%===================================================================== % III-B.tex % % Classical relational descent in a conical Enter laboratory % % Build: % pdflatex -interaction=nonstopmode -halt-on-error III-B.tex % pdflatex -interaction=nonstopmode -halt-on-error III-B.tex % % Self-contained source: % - no external graphics; % - no external bibliography database; % - no shell escape; % - no executable code is required by the proofs. % % Principal debugging corrections: % - The full punctured conical product is not treated as globally % hyperbolic. % - Classical evolution is proved in an explicitly supported regular % laboratory, not transferred to every field on the complete cone. % - Newton inversion is an auxiliary developing-chart prescription, % not an undeclared inverse of the global conical Laplacian. % - The retarded gauge group and the residual-wave group are different. % - C1 denotes the triangular model, not C^1 differentiability. % - A finite register carries a declared evaluation encoding, not an % unconstructed representation of the continuum field quotient. % - Clock readout is separated from event-time and history calibration. % - Enter consumes clock names, not a geometric dimension. % - Remote record-controlled operations require causal record wires. % - Technical XV-B manuscripts are separated from quantum admission. % % Verification disclosure: % No successful pdfLaTeX build, proof-assistant run, numerical execution, % physical compilation, laboratory implementation, or journal acceptance % is asserted in this source. % % Bibliographic consultation: % Linked III and Enter records and published primary-source metadata % were consulted on 16 October 2026. Other companion citations identify % the supplied versioned documentary records. %===================================================================== \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={III-B: Classical relational descent in a conical Enter laboratory}, pdfsubject={A local Grade S adaptation with a complete destructive clock interface} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \setlist{itemsep=3pt,topsep=5pt} \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{1} \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\NN}{\mathbb N} \newcommand{\HH}{\mathcal H} \newcommand{\OO}{\mathcal O} \newcommand{\JJ}{\mathcal J} \newcommand{\XX}{\mathcal X} \newcommand{\SSol}{\mathcal S} \newcommand{\GG}{\mathcal G} \newcommand{\DD}{\mathcal D} \newcommand{\dd}{\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\Glin}{G^{(1)}} \newcommand{\GradeS}{\mathsf{Grade\ S}} \newcommand{\GradeE}{\mathsf{Grade\ E}} \newcommand{\COne}{\mathsf C_1} \newcommand{\Mclock}{\mathcal M_C} \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{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\SourceLabel}[1]{\texttt{\detokenize{#1}}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \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{III-B: Classical relational descent\\ in a conical Enter laboratory}\\[4pt] \large A local Grade S adaptation with destructive clock readout} \author{} \date{} \begin{document} \maketitle \begin{abstract} We construct a local operational adaptation of the classical descent problem of Manuscript III on the prescribed conical spacetime of I-B. A necessary domain correction is made first: the full regular conical product is not globally hyperbolic. Our evolution and quotient results therefore concern an explicitly supported finite-propagation laboratory inside a regular developing chart, with no incoming boundary data and no interaction with the singular worldsheet. Within this laboratory we prove the exhibited Poisson-symbol obstruction, the local Killing classification of massive free-shell tangency, first-order jet descent with a past-trivial harmonic section, and a Hausdorff well-posed quotient for the vacuum-metric triangular model. We construct compact coupled Ward pairs with strictly positive on-shell value and distinguish them from an adjoint-exact family that vanishes on shell. A complete destructive clock instrument is attached through explicit time, rod, and causal dictionaries. We prove its reference-preserving identity and construct a finite evaluation encoding of the descended observables. An explicit three-qubit pilot realizes four compact Ward evaluations with integer values \(j(k+1)\), \(0\le j\le7\), \(0\le k\le3\). Target-adapted SWAP transport and causal classical record wires give an exact finite endpoint-channel certificate. The release is Grade S under the definition given here. The geometry, equations, support restrictions, and finite encoding are inputs. No global conical Cauchy theorem, continuum field instrument, sourced Einstein--scalar completion, derived metric, or inherited quantum coefficient is constructed. \end{abstract} \statementbox{Main result and its two levels}{% \[ \boxed{ \begin{gathered} \text{specified local classical equation spaces} \longrightarrow \text{proved quotient functionals},\\ \text{finite evaluation encoding} +\text{destructive clock instrument} +\text{causal transport}\\ \longrightarrow \text{an exact finite operational comparison}. \end{gathered}} \] The second line does not quantize the first. A register encoding of selected evaluations is not a representation of the entire field quotient.} \statementbox{Domain correction}{% The complete punctured conical spacetime is not assumed to possess a global retarded inverse or a global Cauchy isomorphism. All positive evolution and quotient theorems below have the laboratory support contract of Definition~\ref{def:laboratory}.} \tableofcontents %===================================================================== \section{Introduction: descent before operational interpretation} \label{sec:introduction} %===================================================================== A relational expression must first be a function of the equivalence class that it is intended to describe. A measurement interface cannot repair a failure of that set-theoretic requirement. Conversely, classical descent alone does not construct an operator, a quantum field apparatus, or a record. Manuscript III separates four classical equation-space questions \cite{CheungIII}. I-B supplies a finite experiment on a prescribed conical background \cite{CheungIB}. This paper gives a precise local composition of those structures. Its new interface is deliberately finite: selected values of the proved classical functionals are encoded in finite matrices and read by a complete instrument. The classical results are reproved for the declared laboratory rather than imported as global statements about the cone. The finite instrument results are also proved directly. Companion citations identify origins and comparisons; no unwritten companion theorem is a premise. \subsection{Enter background and philosophy} Cheung's Enter calculus treats destructive readout as an operation that consumes the measured quantum names and returns a classical record, with the surviving quantum context explicitly typed \cite{CheungEnter,CheungI}. The accompanying philosophy distinguishes readout from quantum recovery, a channel from a thermal machine, and an analogy from an identification \cite{CheungPhilosophy}. We adopt finite Born and projective-update semantics. We do not claim a new derivation of those semantics outside the structural class of the Enter characterization. In particular: \begin{enumerate}[label=\textup{(\roman*)}] \item a consumed clock register is not a destroyed coordinate axis; \item a classical time label is not a surviving quantum clock; \item a residual register carrying evaluation data is not automatically a scalar quantum field; \item a causal schedule on a supplied metric is not a derivation of that metric. \end{enumerate} Finite instruments and reference extensions are understood in the usual completely positive framework \cite{DaviesLewis,Watrous}. Every instrument identity needed for the release is established below. \subsection{The grade used in this paper} \begin{definition}[Local Grade S adaptation] \label{def:grade} A local \(\GradeS\) adaptation consists of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed Lorentzian background, local rods, time orientation, and explicit analytical domain; \item named classical equations, gauge classes, topologies, and proved descent statements on that domain; \item a finite named quantum experiment with a complete destructive clock instrument and explicit calibration dictionaries; \item a declared finite encoding of selected descended evaluations, with a complete instrument proof; \item a reference-preserving transport and resource certificate whose complete dependencies are causally embedded. \end{enumerate} This definition does not require geometry, field equations, or the encoding to be inferred from records. \end{definition} The phrase \(\GradeE\) will refer only to the stronger, unconstructed emergence programme described in Section~\ref{sec:emergence}. Neither grade is an external accreditation. \subsection{The four classical rows} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.27\linewidth}XX@{}} \toprule Equation space & Identification tested & Result here\\ \midrule Vacuum metric and unchanged free scalar & Only \(h\) is quotiented & One exhibited Poisson symbol fails to descend.\\ \addlinespace Unchanged massive free scalar shell & Proposed scalar variation \(X_\xi\phi\) & Local tangency for every admitted solution is equivalent to local Killing symmetry.\\ \addlinespace First-order scalar jet & \((h,\phi_0,\phi_1)\mapsto (h+K\xi,\phi_0,\phi_1+X_\xi\phi_0)\) & A past-trivial harmonic section gives descent.\\ \addlinespace Fixed-background triangular model \(\COne\) & Constrained harmonic data modulo residual waves & A Hausdorff Cauchy quotient and nonzero compact coupled Ward observables.\\ \bottomrule \end{tabularx} \end{center} The second row is not a quotient by arbitrary compact gauges: the proposed transformations generally fail to preserve its equation space. The notation \(\COne\) names a model. It does not mean that the fields have only \(C^1\) regularity. All classical fields in this paper are smooth. %===================================================================== \section{Prescribed geometry and the local analytical domain} \label{sec:geometry} %===================================================================== \subsection{The conical product} Fix \[ 0<\mu<2\pi,\qquad \alpha=1-\frac{\mu}{2\pi},\qquad c>0. \] Let \[ C_\mu^\circ=(0,\infty)\times(\RR/2\pi\mathbb Z), \] and prescribe \begin{equation} M=\RR_T\times C_\mu^\circ\times\RR_z, \qquad \bar g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2. \label{eq:metric} \end{equation} The metric completion has singular worldsheet \[ \mathcal W=\RR_T\times\{o\}\times\RR_z, \] where \(o\) is the cone apex. We use only the smooth regular part. The background and its unit lapse are supplied data, as in I-B \cite{CheungIB}. \begin{proposition}[Why a global transfer is unavailable] \label{prop:notGH} The regular product \((M,\bar g)\) is not globally hyperbolic. Consequently, local flatness alone does not justify applying a global globally-hyperbolic Green-operator theorem to all of \(M\). \end{proposition} \begin{proof} Fix \(r_0>0\), one angular coordinate, and \(z=0\). Take \[ p=(-2r_0/c,r_0,\phi,0),\qquad q=(2r_0/c,r_0,\phi,0). \] For any sequence \(r_n\downarrow0\), the events \[ v_n=(0,r_n,\phi,0) \] belong to \(J^+(p)\cap J^-(q)\). Radial travel has length \(r_0-r_n0\) with \begin{equation} r_*+a0,\qquad |\Delta x|\le c\Delta T; \] \item a straight segment with this inequality is future causal; \item a static observer has \(\dd\tau=\dd T\). \end{enumerate} \end{lemma} \begin{proof} The tangent inequality is \(-c^2\dot T^2+|\dot x|^2\le0\), with the future orientation \(\dot T>0\). Apply it to a constant-velocity segment. For a static curve, \(\bar g=-c^2\dd T^2\). \end{proof} Only containment in the supplied causal relation is needed. No global equality between an auxiliary chart past and the full conical causal past is assumed. %===================================================================== \section{Operators, signs, and smooth evolution} \label{sec:operators} %===================================================================== Fix \(m>0\), and define on the auxiliary slab \[ \Box=-\partial_s^2+\Delta,\qquad P=\Box-m^2. \] Here \(m\) has the inverse-length normalization appropriate to \(s=c(T-T_0)\). For a covector \(\xi_\mu\), set \[ (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] For symmetric \(h\), put \[ (\mathsf rh)_{\mu\nu} =h_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta h, \qquad (Dh)_\nu=\partial^\mu(\mathsf rh)_{\mu\nu}, \] \begin{equation} L_hu=-h^{\mu\nu}\partial_\mu\partial_\nu u -(Dh)^\nu\partial_\nu u, \label{eq:Lh} \end{equation} and \begin{equation} \Glin(h)_{\mu\nu} =-\frac12\Box(\mathsf rh)_{\mu\nu} +\partial_{(\mu}(Dh)_{\nu)} -\frac12\eta_{\mu\nu}\partial^\alpha(Dh)_\alpha. \label{eq:Glin} \end{equation} Symmetrization includes the factor \(1/2\). \begin{lemma}[Differential identities] \label{lem:identities} For smooth fields, \[ DK\xi=\Box\xi,\qquad \Glin(K\xi)=0,\qquad \partial^\mu\Glin(h)_{\mu\nu}=0, \] and \begin{equation} [P,X_\xi]u =(\Box\xi^\nu)\partial_\nu u +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu u, \qquad L_{K\xi}=-[P,X_\xi]. \label{eq:commutator} \end{equation} Also, \[ \Box_{\eta+\eps h}-m^2=P+\eps L_h \pmod{\eps^2}. \] \end{lemma} \begin{proof} In \(DK\xi\), the derivatives of \(\partial_\mu\xi^\mu\) cancel. Substitution into \eqref{eq:Glin} gives \(\Glin K=0\). Taking a divergence in \eqref{eq:Glin} cancels the wave and divergence terms pairwise. Apply the product rule twice to \(\Box(\xi^\nu\partial_\nu u)\); the mass term commutes with \(X_\xi\). Substitution of \(K\xi\) into \eqref{eq:Lh} gives the negative commutator. Finally use \[ (\eta+\eps h)^{\mu\nu} =\eta^{\mu\nu}-\eps h^{\mu\nu}, \qquad \sqrt{|\det(\eta+\eps h)|} =1+\frac{\eps}{2}\tr_\eta h \pmod{\eps^2} \] in the divergence formula for the scalar wave operator. \end{proof} These calculations fix the signs in all subsequent gauge actions. \begin{lemma}[Supported smooth Cauchy evolution] \label{lem:evolution} For \(\nu\ge0\), smooth data supported in \(\overline B_{r_*}\), and smooth forcing supported in \(\mathcal C_{r_*}\), the equation \[ (\partial_s^2-\Delta+\nu^2)u=F \] has a unique smooth solution on \(\widehat{\mathcal L}\). It is supported in \(\mathcal C_{r_*}\). Evolution is continuous in the compact-data smooth topology and the compact-open smooth topology on each closed subinterval of \(I\). If \(F=0\) for \(s\le0\), its zero-data solution is zero there. \end{lemma} \begin{proof} Let \(\Omega_\nu=(-\Delta+\nu^2)^{1/2}\). Spatial Fourier transformation gives \[ u(s)= \cos(s\Omega_\nu)u_0+ \frac{\sin(s\Omega_\nu)}{\Omega_\nu}u_1+ \int_0^s \frac{\sin((s-\sigma)\Omega_\nu)}{\Omega_\nu} F(\sigma)\,\dd\sigma. \] The sine quotient has its continuous value at zero frequency. On compact time intervals the source has uniformly compact spatial support, so the formula and its differentiated versions are well defined in Sobolev spaces of every order. For integer \(n\ge0\), an energy containing the \(L^2\) norms of \(\partial_s\partial^\beta u\), \(\nabla\partial^\beta u\), and \(\partial^\beta u\), \(|\beta|\le n\), satisfies \[ E_n'(s)\le C_\nu E_n(s)+C\norm{F(s)}_{H^n}^2 \] in the forward direction, with the corresponding backward estimate. Gronwall's inequality gives uniqueness and continuous dependence. Sobolev embedding and the equation give smooth control. The local energy identity outside the expanding initial ball has nonnegative outward characteristic flux. The source bound \(|x|\le r_*+|\sigma|\) therefore keeps the solution in \(\mathcal C_{r_*}\), in either time direction. Zero forcing and zero data on the past side give zero there. \end{proof} This is a flat supported proof, consistent with the general smooth wave framework \cite{BGP}. No global conical Green operator is invoked. %===================================================================== \section{The unchanged-product obstruction} \label{sec:product} %===================================================================== Let \(\SSol_g\) consist of supported laboratory tensors satisfying \(\Glin(h)=0\), and let \(\SSol_\phi\) be the solutions of \(P\phi=0\) with smooth data supported in \(\overline B_{r_*}\). Let \[ \GG_0=\{K\xi:\xi\in C_c^\infty(I\times B_{r_*};T^*)\}. \] This is a linear subspace of \(\SSol_g\). Define the unchanged product map \[ q_0(h,\phi)=([h],\phi), \qquad [h]\in\SSol_g/\GG_0. \] No topological quotient theorem for this row is needed. \subsection{The chart Poisson prescription} For a compact smooth spatial source \(f\), define \[ (\mathcal Nf)(x) =-\frac1{4\pi}\int_{\RR^3}\frac{f(y)}{|x-y|}\,\dd^3y. \] Then \(\Delta\mathcal N f=f\), and \(\mathcal N\Delta f=f\) for compact smooth \(f\). For the latter assertion, the difference is harmonic and decays at infinity. Apply \(\mathcal N\) at each time and put \[ A_i[h]=\partial_jh_{ij}-\frac12\partial_i h_{jj}, \qquad B[h]=\partial_i h_{0i}-\frac12\partial_s h_{ii}, \] \begin{equation} w_{\rm P}^i[h]=-\mathcal N A_i[h], \qquad w_{\rm P}^0[h]=\mathcal N B[h]. \label{eq:poisson} \end{equation} The potentials may have spatial tails. They are evaluated only in the specified chart functional. \begin{lemma}[Poisson compensation] \label{lem:poisson} For \(\xi\in C_c^\infty(I\times B_{r_*};T^*)\), \[ w_{\rm P}^\mu[h+K\xi] =w_{\rm P}^\mu[h]-\xi^\mu. \] \end{lemma} \begin{proof} Direct differentiation gives \[ A_i[K\xi]=\Delta\xi_i,\qquad B[K\xi]=\Delta\xi_0. \] Use Newton inversion, \(\xi^i=\xi_i\), and \(\xi^0=-\xi_0\). \end{proof} For real \(f\in C_c^\infty(\mathcal L)\), define \[ F_{\rm P}^{\lambda}(f;h,\phi) =\int f\phi\,\dd^4x+ \lambda\int f\,w_{\rm P}^\mu[h]\partial_\mu\phi\,\dd^4x, \] where \(\lambda\) is a real parameter. \begin{theorem}[One fixed local free-product obstruction] \label{thm:product} There exist fixed real \(f,\phi,\xi\) in the preceding classes such that \[ J=\int f\,\xi^\mu\partial_\mu\phi\,\dd^4x>0, \] and \[ F_{\rm P}^{\lambda}(f;K\xi,\phi) -F_{\rm P}^{\lambda}(f;0,\phi)=-\lambda J. \] For \(\lambda\ne0\), this symbol does not descend through \(q_0\). Its first-order coefficient also does not descend. \end{theorem} \begin{proof} Choose a bump \(\zeta\) supported in \(B_{r_*}\), equal to one near \(0\), and scalar data \[ \phi(0,x)=x^1\zeta(x),\qquad \partial_s\phi(0,x)=0. \] Then \(\partial_1\phi(0,0)=1\). On a small relatively compact neighborhood of that event, \(\partial_1\phi>0\). Choose nonnegative nonzero \(f\) supported there and a compact vector \(\xi^\sharp\) equal to \(\partial_1\) near \(\supp f\). Thus \(J>0\). The metric representatives \(0,K\xi\) are equivalent and both solve the vacuum metric equation. Their scalar labels agree. Lemma~\ref{lem:poisson} gives the displayed gap, which no function of their common quotient label can reproduce. \end{proof} This is the local form of the exhibited-symbol obstruction in III \cite{CheungIIIProduct}. It is not a statement about every scalar functional. For example, a metric-independent scalar functional already descends through \(q_0\). The elementary two-representative bound gives, for every candidate \(G\) on the product quotient, \[ \max_{h\in\{0,K\xi\}} \abs{G([h],\phi)-F_{\rm P}^{\lambda}(f;h,\phi)} \ge\frac{|\lambda|J}{2}. \] \subsection{Stability under the stated spatial regulator} Replace \(\mathcal N\) by the Fourier multiplier \[ -\frac{\chi_{\epsilon,\Lambda}(k)}{|k|^2}, \qquad 0\le\chi_{\epsilon,\Lambda}\le1, \] where the cutoff removes a neighborhood of \(0\), has compact frequency support, and tends pointwise to \(1\) as \(\epsilon\downarrow0,\Lambda\uparrow\infty\). For the same representatives, \[ w_{{\rm P},\epsilon,\Lambda}(K\xi) =-\chi_{\epsilon,\Lambda}(D)\xi. \] Their regulated gap is \(-\lambda J_{\epsilon,\Lambda}\), where \[ J_{\epsilon,\Lambda} =\int f\, \bigl(\chi_{\epsilon,\Lambda}(D)\xi^\mu\bigr) \partial_\mu\phi\,\dd^4x \longrightarrow J. \] Indeed, the spatial Fourier transforms of \(\xi\) have a uniform Schwartz bound on the compact time interval of \(\supp f\); dominated convergence gives uniform convergence of the cutoff vectors there. The lower bound therefore tends to \(|\lambda|J/2>0\). This is a regularization of the declared auxiliary classical symbol. It is not a cone-wide inverse prescription, an operator regularization, or a field-measurement theorem. %===================================================================== \section{Massive free-shell tangency} \label{sec:killing} %===================================================================== Use the connected open sublaboratory \[ \mathcal D_0= \{(s,x):|s|0\), that multiple is zero. Differentiating the Killing equation and permuting indices gives \[ 2\partial_\mu\partial_\nu\xi_\rho =\partial_\mu(K\xi)_{\nu\rho} +\partial_\nu(K\xi)_{\mu\rho} -\partial_\rho(K\xi)_{\mu\nu}=0. \] Connectedness gives the affine form. Conversely such an affine Killing field makes the commutator zero. An affine field vanishing on a nonempty open set is zero. \end{proof} This is a local classification in a developing chart \cite{CheungIIIKilling}. It does not assert that every local Poincare generator extends through conical holonomy. No massless version is inferred. The distinction from Theorem~\ref{thm:product} is essential: that theorem fixes \(\phi\) and changes only its metric representative. The present theorem tests whether moving \(\phi\) preserves its unchanged equation. %===================================================================== \section{The retained first-order jet and its retarded section} \label{sec:jet} %===================================================================== Use the dual numbers \[ \mathbb D=\RR[\eps]/(\eps^2). \] Statements in \(\mathbb D\) are coefficient identities, not numerical remainder estimates. \subsection{A named past-trivial class} Let \(\mathscr E_{+,r_*}\) consist of supported smooth fields on \(\widehat{\mathcal L}\) that vanish for \(s\le0\). For \(F\) in this class, let \(R_\Box F\) be the zero-data solution of \[ \Box R_\Box F=F \] at \(s=0\), continued as zero for \(s<0\). Lemma~\ref{lem:evolution} proves existence, smoothness, and preservation of the support class. For \(a\in\mathscr E_{+,r_*}\), \begin{equation} \Box R_\Box F=F,\qquad R_\Box\Box a=a. \label{eq:inverse} \end{equation} The second identity follows by homogeneous-wave uniqueness. Define \[ \GG_R=\mathscr E_{+,r_*}(T^*), \] \[ \mathscr M_R= \{h:\supp h\subseteq\mathcal C_{r_*},\ Dh=0\text{ for }s\le0\}, \] and \begin{equation} w_R(h)=R_\Box(Dh)^\sharp,\qquad \Pi_Rh=h-K(w_R(h)^\flat). \label{eq:ret-section} \end{equation} \begin{lemma}[Section identities] \label{lem:section} For \(\xi\in\GG_R\), \[ w_R(h+K\xi)=w_R(h)+\xi^\sharp, \qquad D\Pi_Rh=0, \] \[ w_R(\Pi_Rh)=0,\qquad \Pi_R^2h=\Pi_Rh,\qquad \Glin(\Pi_Rh)=\Glin(h). \] A homogeneous vector wave in \(\GG_R\) is zero. \end{lemma} \begin{proof} Use \(DK\xi=\Box\xi\) and \eqref{eq:inverse}. Then \[ D\Pi_Rh=Dh-\Box R_\Box Dh=0. \] The other assertions follow from this equation, \(\Glin K=0\), and uniqueness for zero wave data. \end{proof} Notice the sign: \(w_R(K\xi)=+\xi^\sharp\), whereas \(w_{\rm P}(K\xi)=-\xi^\sharp\). The two compensators have different definitions. \subsection{The equation space and its action} Let \(\JJ_R\) consist of triples \[ (h,\phi_0,\phi_1) \] such that \[ h\in\mathscr M_R,\qquad \Glin(h)=0, \qquad P\phi_0=0,\qquad P\phi_1=-L_h\phi_0, \] with the scalar Cauchy data supported in \(\overline B_{r_*}\). The forced equation and Lemma~\ref{lem:evolution} keep the scalars in \(\mathcal C_{r_*}\). For field-independent \(\xi\in\GG_R\), set \begin{equation} T_\xi(h,\phi_0,\phi_1) =(h+K\xi,\phi_0,\phi_1+X_\xi\phi_0). \label{eq:jet-action} \end{equation} \begin{proposition}[The action preserves the retained equations] \label{prop:jet-action} Equation~\eqref{eq:jet-action} defines an additive action on \(\JJ_R\): \[ T_\xi T_\zeta=T_{\xi+\zeta},\qquad T_\xi^{-1}=T_{-\xi}. \] \end{proposition} \begin{proof} The support and past-trivial conditions are preserved. The scalar equation transforms to \begin{align*} P(\phi_1+X_\xi\phi_0)+L_{h+K\xi}\phi_0 &=P\phi_1+L_h\phi_0 +PX_\xi\phi_0-[P,X_\xi]\phi_0\\ &=X_\xi P\phi_0=0. \end{align*} The metric equation follows from \(\Glin K=0\). The group identities are immediate. \end{proof} The additive action concerns retained coefficients. Commutators of the corresponding formal diffeomorphisms start at the discarded second order. \begin{theorem}[Local first-order jet descent] \label{thm:jet-descent} Each orbit of \(\JJ_R/\GG_R\) has exactly one representative with \(Dh=0\), namely \[ \mathfrak S_R(h,\phi_0,\phi_1) = \left(\Pi_Rh,\phi_0,\phi_1-X_{w_R(h)}\phi_0\right). \] For every real compact \(f\) in the laboratory, \begin{equation} \mathcal F_f(h,\phi_0,\phi_1) = \int f\left[ \phi_0+\eps\bigl(\phi_1-X_{w_R(h)}\phi_0\bigr) \right]\dd^4x \label{eq:jet-observable} \end{equation} descends to \(\JJ_R/\GG_R\). Its scalar coefficient satisfies \[ P\bigl(\phi_1-X_{w_R(h)}\phi_0\bigr) =-L_{\Pi_Rh}\phi_0. \] \end{theorem} \begin{proof} Apply \(T_\xi\) with \(\xi^\sharp=-w_R(h)\). Lemma~\ref{lem:section} puts the metric in the required slice, and Proposition~\ref{prop:jet-action} preserves the equations. The section is constant on each orbit because the two changes \(X_\xi\phi_0\) and \(X_\xi\phi_0\) cancel. Two representatives in the slice differ by a past-trivial homogeneous wave, which is zero. The integrand in each coefficient of \eqref{eq:jet-observable} is therefore orbit invariant. For the final identity, writing \(w=w_R(h)\), \[ P(\phi_1-X_w\phi_0)+L_{h-Kw}\phi_0 =P\phi_1+L_h\phi_0-X_wP\phi_0=0. \] \end{proof} The proof retains the forced scalar correction \cite{CheungIIIJet}. Discarding \(\phi_1\), or declaring it independently free, would change the equation space and invalidate this conclusion. The evaluation of a field-dependent section is an orbit calculation. Differentiating that section in a symplectic or Hamiltonian reduction is an additional operation; derivatives of \(w_R(h)\) would then have to be retained. %===================================================================== \section{The supported triangular Cauchy quotient} \label{sec:cauchy} %===================================================================== \subsection{Prescribed background and equations} Choose \(0<\rho0\). Let \[ P\Phi=0,\qquad \Phi(0,x)=A_*\chi_0(x),\qquad \partial_s\Phi(0,x)=0. \] Finite propagation gives \begin{equation} \Phi(s,x)=A_*\cos(ms) \quad\text{when }|x|+|s|<\rho. \label{eq:plateau} \end{equation} The \(\COne\) model is \begin{equation} \Box h=0,\qquad Dh=0,\qquad P\varphi=-L_h\Phi. \label{eq:C1} \end{equation} Its metric equation is vacuum linearized gravity. The scalar background stress is not inserted into that equation. This is not the full Einstein--Klein--Gordon linearization about a nonlinear on-shell background. The latter has different equations and compatibility conditions \cite{Hack}. \subsection{Constrained data and topology} Write \[ A_{\mu\nu}=h_{\mu\nu}|_{s=0},\qquad B_{\mu\nu}=\partial_s h_{\mu\nu}|_{s=0}, \] \[ u=\varphi|_{s=0},\qquad v=\partial_s\varphi|_{s=0}. \] Put \(\bar A=\mathsf rA\), \(\bar B=\mathsf rB\). Let \(\XX_{r_*}\) be the smooth tuples supported in \(\overline B_{r_*}\) satisfying \begin{equation} \bar B_{0\nu}=\partial_i\bar A_{i\nu},\qquad \Delta\bar A_{0\nu}=\partial_i\bar B_{i\nu}. \label{eq:constraints} \end{equation} Give \(\XX_{r_*}\) its inherited fixed-support smooth Fr\'echet topology. Let \(\SSol_{r_*}\) be the solutions of \eqref{eq:C1} supported in \(\mathcal C_{r_*}\), with compact-open smooth topology on the auxiliary slab. Support and differential equations are closed conditions. \begin{theorem}[Supported unreduced Cauchy isomorphism] \label{thm:Cauchy} Evolution and time-zero restriction are continuous inverse linear maps \[ \mathcal E:\XX_{r_*}\longrightarrow\SSol_{r_*}. \] In particular, every admissible datum determines one supported conical-laboratory solution. \end{theorem} \begin{proof} First evolve \(A,B\) componentwise by the wave equation. Then solve the scalar forced equation with data \(u,v\). Both are furnished by Lemma~\ref{lem:evolution}. The field \(d_\nu=(Dh)_\nu\) obeys \(\Box d_\nu=0\), with \[ d_\nu|_0=-\bar B_{0\nu}+\partial_i\bar A_{i\nu}, \] \[ \partial_s d_\nu|_0 =-\Delta\bar A_{0\nu}+\partial_i\bar B_{i\nu}. \] Both vanish by \eqref{eq:constraints}, hence \(Dh=0\). Uniqueness holds first for \(h\), then for \(\varphi\). The source \(L_h\Phi\) is supported in \(\mathcal C_{r_*}\). Multiplication by the fixed smooth derivatives of \(\Phi\) is continuous in the relevant seminorms. The energy estimates prove continuity of evolution. Restriction and differentiation prove continuity of the inverse. The zero collar from \eqref{eq:margin} gives the claimed conical-laboratory interpretation. \end{proof} This is a fixed-support local version of the Cauchy construction in III \cite{CheungIIICauchy}. No inductive limit over supports escaping the developing chart is taken. \subsection{Residual wave gauges} Let \[ \GG_{r_*} =C^\infty_{\overline B_{r_*}}(\RR^3;\RR^4) \oplus C^\infty_{\overline B_{r_*}}(\RR^3;\RR^4) \] be covector initial data \((a_\mu,b_\mu)\). Evolve them by \[ \Box\xi_\mu=0,\qquad \xi_\mu|_0=a_\mu,\qquad \partial_s\xi_\mu|_0=b_\mu. \] The coupled transformation is \[ (h,\varphi)\longmapsto(h+K\xi,\varphi+X_\xi\Phi). \] Its induced data map \(\mathfrak g:\GG_{r_*}\to\XX_{r_*}\) is \begin{align} \delta A_{00}&=2b_0, &\delta B_{00}&=2\Delta a_0,\nonumber\\ \delta A_{0i}&=b_i+\partial_i a_0, &\delta B_{0i}&=\Delta a_i+\partial_i b_0,\nonumber\\ \delta A_{ij}&=\partial_i a_j+\partial_j a_i, &\delta B_{ij}&=\partial_i b_j+\partial_j b_i, \label{eq:gauge-data} \end{align} together with \begin{equation} \delta u=a^\mu(\partial_\mu\Phi)_0,\qquad \delta v=b^\mu(\partial_\mu\Phi)_0+ a^\mu(\partial_s\partial_\mu\Phi)_0. \label{eq:scalar-data} \end{equation} These equations specify the full quotient datum, including the homogeneous scalar data. \subsection{Closedness without admitting Newton tails} For arbitrary compact metric data define \begin{equation} a_0=\frac12\mathcal N B_{00},\qquad a_i=\mathcal N\left( \partial_jA_{ij}-\frac12\partial_iA_{jj}\right), \qquad b_0=\frac12A_{00},\qquad b_i=A_{0i}-\partial_i a_0. \label{eq:reconstruction} \end{equation} Call this auxiliary reconstruction \(\mathfrak R(A,B)\). It may initially be noncompact. \begin{lemma}[Fixed-support closed-range test] \label{lem:closed} For \(x=(A,B,u,v)\in\XX_{r_*}\), \[ x\in\mathfrak g(\GG_{r_*}) \quad\Longleftrightarrow\quad x=\mathfrak g(\mathfrak R(A,B)). \] When the equality holds, every reconstructed descriptor is supported in \(\overline B_{r_*}\). The range \(\mathfrak g(\GG_{r_*})\) is closed. \end{lemma} \begin{proof} For true compact gauge data, \eqref{eq:gauge-data} gives \[ \frac12B_{00}=\Delta a_0,\qquad \partial_jA_{ij}-\frac12\partial_iA_{jj}=\Delta a_i. \] Newton inversion recovers \(a\), and the remaining equations recover \(b\). Conversely suppose the stated equality holds. All Newton potentials in \eqref{eq:reconstruction} decay at infinity. Outside \(\overline B_{r_*}\), \(A_{ij}=0\), so \[ \partial_i a_j+\partial_j a_i=0. \] Differentiation and index permutation give \[ 2\partial_i\partial_j a_k =\partial_i(\partial_j a_k+\partial_k a_j) +\partial_j(\partial_i a_k+\partial_k a_i) -\partial_k(\partial_i a_j+\partial_j a_i)=0. \] The exterior of a ball is connected. Thus the spatial \(a_i\) are affine there, and decay makes them zero. Also \(b_0=0\) outside the ball, and \(A_{0i}=0\) gives \(b_i=-\partial_i a_0\). The equation \(B_{ij}=0\) then gives \(\partial_i\partial_j a_0=0\) there. Again decay forces \(a_0=0\), and hence \(b_i=0\). The scalar equations in the assumed equality provide exactly \eqref{eq:scalar-data}. Finally, Newton convolution from a fixed compact smooth support stage into \(C^\infty(\RR^3)\), with compact-open topology, is continuous. Derivatives can be transferred to the source; the Newton kernel is locally integrable. Therefore \[ x\longmapsto x-\mathfrak g(\mathfrak R(A,B)) \] is a continuous map into a finite product of Hausdorff smooth spaces. Its kernel is precisely the gauge range, so that range is closed. \end{proof} The exterior argument is an auxiliary membership test \cite{CheungIIIClosed}. It neither imports noncompact gauge descriptors nor selects a global conical Poisson boundary condition. \begin{theorem}[Hausdorff reduced Cauchy problem] \label{thm:reduced} Define \[ \XX_{r_*}^{\rm red} =\XX_{r_*}/\mathfrak g(\GG_{r_*}), \] \[ \SSol_{r_*}^{\rm red} =\SSol_{r_*}/ \{(K\xi,X_\xi\Phi):\Box\xi=0,\ (\xi,\partial_s\xi)|_0\in\GG_{r_*}\}. \] Both spaces are Hausdorff locally convex spaces, and evolution induces a topological linear isomorphism \[ \XX_{r_*}^{\rm red}\simeq\SSol_{r_*}^{\rm red}. \] \end{theorem} \begin{proof} Theorem~\ref{thm:Cauchy} maps the data gauge range exactly onto the solution gauge range, by covariance and uniqueness. The former is closed by Lemma~\ref{lem:closed}, so both quotients are Hausdorff. The continuous inverse unreduced maps induce continuous inverse quotient maps. \end{proof} This is a relative theorem for the declared supported representatives. It is not a global TT-section theorem or a classification of all gauge orbits on the cone. \subsection{Why zero scalar data are not generally preserved} Let \(S_0(h)\Phi\) be the zero-scalar-data solution of \[ P S_0(h)\Phi=-L_h\Phi, \] and let \(H_0(q)\) denote the homogeneous Klein--Gordon solution with the Cauchy data of a smooth scalar \(q\). \begin{proposition}[Homogeneous subtraction] \label{prop:subtraction} For every admitted residual wave gauge, \[ S_0(K\xi)\Phi=X_\xi\Phi-H_0(X_\xi\Phi). \] Consequently a zero-scalar-data prescription is preserved only when \(X_\xi\Phi\) has zero scalar Cauchy data. \end{proposition} \begin{proof} Both sides have zero data and satisfy \[ Pq=-L_{K\xi}\Phi=P(X_\xi\Phi). \] Use uniqueness. The last assertion follows by evaluating the transformed scalar data. \end{proof} A residual wave generally has nonzero initial data and is not past trivial. Its intersection with the past-trivial homogeneous-wave class is zero. The section of Section~\ref{sec:jet} therefore does not remove the residual group of the present quotient. %===================================================================== \section{Compact coupled Ward observables} \label{sec:ward} %===================================================================== For compact smooth \(J^{\mu\nu}=J^{\nu\mu}\) and scalar \(q\) supported in the laboratory, define \begin{equation} \OO_{J,q}(h,\varphi) =\int h_{\mu\nu}J^{\mu\nu}\,\dd^4x +\int q\varphi\,\dd^4x. \label{eq:Ward-functional} \end{equation} \begin{proposition}[Compact Ward criterion] \label{prop:Ward} If \begin{equation} -2\partial_\mu J^{\mu\nu} +q\,\partial^\nu\Phi=0, \label{eq:Ward} \end{equation} then \(\OO_{J,q}\) is invariant under the coupled transformation for every smooth descriptor for which that transformation is considered. It descends continuously to \(\SSol_{r_*}^{\rm red}\). If invariance is required for every compact descriptor, the local identity \eqref{eq:Ward} is also necessary. \end{proposition} \begin{proof} Integration by parts, with no boundary term, gives \[ \delta_\xi\OO_{J,q} =\int \xi_\nu \left(-2\partial_\mu J^{\mu\nu} +q\,\partial^\nu\Phi\right)\dd^4x. \] This proves sufficiency and, by compact test separation, the stated necessity. Compact smooth pairings are continuous in the solution topology, so quotient continuity follows. \end{proof} The pointwise identity is sufficient for the smaller residual-wave group. Its necessity for that smaller group is not asserted. \subsection{An exact family that vanishes} For \(\psi\in C_c^\infty(\mathcal L)\), let \[ J_\psi^{\mu\nu} =\partial^{(\mu}\psi\,\partial^{\nu)}\Phi -\frac12\eta^{\mu\nu} \partial_\alpha(\psi\partial^\alpha\Phi), \qquad q_\psi=P\psi. \] \begin{proposition}[Adjoint-exact zero family] \label{prop:exact-zero} The pair \((J_\psi,P\psi)\) satisfies \eqref{eq:Ward}, but \[ \OO_{J_\psi,P\psi}(h,\varphi)=0 \] whenever \(P\varphi=-L_h\Phi\). \end{proposition} \begin{proof} Using \(P\Phi=0\), direct differentiation gives \[ \partial_\mu J_\psi^{\mu\nu} =\frac12(P\psi)\partial^\nu\Phi. \] Also integration by parts in \eqref{eq:Lh} gives \[ \int h_{\mu\nu}J_\psi^{\mu\nu}\,\dd^4x =\int\psi L_h\Phi\,\dd^4x. \] Formal self-adjointness of \(P\) yields \[ \OO_{J_\psi,P\psi} =\int\psi(L_h\Phi+P\varphi)\,\dd^4x=0. \] \end{proof} This exact family does not exhaust compact coupled Ward observables \cite{CheungIIIExact}. \subsection{A strictly positive compact pair} Choose \[ 00. \label{eq:positive} \end{equation} Moreover \(q\notin P C_c^\infty(\mathcal L)\). \end{theorem} \begin{proof} The coefficient support lies in the plateau \eqref{eq:plateau}. There \[ \partial_i\Phi=0,\qquad \partial^0\Phi=A_*m\sin(ms). \] The temporal Ward component is \[ -2a_0'b+qA_*m\sin(ms)=0; \] the spatial components vanish. Since \(h=0\), \(\varphi=\Phi\) solves the triangular equations. Its pairing is \[ \int q\Phi\,\dd^4x =\frac2m\left(\int a_0'(s)\cot(ms)\,\dd s\right)\int b. \] Integrating by parts gives \eqref{eq:positive}. If \(q=P\psi\) for compact smooth \(\psi\), then \(\int q\Phi=\int\psi P\Phi=0\), a contradiction. \end{proof} This reproduces the existence mechanism of III's compact-pair result with a convenient normalization \cite{CheungIIICompact}. It is genuinely coupled: \(\partial_\mu J^{\mu0}=a_0'b\) is not identically zero. The test solution uses nonzero homogeneous scalar data. No complete classification of compact currents follows. %===================================================================== \section{Finite Enter experiments and the clock instrument} \label{sec:instrument} %===================================================================== \subsection{Declared interfaces} A finite experiment specifies: \begin{enumerate}[label=\textup{(\roman*)}] \item the prescribed geometry and laboratory of Section~\ref{sec:geometry}; \item finitely many live quantum names and their qubit tensor factors, with all port locations; \item accessible matrix algebras, acting as the identity on inaccessible names; \item a finite instruction tree containing preparation, allocations, gates, destructive readouts, records, causal classical controls, transport, and storage reservations. \end{enumerate} Every used control input is preinstalled data or an explicitly available causal record. No inaccessible spectator is read. This is the interface discipline of I-B and En03-A \cite{CheungIB,CheungEn03}. It is not inferred from the classical field equations. \subsection{Complete destructive reading} Let \[ \HH_C=(\CC^2)^{\otimes2}, \] with orthonormal basis \(\{\ket{\tau_k}:0\le k\le3\}\), and let \(X\) be a finite surviving register. Put \(P_k=\proj{\tau_k}\) and \[ D_k=\bra{\tau_k}_C\otimes I_X. \] The record \(R\) is classical, with diagonal observable algebra in the basis \(\ket{k}\). \begin{theorem}[Clock instrument] \label{thm:instrument} The complete destructive instrument is \begin{align} \Mclock(Z) &=\sum_{k=0}^3\proj{k}_R\otimes D_kZD_k^\dagger \label{eq:clock}\\ &=\sum_{k=0}^3\proj{k}_R\otimes \Tr_C\!\left[(P_k\otimes I_X)Z(P_k\otimes I_X)\right]. \nonumber \end{align} It is completely positive and trace preserving. For a state \(\rho_{CXE}\), its \(k\)-block is \[ \sigma_{XE,k} =(\bra{\tau_k}\otimes I_{XE})\rho_{CXE} (\ket{\tau_k}\otimes I_{XE}). \] All identities hold with every untouched reference \(E\). The clock names are not returned as quantum inputs. \end{theorem} \begin{proof} Apply the unitary sending \(\ket{\tau_k}\) to computational basis \(\ket{k}\), then perform destructive Enter on both clock qubits. Its recorded Kraus operators are \(L_k=\ket{k}_R\otimes D_k\), and \[ \sum_kL_k^\dagger L_k =\sum_kP_k\otimes I_X=I_{CX}. \] This proves the channel assertion. The second expression follows from the rank-one projectors. Tensoring the Kraus operators with \(I_E\) proves the complete reference identity. Their codomain contains \(X\), not \(C\). \end{proof} The conditional state is \(\sigma_{XE,k}/p_k\) only when \(p_k=\Tr\sigma_{XE,k}>0\). No normalized state is assigned to a zero-probability outcome. This is the clock extraction interface discussed in En09-A \cite{CheungEn09}. \begin{proposition}[No-return, record separation, and finite CCR] \label{prop:clock-limits} The following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item \(\Mclock\) has no universal left inverse. \item Its classical record marginal does not determine its quantum residual. \item No finite-dimensional matrices satisfy \([T_C,H_C]=iI_C\). \end{enumerate} \end{proposition} \begin{proof} The two clock states \[ \ket{\pm}=\frac{\ket{\tau_0}\pm\ket{\tau_1}}{\sqrt2} \] with the same \(X\) input give the same output under \(\Mclock\). A left inverse cannot recover both. Inputs \(P_0\otimes\rho_X\) and \(P_0\otimes\rho_X'\) have the same deterministic classical record but different residuals when \(\rho_X\ne\rho_X'\). Finally, \(\Tr[T_C,H_C]=0\), whereas \(\Tr(iI_C)=4i\). \end{proof} \subsection{A finite conditional-evolution lemma} Let \(S_C\ket{\tau_k}=\ket{\tau_{k+1}}\), with indices modulo four, and let \(U\) be a unitary on \(X\). \begin{proposition}[Sampled stationarity and conditioning] \label{prop:stationarity} If \[ (S_C\otimes U)\rho(S_C^\dagger\otimes U^\dagger)=\rho, \] then \[ \sigma_{k+1}=U\sigma_kU^\dagger,\qquad p_{k+1}=p_k. \] Thus supported normalized conditional states obey the same sampled unitary law. The cyclic condition also requires \(U^4\sigma_0U^{\dagger4}=\sigma_0\). If \(U^4=I\), an inhabited class is supplied by the isometry \[ W\ket{\psi} =\frac12\sum_{k=0}^3\ket{\tau_k}\otimes U^k\ket{\psi}, \] including arbitrary input references. \end{proposition} \begin{proof} Take the \((k+1,k+1)\) clock block of the stationarity equation. Trace gives equality of probabilities. For the example, \(W^\dagger W=I\), and \((S_C\otimes U)W=W\). These are isometry identities and hence preserve references. \end{proof} This is a finite conditioning statement, related in form to Page--Wootters constructions \cite{PageWootters}. Its preparation, reading event, and schedule still use the externally supplied time coordinate. It is not a construction of quantum evolution without external time. %===================================================================== \section{The time, rod, and causal dictionaries} \label{sec:dictionary} %===================================================================== \subsection{Four compact classical reading regions} Choose four equally spaced centers \[ 00\Rightarrow t_k=T(e)\); \item history calibration means that a declared map \(\kappa\) satisfies \(T(\kappa(k))=t_k\). \end{enumerate} The latter need not identify the labelled event with the readout event. \end{definition} \begin{proposition}[Exact event-time calibration is deterministic] \label{prop:deterministic} At one fixed event and with distinct \(t_k\), exact event-time calibration permits at most one supported outcome. If no \(t_k=T(e)\), no normalized exactly calibrated input exists. \end{proposition} \begin{proof} At most one code has the required coordinate. Normalization forces its probability to be one. For a state \(\rho\) and the complementary projector \(Q\), \[ \Tr(Q\rho)=\norm{Q\rho^{1/2}}_{\rm HS}^2=0 \] forces support in that single clock subspace. \end{proof} The nontrivial multi-outcome model below uses history calibration. \subsection{The other two maps} At every cut, a name \(a\) has a prescribed port location \(\ell(a)\in U\). The name-to-rod map is \[ a\longmapsto\chi(\ell(a))\in\RR^3. \] Quantum factors may be grouped into three named collections for bookkeeping, but this grouping does not derive three-dimensional geometry. The schedule has a port-event map \[ \iota:\{\text{scheduled ports}\}\longrightarrow\mathcal L. \] Every permitted dependency edge is required to map to a future-causal pair. Therefore the scheduled ancestors of an output map into its metric past. This is an order-preserving containment, not an equality between ancestry and actual influence. \begin{proposition}[Agreement and nonuniqueness] \label{prop:dictionary} On the declared finite events: \begin{enumerate}[label=\textup{(\roman*)}] \item code \(k\) has history time \(t_k=T(\kappa(k))\); \item named carriers have the prescribed rod coordinates; \item scheduled ancestry is mapped into the output metric past. \end{enumerate} None of these dictionaries is selected uniquely by \(\Mclock\). \end{proposition} \begin{proof} The agreements are the evaluations of the three specified maps, with causal transitivity for (iii). For nonuniqueness, keep the complete instrument and its input unchanged, but exchange the history assignments of codes \(0\) and \(1\). Every matrix and probability in \eqref{eq:clock} is unchanged, while the interpreted event changes. One can likewise install the same matrices at different allowed regular port positions and schedules. The instrument formula alone does not distinguish those geometric specifications. \end{proof} A change of dictionary is a change of interpretation data. A dictionary-dependent later probe must be correspondingly specified; it is not silently kept fixed. %===================================================================== \section{A finite operational encoding of descended evaluations} \label{sec:encoding} %===================================================================== \subsection{Four normalized compact Ward pairs} Fix the nonnegative spatial bump \(b\) of Section~\ref{sec:ward}, and put \[ B_b=\int b(x)\,\dd^3x>0. \] For each \(k\), choose nonnegative nonzero \(\widetilde a_k\in C_c^\infty(I_k)\), and define \[ C_k=2B_b\int\widetilde a_k(s)\csc^2(ms)\,\dd s>0, \qquad a_k=\frac{k+1}{C_k}\widetilde a_k. \] Use \eqref{eq:positive-pair} with \(a_0\) replaced by \(a_k\); call the resulting pair \((J_k,q_k)\). Then \begin{equation} F_k:=\OO_{J_k,q_k}, \qquad F_k(0,\Phi)=k+1. \label{eq:Fk} \end{equation} All four functionals descend on \(\SSol_{r_*}^{\rm red}\). Take the finite classical catalogue \[ z_j=[(0,j\Phi)],\qquad j=0,\ldots,7. \] The classes are distinct because \(F_0(z_j)=j\). Their table is \begin{equation} F_k(z_j)=j(k+1). \label{eq:integer-table} \end{equation} This normalization produces exact integer evaluation data without claiming numerical quadrature. The positive normalization integrals are part of the displayed classical definitions. \subsection{Three-qubit residual and its reading instrument} Let \[ \HH_X=(\CC^2)^{\otimes3}, \] with computational basis \(\ket{j}\), \(0\le j\le7\). Declare the encoding \(z_j\mapsto\ket{j}\), and set \[ A_k=\sum_{j=0}^7j(k+1)\proj{j}, \qquad B_*=28. \] The largest eigenvalue is \(28\). Define effects and Kraus operators \[ E_{k,y}=\frac12\left(I+\frac{yA_k}{B_*}\right), \qquad M_{k,y}=E_{k,y}^{1/2}, \qquad y\in\{+1,-1\}. \] These are positive and \(\sum_yM_{k,y}^\dagger M_{k,y}=I\). After receiving the classical clock code, read this binary instrument and retain the classical outcome \(y\). The complete clock--evaluation instrument is \begin{equation} \mathcal I(Z) = \sum_{k=0}^3\sum_{y=\pm1} \proj{k}_R\otimes\proj{y}_Y\otimes M_{k,y}D_kZD_k^\dagger M_{k,y}^\dagger. \label{eq:complete-pilot} \end{equation} \begin{theorem}[Exact finite descended-evaluation pilot] \label{thm:pilot} The following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item Equation~\eqref{eq:complete-pilot} is a complete instrument on every input, with every untouched reference. \item For \(\sigma_k=D_k\rho D_k^\dagger\) and \(p_k>0\), \[ \mathbb E[B_*y\mid k] =\frac{\Tr(A_k\sigma_k)}{p_k}. \] \item On the encoded classical input \(\ket{j}\), \[ \mathbb E[B_*y\mid k,j]=F_k(z_j)=j(k+1). \] \item The table is independent of the representative used to name each \(z_j\). \end{enumerate} \end{theorem} \begin{proof} A recorded Kraus family is \[ L_{k,y}=\ket{k}_R\otimes\ket{y}_Y\otimes M_{k,y}D_k. \] Its completeness is \[ \sum_{k,y}L_{k,y}^\dagger L_{k,y} =\sum_kD_k^\dagger I_XD_k=I_{CX}. \] Tensoring with \(I_E\) proves the complete reference statement. Since \[ \sum_{y=\pm1}B_*yE_{k,y}=A_k, \] the conditional score identity follows. The diagonal matrix entries and \eqref{eq:integer-table} give (iii). Classical Ward descent gives (iv). \end{proof} A direct finite implementation uses one fresh probe qubit \(Q\). For fixed \(k,j\), apply the real rotation whose first column is \[ \begin{pmatrix} \sqrt{(1+j(k+1)/28)/2}\\ \sqrt{(1-j(k+1)/28)/2} \end{pmatrix}. \] Control it diagonally by \(j\), then destructively read \(Q\). There are only \(32\) specified controlled rotations. They are admitted finite gates of this model; no hardware compilation or gate-synthesis error is asserted. The instrument is defined on coherent superpositions of \(\ket{j}\) as well. Such superpositions are probe states. They are not identified with superpositions of the continuum classical field configurations. \subsection{What this interface does not construct} The encoding is a finite map from selected classical classes to probe labels. It is not an injective linear representation of \(\SSol_{r_*}^{\rm red}\), a quantization of its symplectic structure, or an apparatus coupling to an unknown field. A physical field-measurement bridge would additionally need: \begin{enumerate}[label=\textup{(\roman*)}] \item a field representation, admitted states, and input class; \item a specified field--probe coupling and localized access; \item a complete selective-instrument comparison; \item a controlled approximation topology and error bound. \end{enumerate} Those data are not supplied by classical descent or by naming \(X\) a matter residual. For a finite catalogue of jet configurations, the two real coefficients of \eqref{eq:jet-observable} can similarly be encoded as separate settings. A dual-number coefficient is not inserted as a probability or a numerical quantum amplitude. \subsection{Enter cannot repair a failed descent} \begin{proposition}[Faithful scalar coding preserves the descent question] \label{prop:faithful} Let \(q:Z\to Q\) be a quotient map and \(F:Z\to\RR\). Put \[ b(v)=\frac{v}{\sqrt{1+v^2}}, \qquad p_\pm(z)=\frac{1\pm b(F(z))}{2}. \] The associated binary law factors through \(q\) if and only if \(F\) does. The same statement holds for a finite family selected by a retained clock code, when equality is required on all clock basis inputs. \end{proposition} \begin{proof} If \(F\) is constant on quotient fibers, so are the probabilities. Conversely equality of the two probabilities on a fiber implies equality of \(b(F)\). The real function \(b\) is injective, so \(F\) is equal on that fiber. For a retained clock code, prepare each clock basis state in turn. This isolates the corresponding binary law. \end{proof} Thus a faithful encoding of the two representatives in Theorem~\ref{thm:product} retains their discrepancy. Destructive readout does not turn that non-descending symbol into a quotient observable. Proposition~\ref{prop:faithful} is an interface statement, not a field apparatus construction. %===================================================================== \section{Causal schedules, records, and payload transport} \label{sec:transport} %===================================================================== \subsection{Complete schedule locality} Use a finite reserved site set in the developing patch. At each tick, a partition \(\mathcal P_s\) specifies complete block channels. Include classical flags, records, and auxiliary names in the dependency network. Every block dependency between times \(T_s,T_{s+1}\) must satisfy \[ \diam_\chi(b)\le c(T_{s+1}-T_s), \] or have an explicitly supplied causal endpoint route. Adaptive instruction trees may be unrolled into such a finite network using declared local classical flags and identity maps on inactive branches. For a terminal register set \(Y\), define the backward sets \[ N_n(Y)=Y,\qquad N_s(Y)= \bigcup\{b\in\mathcal P_s:b\cap N_{s+1}(Y)\ne\varnothing\}. \] \begin{theorem}[Complete locality and causal containment] \label{thm:locality} For the complete scheduled channel \(\Phi_{\rm sch}\), \[ \Phi_{\rm sch}^*(\mathcal A(Y)) \subseteq\mathcal A(N_0(Y)). \] The terminal reduction to \(Y\) factors through the initial reduction to \(N_0(Y)\). With a retained reference \(E\), the joint output factors through the initial reduction to \(N_0(Y)E\). Replacing a complete channel outside the backward dependency region does not change the terminal law. Every scheduled ancestor event maps into the terminal metric past. \end{theorem} \begin{proof} An output observable is the identity on each block disjoint from its current support. A channel adjoint is unital, so it preserves that identity. Other blocks enlarge support only to their union. Backward induction proves the inclusion. Append any fixed state on the unused initial complement, apply the channel, and reduce to \(Y\). The adjoint inclusion proves that this defines a reduced channel with the same action even on initially correlated inputs. Tensoring the argument with \(I_E\) proves the reference statement. An operation outside the dependency region acts on an identity in the backward calculation and is invisible. Each dependency edge is causally embedded, and concatenation proves the final containment. \end{proof} This is the finite support argument used in En03-A and En05-A \cite{CheungEn03,CheungEn05}. Scheduled ancestry is an upper bound, not necessarily the minimal influence region. The statements concern complete instruments with records retained, not selectively normalized postselection maps. \subsection{Target-adapted null transport} Choose \(x_B\in U\) with \[ L=\norm{\chi(x_B)-\chi(x_A)}_2>0. \] Choose the departure time and distance so that the entire route and final local probe reading lie within the laboratory. For example, use departure \(s_{\rm d}=3a/4\) and \(L