-J-ST development \cite{IB,En09,XEnterST,XHST,XIST,XJST}. They are documentary precedents, not unexamined theorem premises. The finite formulas needed here are stated and proved below. The background host and operation permissions are nevertheless explicit assumptions. Calling a citation a separation does not remove a hypothesis actually used in a proof. \subsection{The philosophy of Enter} The supplied introductory essay distinguishes four interfaces and interpretations \cite{EnterPhilosophy}: \begin{enumerate} \item destructive readout is not quantum recovery; \item a classical record is not a quantum-return decoder; \item a channel is not a thermal machine; \item an analogy is not an identification. \end{enumerate} Here destructive readout consumes a measured quantum name from the accessible quantum output and returns a classical code together with unmeasured surviving systems. This is a statement about access and output type. It is not physical annihilation of every implementing degree of freedom. We adopt specified Born semantics. We do not rederive the class-relative characterization of Manuscript I from an abbreviated hypothesis list \cite{EnterI}. In particular, record probabilities and complete residual instruments remain different comparison targets. The mathematical setting of instruments, finite channels, and sequential networks is established \cite{DaviesLewis,Watrous,CDP}. No priority claim is made for Kraus composition, partial trace, or the channel-error telescope. \subsection{Three corrections to the proposed statement} First, a fresh clock means a new admissible live input name, not a statistical independence hypothesis. Correlated clock supplies are allowed when their joint preparation and storage are declared. Second, an error theorem must distinguish: \[ \text{uniform complete-channel error}, \qquad \text{error on one specified referenced state}. \] The latter does not imply the former. Third, no-enlargement is a resource-qualified assertion. Finiteness alone does not prove that a protocol fits a previously fixed capacity or time window. The certificate must pass those checks before a host-level realization is released. %===================================================================== \section{The retained host and the fit contract} \label{sec:host} %===================================================================== \begin{assumption}[Unchanged supported host] \label{ass:host} Retain the I-B/X-A-ST instance, including its deficit, time orientation, supported developing chart, regular laboratory reserve, time calibration, and finite-operation permissions. In the inherited conical coordinates, \[ \mathcal M^\circ =\RR_T\times(C_\alpha\setminus\{o\})\times\RR_z, \qquad G=-c_{\mathrm h}^2\dd T^2+\dd r^2 +\alpha^2r^2\dd\varphi^2+\dd z^2, \] where \(0<\alpha<1\) and \(c_{\mathrm h}>0\) are fixed. The completion defect is the worldsheet \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] The independent analytical completion and supported chart are \[ \widehat M=\RR^{1+3},\qquad \eta=\diag(1,-1,-1,-1),\qquad \iota:\mathcal L\longrightarrow\mathcal M^\circ, \] \begin{equation} \iota^*G=-\eta,\qquad t=c_{\mathrm h}(T-T_\Sigma). \label{eq:signature} \end{equation} The retained compact laboratory reserve is contained in \(\iota(\mathcal L)\) and is disjoint from \(\mathcal W\). Retain the same smooth switches and fixed positive width: \[ \sigma_\varepsilon(t)=\chi(t/\varepsilon),\qquad 0<\varepsilon<\delta, \] and the same declared plateau and regulator prescription. No regulator profile or width is changed in this manuscript. Every global free field equation and Green operator, if used in a predecessor comparison, belongs to \(\widehat M\), not to the regular conical product. The present instrument construction uses no Green operator. \end{assumption} This is a prescribed spacetime with a finite operational interface, not a spacetime reconstructed from records \cite{IB,XAST}. \begin{proposition}[No global conical Cauchy inference] \label{prop:nonGH} The regular conical product is not globally hyperbolic. The finite instrument construction does not require a global conical inverse. \end{proposition} \begin{proof} Fix a radial ray, \(r_0>0\), and \(z_0\). For \(A>r_0/c_{\mathrm h}\), the causal diamond between \((-A,r_0,\varphi_0,z_0)\) and \((A,r_0,\varphi_0,z_0)\) contains \[ (0,r_0/n,\varphi_0,z_0),\qquad n\ge1. \] The sequence approaches the excluded apex and has no convergent subsequence in \(\mathcal M^\circ\). The diamond is not compact. Every new calculation below concerns finite-dimensional operators and explicitly located finite ports. \end{proof} \begin{definition}[Resource and location fit] \label{def:fit} The retained laboratory supplies a declared spatial and time reserve, permitted port operations, and capacities \(\Qres\) for simultaneously occupied laboratory qubits and \(\Bres\) for classical storage. A finite protocol \emph{fits} when: \begin{enumerate}[label=\textup{(\roman*)}] \item all its ports and certified paths lie in the retained regular reserve; \item all operations, waits, preparation accesses, and communication durations lie in the supplied time window; \item its cut-by-cut occupancy and storage counts obey the retained capacities; \item its gates, allocations, releases, and communication maps are among the declared permissions; \item every control record reaches its control port before use. \end{enumerate} If a capacity is not specified numerically in the retained instance, the certificate states the required inequality rather than asserting that it has passed. \end{definition} Classical geometry may be prescribed by real parameters. This does not assert finite-bit physical calibration of arbitrary real numbers. %===================================================================== \section{Finite interfaces, fresh clocks, and complete memory} \label{sec:types} %===================================================================== \subsection{Classical--quantum cuts} A cut with finite public history set \(H\) and branch spaces \(\HH_h\) has algebra \[ \Alg_H=\bigoplus_{h\in H}\BB(\HH_h). \] A state is a family of positive blocks \((\rho_h)_h\) with \(\sum_h\Tr\rho_h=1\). Equivalently, it is the block-diagonal operator \[ \sum_h\proj h\otimes\rho_h. \] The quantum names in \(\HH_h\), their locations, and their ownership are part of the cut specification. Histories may have different live spaces. Direct sums keep those types distinct. Whenever a full matrix extension is needed, the specified extension first pinches between the classical history sectors. It does not interpret a coherent superposition of controller histories as an ordinary classical control. An untouched reference \(E\) has no execution or communication wire. All channel identities below hold after tensoring with \(\id_E\). \begin{definition}[Fresh or separately supplied clock input] \label{def:fresh} At a clock-readout node, the clock factor \(C_v\) must be present in the live input context. Its provenance is one of: \begin{enumerate}[label=\textup{(\roman*)}] \item a new allocation with a specified preparation; \item a previously installed, still unconsumed supplier factor; \item a new output factor of a separately declared preparation or restoration channel on available inputs. \end{enumerate} The measured quantum names are removed at readout and never become available again merely by reusing their spelling. A later allocation in the same physical memory slot receives a new logical name and requires a new preparation permission. It is not a second measurement of the old input. A restoration claim includes its actual input access, output factor, correctness domain, errors, and resource costs. No universal restoration of a destructively consumed unknown clock is assumed. \end{definition} \begin{remark}[Freshness is not independence] A new clock name may be correlated with other live systems through an explicitly specified joint preparation or encoder. Alternatively, a supplier may install several mutually correlated clock factors at the initial cut. What is forbidden is an unspecified assignment of a joint extension from a reduced state, or access to a supplier or reference not present in the declared interface. \end{remark} For instance, allocation of a fixed state \(\omega_C\) is the channel \[ \mathcal A_\omega(X)=\omega_C\otimes X. \] A correlated preparation is instead a named linear CPTP map \[ \mathcal E:\BB(\HH_{\mathrm{in}}) \longrightarrow\BB(\HH_C\otimes\HH_S). \] It is not introduced by writing an arbitrary state-dependent extension \(\rho_S\mapsto\rho_{CS}\). \subsection{Declared instruments and their Kraus refinements} \begin{definition}[Complete node instrument] \label{def:node} At a node \(v\), a finite outcome set \(A_v\) and finite Kraus families \[ L_{v,a,\mu}:\HH_v\longrightarrow\HH_{va} \] are supplied, with \begin{equation} \sum_{a\in A_v}\sum_\mu L_{v,a,\mu}^\dagger L_{v,a,\mu}=I_{\HH_v}. \label{eq:nodecomplete} \end{equation} The public outcome is \(a\), while \(\mu\) is an unobserved Kraus index unless separately reported. The selective map is \[ \Gamma_{v,a}(X) =\sum_\mu L_{v,a,\mu}XL_{v,a,\mu}^\dagger, \] and the recorded node channel appends \(a\) to the public history. \end{definition} A failure is an ordinary member of \(A_v\) with a complete continuation or a complete terminal output. An unavailable preparation or rejected task must have an explicit failure convention if it belongs to the released input domain. Adaptive control may depend on the public history. It may not depend on an unobserved Kraus label or an unknown state through an undeclared nonlinear rule. \subsection{Release and storage} \begin{definition}[Declared release] \label{def:release} A release has one of three explicit meanings: \begin{enumerate} \item a register is returned as an output and thereafter is acted on by the identity; \item a register is transferred to a named storage owner, with its subsequent storage ledger retained; \item a specified subsystem is traced out at a declared output boundary. \end{enumerate} A data-dependent surviving output is not omitted while the remaining marginal is called the complete output. An independent fixed blank may be discarded after its decoupling has been proved. The discard does not establish a thermodynamic reset cost. \end{definition} ``Complete'' means complete on the declared comparison boundary, including every declared survivor, record, and failure. It does not automatically mean access to every microscopic environment in a possible physical implementation. %===================================================================== \section{One destructive clock and its exact limitations} \label{sec:oneclock} %===================================================================== Let \[ \HH_C\cong\CC^{2^q},\qquad q\ge1, \] have a specified orthonormal basis \(\{\ket{\tau_k}\}_{k=0}^{2^q-1}\), and put \[ P_k=\proj{\tau_k},\qquad D_k=\bra{\tau_k}_C\otimes I_S. \] The calibrated basis change satisfies \(B_C\ket{\tau_k}=\ket k\). Define \begin{equation} \Mclock_C(X) =\sum_k\proj k_R\otimes D_kXD_k^\dagger. \label{eq:oneclock} \end{equation} \begin{theorem}[Complete destructive-clock identity] \label{thm:oneclock} Equation~\eqref{eq:oneclock} is CPTP and equals \[ \sum_k\proj k_R\otimes \Tr_C\bigl[(P_k\otimes I_S)X(P_k\otimes I_S)\bigr] \] on every input operator. For every reference extension, \begin{equation} \Tr_R[(\Mclock_C\otimes\id_E)(X)] =\Tr_CX. \label{eq:drop} \end{equation} For a state \(\rho\), the \(k\)-block is \[ \sigma_k=D_k\rho D_k^\dagger,\qquad p_k=\Tr\sigma_k. \] Its normalized conditional state is \(\sigma_k/p_k\) only when \(p_k>0\). No normalized state is assigned to a zero block. The output quantum context contains \(S\), not the measured names in \(C\). \end{theorem} \begin{proof} The recorded Kraus maps are \[ \ket k_R\otimes D_k, \qquad \sum_kD_k^\dagger D_k =\sum_kP_k\otimes I_S=I_{CS}. \] This proves complete positivity and trace preservation. Expand \[ X=\sum_{k,l}\kb{\tau_k}{\tau_l}\otimes X_{kl}. \] Both formulas return the \(kk\)-block. Their sum is \(\Tr_CX\), proving \eqref{eq:drop}. Replacing \(I_S\) by \(I_S\otimes I_E\) proves the reference statements. The output type gives consumption. \end{proof} This local calculation restates the I-B/En09-A pattern \cite{IB,En09}; it is not an independent verification of those entire manuscripts. \begin{proposition}[No universal return from the destructive output] \label{prop:noreturn} The clock channel has no left inverse on all clock inputs. No fixed continuation supplied only with that output and input-independent new resources can universally restore the consumed unknown clock. \end{proposition} \begin{proof} For a fixed surviving state \(\rho_S\), the two inputs \[ \proj{\psi_+}\otimes\rho_S,\qquad \proj{\psi_-}\otimes\rho_S,\qquad \ket{\psi_\pm} =\frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2}, \] are different and have identical outputs. Every fixed continuation gives identical outputs on identical inputs, so it cannot restore both. \end{proof} This excludes a universal inverse on the full input space. It does not exclude recovery on a separately declared restricted code with additional accessible side information. \begin{remark}[What reference preservation does not say] Every local CPTP map leaves the unconditional marginal of an untouched reference unchanged. That property alone does not establish preservation of correlations between a message and the reference. The route theorem below proves the stronger identity on the complete payload--reference state. The destructive clock does not preserve arbitrary coherences between its measured outcomes. \end{remark} \begin{proposition}[Records are not surviving states] \label{prop:recordresidual} Equality of clock record laws does not determine equality of surviving quantum outputs. \end{proposition} \begin{proof} Inputs \(P_0\otimes\rho_S\) and \(P_0\otimes\rho'_S\), with \(\rho_S\ne\rho'_S\), have the same deterministic record and different surviving outputs. \end{proof} %===================================================================== \section{Regular transport and record readiness} \label{sec:route} %===================================================================== Choose a straight developed spatial segment of length \(\ell>0\) inside the retained convex regular tube. For \(n\ge1\), set \[ \mathbf x_j=\mathbf x_0+ \frac jn(\mathbf x_n-\mathbf x_0), \qquad j=0,\ldots,n. \] At station \(j\) install an equal \(p\)-qubit payload register \(A_j\). Prepare \(A_1,\ldots,A_n\) in independent blanks \(\omega\), independent of \(A_0\) and every reference. Fix a supplied developed speed parameter \(0<\upsilon\le1\). At hop \(j\), apply the permitted tick-boundary SWAP between \(A_{j-1}\) and \(A_j\), over developed duration \[ \Delta t=\frac{\ell}{n\upsilon}. \] The cross-port dependencies have the specified straight future-causal routes. Same-station dependencies are future timelike. \begin{theorem}[Complete regular route] \label{thm:route} For every input \(\rho_{A_0E}\), after hop \(j\) the complete route state is \begin{equation} \rho_{A_jE}\otimes \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}. \label{eq:route} \end{equation} The logical payload channel is the identity, including all reference correlations. All non-output route registers are fixed independent blanks. With departure at \(t_{\mathrm d}\), arrival is \begin{equation} t_{\mathrm d}+\frac{\ell}{\upsilon}, \qquad T_{\mathrm d}+\frac{\ell}{c_{\mathrm h}\upsilon}. \label{eq:arrival} \end{equation} For \(\upsilon=1\), the cross-port paths are null. For \(\upsilon<1\), they are timelike. \end{theorem} \begin{proof} A SWAP moves the entire current payload, including its reference correlations, into the next independent blank and leaves the old factor blank. Induction proves \eqref{eq:route}. Each spatial displacement has length \(\ell/n\) and duration \(\ell/(n\upsilon)\), so its developed slope is at most one. The supplied calibration \eqref{eq:signature} converts the duration into host units, giving \eqref{eq:arrival}. \end{proof} The route is the target-adapted I-B/X-Enter-ST mechanism \cite{IB,XEnterST}. It is not exact arbitrary-null attainment on an independently fixed lattice. The different lattice-family contract of En08-B remains a comparison \cite{En08B}. No continuous-time microscopic Hamiltonian for a spatially separated SWAP is derived. \subsection{A mathematical reference is not a delivered control} A record \(R\) can be included in \(E\) when proving \eqref{eq:route}. This proves preservation of its correlations with the transported payload. It does not move \(R\) to the receiver. For a diagonal classical register, copying has the canonical channel \[ \mathcal C_{\mathrm{copy}}(X) =\sum_k\bra kX\ket k\,\proj{k,k}. \] On a classical--quantum input it preserves the full correlation with the quantum residual while creating a new classical copy. Delivery of that copy requires a separately permitted causal wire. \begin{definition}[Control-readiness condition] \label{def:readiness} A conditional operation at a receiver may use a code only after: \begin{enumerate} \item a copy or transport map for that classical code has been specified; \item its actual communication path and storage are included in the certificate; \item its arrival precedes the control input in the external event order. \end{enumerate} Otherwise the purported adaptive operation is not a host-realizable instruction. \end{definition} The alternative is to perform the controlled operation at the sender before departure, or to include the record in the transported payload and charge the enlarged payload dimension. None is automatic. %===================================================================== \section{The multi-instrument composition law} \label{sec:composition} %===================================================================== \subsection{A finite instruction certificate} \begin{definition}[X-K finite composition certificate] \label{def:certificate} A certificate consists of: \begin{enumerate}[label=\textup{(\roman*)}] \item a finite rooted tree of public outcomes and a finite quantum-name context at every node; \item each node instrument of Definition~\ref{def:node}, with all allocations, memory, routes, releases, and classical controls included; \item the provenance of each clock under Definition~\ref{def:fresh}; \item every input, returned output, stored register, failure flag, and declared discard; \item a timed port graph satisfying Definition~\ref{def:readiness}; \item the branchwise occupancy, storage, operation-count, and duration ledgers; \item any ideal--implemented comparisons on explicitly matching interfaces. \end{enumerate} Every leaf has a complete output convention. \end{definition} The tree is indexed by accessible classical outcomes. It is not indexed by hidden environmental information. A public failure may terminate immediately or continue with a declared recovery or reporting branch. \subsection{Fine path maps and public history maps} Let a leaf public history be \(a=(a_1,\ldots,a_m)\), with path nodes \(v_0,\ldots,v_{m-1}\). For its Kraus refinement \(\mu=(\mu_1,\ldots,\mu_m)\), define \begin{equation} K_{a,\mu} =L_{v_{m-1},a_m,\mu_m}\cdots L_{v_0,a_1,\mu_1}. \label{eq:path} \end{equation} Each product is between its stated successive spaces. The fine index \((a,\mu)\) is not necessarily a list of physically observed outcomes. \begin{theorem}[Complete finite adaptive composition] \label{thm:compose} Every certificate of Definition~\ref{def:certificate} defines the complete history channel \begin{equation} \mathfrak C(X) =\bigoplus_{a\ \mathrm{leaf}} \sum_\mu K_{a,\mu}XK_{a,\mu}^\dagger, \qquad \sum_{a,\mu}K_{a,\mu}^\dagger K_{a,\mu}=I. \label{eq:completehistory} \end{equation} It is CPTP, including every untouched reference. For a finite reported label \(y=g(a)\), the selective output is the sum of the corresponding leaf blocks. Every failure leaf remains in that sum or in an explicitly retained failure class. The construction uses no consumed quantum name as a later input. \end{theorem} \begin{proof} At a leaf, the remaining channel is the declared terminal output map, which may be included as a final node. Suppose every child continuation is complete. Pull its Kraus completeness identity back through the parent maps: \[ \sum_{a,\mu} L_{v,a,\mu}^\dagger I_{\HH_{va}}L_{v,a,\mu} =I_{\HH_v}. \] Backward induction over the finite tree proves the second identity in \eqref{eq:completehistory}. The first expression is therefore completely positive and trace preserving. Tensor each path map with \(I_E\); the same completeness calculation proves the reference statement. Classical coarse-graining sums CP blocks and preserves their total trace. The declared failure convention excludes omitted leaves. The type checks and Definition~\ref{def:fresh} prevent any later factor from referring to a consumed name. \end{proof} A single Kraus path map per public history is generally insufficient for mixed preparations, discarded systems, or non-rank-one instruments. The sum over \(\mu\) in \eqref{eq:completehistory} is essential. \subsection{Typed concatenation} For a nonadaptive list of complete stage channels, with the full history and memory included in the successive spaces, \[ \Gamma_j:\Alg_{j-1}\longrightarrow\Alg_j, \qquad j=1,\ldots,m, \] the complete channel is \[ \mathfrak C=\Gamma_m\circ\cdots\circ\Gamma_1. \] For compatible cut indices \(i\le j\le k\), define \[ \mathfrak T(k,i)=\Gamma_k\cdots\Gamma_{i+1}. \] Then \begin{equation} \mathfrak T(k,j)\mathfrak T(j,i) =\mathfrak T(k,i). \label{eq:typedcomposition} \end{equation} This is an operational channel identity between specified cuts. A readout of \(C_1\), a new supply \(\mathcal E_2\), and a readout of \(C_2\) can occur in a product such as \[ \mathcal M_{C_2}\,\mathcal E_2\, \mathcal V\,\mathcal M_{C_1}. \] By contrast, \(\mathcal M_C\mathcal M_C\) is not well-typed on the destructive interfaces. The first map has no quantum \(C\)-output \cite{En09}. \subsection{An exact many-clock law} \begin{corollary}[Several supplied clocks without intervening coupling] \label{cor:manyclock} Let \(C_1,\ldots,C_m,S\) be supplied jointly at the initial cut, with no independence assumption. For successive destructive readout in the specified bases and no intervening coupling, the complete channel is \[ X\longmapsto \sum_{\mathbf k}\proj{\mathbf k}_R\otimes D_{\mathbf k}XD_{\mathbf k}^\dagger, \qquad D_{\mathbf k} =\bra{\tau^{(1)}_{k_1}}\cdots \bra{\tau^{(m)}_{k_m}}\otimes I_S. \] Discarding every record gives \(\Tr_{C_1\cdots C_m}X\), including all references. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:oneclock} to different live factors in succession. The resulting Kraus bra is \(D_{\mathbf k}\). Summing its diagonal blocks gives the joint partial trace. \end{proof} The joint law is not generally the product of the individual clock marginals. For example, \[ \rho_{\mathrm{same}} =\tfrac12(\proj{00}+\proj{11}),\qquad \rho_{\mathrm{opp}} =\tfrac12(\proj{01}+\proj{10}) \] have identical one-clock marginals and disjoint two-clock record laws. Their joint record TV distance is one. \subsection{Independent scheduling changes} \begin{proposition}[Interchange of independent instructions] \label{prop:interchange} Two adjacent instructions acting on disjoint live factors, with no control or shared-resource dependency, may be interchanged without changing the complete channel, after the fixed output permutation. This equality does not assert equality of their physical schedules or peak-storage ledgers. \end{proposition} \begin{proof} The two maps have tensor forms \(\Phi_A\otimes\id_B\) and \(\id_A\otimes\Psi_B\), whose compositions agree. Classical outcomes are retained in their respective factors. Occupancy and timing are separate data and can differ between the two schedules. \end{proof} This is finite schedule algebra, not a reconstruction of the prescribed cone \cite{En08A}. %===================================================================== \section{Occupancy, storage, and no enlargement} \label{sec:resources} %===================================================================== \subsection{Cut-by-cut ledgers} Let \(c\) range over all semantic cuts of all branches. For each declared qubit register \(A\), let \(q(A)\) be its qubit count and \(\mathbf1_c(A)\) indicate whether it is live and laboratory-owned at that cut. Define \begin{align} \Qlab(c)&=\sum_Aq(A)\mathbf1_c(A), &Q_{\mathrm{peak}}&=\max_c\Qlab(c), \label{eq:peak}\\ B_{\mathrm{peak}}&=\max_c B_{\mathrm{lab}}(c). \notag \end{align} Reserved accessible blanks count when they are allocated and held. Preinstalled supplier factors count from their actual supply time, not only from the moment of use. Returned registers and remote supplier holdings have their own storage function \(Q_{\mathrm{ret/sup}}(c)\). The untouched reference ledger is \[ \Qext(c)=q(E) \] when the reference consists of \(q(E)\) qubits. A total convention, if desired, is \[ Q_{\mathrm{total}}(c) =\Qlab(c)+Q_{\mathrm{ret/sup}}(c)+\Qext(c). \] These are different ledgers, not interchangeable bounds. A useful conservative specialization is \begin{equation} \Qlab(c) \le q_{\mathrm{clock}}(c)+q_{\mathrm{memory}}(c) +q_{\mathrm{work}}(c) +\sum_{\rho\in\mathcal R(c)}(n_\rho+1)p_\rho, \label{eq:reserve} \end{equation} where the summands refer to disjoint currently allocated registers. If two roles use the same physical allocation at different times, the actual indicator ledger, not simultaneous double counting, determines the peak. The cumulative clock supply count is separate: \[ Q_{\mathrm{clock,used}}(a) =\sum_{\text{clock uses on }a}q(C_v). \] It can be much larger than the maximum number of simultaneously live clock qubits. \begin{proposition}[Resource soundness] \label{prop:resource} The maxima in \eqref{eq:peak} are finite. A legal execution obeys the stated peak and storage bounds when every allocation, communication buffer, release, and returned output is included in its cut specification. \end{proposition} \begin{proof} There are finitely many branches and finitely many cuts, each with finitely many finite registers. Their specified indicator sums are therefore finite. Each allowed instruction changes the live-name and ownership lists by its declared input/output rule. Induction over an execution identifies the actual lists with the certificate lists at every cut. Their maxima are the asserted bounds. \end{proof} This is accounting, not an energy, work, heat, or Landauer theorem. \subsection{Fixed host parameters} Let \[ \mathsf H =(\mathcal M^\circ,G,\mathcal W,\mathcal L,\iota, c_{\mathrm h},\varepsilon,\chi, \text{switches},\text{regulator prescription}) \] denote the retained metadata. \begin{theorem}[No enlargement under a fitting certificate] \label{thm:noenlarge} If a certificate passes Definition~\ref{def:fit}, its composition uses only its declared finite registers, ports, and operations inside the retained laboratory. Throughout execution, \[ \mathsf H_{\mathrm{after}}=\mathsf H_{\mathrm{before}}. \] In particular: \begin{enumerate} \item the conical deficit, metric, worldsheet, and chart are unchanged; \item the host and analytical signature conventions and the time calibration are unchanged; \item the width remains the same positive \(\varepsilon\), with the same regulator and switches; \item no second cone, sharp-width limit, or global conical Green operator is introduced. \end{enumerate} \end{theorem} \begin{proof} The command types act on finite register interfaces. None has a host-update or regulator-update output. By induction, every command leaves \(\mathsf H\) unchanged. The location, timing, and capacity clauses of the fit contract guarantee that the finite register operations stay inside the supplied reserve. There is no limiting operation in this argument. \end{proof} A failed fit is a failed host-level admission of that protocol, not permission to enlarge the chart or increase a frozen capacity. This theorem does not claim that all finite trees fit one bounded reserve. %===================================================================== \section{Scheduled locality and hopwise metric containment} \label{sec:locality} %===================================================================== For a finite site set \(X\), represent every controller, record buffer, and quantum memory at its actual site. For a fixed padded schedule, let the complete tick channel be \[ \Gamma_s=\bigotimes_{B\in\mathcal B_s}\Gamma_{s,B}, \] where \(\mathcal B_s\) is a supplied partition. Inactive slots may be represented by placeholders, but those placeholders are not consumed quantum names available for later use. For output sites \(Y\), define \begin{equation} N_{t,t}(Y)=Y,\qquad N_{s,t}(Y)= \bigcup_{\substack{B\in\mathcal B_{s+1}\\ B\cap N_{s+1,t}(Y)\ne\varnothing}}B. \label{eq:ancestors} \end{equation} Outcome-dependent choices can be represented by controlled local channels. If alternative routes are padded into a common schedule, the dependency graph includes all possible active control wires. \begin{theorem}[Complete scheduled locality] \label{thm:locality} The complete network satisfies \begin{equation} (\Gamma_{s+1}^*\circ\cdots\circ\Gamma_t^*)(\Alg(Y)) \subseteq\Alg(N_{s,t}(Y)). \label{eq:locality} \end{equation} Equal input reductions on \(N_{s,t}(Y)\) give equal output reductions on \(Y\). For a conclusion involving an untouched reference \(E\), equality is required on the joint ancestor--reference input reduction. Replacing a complete channel outside the backwards dependency region cannot change the corresponding unconditional output, with the retained wiring fixed. If every dependency edge has a certified regular future-causal host path, each scheduled ancestor lies in the prescribed metric past of the output. \end{theorem} \begin{proof} A block disjoint from the pulled-back observable support acts on an identity factor. Its adjoint is unital, so that factor remains the identity. A meeting block enlarges support only within its own input interface. Backward induction proves \eqref{eq:locality}. Testing all output observables proves the reduced-state claim. Tensoring with the reference algebra gives the joint version. The same identity argument permits replacement of an excluded complete channel. Concatenate the separately certified future-causal paths along each dependency path to obtain metric containment. \end{proof} The same proof works directly on a finite acyclic port graph by reversing a topological order. This is the prescribed-support mechanism of I-B, En03-A, VII-E, and the complete-network comparisons \cite{IB,En03,VIIE,VIIJST}. Scheduled ancestry may strictly exceed actual influence. For example, two successive SWAPs on the same pair have cross-site dependency paths but total identity channel. \begin{remark}[Three excluded shortcuts] A cycle-averaged transport velocity does not certify every short-time dependency. A small Hamiltonian tail is not an exact causal cutoff. A small diamond error does not prove exact metric containment of an erroneous device. The present exact ideal containment uses the supplied path of every actual ideal dependency. \end{remark} Scheduler consistency does not select or rebuild a Lorentz cone \cite{En08A}. No cone-classification theorem is used here. %===================================================================== \section{Complete-channel and state-specific error bounds} \label{sec:errors} %===================================================================== For states and compatible finite channels use \[ \dtr(\rho,\sigma)=\frac12\norm{\rho-\sigma}_1, \qquad \ddia(\Phi,\Psi)=\frac12\norm{\Phi-\Psi}_\diamond. \] The diamond norm includes arbitrary untouched references. Channel pre- and postcomposition are contractive in this norm \cite{Watrous}. \subsection{Matching interfaces are part of the hypothesis} Ideal and implemented stages must include the same compared histories, memory, residual outputs, failure flags, and ownership boundary. Padding, pinching, encoding, and final coarse-graining are explicit maps, not invisible identifications. If a device has correlated memory, that memory belongs to the successive stage interface or to an enlarged declared process input. It is not an independent fresh per-stage error term. \begin{theorem}[Sequential complete-channel telescope] \label{thm:error} Let \[ \mathfrak C=D\Gamma_m\cdots\Gamma_1E, \qquad \widetilde{\mathfrak C} =\widetilde D\widetilde\Gamma_m\cdots \widetilde\Gamma_1\widetilde E \] be matched compatible channel compositions. If \[ \ddia(\widetilde E,E)\le e_{\mathrm{in}},\quad \ddia(\widetilde\Gamma_j,\Gamma_j)\le e_j,\quad \ddia(\widetilde D,D)\le e_{\mathrm{out}}, \] then \begin{equation} \ddia(\widetilde{\mathfrak C},\mathfrak C) \le \min\left\{1,\, e_{\mathrm{in}}+\sum_{j=1}^m e_j+e_{\mathrm{out}}\right\}. \label{eq:telescope} \end{equation} The same allowance controls every input--reference output state and every retained record marginal. \end{theorem} \begin{proof} Insert intermediate compositions replacing one factor at a time. Each difference has one changed factor, with channels before and after it. Their diamond norms are one, so each contribution is bounded by the corresponding factor difference. Sum, divide by two, and use that half-diamond distance between channels is at most one. Tensoring with a reference and applying to a state gives the complete trace-distance bound. Partial trace gives the record marginal bound. \end{proof} This is the standard telescope also printed in the supplied I-B, En08-B, X-J-ST, and finite transport comparisons \cite{IB,En08B,XJST,Oai283}. No independent synthesis or error estimate for a physical device is supplied by the telescope. \subsection{Adaptive errors: mutually exclusive branches} \begin{theorem}[Worst-path adaptive bound] \label{thm:adaptiveerror} Consider ideal and implemented versions of the same finite public-outcome tree, with matched node and terminal interfaces. At each nonterminal node \(v\), suppose its complete recorded node channels obey \[ \ddia(\widetilde\Gamma_v,\Gamma_v)\le e_v. \] Include terminal comparison maps as final nodes when needed. Then \begin{equation} \ddia(\widetilde{\mathfrak C},\mathfrak C) \le \min\left\{1,\, \max_{a\ \mathrm{leaf}}\sum_{v\prec a}e_v\right\}. \label{eq:patherror} \end{equation} An input preparation allowance is added once, before the tree. The bound does not require independent outcome laws. \end{theorem} \begin{proof} At a leaf let the remaining error bound be zero. At an internal node define recursively \[ b_v=e_v+\max_{a\in A_v}b_{va}. \] Let \(C_{va},\widetilde C_{va}\) be the complete continuations at its children. On the classical--quantum direct-sum output, the controlled continuation difference has half-diamond norm at most \(\max_a b_{va}\). Indeed, after adjoining a reference, a block-diagonal operator \(Z=\bigoplus_a Z_a\) has \(\norm Z_1=\sum_a\norm{Z_a}_1\), and \[ \sum_a \norm{((\widetilde C_{va}-C_{va})\otimes\id)(Z_a)}_1 \le 2\max_a b_{va}\sum_a\norm{Z_a}_1. \] The canonical pinching extension is trace-norm contractive, so the same bound applies on its full matrix presentation. Insert the ideal node channel between the two complete node-plus-continuation compositions. Contraction gives the node error \(e_v\), and the controlled-continuation estimate gives \(\max_a b_{va}\). Backward induction proves the recursion. Its root value is the maximum root-to-leaf sum. Cap at one. \end{proof} The comparison at a node is for its whole instrument, not an arbitrarily normalized successful branch. Replacing \(\max\) by favorable outcome probabilities would require a separate justified estimate. \subsection{What trace-distance estimates can prove} \begin{proposition}[State-specific telescope] \label{prop:stateerror} Fix a complete referenced input. Let \(\rho_j,\widetilde\rho_j\) be the ideal and implemented states on the matching successive interfaces. If \[ \dtr\bigl( (\widetilde\Gamma_j\otimes\id_E)(\rho_{j-1}), (\Gamma_j\otimes\id_E)(\rho_{j-1})\bigr)\le a_j \] and \(\dtr(\widetilde\rho_0,\rho_0)\le a_0\), then \[ \dtr(\widetilde\rho_m,\rho_m) \le\min\{1,a_0+\cdots+a_m\}. \] This is a bound for the specified input, not a diamond-norm claim. \end{proposition} \begin{proof} At each step insert \((\widetilde\Gamma_j\otimes\id_E)(\rho_{j-1})\) between the implemented and ideal next states. Contractivity bounds the propagated preceding error; the displayed hypothesis bounds the new term. Induct. \end{proof} The same argument applies to a specified prepared clock-state error, provided the compared error is on its actual joint state with any correlated memory, not merely on the clock marginal. \subsection{Two explicit error pitfalls} \begin{proposition}[Calibration on one state is not a channel bound] \label{prop:stateversusdiamond} Two channels can agree exactly on one tested state and have half-diamond distance one. \end{proposition} \begin{proof} On a qubit, \(\id\) and \(\Ad_Z\) agree on \(\proj0\). Their outputs on \(\proj+\) are \(\proj+\) and \(\proj-\), which are orthogonal. Thus their half-diamond distance is one. \end{proof} As a separate memory example, suppose an unreported classical memory bit \(M\) is uniform. A device emits \(M\) at two successive calls. Each one-call marginal is a uniform bit, just as for two independent fair coins. The two-call laws differ: the memory device returns only \(00\) and \(11\), while the independent device returns all four words. Their joint TV distance is \(1/2\). There is no zero-error second-stage comparison once the preceding record and the relevant memory are included. One-call marginal agreement cannot price the history channel. \subsection{Failure events, robust margins, and conditioning} \begin{corollary}[Retained-event and TV margins] \label{cor:events} If the complete ideal and implemented outputs differ by trace distance at most \(e\), then the probability of any retained event differs by at most \(e\). In particular, an event of ideal probability zero has actual probability at most \(e\). If two ideal record laws have gap \(\Delta\), and their implementations have complete output errors at most \(e_0,e_1\), then \[ \TV(\widetilde P_0,\widetilde P_1) \ge\max\{0,\Delta-e_0-e_1\}. \] \end{corollary} \begin{proof} An event is an effect on the compared output. The variational characterization of trace distance bounds its probability difference. For the second claim, take the record marginals and apply the reverse triangle inequality. \end{proof} An arrival, timeout, or enclosure-status event is covered only if its actual flag and interpretation are part of the compared complete output. A logical payload bound alone does not certify an unobserved arrival flag. \begin{proposition}[Conditioning requires a probability margin] \label{prop:conditioning} Let ideal and implemented complete recorded states have trace distance at most \(e\). For a retained event let its subnormalized blocks be \(X,Y\), of probabilities \(p,q\). Then \[ |p-q|\le e. \] If \(p>e\), then \(q>0\) and \[ \dtr(X/p,Y/q)\le\min\{1,e/p\}. \] No uniform normalized-branch bound follows when \(p\) can approach zero. \end{proposition} \begin{proof} Group the record into event and complement. Contraction gives \[ \norm{X-Y}_1+\norm{X_{\mathrm c}-Y_{\mathrm c}}_1\le2e. \] The second norm is at least \(|p-q|\), proving both \(|p-q|\le e\) and \(\norm{X-Y}_1+|p-q|\le2e\). When \(q>0\), \[ \norm{X/p-Y/q}_1 \le\frac{\norm{X-Y}_1+|p-q|}{p} \le\frac{2e}{p}. \] Divide by two and cap at one. \end{proof} None of these bounds validates a false supplied numerical enclosure. None is a finite-sample confidence statement. Such statements require a separate sampling and calibration model. %===================================================================== \section{An explicit two-readout, two-route certificate} \label{sec:example} %===================================================================== This section exhibits a nontrivial multi-instrument law rather than merely asserting that a tree exists. Its capacity requirements are stated, not inferred from finiteness. \subsection{Installed registers and schedule} Choose two regular ports \(A,B\), separated by a straight developed distance \(\ell>0\) in the existing tube. Let the available developed time window have length \(D>0\). Choose \(\beta>0\) and \(\ell>0\) small enough that \begin{equation} 6\beta+4\ell0\Longrightarrow\theta_k=T(e). \] Then at most one outcome has positive probability. A normalized exactly calibrated input exists only when one label equals \(T(e)\); the corresponding outcome is deterministic. For several fixed readout events, the same statement applies separately at each event. \end{proposition} \begin{proof} Distinctness permits at most one matching label. Normalization then gives its probability one. If no label matches, normalization is impossible. \end{proof} Nontrivial stochastic readout may instead use a history-event dictionary, or a classically selected reading event. In the latter case the event selector is itself part of the input and control certificate. This is the I-B distinction \cite{IB}. \subsection{The composition law does not imply relational dynamics} The identity \eqref{eq:typedcomposition} concerns external operational cuts. It does not prove that clock-conditioned matter states obey a common Schr\"odinger evolution. The single-clock covariance interpretation in En09-A requires its preparation stationarity and clock-basis covariance hypotheses \cite{En09}. Repeating a channel does not impose those hypotheses on subsequent prepared clocks or on their joint reference extensions. There is also no finite canonical time commutator. \begin{proposition}[Finite CCR limitation] \label{prop:CCR} Finite matrices \(T_C,H_C\) cannot satisfy \[ [T_C,H_C]=i\hbar I \qquad\text{when }\hbar\ne0. \] \end{proposition} \begin{proof} A finite matrix commutator has trace zero, whereas the right side has trace \(i\hbar\dim\HH_C\ne0\). \end{proof} This excludes that exact finite-dimensional relation. It is not a theorem excluding every possible clock observable or every future relational-time model. \subsection{Finite cuts do not determine a continuum interpolation} \begin{proposition}[A simple finite-cut non-determination witness] \label{prop:interpolation} A finite list of channel values at externally specified cuts does not in general determine the intermediate continuous-time family. \end{proposition} \begin{proof} Take one co-located qubit and finitely many sampling times \(s_0<\cdots0\) and its frozen operator and gauge contract remain independent and unrepaired \cite{XConvST,XEnterST}. A delivered report about that witness is not a new covariance identity. The compact scalar moment constructed in the supplied X-J-ST development is not identified with the missing whole-algebra convolution reference \cite{XJST}. Finite composition does not supply that identification. The relative classical BV structure of X-H-ST and X-I-ST is a separate comparison object \cite{XHST,XIST}. No interacting quantum differential, equation-space transport, or quantum master operation is imported into this register package. The X-D-ST sharp-limit question is likewise untouched: there is no width-removal or metric-admission operation in the command grammar. \subsection{Version-qualified standing stops} The plan's shorthand that H2, I3, and J2 remain unresolved needs a qualification already present in the supplied source material. XIV-H's H2 verdict is a historical unresolved ending for its named internal amplitude requirement. XIV-I subsequently constructs the narrower I2 internal retarded-pole amplitude package. That progress is not withdrawn here. The full joint characteristic map at I3 and J2 remains unconstructed in the supplied records \cite{CharacteristicRecords}. Thus no full characteristic coefficient is admitted, and no downstream quantum-admission exit is opened. The technical existence of a manuscript called XV-B is not denied. Its reported lifting and extension interfaces do not supply the absent graph-specific map or its quantum admission \cite{XVB}. The conditional prepared-insertion obstruction of XVI-N remains in its stated class; its Route B is not discharged by this composition theorem \cite{XVIN}. It is not a general prohibition of every prepared observable. The statement \[ \loc{O.1 = NOT_CLOSED} \] is retained for the XVI-O prescription and standing boundary supplied with the plan \cite{XVIO}. It is not generalized into a denial of every version-qualified subsector result under a different counterterm domain or prescription. \subsection{Other time-related and documentary comparisons} The harmonic-coordinate descent, lapse, incidence, source-weight, and scheduled-duration records named in the plan concern different objects. A classical lapse or one represented effect is not the finite clock of this paper. The supplied comparisons to VI/VI-B, XIX-A/C, XX-A, Em02-E, En10-A/B, and the Hawk07-A description remain documentary, at the level of the supplied materials \cite{TimeNeighbours}. No unsupplied full proof or version is invented. Oai283-A and Oai212-A provide documentary examples of explicit transport allowances and externally calibrated timers; their target models are not this conical host \cite{Oai283,Oai212}. Oai282-A's Ward certificate is not an instrument-composition theorem. OpenAI Families 282, 362, and 376 are documentary comparisons only \cite{OpenAIRecords}. No OpenAI family theorem, reported formalization, or catalogue verification is a premise of the present composition or error proofs. The Companion is a source interface, not a proof checker. %===================================================================== \section{Local Grade S release and prospective use} \label{sec:release} %===================================================================== \begin{definition}[Local Grade S for this manuscript] \label{def:grade} A released X-K-ST package consists of: \begin{enumerate} \item the retained host and regulator metadata; \item a finite typed instruction certificate; \item the complete public-history instrument with its Kraus refinements and all failures; \item the actual clock-supply, memory, output, and release declarations; \item causal control delivery and certified regular routes; \item the occupancy, external storage, duration, and operation-count ledgers; \item a resource and location fit, or explicitly stated unpassed fit inequalities; \item matching complete-channel or state-specific error hypotheses, with the appropriate conclusions. \end{enumerate} Only certificates that pass the host-level fit are called realized local Grade S experiments. \end{definition} \begin{theorem}[X-K-ST mathematical release] \label{thm:release} For the retained host of Assumption~\ref{ass:host}: \begin{enumerate}[label=\textup{(\roman*)}] \item each declared destructive clock has the complete channel identity of Theorem~\ref{thm:oneclock}; \item every finite well-typed instruction certificate has the complete adaptive history law \eqref{eq:completehistory}; \item no later quantum use of a clock follows from the consumed name or from its classical record; \item the regular route is the identity on the entire payload--reference state, while remote use of a record requires its own readiness certificate; \item every fitting certificate stays on the same host, chart, signature ledger, fixed positive width, and regulator; \item the complete network obeys scheduled locality, and metric containment follows only from its separately certified causal edges; \item matched complete-channel errors obey \eqref{eq:telescope} and the adaptive bound \eqref{eq:patherror}; specified-input trace errors have only the qualified state-specific conclusion; \item the explicit two-readout, two-route experiment has the complete law \eqref{eq:twoclock}, the stated nonzero failure branch, and the proved finite resource requirements; \item no classical obstruction, sharp-limit question, BV transport problem, or characteristic quantum admission is repaired by the composed instrument. \end{enumerate} A resource-fitting instance with these data is local Grade S. It is not continuum field acquisition, not gravity-generated entanglement, not a Page curve, not ultraviolet completion, and not spacetime emergence. In particular, it is not quantized relational time without an external clock or external scheduling time. \end{theorem} \begin{proof} Theorem~\ref{thm:oneclock} proves (i). Theorem~\ref{thm:compose} proves (ii), and its type audit with Proposition~\ref{prop:noreturn} proves (iii). Theorem~\ref{thm:route} and Definition~\ref{def:readiness} prove (iv). Theorem~\ref{thm:noenlarge} proves (v). Theorem~\ref{thm:locality} proves (vi). Theorems~\ref{thm:error} and \ref{thm:adaptiveerror}, with Proposition~\ref{prop:stateerror}, prove (vii). Theorem~\ref{thm:twoclock} and Corollary~\ref{cor:examplefit} prove (viii). Section~\ref{sec:boundaries} records precisely the non-interference in (ix). The external-time limitation follows from the declared inputs and Section~\ref{sec:time}; no stronger physical identification appears in the construction. \end{proof} \subsection{Novelty and potential impact} The general instrument and finite-channel results are established mathematics. The scoped advance is a single auditable package tying their exact finite semantics to consumed names, causal record readiness, complete memory, actual occupancy, and a fixed host. The worst-path error bound is useful when adaptive branches are mutually exclusive: it avoids charging all unused branches as if they executed together. Its force still depends on complete node comparisons and matched memory interfaces. The package can serve as a finite operational layer for later supplied tasks. Its usefulness is that the interface refuses unsupported promotions: a later author must provide an acquisition channel, an interacting product, a physical time model, or a geometric identification separately. \subsection{Successor requirements} A stronger successor must supply any additional construction it claims, rather than renaming \(\mathfrak C\): \begin{enumerate} \item a physical continuum-to-register acquisition channel and complete comparison, if field acquisition is claimed; \item an actual joint memory model, if stagewise marginal estimates are to be used; \item resource and causal certificates for additional routes or readouts; \item an externally independent time construction, if Problem 8 is to be advanced beyond this boundary; \item the separate analytic and admission chain, if a characteristic quantum coefficient is claimed. \end{enumerate} %===================================================================== \appendix \section{Debugging audit} \label{app:debug} %===================================================================== \begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}} \toprule Potential overstatement & Correction\\ \midrule \endfirsthead \toprule Potential overstatement & Correction\\ \midrule \endhead Finite adaptive composition is first proved here & X-I-ST already gives the skeleton. The present claim is the assembled typed, routed, resource-fitting certificate.\\[4pt] A new clock name is an independent random clock & Freshness is a type/provenance condition. Joint preparations may be correlated.\\[4pt] A consumed name can be restored by relabelling & A new output factor and its actual preparation or restoration map are required.\\[4pt] The record universally restores the clock & Two different clock superpositions have identical destructive outputs.\\[4pt] One path operator exists for every public history & General instruments require a Kraus refinement inside each public history.\\[4pt] Hidden Kraus labels may control later commands & Only declared accessible outcomes can select a continuation.\\[4pt] Single-clock marginals determine a multi-clock law & The correlated and anticorrelated examples have the same marginals and disjoint joint laws.\\[4pt] A successful branch is a deterministic process & Every failure continuation and terminal output is retained.\\[4pt] A joint state formula delivers the record & Remote control requires a causal record wire and arrival before use.\\[4pt] Identity of the reference marginal proves identity transport & Even replacement channels leave an untouched reference marginal unchanged. The route proves the full payload--reference identity.\\[4pt] A release removes the register from every cost & Returned and supplier storage remain in their own ledgers.\\[4pt] Every finite protocol fits a fixed reserve & Capacity, duration, permissions, and location fit are explicit additional checks.\\[4pt] Cumulative clock supply equals simultaneous peak & They are different counts. Every actual holding is charged at the cut where it occurs.\\[4pt] One tested trace error gives a diamond estimate & The identity/Pauli example disproves this. State-specific and uniform bounds are separated.\\[4pt] Per-stage marginals price correlated device memory & The memory and history must be in the step interfaces.\\[4pt] All branches should be added in the adaptive budget & The proved worst-path recursion charges only one root-to-leaf chain in the uniform bound.\\[4pt] A favorable branch probability discounts any node error & Such weighting needs a separate justified estimate; it is not assumed.\\[4pt] A small complete error controls normalized rare branches uniformly & Conditioning requires a positive event-probability margin.\\[4pt] A logical-only bound certifies arrival & The arrival or timeout flag must be included in the complete compared output.\\[4pt] A complete-channel bound validates an enclosure & The enclosure needs its own independent mathematical or numerical justification.\\[4pt] A telescope is a sampling confidence theorem & No sampling model or finite-shot theorem is introduced.\\[4pt] A schedule-derived cone is the retained cone & The metric and every causal edge certificate are supplied independently.\\[4pt] Signature reversal alone calibrates a duration & The time conversion uses the separately fixed relation \(t=c_{\mathrm h}(T-T_\Sigma)\).\\[4pt] No-enlargement is a width-limit theorem & It is an invariant-metadata statement under finite fitting commands.\\[4pt] Several stochastic labels equal one event time & Distinct exact event-time labels force a deterministic readout at that event.\\[4pt] The composition law removes external time & It is indexed by external cuts, locations, and durations.\\[4pt] H2 is globally unresolved despite the I2 advance & H2 is retained as XIV-H's historical ending; XIV-I's narrower constructed I2 result is not withdrawn.\\[4pt] O.1 is declared open in every prescription & Only the stated XVI-O/standing boundary is retained; other qualified subsectors are not reclassified.\\[4pt] A proposed source is a registered transaction & All integration fields below remain proposed or false.\\ \bottomrule \end{longtable} %===================================================================== \section{Open separation obligations} \label{app:obligations} %===================================================================== The following labels identify boundaries that X-K-ST does not cross. They are not unfinished steps needed for the finite composition proof itself. \begin{description}[style=nextline,leftmargin=1.5em] \item[\loc{composed-clock-not-relational-time}] X-K-ST does not construct quantized relational time without an external clock or external schedule. \item[\loc{fresh-clock-not-reuse}] X-K-ST does not return a consumed unknown clock as a new quantum input. A separately supplied factor is required. \item[\loc{transport-not-host-enlargement}] X-K-ST does not change the cone, chart, width, regulator, or signature ledger through transport. \item[\loc{route-not-acquisition}] X-K-ST does not turn a supplied finite report into continuum field acquisition. \item[\loc{composition-not-emergence}] X-K-ST does not derive the metric, causal cone, or Einstein dynamics from its composed instrument. \end{description} For each obligation, the technical XV-B release and the conditional XVI-N massless obstruction remain separate documentary boundaries \cite{XVB,XVIN}. No finite channel identity here supplies the missing quantum admission or repairs the prepared insertion. %===================================================================== \section{Proposed companion integration record} \label{app:integration} %===================================================================== This is proposed metadata, not an executed repository transaction. The actual schema and identifiers must be checked during any later integration. \paragraph{Proposed routing note.} X-K-ST assembles a finite multi-instrument law for successive fresh destructive clocks, complete histories, causal record delivery, regular routes, resource ledgers, and matched complete-channel errors on the unchanged I-B/X-A-ST host. It is not quantized relational time without an external clock, not continuum acquisition, not quantum BRST, and not a characteristic coefficient. The convolution obstruction is unrepaired and the three trees remain counted once. Documentary connections are not premises of the new proofs. \begin{verbatim} identifier: X-K-ST sourceStatus: proposed placement: after X-J-ST grade: local Grade S, subject to actual fit certificate track: ramp problemConnections: - relational - products host: retained: I-B / X-A-ST defect: worldsheet laboratory: regular supported reserve analyticalCompletion: declared Minkowski completion signatureLedger: iota-star G = -eta width: same fixed positive width regulatorReplaced: false hostUpdate: not constructed composition: finitePublicOutcomeTree: true krausRefinementsDistinguished: true freshClockProvenanceRequired: true consumedNameReuse: forbidden correlatedSupply: allowed only with declared joint interface failuresRetained: true untouchedReferencesIncluded: true causalRecordReadinessRequired: true errors: completeChannels: matched half-diamond comparisons adaptiveBudget: worst root-to-leaf sum stateSpecificErrors: separate qualified theorem hiddenMemoryMarginals: insufficient confidenceTheorem: not constructed resources: cutByCutOccupancy: required returnedStorage: separately charged supplierStorage: separately charged externalReferenceStorage: separate ledger actualCapacityAndTimeFit: required obligations: - composed-clock-not-relational-time - fresh-clock-not-reuse - transport-not-host-enlargement - route-not-acquisition - composition-not-emergence standingBoundaries: XV-B-quantum-admission-exit: not opened XVI-N: not discharged XVI-N-Route-B: not discharged XVI-O-O.1: NOT_CLOSED in its stated prescription XIV-H-H2: historical unresolved ending retained XIV-I-I2: narrower constructed advance not withdrawn I3-full-joint-map: not constructed here J2-full-joint-map: not constructed here characteristicQuantumCoefficient: not admitted timeLandmark: proposed: true basis: complete finite clock and multi-readout certificate externalScheduleRemoved: false emergenceStatus: not constructed includeInEmergenceSources: false registryEditsExecuted: false repositoryTestsExecuted: false compilationAsserted: false numericalExecutionAsserted: false machineVerificationAsserted: false independentExecutionAsserted: false \end{verbatim} No passing test count, connection-array length, time-index shift, hash, or post-registration freeze is asserted before an actual integration transaction exists. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship metadata.} The author field is omitted for anonymous review. Citations to N.~Cheung attribute the identified source records; they do not by themselves assign authorship of this submission draft. \paragraph{Scope and novelty.} The claimed contribution is the explicit assembly of finite instrument semantics, fresh-name discipline, causal record readiness, resource fit, and complete error comparison on a retained host. Standard instrument and channel-composition mathematics is not claimed as new. \paragraph{Source status.} The supplied X-I-ST source and X-J-ST development are cited as supplied texts, not as independently certified registrations. Internal titles, versions, and labels are used only at the documentary level available in the supplied materials or named source records. No internal publication date, DOI, or catalogue-wide verification is invented. \paragraph{Operational boundary.} Every declared surviving factor, public record, failure, and untouched reference is included in the stated comparison. Unobserved Kraus indices are not automatically accessible classical outcomes. A finite channel is not a microscopic apparatus, thermodynamic machine, or continuum detector. \paragraph{Verification.} No successful \loc{pdflatex} execution, numerical evaluation, proof-assistant verification, symbolic execution, repository installation, hardware synthesis, experiment, independent execution, independent referee approval, or journal acceptance is asserted. The mathematical claims are the explicit definitions and written proofs in this file. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{EnterPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \url{https://enter.grok.me/enter}. Interpretive map, not a theorem premise. \bibitem{EnterI} N.~Cheung, \emph{Manuscript I}. \loc{I.tex}, version 2. Enter Calculus Companion. \url{https://enter.grok.me/browse/I}. Class-relative characterization of destructive measurement; its hypotheses are not abbreviated into a new characterization here. \bibitem{IB} N.~Cheung, \emph{I-B: A Finite Enter Spacetime: Calibrated Destructive Time Readout on a Prescribed Conical Product}. \loc{I-B-v1.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/I-B}. Especially \loc{thm:instrument}, \loc{prop:fixed-event}, \loc{thm:transport}, \loc{thm:locality}, \loc{thm:metric-containment}, \loc{eq:peak}, and \loc{prop:noise}. \bibitem{XAST} N.~Cheung, \emph{X-A-ST}. Version 1, supplied predecessor record. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-A-ST}. Supported completion chart, fixed-width classical host contract, and target-adapted finite route. \bibitem{En09} N.~Cheung, \emph{Realization of Relational Time by Destructive Enter on a Clock Register}. \loc{En09-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En09-A}. Complete clock identity, typed one-event hybrid composition, separate covariance hypotheses, and retention of external scheduling time. \bibitem{XEnterST} N.~Cheung, \emph{X-Enter-ST: Readout closure of an obstructed convolution coefficient on a prescribed conical host}. \loc{X-Enter-ST.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/X-Enter-ST}. Complete supplied-report instruments, regular transport, and the separation between reporting and continuum acquisition. \bibitem{XHST} N.~Cheung, \emph{X-H-ST: Relative classical BV structure and complete instruments for supplied residual reports}. Version 1, supplied manuscript record. \url{https://enter.grok.me/browse/X-H-ST}. Documentary relative-classical and complete-interface comparison; no interacting quantum differential is imported. \bibitem{XIST} N.~Cheung, \emph{X-I-ST: A supported finite-order classical BV deformation on a prescribed conical Enter spacetime}. Supplied source \loc{X-I-ST.tex}. Especially \loc{thm:composition}, \loc{prop:reporterror}, \loc{thm:locality}, and \loc{thm:route}. Cited as the supplied source, without independent registration or compilation claims. \bibitem{XJST} \emph{X-J-ST: A second charged coefficient and a selective amplitude package}. Supplied development source in the present exchange. Especially \loc{lem:lipschitz}, \loc{thm:robust}, \loc{thm:composition}, and its distinction between a compact scalar moment and a whole-algebra convolution reference. No independent registration, execution, or authorship authorization is inferred from that development. \bibitem{En08A} N.~Cheung, \emph{Scheduler Consistency and Lorentz-Cone Representation: Countermodels, Explicit Geometric Hypotheses, and Causal Typing}. \loc{En08-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En08-A}. Schedule algebra does not by itself select a Lorentz cone. \bibitem{En08B} N.~Cheung, \emph{Resource-controlled operational approximation of null boundary transport}. \loc{En08-B.tex}, version 1. Version-qualified supplied manuscript passages. \url{https://enter.grok.me/browse/En08-B}. Lattice-family transport, complete references, and matched finite-channel telescoping; its permission contract is not silently identified with the target-adapted route. \bibitem{En03} N.~Cheung, \emph{En03-A}. \loc{En03-A.tex}, version 1. Enter Calculus Companion. \url{https://enter.grok.me/browse/En03-A}. Complete scheduled block interfaces, destructive readout distinctions, and backwards support. \bibitem{VIIE} N.~Cheung, \emph{VII-E}. Version 1, supplied manuscript record. \url{https://enter.grok.me/browse/VII-E}. Prescribed-schedule locality and observation boundaries. \bibitem{VIIJST} N.~Cheung, \emph{VII-J-ST}, version 1, and \emph{VII-J}, version 1. Version-qualified supplied passages. \url{https://enter.grok.me/browse/VII-J-ST}. Complete dependency networks and separate physical causal embedding. \bibitem{XFST} N.~Cheung, \emph{X-F-ST: Three fixed-width retarded trees and the background identity}. Version 1, supplied manuscript record. \url{https://enter.grok.me/browse/X-F-ST}. The earlier tree packet is retained once, not recomputed or duplicated by finite composition. \bibitem{XConvST} N.~Cheung, \emph{X-A-conv-ST}, version 1, with the separately identified \emph{X-A-conv} source. Version-qualified supplied manuscript records. \url{https://enter.grok.me/browse/X-A-conv-ST}. The frozen classical convolution obstruction remains independent. \bibitem{XIXC} N.~Cheung, \emph{XIX-C}. Version 1, supplied manuscript passages. \url{https://enter.grok.me/browse/XIX-C}. Finite interfaces and support certificates do not determine an unspecified continuum evolution. \bibitem{Oai283} N.~Cheung, \emph{Oai283-A}. Version 1, supplied manuscript passages. \url{https://enter.grok.me/browse/Oai283-A}. Resource-accounted transport and complete-output error allowances. No underlying OpenAI family proof is imported or verified. \bibitem{Oai212} N.~Cheung, \emph{Oai212-A}. Version 1, supplied manuscript passages. \url{https://enter.grok.me/browse/Oai212-A}. External regularity assertions and calibrated timers are separate from a physical time construction. \bibitem{CharacteristicRecords} N.~Cheung, \emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J}, version 1. Version-qualified supplied passages. Historical H2 boundary, constructed I2 internal amplitude, and unconstructed full joint maps at I3/J2. These are distinct documentary verdicts. \bibitem{XVB} N.~Cheung, \emph{XV-B}. Version 2, supplied technical manuscript passages, especially \loc{thm:release}. Lifting, reconstruction-null, and extension interfaces are not admission of the inherited characteristic quantum coefficient. \bibitem{XVIN} N.~Cheung, \emph{XVI-N}. Revision r12, supplied passages, especially \loc{thm:massless-obstruction}. The conditional prepared-insertion obstruction and its Route B boundary retain their stated scope. \bibitem{XVIO} N.~Cheung, \emph{XVI-O}. Revision o70-sync, supplied source descriptions. Kernel-counterterm construction under its prescription, with the stated O.1 \loc{NOT_CLOSED} boundary. No statement about every other prescription is inferred. \bibitem{TimeNeighbours} N.~Cheung, \emph{VI/VI-B}, \emph{XIX-A}, \emph{XX-A}, \emph{Em02-E}, \emph{En10-A/En10-B}, and the \emph{Hawk07-A} description. Documentary comparisons at the level supplied with the plan. Unavailable full texts, versions, and theorem statements are not reconstructed. These records are not folded into the composition proof. \bibitem{OpenAIRecords} OpenAI, \emph{Mathematics catalogue and related finite-interface descriptions}. Supplied documentary records, including Families 282, 362, and 376. No catalogue theorem, formalization claim, or family proof is a premise of this manuscript or independently verified here. \bibitem{DaviesLewis} E.~B.~Davies and J.~T.~Lewis, ``An operational approach to quantum probability,'' \emph{Communications in Mathematical Physics} \textbf{17} (1970), 239--260. doi:10.1007/BF01647093. \url{https://link.springer.com/article/10.1007/BF01647093}. External instrument framework. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author manuscript and errata: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. Finite channels, reference extensions, trace distance, and diamond-norm comparison. \bibitem{CDP} G.~Chiribella, G.~M.~D'Ariano, and P.~Perinotti, ``Theoretical framework for quantum networks,'' \emph{Physical Review A} \textbf{80} (2009), 022339. doi:10.1103/PhysRevA.80.022339. \url{https://arxiv.org/abs/0904.4483}. External sequential-network framework. No general network representation theorem is needed for the finite path-map proof here. \end{thebibliography} \end{document}