% !TEX program = pdflatex %===================================================================== % Landmark_XEnter.tex % % Standalone source for pdfLaTeX. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error Landmark_XEnter.tex % pdflatex -interaction=nonstopmode -halt-on-error Landmark_XEnter.tex % % Separate conclusions: % A: Closed, negative, for encodings of the convolution coefficient % a alone. % B: Conditional relational construction proved; neither classical % source supplies the independently charged reference required here. % C: Conditional quantitative trade-off; no amplitude cutoff follows % from either source's support condition. % D: Sharp limits and coordinate counterterms are excluded. % R: Comparison only. The ramp identity is a classical source result, % not a fifth closure and not the source of c_epsilon. % % Logical corrections retained or incorporated: % - Nonconstant encodings need not have visible computational readout. % - Endpoint coincidence does not establish orbit invariance. % - Classical laws do not determine selective quantum residuals. % - The relational circuit must merge its even-parity outcomes; % reporting both raw measurement bits is not invariant. % - Witness-orbit charge matching does not imply whole-algebra matching. % - Relative-phase readout is informative but not globally injective. % - Quantitative visibility requires specified tolerance and target. % - The surjective-orbit lemma does NOT imply that a function of an % already invariant scalar is constant across physically distinct % inputs: its surjective-orbit hypothesis is absent in that case. % - The ramp coordinate response Co_1[K xi] is not itself an invariant % scalar; it supplies the classical cancellation in the observable. % - A first-order identity is not promoted to an all-orders identity. % - No field-to-qubit preparation or physical reference is invented. % % No successful compilation or mechanized verification is claimed. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{2pt} \setlist{itemsep=3pt,topsep=5pt,leftmargin=*} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{hypothesis}[theorem]{Hypothesis} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sech}{sech} \DeclareMathOperator{\artanh}{artanh} \DeclareMathOperator{\Ad}{Ad} \newcommand{\R}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\Q}{\mathsf Q} \newcommand{\Bit}{\mathsf{Bit}} \newcommand{\Unit}{\mathbf 1} \newcommand{\Enter}{\mathsf{Enter}} \newcommand{\New}{\mathsf{new}} \newcommand{\CNOT}{\mathsf{CNOT}} \newcommand{\lett}{\mathsf{let}} \newcommand{\inn}{\mathsf{in}} \newcommand{\lto}{\multimap} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Law}{\mathsf{Law}} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\Class}{\mathscr P_4} \newcommand{\gC}{\mathfrak g_U^{\mathrm C}} \newcommand{\gR}{\mathfrak g_U^{\mathrm R}} \newcommand{\Co}{\mathrm{Co}_1} \newcommand{\ObsR}{\mathrm{Obs}_{f,g,\eps}^{\mathrm R}} \newcommand{\sigR}{\sigma_{\mathrm R}} \newcommand{\Irel}{I_{\mathrm{rel}}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\tnorm}[1]{\left\lVert #1\right\rVert_1} \newcommand{\ifte}[3]{% \mathsf{if}\ #1\ \mathsf{then}\ #2\ \mathsf{else}\ #3} \newcommand{\scopebox}[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{Landmark X-Enter}\\[0.4em] Readout closure tests for the obstructed convolution coefficient:\\ a named comparison with the ramp prescription} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We separate four closure questions for the fixed-regulator coefficient inherited from X-A--Convolution, the convolution manuscript. An encoding-independent orbit lemma proves that every invariant readout of that coefficient alone is constant: its inherited gauge orbit covers the entire real input domain. The retained amplitude encoding gives an explicit finite Enter witness, including its selective-record formula and second-order obstruction. Conditional on an independently supplied reference coefficient with a proved gauge charge, we construct a two-qubit relational readout. Its CNOT--Hadamard circuit has an invariant declared record after its even-parity outcomes are merged into a retained failure outcome. We prove the complete selective maps. Whole-algebra invariance requires an additional charge-matching hypothesis. A separately imposed bounded gauge segment gives quantitative readout bounds and an exact tolerance--visibility trade-off; no amplitude cutoff is inherited from the support condition. Sharp-regulator limits and coordinate counterterms remain excluded. A separate comparison names the Ramp Prescription manuscript. Its classical invariant-observable identity is not an Enter theorem, and its excluded-edge defect is not the convolution coefficient \(c_\eps\). No ramp identity is used as a premise of the four convolution releases. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Convolution coefficient}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Inherited gauge orbit}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A nontrivial invariant record}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {No encoding of that coefficient alone keeps a nontrivial invariant record.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{An invariant of the coefficient alone is constant. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:scope-X-Enter} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Convolution coefficient}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A nontrivial invariant record}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {No encoding of that coefficient alone keeps a nontrivial invariant record.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{An invariant of the coefficient alone is constant. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-X-Enter} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Convolution coefficient}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Inherited gauge orbit}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A nontrivial invariant record}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {No encoding of that coefficient alone keeps a nontrivial invariant record.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (c) -- (s); \end{tikzpicture} \caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:chain-X-Enter} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Convolution coefficient}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A nontrivial invariant record}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Inherited gauge orbit. No encoding of that coefficient alone keeps a nontrivial invariant record.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Convolution coefficient is not A nontrivial invariant record. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-X-Enter} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Convolution coefficient}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Inherited gauge orbit}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A nontrivial invariant record}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {No encoding of that coefficient alone keeps a nontrivial invariant record.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:gate-X-Enter} \end{figure} Figure~\ref{fig:split-X-Enter} records one view of the same separation. Figure~\ref{fig:chain-X-Enter} records one view of the same separation. Figure~\ref{fig:pair-X-Enter} records one view of the same separation. Figure~\ref{fig:gate-X-Enter} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-X-Enter} records that separation. The drawing adds no hypothesis. \scopebox{Four convolution scope lines and one comparison line}{% \textbf{A --- Closed, negative.} No encoding of the convolution coefficient \(a\) alone has an invariant record on its inherited whole orbit while retaining dependence on \(a\). \smallskip \textbf{B --- Conditional construction proved; application unresolved.} The relational programs work under the stated reference hypotheses. Neither named classical source supplies that reference. \smallskip \textbf{C --- Conditional trade-off; application unresolved.} The bounds give tolerance-based closure only under an independently justified amplitude cutoff. \smallskip \textbf{D --- Excluded scope entry, not a release.} No sharp-regulator limit or coordinate counterterm is performed. \smallskip \textbf{R --- Comparison only, not a fifth closure.} The ramp observable is already classically invariant on its stated algebra, at the order covered by its source theorem. That identity is not proved by Enter, and the ramp edge defect is not \(c_\eps\).} \tableofcontents %===================================================================== \section{Split classical premises and the readout boundary} \label{sec:premises} %===================================================================== Two classical manuscripts enter with different roles. We call them \emph{X-A--Convolution}, the convolution manuscript \cite{Convolution}, and \emph{Ramp Prescription}, the ramp manuscript \cite{Ramp}. These are descriptive source identifiers. The former supplies the obstruction tested in Options A--D; the latter supplies only the comparison in Section~\ref{sec:R}. We distinguish their source-specific supported gauge algebras by \(\gC\) and \(\gR\). Each superscript records which manuscript supplies the prescription and support conditions. Repeated generic symbols do not identify the two prescriptions, their tests, or their coordinate responses. \subsection{Convolution premises: the inputs of Options A--D} \label{subsec:convolution} Fix the regulator, switch, Cauchy prescription, observation test, and supported gauge algebra of X-A--Convolution. Its copied identities and positive witness are \begin{equation} \begin{gathered} \mathsf E_0 C T_{g,\eps}(K\xi)=S_\eps\xi, \qquad \xi\in\gC,\\ c_\eps= \int f(\xi_*-S_\eps\xi_*)^\mu\partial_\mu x^1>0, \qquad 0<\eps\leq1/8. \end{gathered} \label{eq:copied-convolution} \end{equation} Here the operators and tests retain their meanings in the convolution source; their kernels are not needed below. The action Noether identity is already zero on \(\gC\). The inherited first-order relational gap is \(X_{\xi-S_\eps\xi}\), and no term at order \(\kappa^2\) repairs that first coefficient \cite{Convolution}. Let \(a\) be its first evaluated smeared coefficient. The inherited witness orbit is \begin{equation} a(\lambda)=\lambda c_\eps,\qquad \lambda\in\R. \label{eq:orbit} \end{equation} Every real multiple \(\lambda\xi_*\) remains in \(\gC\). Consequently this coefficient orbit covers \(\R\). It is an orbit of the first-order coefficient, not a claim that a coupling expansion is uniform in arbitrarily large \(\lambda\). \subsection{Ramp premises: comparison only} \label{subsec:ramp} The separate ramp main theorem supplies the first-order coordinate identity \begin{equation} \Co[K\xi]=\xi \quad\text{for every }\xi\in\gR, \qquad \overline U\subset P_g\cap\{\eps0\) below. The ramp signature is \((+,-,-,-)\). The VII-A and VI materials quoted inside the ramp manuscript use the opposite signature and are not inputs here. No ramp kernel, scalar-vertex calculation, or handoff calculation is imported into Options A--D. \subsection{Declared output and exact closure} Landmark I characterizes destructive Enter as Born readout with destructive projective update within its stated strategy class \cite{I}. We retain this semantics. Its record theorem does not supply a preparation of an external field functional or a classical gravitational gauge identity. For an encoding \(\iota\) and a program \(P\), the closure boundary is the declared classical output together with every returned quantum output. Write its complete selective record as \(\Rec_P(\iota(a))\). Exact convolution-orbit invariance means \begin{equation} \Rec_P(\iota(a(\lambda))) = \Rec_P(\iota(a(0))) \quad\text{for every }\lambda\in\R. \label{eq:invariance} \end{equation} For differentiable records this is equivalent to vanishing orbit derivative everywhere, not merely at one representative. A record is \emph{informative} about its scalar input if it is nonconstant as a function of that scalar. Exact recovery is stronger. Whole-algebra invariance implies \eqref{eq:invariance}; failure on the convolution witness orbit therefore already defeats whole-algebra closure. No invariant convolution handoff is assumed. On this inherited coefficient orbit, \(a\) is pure gauge at fixed \(\eps\), not an observable that Enter has failed to invariantize. The separately quoted ramp cancellation does not change this premise. %===================================================================== \section{Frozen finite interface and selective formula} \label{sec:interface} %===================================================================== \subsection{Programs and records} The quantum type \(\Q\) denotes an owned qubit; \(\Bit\) denotes a classical bit; \(\Unit\) denotes the unit output. Tensor contexts split ownership. Conditionals use the same resources in their alternatives, exactly one of which executes. Let \(\Class\) be the class of finite first-order programs \[ P:\Q\lto\Bit^{\otimes m}\otimes\Q^{\otimes k} \] whose executions use at most four live qubits, including the input. Here \(m,k\) are nonnegative integers and \(\Q^{\otimes0}=\Unit\). Programs use fixed finite-register unitaries, fresh zero-state allocation, finite classical control, and destructive Enter only. Every quantum name is consumed or returned. Code and gates remain fixed during gauge comparisons. Complete histories are finite. Their path maps satisfy \begin{equation} \Phi_y^P(\rho) = \sum_{h:\,\operatorname{out}(h)=y}K_h\rho K_h^\dagger, \qquad \sum_hK_h^\dagger K_h=I. \label{eq:instrument} \end{equation} Indeed, fix a history and compose allocation isometries, gates, and destructive measurement bras. Branch normalizations cancel against their probabilities. Backward induction on the finite outcome tree, using \(\sum_b\ketbra b b=I\), gives completeness. Thus the \(\Phi_y^P\) are completely positive selective maps whose sum is trace preserving. Define \begin{equation} \widehat\Phi_P(\rho) = \sum_y\ketbra y y\otimes\Phi_y^P(\rho), \qquad \Rec_P(\rho)=\widehat\Phi_P(\rho). \label{eq:recordchannel} \end{equation} This instrument notation includes probabilities and conditional quantum outputs \cite[Section~2.3.2]{Watrous}. Failure branches are retained. No postselected renormalization defines the record. \subsection{Explicit amplitude encodings} Fix a scale \(\sigma>0\) before the convolution witness evaluation. Freeze the two encoding trials \begin{equation} p_j(a)=\frac{1+j\tanh(a/\sigma)}2,\qquad \iota_j(a)= \sqrt{1-p_j(a)}\ket0+\sqrt{p_j(a)}\ket1, \qquad j\in\{0,1\}. \label{eq:encoding} \end{equation} An attempt selects one trial once. The input register size is \(n=1\). The constant trial is \(\iota_0(a)=\ket+\). These are mathematical preparation interfaces, not field-to-qubit implementation certificates. Set \[ t(a)=\tanh(a/\sigma),\qquad s(a)=\sech(a/\sigma),\qquad D=\ketbra11-\ketbra00,\qquad X=\ketbra01+\ketbra10. \] For \(\tau_1(a)=\ketbra{\iota_1(a)}{\iota_1(a)}\), \begin{equation} \tau_1(a)=\tfrac12\bigl(I+t(a)D+s(a)X\bigr). \label{eq:state} \end{equation} Define \[ A_y=\Phi_y^P(I/2),\qquad B_y=\Phi_y^P(D/2),\qquad C_y=\Phi_y^P(X/2). \] The retained selective formula is \begin{equation} \Phi_y^P(\tau_1(a))=A_y+B_y t(a)+C_y s(a). \label{eq:selective} \end{equation} Its traces give the classical law: \begin{equation} \Law_P(a)(y)=\alpha_y+\beta_y t(a)+\gamma_y s(a), \quad (\alpha_y,\beta_y,\gamma_y) =(\tr A_y,\tr B_y,\tr C_y). \label{eq:law} \end{equation} Equal laws need not mean equal selective records. %===================================================================== \section{Option A: the encoding-independent negative closure} \label{sec:A} %===================================================================== \begin{lemma}[Surjective-orbit obstruction] \label{lem:orbit} Let \(\iota:\R\to\mathcal S\) be any encoding into a state space and let \(\Phi\) be any instrument with a declared record map. If \[ \lambda\longmapsto\Phi(\iota(\lambda c_\eps)) \] is constant, then \(\Phi\circ\iota\) is constant on \(\R\). \end{lemma} \begin{proof} For each \(a\in\R\), choose \(\lambda=a/c_\eps\). This is possible because \(c_\eps>0\). Orbit constancy then gives \(\Phi(\iota(a))=\Phi(\iota(0))\). \end{proof} No Born rule, dimensional bound, continuity, linearity, or amplitude encoding enters this proof. It applies equally to an encoding by density operators. A stochastic encoding, after averaging its input-dependent preparation distribution, still gives a state-valued function of \(a\); its output law is still a function of \(a\). Retaining a preparation label does not alter the argument if that label is itself supplied only as a function of \(a\). Consequently, exact readout closure for the convolution coefficient \(a\) alone is closed in the negative. Any informative invariant input must be supplied upstream by changing the input object, not by choosing a different readout of the same scalar. This conclusion does not authorize a coordinate counterterm. The lemma does not obstruct functions of several coefficients whose gauge orbit occupies only part of their joint domain. Option B changes the input in precisely that way. Nor does the lemma force a function of an already invariant scalar to be constant across different invariant values: such a scalar does not satisfy the lemma's surjective gauge-orbit hypothesis. %===================================================================== \section{The explicit convolution amplitude witness} \label{sec:witness} %===================================================================== The retained typed trial is \begin{equation} P_{\mathrm{read}} = \lambda q{:}\Q.\, \lett\,b=\Enter(q)\ \inn\ \langle b,*\rangle, \label{eq:read} \end{equation} of type \(\Q\lto\Bit\otimes\Unit\). It uses one live qubit. Born readout gives \begin{equation} \Pr(P_{\mathrm{read}}=1\mid a) = \frac{1+\tanh(a/\sigma)}2. \label{eq:readlaw} \end{equation} On the convolution witness, \begin{equation} \TV\bigl(\Law_{\mathrm{read}}(c_\eps), \Law_{\mathrm{read}}(0)\bigr) = \tfrac12\tanh(c_\eps/\sigma)>0. \label{eq:TVgap} \end{equation} For a classical displacement \(d\), the actual encoding difference is \begin{align} \iota_1(a+d)-\iota_1(a) ={}& \bigl(\sqrt{1-p_1(a+d)}-\sqrt{1-p_1(a)}\bigr)\ket0 \nonumber\\ &+ \bigl(\sqrt{p_1(a+d)}-\sqrt{p_1(a)}\bigr)\ket1. \label{eq:encdifference} \end{align} The induced record derivative is \begin{equation} \frac{\mathrm d}{\mathrm d\lambda} \Pr(P_{\mathrm{read}}=1\mid\lambda c_\eps) = \frac{c_\eps}{2\sigma} \sech^2(\lambda c_\eps/\sigma)>0. \label{eq:readderivative} \end{equation} At the flat representative its coefficient is \(c_\eps/(2\sigma)\). These calculations establish visibility for this encoding. Nonconstancy of a state encoding alone would not do so: \((\ket0+e^{ia}\ket1)/\sqrt2\) has constant direct Enter statistics. Likewise, a normalized pure-state encoding of all real inputs cannot be a faithful linear vector map \(a\mapsto av\). %===================================================================== \section{Selective obstruction and retained-order test} \label{sec:amplitudetheorem} %===================================================================== \begin{theorem}[Amplitude visibility or erasure] \label{thm:amplitude} For \(\iota_1\) and \(P\in\Class\), the following are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item The complete record is invariant on \eqref{eq:orbit}. \item \(B_y=C_y=0\) for every reported output \(y\). \item The complete record is independent of \(a\) on \(\R\). \end{enumerate} For the classical law alone, the corresponding condition is \(\beta_y=\gamma_y=0\) for all \(y\). \end{theorem} \begin{proof} Differentiate \eqref{eq:selective}: \begin{equation} \frac{\mathrm d}{\mathrm d\lambda} \Phi_y^P(\tau_1(\lambda c_\eps)) = \frac{c_\eps}{\sigma} \bigl(B_y s^2-C_y st\bigr), \label{eq:variation} \end{equation} where \(s,t\) are evaluated at \(\lambda c_\eps\). Distinct classical blocks cannot cancel. At \(\lambda=0\), invariance forces \(B_y=0\). At any nonzero \(\lambda\), \(st\ne0\), so it then forces \(C_y=0\). The converse follows from \eqref{eq:selective}. Taking traces proves the law statement. \end{proof} Endpoint coincidence is weaker than orbit invariance: \[ \Phi_y^P(\tau_1(c_\eps))-\Phi_y^P(\tau_1(0)) = B_y\tanh(c_\eps/\sigma) +C_y\bigl(\sech(c_\eps/\sigma)-1\bigr). \] These terms may cancel for a particular endpoint. \begin{theorem}[Second-order test] \label{thm:secondorder} Keep the encoding and program fixed. Suppose \[ a(\kappa,\lambda) = \kappa\lambda c_\eps+\kappa^2 b(\lambda)+O(\kappa^3), \qquad b(0)=0. \] If \[ \Rec_P(\tau_1(a(\kappa,\lambda))) - \Rec_P(\tau_1(0)) = O(\kappa^3) \] for every \(\lambda\), then \(B_y=C_y=0\) for every \(y\). The remainder assertion is pointwise in \(\lambda\), in trace norm on the finite-dimensional record space. \end{theorem} \begin{proof} Each selective block has expansion \begin{align} \Phi_y^P(\tau_1(a))-\Phi_y^P(\tau_1(0)) ={}& \frac{\kappa\lambda c_\eps}{\sigma}B_y \nonumber\\ &+\kappa^2\left( \frac{b(\lambda)}{\sigma}B_y -\frac{\lambda^2c_\eps^2}{2\sigma^2}C_y \right)+O(\kappa^3). \label{eq:secondorder} \end{align} The first coefficient forces \(B_y=0\); the second forces \(C_y=0\), independently of \(b\). \end{proof} This is a small-signal encoding test, not a new expansion of the X-A--Convolution prescription. It also detects records missed by a first-derivative test: Hadamard followed by Enter has \[ \Pr(1\mid a)=\frac{1-\sech(a/\sigma)}2 =\frac{a^2}{4\sigma^2}+O(a^4). \] Its linear response vanishes at zero, but it is not invariant. %===================================================================== \section{R: comparison with the ramp prescription} \label{sec:R} %===================================================================== The four statements in this section compare the named classical sources. They are not an additional closure release. Throughout, the ramp result is understood at the order covered by its main theorem; no higher-order conclusion is inferred from \eqref{eq:copied-ramp} alone. \subsection*{R1. An already invariant observable is not the convolution gap} On \(\gR\), the ramp coordinate response equals the gauge parameter: \(\Co[K\xi]=\xi\), both off shell and on the switched solution. Thus the first-order mismatch \(\xi-\Co[K\xi]\) vanishes on that algebra, supplying the classical cancellation in the ramp observable \cite[Theorem (main)]{Ramp}. The coordinate response itself is not an invariant numerical observable; it is the compensating response used in the cancellation. There is no inherited convolution-type defect orbit \(a(\lambda)=\lambda c_\eps\) for this invariant ramp coefficient. Lemma~\ref{lem:orbit} is therefore inapplicable to it. A function of an invariant number may vary across physically different invariant inputs while remaining constant along each gauge orbit. Invariance alone also does not prove that the ramp observable takes different physical values; that is a separate classical question. \subsection*{R2. Reading an invariant input does not prove its invariance} Let \(o_{\mathrm R}\) be a real numerical coefficient of \(\ObsR\) to which the ramp theorem's invariance conclusion applies. If that coefficient is supplied as an input, encoding it in \(\iota_1\) and running \(P_{\mathrm{read}}\) gives \[ \Pr(1\mid o_{\mathrm R}) = \frac{1+\tanh(o_{\mathrm R}/\sigma)}2. \] For gauge-related inputs with the same \(o_{\mathrm R}\), the encoded states, and hence their complete records, are equal. This is preservation of an upstream identity, not a proof of \eqref{eq:copied-ramp}, not cancellation of a defect by Enter, and not Option B. It asserts invariance of the numerical coefficient actually covered by the source result, not an unproved all-orders field functional. The field-to-qubit preparation remains missing; none is constructed. \subsection*{R3. The excluded edge supports the same negative lemma} Suppose, as a new comparison, that the ramp gauge algebra is enlarged to include the tested edge vector while its coordinate prescription is kept fixed. Real multiples of that vector remain in the enlarged algebra. For the tested first-order discrepancy, linearity and \eqref{eq:copied-edge} then give the coefficient orbit \[ e(\lambda)=e_0+\lambda d_{\mathrm{edge}}, \qquad d_{\mathrm{edge}}\ne0. \] Its image is \(\R\): for any \(x\), take \(\lambda=(x-e_0)/d_{\mathrm{edge}}\). The proof of Lemma~\ref{lem:orbit} therefore applies unchanged in substance. No readout of this gap alone can be both invariant on that enlarged orbit and informative about the gap. This is a comparison outside the frozen ramp algebra. The ramp manuscript did not admit the edge vector and then remove it after readout; its support restriction is part of its stated prescription. The edge integral is neither \(c_\eps\) nor an independently charged reference coefficient \(r\). \subsection*{R4. Neither a reference nor a repair transfers} The ramp handoff's single scalar vertex does not supply a second coefficient with a proved charge \(r_\eps\) \cite{Ramp}. The two-qubit program in Option B still waits for an independent reference and, for whole-algebra invariance, the matching condition \eqref{eq:matching}. It is not retargeted at \(\ObsR\). The ramp source likewise excludes the sharp-regulator limit. Its smooth ramp kernel is not the convolution defect. No Enter operation changes either \[ w^\mu=\mathsf E_0 C T_{g,\eps}h \qquad\text{or}\qquad \Co. \] A change to either coordinate prescription requires a new classical covariance argument, not a transfer of a readout theorem. %===================================================================== \section{Option B: conditional relational inputs} \label{sec:B} %===================================================================== \subsection{The missing classical reference} \begin{hypothesis}[Reference on the convolution witness orbit] \label{hyp:reference} An independently defined second smeared coefficient \(r\) is supplied, with a proved nonzero charge \(r_\eps\), such that \[ a(\lambda)=a_{\mathrm{phys}}+\lambda c_\eps,\qquad r(\lambda)=r_{\mathrm{phys}}+\lambda r_\eps. \] The definition and charge of \(r\) belong to the upstream classical model, not to the Enter program. \end{hypothesis} The subscript ``phys'' labels the chosen baseline values in this hypothesis; it does not separately prove their gauge invariance. Under the hypothesis, \begin{equation} \Irel=r_\eps a-c_\eps r =r_\eps a_{\mathrm{phys}}-c_\eps r_{\mathrm{phys}} \label{eq:invariant} \end{equation} is invariant on the witness orbit. This does not automatically prove invariance on all of \(\gC\). If the first-order charges for arbitrary \(\xi\in\gC\) are \(q_a(\xi)\) and \(q_r(\xi)\), the required stronger condition is \begin{equation} r_\eps q_a(\xi)-c_\eps q_r(\xi)=0 \quad\text{for every }\xi\in\gC. \label{eq:matching} \end{equation} Only then does the same combination satisfy the whole-algebra record identity in the first-order coefficient model considered here. Neither the reference nor this matching condition is supplied by the copied convolution premises or by the ramp comparison. The defect \(S_\eps\xi-\xi\) is not declared to be \(r\). A freely chosen compensating number is not an independently established reference coefficient. Nor is that convolution defect identified with the ramp observable or its edge integral. \subsection{Classical pre-combination} Conditionally on the reference, encode \(\iota_1(\Irel)\) with scale \(\sigma_I>0\), and use \(P_{\mathrm{read}}\). Its complete record, with unit quantum output, has law \[ \Pr(1)=\frac{1+\tanh(\Irel/\sigma_I)}2. \] It is invariant wherever \(\Irel\) is invariant and distinguishes different values of \(\Irel\) statistically. The scale \(\sigma_I\) has the units of \(\Irel\). All relational work occurs in the classical pre-combination. Enter supplies no new covariance identity. Informative here means dependence across different invariant values, not variation along one gauge orbit. \subsection{A two-qubit program without pre-combination} Fix the phase scales \[ \sigma>0,\qquad \sigma'=\frac{\sigma r_\eps}{c_\eps}\ne0. \] The second scale may be signed. Freeze the normalized input encoding \begin{equation} \ket{\psi(a,r)} = \frac12 \bigl(\ket0+e^{ia/\sigma}\ket1\bigr) \otimes \bigl(\ket0+e^{ir/\sigma'}\ket1\bigr). \label{eq:phaseencoding} \end{equation} Write \[ \alpha=a/\sigma,\qquad \beta=r/\sigma',\qquad \theta=\alpha-\beta=\frac{\Irel}{\sigma r_\eps}. \] Gauge translation adds the same phase \(\lambda c_\eps/\sigma\) to both \(\alpha\) and \(\beta\). Let \[ H=\frac1{\sqrt2} \begin{pmatrix}1&1\\1&-1\end{pmatrix}. \] CNOT uses the first qubit as control and the second as target. Define the finite decoder \[ d(b,e)= \ifte e {\ifte b{\langle1,0\rangle}{\langle0,1\rangle}} {\ifte b{\langle0,0\rangle}{\langle0,0\rangle}}. \] Its reported codes are \[ F=(0,0),\qquad +=(0,1),\qquad -=(1,0). \] Both input bits are consumed by guards, including in the failure branch. The program is \begin{equation} \begin{aligned} P_{\mathrm{rel}} ={}&\lambda z{:}\Q\otimes\Q.\, \lett\,\langle q_a,q_r\rangle=z\ \inn\\ &\lett\,\langle u,v\rangle =\CNOT(\langle q_a,q_r\rangle)\ \inn\\ &\lett\,b=\Enter(H(u))\ \inn\ \lett\,e=\Enter(v)\ \inn\ \langle d(b,e),*\rangle . \end{aligned} \label{eq:relprogram} \end{equation} Its type is \[ \Q\otimes\Q\lto\Bit^{\otimes2}\otimes\Unit. \] It uses two live qubits and the same four-qubit budget, but changes the input interface and is not a member of the original single-input class \(\Class\). \begin{theorem}[Conditional selective relational readout] \label{thm:relational} On the encoded family \eqref{eq:phaseencoding}, \(P_{\mathrm{rel}}\) has complete selective record \[ \Pr(F)=\frac12,\qquad \Pr(+)=\frac{1+\cos\theta}{4},\qquad \Pr(-)=\frac{1-\cos\theta}{4}, \] with the unit residual on every positive branch. It is invariant under Hypothesis~\ref{hyp:reference}. Under \eqref{eq:matching}, it is invariant on the whole convolution gauge algebra in the stated coefficient model. \end{theorem} \begin{proof} The amplitudes after CNOT and Hadamard, in raw output order \((b,e)\), are \[ \frac1{2\sqrt2} \begin{array}{c|c} 00&1+e^{i(\alpha+\beta)}\\ 10&1-e^{i(\alpha+\beta)}\\ 01&e^{i\beta}+e^{i\alpha}\\ 11&e^{i\beta}-e^{i\alpha} \end{array}. \] The raw \(e=0\) probabilities depend on \(\alpha+\beta\). Thus reporting both raw bits would not be invariant. For arbitrary two-qubit density operators, put \[ \ket{\phi_\pm}=\frac{\ket{00}\pm\ket{11}}{\sqrt2}, \qquad \ket{\chi_\pm}=\frac{\ket{01}\pm\ket{10}}{\sqrt2}, \qquad \omega_{\Unit}=\ketbra{*}{*}. \] The path formula gives the actual selective maps \begin{align} \Phi_F(\rho) &= \bigl(\bra{00}\rho\ket{00} +\bra{11}\rho\ket{11}\bigr)\omega_{\Unit}, \nonumber\\ \Phi_+(\rho) &=\bra{\chi_+}\rho\ket{\chi_+}\,\omega_{\Unit}, \nonumber\\ \Phi_-(\rho) &=\bra{\chi_-}\rho\ket{\chi_-}\,\omega_{\Unit}. \label{eq:relmaps} \end{align} In the first line the two fine maps with bras \(\bra{\phi_+},\bra{\phi_-}\) are added, not postselected. Their effects sum to \(\ketbra{00}{00}+\ketbra{11}{11}\). All three maps sum to a trace-preserving map. For the encoded input, the first line has weight \(1/2\). The other weights are \(\abs{e^{i\beta}\pm e^{i\alpha}}^2/8\), giving the stated law. More generally, let \[ G_\gamma=\operatorname{diag}(1,e^{i\gamma})^{\otimes2}, \qquad \Ad_{G_\gamma}(\rho)=G_\gamma\rho G_\gamma^\dagger. \] The failure effect commutes with \(G_\gamma\), and both \(\chi_\pm\) acquire the same phase \(e^{i\gamma}\). Consequently \[ \Phi_y\circ\Ad_{G_\gamma}=\Phi_y,\qquad y\in\{F,+,-\}. \] This proves equality of the displayed selective maps, not just an unexamined equality of effects. The reference hypothesis induces exactly this collective phase shift. Condition \eqref{eq:matching} supplies the same conclusion for arbitrary convolution gauge directions. \end{proof} Failure has probability \(1/2\) and remains reported. No branch is removed. The invariant boundary is the declared coarse record and unit residual, not an exported log of every raw bit. The circuit is informative about \(\cos\theta\), not globally faithful to \(\Irel\): phase periodicity and the evenness of cosine create aliases. Physical informativeness requires admissible invariant values with different cosines. Classical pre-combination avoids those statistical aliases. Neither realization is an exact single-run decoder of a real coefficient. \scopebox{Option B status}{% The conditional programs and their selective maps are proved. A named physical \(r\), its witness charge, and, for whole-algebra closure, the matching condition \eqref{eq:matching} remain missing. The convolution application is unresolved. The ramp comparison does not supply the missing input.} %===================================================================== \section{Option C: bounded-segment tolerance and visibility} \label{sec:C} %===================================================================== Declare a new restriction \[ |\lambda|\leq\Lambda,\qquad \Lambda>0,\qquad d=\Lambda c_\eps. \] This is not the original convolution gauge algebra. Plateau support alone cannot supply such a bound: multiplying a supported gauge vector by any real number preserves its support. The ramp algebra likewise supplies no \(\Lambda\), since every real multiple of a vector supported in its \(U\) remains supported there. This observation does not introduce a convolution defect into the already invariant ramp observable. For the direct convolution amplitude trial, \begin{equation} \sup_{|\lambda|\leq\Lambda} \TV\bigl(\Law_{\mathrm{read}}(\lambda c_\eps), \Law_{\mathrm{read}}(0)\bigr) = \frac12\tanh(d/\sigma) \leq\frac{d}{2\sigma}. \label{eq:segment} \end{equation} This is approximate closure, not invariance. To make the trade-off precise, prescribe a tolerance \(0<\delta<1/2\) and a desired visibility \(00\). Its flat-baseline visibility is \[ V_\sigma(A)=\tfrac12\tanh(A/\sigma). \] Both requirements hold exactly when \begin{equation} \frac{d}{\artanh(2\delta)} \leq\sigma\leq \frac{A}{\artanh(2v)}. \label{eq:scaleinterval} \end{equation} Indeed, monotonicity of \(\tanh\) makes the tolerance condition equivalent to the lower bound and the visibility condition equivalent to the upper bound. Thus a suitable scale exists if and only if \begin{equation} \frac{A}{d} \geq \frac{\artanh(2v)}{\artanh(2\delta)}. \label{eq:tradeoff} \end{equation} There is no unconditional statement that the signal must dominate the defect: the required ratio depends on the visibility target relative to the tolerated variation. At \(\delta=0\), no finite scale gives exact closure for \(d>0\). Calling \(A\) a physical signal requires independent provenance or a justified choice of representatives. A scalar input alone does not distinguish a physical displacement from a gauge displacement. \subsection{General programs and selective distance} Define \[ \Delta_P(a,a_0) = \tfrac12\tnorm{ \widehat\Phi_P(\tau_1(a))- \widehat\Phi_P(\tau_1(a_0))}. \] For \(J=[a_0-d,a_0+d]\), equation~\eqref{eq:variation} gives \begin{equation} \Delta_P(a_0+\lambda c_\eps,a_0) \leq |\lambda|c_\eps L_P(J), \qquad |\lambda|\leq\Lambda, \label{eq:programbound} \end{equation} where \[ L_P(J)= \sup_{a\in J} \frac1{2\sigma} \sum_y\tnorm{B_y s(a)^2-C_y s(a)t(a)}. \] This follows by integrating the derivative block by block. The trace norm of a block-diagonal Hermitian record is the sum of its block trace norms. For classical laws, replace the trace norms in \(L_P(J)\) by \(\abs{\beta_y s^2-\gamma_y st}\). There is also a program-independent estimate: \begin{equation} \Delta_P(a_0+h,a_0)\leq\frac{|h|}{2\sigma}. \label{eq:contractive} \end{equation} Indeed, a positive trace-preserving map contracts the trace norm on Hermitian differences. For \(Z=Z_+-Z_-\), positivity, the triangle inequality, and trace preservation give \[ \tnorm{\widehat\Phi_P(Z)} \leq \tr\widehat\Phi_P(Z_+)+\tr\widehat\Phi_P(Z_-) = \tnorm Z. \] This standard contraction principle is also recorded in \cite[Corollary~3.40]{Watrous}. Direct differentiation of \eqref{eq:state} gives \[ \tfrac12\tnorm{\tau_1'(a)} = \frac{s(a)}{2\sigma} \leq\frac1{2\sigma}. \] Integration proves \eqref{eq:contractive}. For arbitrary encodings, constancy on a bounded segment need not imply constancy outside it. For the retained amplitude family, however, exact selective constancy on any nontrivial segment still forces \(B_y=C_y=0\). After dividing its zero derivative by \(s(a)^2>0\), one obtains \[ B_y=C_y\sinh(a/\sigma) \] throughout the segment; two different values of \(a\) force both operators to vanish. The useful alternative here is approximate closure, not an exact loophole in Option A. \scopebox{Option C status}{% The convolution estimates are proved under a declared cutoff. Neither source's support condition justifies \(\Lambda\). Until an amplitude restriction is independently established, the cutoff application remains unresolved. The bounds are not a new closure test for the ramp observable and do not reopen the convolution whole-orbit conclusion.} %===================================================================== \section{Option D: excluded changes of prescription} \label{sec:D} %===================================================================== The convolution regulator remains fixed. Positivity of \(c_\eps\) at fixed \(\eps\) does not itself prove any limit as \(\eps\to0\). If a later classical analysis establishes a vanishing-defect limit, that analysis concerns a different release. The ramp source also excludes a sharp-regulator limit. No such limit enters the present arguments. Subtracting a term based on \(S_\eps\xi-\xi\) from the convolution coordinate prescription changes \[ w^\mu=\mathsf E_0 C T_{g,\eps}h. \] Changing the ramp coordinate coefficient likewise changes its classical prescription. No Enter operation acts on either definition. Either counterterm requires its own upstream construction and covariance proof; it is not a measurement update. These possibilities are excluded scope entries, not conditional readout releases of this paper. The ramp identity provides no permission to replace one prescription with the other. %===================================================================== \section{Retained diagnostic branches and exact decoding} \label{sec:diagnostics} %===================================================================== The programs in this section retain the convolution input \(\iota_j(a)\). They are not retargeted at \(\ObsR\). \subsection{Erasure} With \(\iota_0\), every program receives the same state. Invariance is bought by ignoring the coefficient. With \(\iota_1\), deliberate erasure remains well typed: \[ P_{\mathrm{erase}} = \lambda q{:}\Q.\, \lett\,b=\Enter(q)\ \inn\ \ifte b{\langle0,*\rangle}{\langle0,*\rangle}. \] The input qubit and its measured bit are consumed. The constant output does not invariantize an informative record. \subsection{A carried qubit is not a comparison} The retained program \[ P_{\mathrm{keep}} = \lambda q{:}\Q.\, \lett\,r=\New\ \inn\ \lett\,b=\Enter(r)\ \inn\ \langle b,q\rangle \] uses two live qubits. Its classical law is surely \(b=0\), but \[ \widehat\Phi_{\mathrm{keep}}(\tau_1(a)) = \ketbra00\otimes\tau_1(a) \] varies. It carries the input as a quantum reference without performing a relational comparison. Its constant bit does not establish selective closure or supply Option B's missing classical reference. The locally bound register name \(r\) in this program is not the second classical coefficient required in Hypothesis~\ref{hyp:reference}. \subsection{No exact decoder of the convolution witness pair} \begin{proposition}[Exact-decoder obstruction] \label{prop:decoder} No program in \(\Class\) with a fixed classical decoder returns \(0\) surely at \(a=0\) and \(c_\eps\) surely at \(a=c_\eps\) under encoding \(\iota_1\). \end{proposition} \begin{proof} Let \(E\) be the sum of effects for outcomes decoded as \(c_\eps\). Then \(0\leq E\leq I\), and exact decoding would require \[ \bra{\iota_1(0)}E\ket{\iota_1(0)}=0,\qquad \bra{\iota_1(c_\eps)}E\ket{\iota_1(c_\eps)}=1. \] Positivity implies \[ E\iota_1(0)=0,\qquad E\iota_1(c_\eps)=\iota_1(c_\eps). \] Since \(E\) is Hermitian, their inner product would vanish. Both vectors have strictly positive real components, so their inner product is positive, a contradiction. \end{proof} Statistical visibility therefore does not mean literal real-valued output in one execution. This distinction also applies to the conditional relational realizations and to readout of an already invariant ramp input. %===================================================================== \section{Decision table and non-identification ledger} \label{sec:decision} %===================================================================== \begin{center} \small \renewcommand{\arraystretch}{1.13} \begin{tabularx}{\linewidth}{@{}p{.16\linewidth}p{.24\linewidth}XX@{}} \toprule Line & Input & Result & Release or scope status\\ \midrule A & Convolution \(a\) alone, any encoding & Whole-orbit invariance implies constancy & Closed, negative\\ A witness & Convolution \(\iota_1\), \(\Class\) & \(B_y=C_y=0\), or selective record varies & Explicit finite special case\\ R, ramp algebra & Already invariant coefficient of \(\ObsR\) & No defect orbit to close & Comparison; not an Enter theorem\\ R, edge enlargement & Ramp edge gap with a surjective real orbit & Same negative orbit lemma & Comparison only; not the frozen algebra\\ B & Independent \(r\) with proved charge & Invariant combination; circuit proved & Conditional; ramp does not supply \(r\)\\ B, full algebra & Reference satisfying \eqref{eq:matching} & Convolution whole-algebra record invariance & Additional conditional result\\ B without \(r\) & Only the copied classical premises & Reference input absent & Unresolved application\\ C & Justified new bound \( |\lambda|\leq\Lambda \) & Convolution tolerance--visibility trade-off & Approximate closure only\\ C without bound & Either original supported algebra & No amplitude cutoff follows & Unresolved cutoff application\\ D & Either sharp limit or either counterterm & Changes a prescription & Excluded; no release\\ \bottomrule \end{tabularx} \end{center} The conclusions are not combined into one ``closed'' statement. Option A is universal for inputs consisting of the convolution coefficient \(a\) alone. Option B changes that input and places the necessary relational information upstream. Option C changes the allowed comparisons. Option D changes a classical prescription. Line R compares source conclusions and does none of these things. In particular, the ramp edge integral is not the convolution positive witness, is not a reference charge, and is not encoded as the ramp observable. The ramp identity neither weakens the convolution obstruction nor supplies the matching condition for the relational circuit. No new regulator, quantum field product, cone, token, or coordinate inverse is introduced. Destructive updates are not metric counterterms. Adaptive control is covered by the finite path maps and receives no defect oracle. Replacing the Born strategy would leave the frozen Enter semantics. Equality is always asserted at a declared output boundary. Coarse-graining internal classical outcomes is allowed and is explicitly represented by summing selective maps. Unreturned quantum names are not silently discarded. No claim is made that every possible apparatus log is invariant. \paragraph{Conclusion.} The convolution single-coefficient closure question has a complete negative answer. The amplitude trial exposes that answer operationally. An invariant relational readout is available conditionally when an independently established reference supplies the required charges; the circuit is proved, but that classical input has not been supplied. Bounded-segment estimates give approximate closure only under an additional restriction. The ramp manuscript supplies a separate classical cancellation on its frozen algebra and a nonzero defect outside it. Enter neither proves the former nor cancels the latter. Neither comparison repairs the frozen X-A--Convolution prescription, and neither is a fifth closure release. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Logical dependencies.} The convolution intertwining identity, zero action identity, positive witness, and real coefficient orbit are copied from X-A--Convolution. They are the classical premises of Options A--D. The ramp coordinate identity and edge integral are separately quoted from Ramp Prescription, Theorem (main) and its edge proposition, for comparison only. No ramp identity is reproved or used as a premise of Options A--D. Landmark I supplies the characterized Enter semantics. The orbit lemma, selective amplitude tests, conditional relational circuit, and quantitative bounds are proved here. \paragraph{Perturbative scope.} The surjective orbit is an orbit of a first-order evaluated coefficient. The second-order readout test is pointwise in the gauge amplitude. No uniform coupling expansion for unbounded gauge amplitudes and no unquoted all-orders ramp identity is asserted. \paragraph{Implementation boundary.} The encodings are explicit mathematical interfaces. No physical device evaluating either external field functional, preparing its encoded state from fields, or supplying the independent reference coefficient is constructed. Reading an already invariant input is distinguished from proving its classical invariance. \paragraph{Source identification.} The convolution and ramp bibliography entries use separate descriptive manuscript identifiers. Their quoted mathematical premises are reproduced above. Publication dates, definitive titles, and other bibliographic metadata not supplied with those excerpts are not invented. \paragraph{Verification status.} No empirical data, numerical experiment, mechanized verification, or successful source compilation is claimed. The source uses standard \LaTeX{} packages and requires no external graphics or bibliography database. %===================================================================== \begin{thebibliography}{9} \addcontentsline{toc}{section}{References} \bibitem{Convolution} \emph{X-A--Convolution: convolution manuscript}. \newblock Companion source for the convolution prescription, its intertwining identity, zero action identity, and positive witness coefficient \(c_\eps\). \newblock The premises used here are reproduced in Section~\ref{subsec:convolution}. \newblock Descriptive manuscript identifier; additional publication metadata were not supplied. \bibitem{Ramp} \emph{Ramp Prescription: ramp manuscript}. \newblock Companion source, Theorem (main) and edge proposition. \newblock Used only for the supported identity \(\Co[K\xi]=\xi\), the stated classical observable cancellation, and the excluded-edge integral. \newblock The copied premises appear in Section~\ref{subsec:ramp}; their comparison appears in Section~\ref{sec:R}. \newblock Descriptive manuscript identifier; additional publication metadata were not supplied. \bibitem{I} N.~Cheung. \newblock \emph{Born--L\"uders as the unique measurement strategy of a strictly linear quantum \(\lambda\)-calculus: The Enter Calculus}. \newblock Landmark I, source \texttt{I.tex}. \newblock Used for the stated strategy characterization, ownership typing, and finite Born path semantics. \bibitem{Watrous} J.~Watrous. \newblock \emph{The Theory of Quantum Information}. \newblock Cambridge University Press, 2018. \newblock Section~2.3.2 and Corollary~3.40. \newblock Standard background for selective instruments and trace-norm contraction. \end{thebibliography} \end{document}