\documentclass[11pt,a4paper]{article} % Psy02-B.tex -- revised standalone source. % % Compile with: % pdflatex Psy02-B.tex % pdflatex Psy02-B.tex % % No external figures, bibliography database, source files, or shell escape. % An executed compilation, numerical reproduction, biological validation, % and machine-checked proof are not represented as having been performed. % % IMPORTANT SOURCE BOUNDARY: % The supplied material contains excerpts, not the complete XII--XV files. % This revision reproduces available definitions and proves independent % interface results. It does not invent unavailable tensor components, % the exact inherited shell ideal, or the missing characteristic map. \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,bm} \usepackage{mathrsfs} \usepackage{booktabs,array,tabularx,longtable} \usepackage{enumitem} \usepackage{xcolor} \usepackage{url} \usepackage[ colorlinks=true, linkcolor=blue!55!black, citecolor=blue!55!black, urlcolor=blue!55!black, pdftitle={Psy02-B: Branch-Matched Reduction Pilots and a Self-Contained Analytic Interface}, pdfauthor={Ngo Cheung} ]{hyperref} \setlength{\emergencystretch}{3em} \setlist[enumerate]{itemsep=0.25em,topsep=0.4em} \setlist[itemize]{itemsep=0.25em,topsep=0.4em} \renewcommand{\arraystretch}{1.12} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \newtheorem{record}[theorem]{Source record} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\E}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\sd}{\operatorname{sd}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\id}{\mathrm{id}} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\mathrm d} \newcommand{\One}{\mathbf 1} \newcommand{\Diss}[1]{\mathcal D_{#1}} \newcommand{\Ad}{\operatorname{Ad}} \newcommand{\dtr}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcommand{\TV}{d_{\mathrm{TV}}} \newcommand{\Wtwo}{W_2} \newcommand{\Fock}{\mathcal F_{\mathrm s}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Ksup}{K_{\mathrm{supp}}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Rker}{R_{\mathrm{ret}}} \newcommand{\Bker}{B_{\mathrm{ret}}} \newcommand{\Lcoord}{L_{\mathrm{coord}}} \newcommand{\Pmet}{\mathcal P} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\Aone}{\mathcal A_1} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\Ocal}{\mathcal O} \newcommand{\Vcal}{\mathcal V_g} \newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}} \newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}} \newcommand{\Fmc}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Eif}{\mathscr E_H^{\mathrm{if}}} \newcommand{\Qif}{Q_H^{\mathrm{if}}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\ketbra}[2]{\lvert #1\rangle\!\langle #2\rvert} \newcommand{\ip}[2]{\langle #1,#2\rangle} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\PV}{PV} \DeclareMathOperator{\gh}{gh} \DeclareMathOperator{\EL}{EL} \DeclareMathOperator{\Hom}{Hom} \DeclarePairedDelimiter{\norm}{\lVert}{\rVert} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} \title{ \textbf{Psy02-B}\\[0.4em] Quantitative Viability and Identifiability of a Branch-Matched Reduction Pilot:\\ Complete Records, Collective Mass Energies, and a Self-Contained Interface to the Inherited Analytic Problem } \author{ Ngo Cheung, MBBS, FHKAM(Psychiatry)\\ Independent Researcher, Hong Kong SAR, China\\ \texttt{info@cheungngomedical.com} } \date{} \begin{document} \maketitle \begin{abstract} We study a finite branch-matched reduction pilot and construct an independent analytic interface for assessing an inherited characteristic-coefficient problem. The pilot uses the same two conformational branch labels for its coherence witness and regularized gravitational energy. An exact finite thermal-bath calculation is separated from a memoryless loss--dephasing model. Observation-specific, channel, instrument, and energy-constrained comparison bounds are proved. Gaussian mass profiles give explicit aligned, alternating, localized, and correlated-domain energies, including cross terms and geometry-dependent stability constants. A complete-process nuisance-overlap theorem gives an exact identification obstruction; a calibrated forward instrument gives a positive robust gap against explicitly restricted alternatives. For the analytic track, we supply spacetime and distributional conventions, a weighted Euler-source identity, response and compact-root interfaces, a tree-composition theorem, a fixed retarded pole, presentation reconstruction, supported normal jets, contact obstructions, and a conditional microcausal admission criterion. A dependency and notation index distinguishes definitions reproduced here from inherited components unavailable in the supplied excerpts. The reported internal-amplitude construction at XIV-I/I2 is not confused with the full map unresolved at I3, XIV-J/J2, and the subsequent XV-B interface. The finite pilot and independent interface theorems are self-contained; an exact reconstruction or completion of the unavailable full inherited model is not claimed. No biological viability finding, unique gravity-specific identification, inherited quantum-coefficient admission, or consciousness interpretation is asserted. \end{abstract} \noindent\textbf{Keywords:} Orch-OR; reduction pilot; instruments; complete quantum process; Coulomb energy; reconstruction descent; microcausal functionals; characteristic distributions; source audit. \section{Question, contribution, and evidence boundary} \label{sec:scope} Orchestrated objective reduction combines a proposed biological carrier, environmental dynamics, an additional reduction mechanism, and an interpretation in terms of experience \cite{HameroffPenrose,Penrose,Diosi}. A mathematical treatment must distinguish these commitments. Psy02-A separated concentration, coherence, measurement, response normalization, and causal support \cite{CheungPsy02A}. The present paper provides two different developments: \begin{enumerate} \item \textbf{Track A: a branch-matched finite pilot.} Its carrier, branches, environmental laws, mass energies, forward instrument, and calibrated discrimination test are defined and calculated here. \item \textbf{Track B: an analytic interface and source-alignment audit.} Its independent distributional theorems are fully specified here. The separate frozen XII--XV instance is documented using the supplied excerpts, without claiming that unavailable definitions or the missing joint characteristic map have been reconstructed. \end{enumerate} \paragraph{What ``self-contained'' means here.} Every independent theorem in this paper has printed objects and hypotheses. Standard analytic inputs are identified. Source records are not relabeled as newly proved theorems. Parameters such as a selected cubic density or a complete contact catalogue are explicitly declared inputs when their exact inherited components were not supplied. \paragraph{What the supplied material does not contain.} The full sources of XII-A, XII-C, XII-D, XII-E, XIII-C, XIV-H, XIV-I, XIV-J, and XV-B were not supplied. Selected passages from them were supplied. The additional information therefore improves the definition sheet, but does not permit a byte-complete reproduction or independent verification of those manuscripts. The four attempted companion pages did not provide accessible full sources during preparation of this revision. \paragraph{Submission scope.} The defensible mathematical submission is the pilot plus the independent analytic interface and its explicit source limitations. A submission claiming the \emph{complete inherited construction} would additionally require the exact full source package and verification of its defining data. That stronger claim is not made. \paragraph{Biological boundary.} The conformational code is tubulin-inspired, not an identified neuronal subsystem. Cheung's open-XXZ study also expressly treats its microtubule interpretation as a toy-model analogy \cite{CheungSpinChains}. Neither that spin carrier nor a free scalar carrier is silently identified with the conformational code below. \begin{table}[ht] \centering \small \caption{Transfers excluded from the mathematical claims.} \begin{tabularx}{\linewidth}{@{}p{0.34\linewidth}X@{}} \toprule Proposed transfer & Required separate certificate\\ \midrule Concentration to biological channel equivalence & A reference-stable channel or complete-instrument comparison.\\ Equal-time variance to noise spectrum & Temporal covariance, filtering, and a declared spectral convention.\\ Scalar input stability to bath or mass stability & The corresponding parameter-to-bath and parameter-to-density estimates.\\ Wick-square contacts to Coulomb contacts & A distribution-specific ambiguity classification.\\ One-time channel equality to complete-process equality & An intervention-dependent process specification.\\ Outcome effects to conditional residuals & Equality or approximation of selective maps.\\ Internal tree kernel to inherited quantum coefficient & The complete grouped cross-line construction and admission theorem.\\ Relative action symmetry to observable closure & An independent observable and carrier-compatibility argument.\\ \bottomrule \end{tabularx} \end{table} \section{Track A1: carrier, matched branches, and exact dynamics} \label{sec:carrier} \subsection{Code and mass assignments} Let \[ \mathcal H_{\mathrm C} =\C\ket{\Omega}\oplus \operatorname{span}\{\ket0,\ket1\}, \] and define \[ P=\ketbra00+\ketbra11,\qquad Z=\ketbra00-\ketbra11,\qquad X=\ketbra01+\ketbra10. \] They vanish on \(\Omega\). The level \(\Omega\) denotes loss from the prepared code, not disappearance of matter. Successful preparation is \[ \rho_+=\ketbra{+}{+},\qquad \ket+=\frac{\ket0+\ket1}{\sqrt2}. \] The comparison preparation is \[ \Delta\rho_+=\tfrac12(\ketbra00+\ketbra11). \] Preparation succeeds with probability \(p\); failure is recorded. For centers \(R_i\in\R^3\), masses \(m_i>0\), displacement \(u\in\R^3\), pattern \(s_i\in\{-1,0,1\}\), and \(a>0\), put \begin{align} g_a(x)&=(2\pi a^2)^{-3/2}e^{-\abs x^2/(2a^2)},\\ \mu_0(x)&=\mu_{\mathrm{rest}}(x)+ \sum_i m_i g_a(x-R_i-s_i u/2),\\ \mu_1(x)&=\mu_{\mathrm{rest}}(x)+ \sum_i m_i g_a(x-R_i+s_i u/2). \label{eq:massbranches} \end{align} Use \(\delta\mu=\mu_0-\mu_1\) and \(\mu_\Omega=(\mu_0+\mu_1)/2\). The common rest density cancels. \begin{assumption}[Physical branch matching] \label{ass:physical} A biological application must identify actual preparations, mass changes, coherent controls, bath couplings, leakage, measurement maps, and a time horizon corresponding to these same branch labels. An optical excitation witness cannot be combined with an unrelated conformational displacement energy without a separate state-identification and coupling theorem. \end{assumption} \subsection{Finite thermal bath} During sensing, define \begin{equation} H_{\mathrm{tot}} = \frac{\hbar\omega}{2}Z\otimes\One +\One\otimes\hbar\sum_{j=1}^M\nu_jb_j^\dagger b_j +\hbar Z\otimes\sum_{j=1}^M g_j(b_j+b_j^\dagger), \label{eq:Htot} \end{equation} with finite \(M\), \(\nu_j>0\), \(g_j\in\R\), and an independent thermal bath of inverse temperature \(\beta>0\): \[ \tau_{\mathrm B}= \bigotimes_j(1-e^{-\beta\hbar\nu_j}) \sum_{n=0}^\infty e^{-\beta\hbar\nu_j n}\ketbra nn. \] Conditional oscillator Hamiltonians are defined by displacement of the free oscillator Hamiltonians. \begin{theorem}[Exact colored dephasing] \label{thm:colored} For code inputs, \[ \rho_{01}(t)=e^{-\ii\omega t-\chi(t)}\rho_{01}(0), \] where \begin{equation} \chi(t)=4\sum_{j=1}^M \frac{g_j^2}{\nu_j^2}(1-\cos\nu_jt) \coth\!\left(\frac{\beta\hbar\nu_j}{2}\right). \label{eq:chi} \end{equation} Populations are preserved. The reduced maps are CPTP. At zero coupling they recover the carrier's own free dynamics. \end{theorem} \begin{proof} For branch \(z=\pm1\), \[ \nu_jb_j^\dagger b_j+zg_j(b_j+b_j^\dagger) = \nu_j(b_j^\dagger+zg_j/\nu_j)(b_j+zg_j/\nu_j) -g_j^2/\nu_j. \] The branch-independent scalar phases cancel. The relative displacement has squared modulus \(8g_j^2(1-\cos\nu_jt)/\nu_j^2\). Thermal displacement expectation is \[ \Tr(\tau_jD(\alpha)) =\exp[-\abs\alpha^2\coth(\beta\hbar\nu_j/2)/2]. \] Multiplying the mode factors proves the formula. Unitary evolution and partial trace establish physicality. \end{proof} This is a finite Gaussian-dephasing calculation, consistent with standard methods \cite{Riberi}. A finite bath may recur; no exponential Markov approximation is inferred from \eqref{eq:chi}. \subsection{Separate memoryless loss--dephasing model} Write \[ \Diss{L}(\rho)=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}. \] The pilot generator is \begin{equation} \mathcal L_R(\rho)= -\frac{\ii\omega}{2}[Z,\rho] +\sum_{a=0}^1\Diss{\sqrt\kappa\,\ket{\Omega}\bra a}(\rho) +\Diss{\sqrt{(\gamma+R)/2}\,Z}(\rho), \label{eq:generator} \end{equation} with \(\kappa,\gamma,R\ge0\). The extra rate \(R=E_G/\hbar\) is a phenomenological hypothesis. A fresh bath qubit is prepared in \(\ket0\), with \(\bra0X_{\mathrm B}\ket0=0\), before each collision \[ U_{\delta t}= \exp[-\ii\sqrt{\gamma\delta t/2}\,Z\otimes X_{\mathrm B}]. \] Its code-coherence multiplier is \(\cos(\sqrt{2\gamma\delta t})\). Repeated fresh collisions converge to \(e^{-\gamma t}\). Loss is realized by the Kraus family \[ K_0=\ketbra{\Omega}{\Omega}+e^{-\kappa\delta t/2}P,\qquad K_a=\sqrt{1-e^{-\kappa\delta t}}\ket{\Omega}\bra a. \] These statements specify the Markov model rather than deriving it from the thermal bath. \begin{theorem}[Complete state and matched coherence witness] \label{thm:pilot} The generator \eqref{eq:generator} defines a CPTP semigroup. For \(\rho_+\), \[ \rho_R(t)=(1-q)\ketbra{\Omega}{\Omega} +\frac q2 \begin{pmatrix} 1 & ce^{-\ii\omega t}\\ ce^{\ii\omega t} & 1 \end{pmatrix}_{\{0,1\}}, \quad q=e^{-\kappa t},\quad c=e^{-(\gamma+R)t}. \] The comparison input \(\Delta\rho_+\) has the same populations and zero code coherence. The complete-state witness is \begin{equation} D_R(t)= \tfrac12\norm{\Lambda_R(t)(\rho_+-\Delta\rho_+)}_1 =\tfrac12e^{-(\kappa+\gamma+R)t}. \label{eq:witness} \end{equation} With a common recorded preparation failure, \(D_0=p/2\) and \(D_R(t)=D_0e^{-(\kappa+\gamma+R)t}\). \end{theorem} \begin{proof} The finite generator has GKSL form \cite{GKS,Lindblad}. Its code block satisfies \[ \dot B=-\frac{\ii\omega}{2}[Z,B]-\kappa B +\frac{\gamma+R}{2}(ZBZ-B). \] Trace gives \(q=e^{-\kappa t}\); the off-diagonal equation gives \(c\). The omitted trace appears in \(\Omega\). The coherent-minus-dephased output has eigenvalues \(\pm qc/2\). Multiplication by the common preparation-success probability gives the last claim. \end{proof} For a code Hamiltonian \(\hbar h\), the conditional state obeys \[ \dot S=-\ii[h,S]+(\gamma+R)(\Delta S-S). \] The explicit exponential witness requires \([h,Z]=0\). The vacuum-plus-code architecture is motivated by Em04-A \cite{CheungEm04A}; its biological matching is not inherited. \section{Track A2: observation, channel, and instrument comparisons} \label{sec:bridges} Use \[ \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1,\qquad \ddiam(\Phi,\Psi)=\tfrac12\norm{\Phi-\Psi}_\diamond. \] Channels and their reference-stable distances are standard \cite{Watrous}. \begin{proposition}[Named-effect transfer] \label{prop:effect} For an encoding channel \(\mathcal E\), suppose \[ \ddiam(\Lambda_{\mathrm b,t}\mathcal E, \mathcal E\Lambda_{\mathrm m,t})\le e(t), \qquad \norm{\mathcal E^*(F_{\mathrm b})-F_{\mathrm m}}\le a_F \] for effects \(F_{\mathrm b},F_{\mathrm m}\). Then the corresponding expectation error is at most \(e(t)+a_F\). The converse is false. \end{proposition} \begin{proof} Insert the expectation in \(\mathcal E\Lambda_{\mathrm m,t}(\rho)\). An effect changes by at most trace distance, and the operator mismatch contributes at most \(a_F\). Identity and complete \(Z\)-dephasing agree on the effect \(\ketbra00\) for every input but are different channels. \end{proof} \begin{theorem}[Matrix-unit channel and instrument certificate] \label{thm:matrixunits} For channels on a common \(d\)-dimensional input, \[ \max_{i,j}\norm{(\Phi-\Psi)(\ketbra ij)}_1\le\varepsilon \quad\Longrightarrow\quad \ddiam(\Phi,\Psi)\le d\varepsilon/2. \] For complete instruments, replace the matrix-unit norm by \[ \max_{i,j}\sum_r \norm{(\Phi_r-\Psi_r)(\ketbra ij)}_1. \] \end{theorem} \begin{proof} For a referenced state \(\rho=\sum_{i,j}\ketbra ij\otimes\rho_{ij}\), positivity gives \[ \norm{\rho_{ij}}_1\le\sqrt{p_ip_j},\qquad p_i=\Tr\rho_{ii}. \] A purification or positive-block factorization proves this bound. Therefore \[ \sum_{i,j}\norm{\rho_{ij}}_1 \le(\sum_i\sqrt{p_i})^2\le d. \] The referenced output trace norm is at most \(d\varepsilon\). For channel differences, maximization over referenced states gives the diamond norm \cite{Watrous}. For instruments use the channel \(\rho\mapsto\bigoplus_r\Phi_r(\rho)\). Its trace norm is the sum of block trace norms. \end{proof} Equal outcome effects do not determine residual states. For example, \[ \mathcal I_x(\rho)=\Tr(P_x\rho)\ketbra00,\qquad \mathcal J_x(\rho)=\Tr(P_x\rho)\ketbra11 \] have the same effects but different later \(Z\)-records. \begin{proposition}[Dynamical residual bounds] \label{prop:dynamic} For common bounded Hamiltonians and independent bath states, \[ \ddiam(\Lambda_t,\Lambda_t') \le\dtr(\tau,\tau') +\hbar^{-1}\int_0^t\norm{H(s)-H'(s)}\,\dd s. \] For bounded generators and a common encoding, if \[ \tfrac12\norm{ \mathcal L_{\mathrm b,s}\mathcal E -\mathcal E\mathcal L_{\mathrm m,s}}_\diamond\le r(s), \] then \[ \ddiam(\Lambda_{\mathrm b,t}\mathcal E, \mathcal E\Lambda_{\mathrm m,t}) \le\int_0^t r(s)\,\dd s. \] Preparation errors are added separately. \end{proposition} \begin{proof} Duhamel's formula bounds the unitary difference by the displayed Hamiltonian integral. Unitary-output trace distance is at most that operator norm. Bath-state and partial-trace contraction give the first assertion. Variation of constants expresses the second defect as \[ \int_0^t\Lambda_{\mathrm b}(t,s) (\mathcal L_{\mathrm b,s}\mathcal E- \mathcal E\mathcal L_{\mathrm m,s}) \Lambda_{\mathrm m}(s,0)\,\dd s. \] Channel composition contracts the diamond norm. \end{proof} The bounded estimate is not applied to an unbounded oscillator interaction without a cutoff or an appropriate domain argument. \begin{theorem}[Energy-constrained finite-code route] \label{thm:energy} Let \(H_{\mathrm E}\ge0\), let \(P_K\) be finite rank, and suppose \(H_{\mathrm E}\ge\Delta_K(\One-P_K)\), \(\Delta_K>0\). Reset the high block to a fixed low-energy vector: \[ \mathcal T_K(\rho)=P_K\rho P_K+ \Tr((\One-P_K)\rho)\ketbra00. \] For referenced inputs of energy at most \(E\), \[ t_K=\Tr((\One-P_K)\rho)\le E/\Delta_K, \qquad \dtr(\rho,(\mathcal T_K\otimes\id)(\rho)) \le\sqrt{t_K}+t_K/2. \] If two channels have finite-code error \(e_K\), then \[ \ddiam_E(\Phi,\Psi) \le e_K+2\sqrt{E/\Delta_K}+E/\Delta_K. \] The bound may be truncated at one. \end{theorem} \begin{proof} The energy inequality proves the tail estimate. For a purification \(\Psi\), projection gives a pure-state versus subnormalized-pure-state trace norm \[ \sqrt{4t_K-3t_K^2}\le2\sqrt{t_K}. \] The reset block contributes trace \(t_K\). Partial trace proves the state bound. Insert the reset input in both channel outputs and apply the finite-code bound to the reset state. \end{proof} The energy-constrained norm has a named input-energy restriction \cite{Shirokov}. Leakage flags and their reference blocks are retained; conditional renormalization is a different comparison. \section{Track A3: concentration and temporal fluctuations} \label{sec:fluctuations} \subsection{Separate scalar benchmark} For \(\Phi(v)=a^\dagger(v)+a(v)\) on bosonic Fock space, suppose \[ U_{\epsilon,t}^\dagger (\widehat O_{\epsilon,t}-o_t\One)U_{\epsilon,t} =\sqrt\epsilon\,\Phi(v_t). \] \begin{theorem}[Exact uniform concentration criterion] \label{thm:concentration} For the spectral laws in the displaced states, \[ \Wtwo(\mu_{\epsilon,t},\delta_{o_t})^2 =\epsilon\Tr(\rho_{\epsilon,t}\Phi(v_t)^2). \] Uniform convergence on \(0\le t\le T\) is equivalent to \[ \sup_{t\le T}\epsilon\Tr(\rho_{\epsilon,t}\Phi(v_t)^2)\to0. \] A sufficient bound is \[ \epsilon\Tr(\rho_{\epsilon,t}\Phi(v_t)^2) \le\epsilon(4n_{\epsilon,t}+2)\norm{v_t}^2. \] \end{theorem} \begin{proof} The displacement identity gives the target-centered second moment. Distance to a point mass has that exact squared Wasserstein value. The number estimate follows from \[ \norm{\Phi(v)\psi}^2\le 2\norm v^2 \bigl(\norm{N^{1/2}\psi}^2+ \norm{(N+\One)^{1/2}\psi}^2\bigr). \] Sum over a normal-state decomposition. \end{proof} This is the observable-specific Em06-B interface \cite{CheungEm06B,CheungPsy02A}, not a conformational-channel bridge. Normalized fluctuations retain the residual field law. Only vanishing pairwise symplectic form gives a common joint spectral law; ordered Weyl expectations do not specify apparatus instruments. \subsection{Bath covariances and filters} For the thermal bath, let \[ B(t)=\sum_jg_j(b_je^{-\ii\nu_jt}+b_j^\dagger e^{\ii\nu_jt}), \qquad F_k=\int_0^T k(t)B(t)\,\dd t \] for a real integrable filter. Write \(\widehat k(\nu)=\int_0^T k(t)e^{\ii\nu t}\,\dd t\). \begin{theorem}[Exact filtered variance] \label{thm:filters} The symmetrized covariance is \[ C_B(\tau)= \sum_jg_j^2\coth(\beta\hbar\nu_j/2)\cos\nu_j\tau, \] and \[ \Var(F_k)= \sum_jg_j^2\coth(\beta\hbar\nu_j/2) \abs{\widehat k(\nu_j)}^2. \] For \(S_B=\int e^{\ii\nu\tau}C_B(\tau)\,\dd\tau\), \[ S_B=\pi\sum_jg_j^2\coth(\beta\hbar\nu_j/2) (\delta_{\nu_j}+\delta_{-\nu_j}), \qquad \Var(F_k)=\frac1{2\pi}\int\abs{\widehat k}^2S_B(\dd\nu). \] \end{theorem} \begin{proof} Thermal anomalous moments vanish and different modes are independent. Expand the anticommutator, then integrate against \(k(t)k(s)\). Fourier transformation of a cosine gives the two spectral atoms. \end{proof} This convention is consistent with standard noise analysis \cite{Clerk}. Equal-time variance alone does not determine the frequency placement or filtered variance. For the memoryless carrier, put \(r=\kappa+\gamma+R\). In its specified Markov dilation, the phase-compensated \(X_t\) has \[ \E X_t=e^{-rt},\qquad \tfrac12\E\{X_t,X_s\} =e^{-\kappa\min(t,s)}e^{-r\abs{t-s}}. \] Consequently \[ C_X(t,s)= e^{-\kappa\min(t,s)}e^{-r\abs{t-s}}-e^{-rt}e^{-rs}. \] \begin{proposition}[Windowed carrier variance] \label{prop:codevariance} For the named Markov process, \[ \Var\!\left(\int_0^T k(t)X_t\,\dd t\right) =\int_0^T\int_0^T k(t)k(s)C_X(t,s)\,\dd t\,\dd s \le\norm{k}_{L^1}^2. \] The displayed covariance is generally nonstationary. \end{proposition} \begin{proof} For \(t\ge s\), the adjoint evolution multiplies the compensated \(X\) by \(e^{-r(t-s)}\). The symmetrized regression expression becomes \[ \Tr\!\left[ X\Lambda_R(t-s)\frac{X\rho_R(s)+\rho_R(s)X}{2} \right] =e^{-r(t-s)}\Tr(P\rho_R(s)), \] because \(X^2=P\). Integrate the centered covariance. The dilation observable has norm at most one, giving the bound. \end{proof} This regression calculation uses the declared Markov process. It is not substituted into the memory-bearing thermal bath. Let \[ \zeta(\dd\nu)= \sum_jg_j^2\coth(\beta\hbar\nu_j/2)\delta_{\nu_j}(\dd\nu). \] For two admissible measures, \[ \sup_{t\le T}\abs{\chi_\zeta(t)-\chi_{\zeta'}(t)} \le4\int\min\{T^2/2,2/\nu^2\}\, \abs{\zeta-\zeta'}(\dd\nu), \] \[ \abs{\Var_\zeta(F_k)-\Var_{\zeta'}(F_k)} \le\int\abs{\widehat k(\nu)}^2 \abs{\zeta-\zeta'}(\dd\nu). \] These follow by integrating the exact formulas and using \(1-\cos x\le\min\{x^2/2,2\}\). They are bath-specific estimates, not renamed scalar input bounds. \section{Track A4: Gaussian mass energies and stability} \label{sec:mass} For real Schwartz densities, \[ Q(f,h)=\iint\frac{f(x)h(y)}{\abs{x-y}}\,\dd^3x\,\dd^3y, \qquad \norm f_{\mathrm C}^2=Q(f,f). \] With \(\widehat f(k)=\int e^{-\ii kx}f(x)\,\dd^3x\), \[ Q(f,f)=\frac1{(2\pi)^3} \int\frac{4\pi}{\abs k^2}\abs{\widehat f(k)}^2\,\dd^3k. \] This form is finite and positive. A wider admissible class can be controlled by Coulomb-energy or Hardy--Littlewood--Sobolev estimates \cite{Lieb}. Our convention is \[ E_G[f]=GQ(f,f)/2,\qquad R[f]=E_G[f]/\hbar. \] Define \[ U_a(r)= \begin{cases} \erf(r/(2a))/r,&r>0,\\ 1/(\sqrt\pi a),&r=0. \end{cases} \] Fourier transformation of the two Gaussians gives \[ Q(g_a(\cdot-x),g_a(\cdot-y))=U_a(\abs{x-y}). \] Indeed the radial integral uses \[ \int_0^\infty e^{-a^2k^2}\frac{\sin(kr)}k\,\dd k =\frac\pi2\erf(r/(2a)), \] obtained by differentiating in \(r\) and integrating back from zero. For equal masses, put \[ f_i=m[g_a(\cdot-R_i-u/2)-g_a(\cdot-R_i+u/2)], \qquad K_{ij}=GQ(f_i,f_j). \] \begin{theorem}[Four exact pattern calculations] \label{thm:patterns} The matrix \(K\) is positive semidefinite and \[ K_{ij}=Gm^2\left[ 2U_a(\abs{R_i-R_j}) -U_a(\abs{R_i-R_j+u}) -U_a(\abs{R_i-R_j-u})\right]. \] For \(\delta\mu_s=\sum_i s_if_i\), \(E_G(s)=s^{\mathsf T}Ks/2\). In particular: \begin{enumerate} \item aligned: \[ E_{\mathrm{al}}=\tfrac12\sum_{i,j}K_{ij}; \] \item alternating: \[ E_{\mathrm{alt}}=\tfrac12\sum_{i,j}(-1)^{i+j}K_{ij}; \] \item localized: \[ E_{\mathrm{loc}}=\tfrac12K_{jj} =Gm^2\left[ \frac1{\sqrt\pi a}-\frac{\erf(\abs u/(2a))}{\abs u} \right], \] with value zero at \(u=0\), and \[ E_{\mathrm{loc}} =\frac{Gm^2\abs u^2}{12\sqrt\pi a^3} +O(Gm^2\abs u^4/a^5); \] \item domain-correlated: \[ K^{\mathrm D}_{cd} =\sum_{i\in D_c,j\in D_d}K_{ij},\qquad E_{\mathrm D}=\tfrac12\xi^{\mathsf T}K^{\mathrm D}\xi. \] For independent symmetric domain signs, \[ \E E_{\mathrm D}=\tfrac12\sum_cK^{\mathrm D}_{cc}, \qquad \Var(E_{\mathrm D})=\sum_{c0\) and \[ \max_i\sum_{j\ne i}\abs{K_{ij}}\le\varepsilon k_0, \qquad \varepsilon<1, \] then \[ \frac{(1-\varepsilon)k_0}{2}\norm s_2^2 \le E_G(s)\le \frac{(1+\varepsilon)k_0}{2}\norm s_2^2. \] This follows from \(2\abs{s_is_j}\le s_i^2+s_j^2\). It gives linear scaling for full patterns in that regime. If all \(f_i=f_*\), aligned energy is quadratic: \(n^2G\norm{f_*}_{\mathrm C}^2/2\). If each domain contains \(d\) identical profiles and every domain profile has the same Coulomb norm, independent domain signs give mean \(ndG\norm{f_*}_{\mathrm C}^2/2\). Neither regime is an asserted molecular geometry. \begin{theorem}[Energy and geometry stability] \label{thm:massstability} For admissible \(f,h\), \[ \abs{E_G[f+h]-E_G[f]} \le G\norm f_{\mathrm C}\norm h_{\mathrm C} +G\norm h_{\mathrm C}^2/2. \] For Gaussian profiles with widths at least \(a_{\min}>0\), \[ \norm{g_a}_{\mathrm C}^2=\frac1{\sqrt\pi a},\quad \norm{\partial_e g_a}_{\mathrm C}^2=\frac1{6\sqrt\pi a^3},\quad \norm{\partial_a g_a}_{\mathrm C}^2=\frac3{4\sqrt\pi a^3}, \qquad \abs e=1. \] Writing \[ c_x=(6\sqrt\pi a_{\min}^3)^{-1/2},\qquad c_a=(3/(4\sqrt\pi a_{\min}^3))^{1/2}, \] fixed masses and patterns satisfy \[ \norm{\delta\mu'-\delta\mu}_{\mathrm C} \le\sum_i m_i\left[ c_x(2\abs{\delta R_i}+\abs{s_i\delta u}) +2c_a\abs{\delta a}\right]. \] \end{theorem} \begin{proof} Expand the positive quadratic form and use Cauchy--Schwarz. The Gaussian derivative norms follow from radial Fourier integrals; rotational averaging contributes a factor \(1/3\) for a directional derivative. Integrate translation and width derivatives along their parameter segments and sum the two branch changes. \end{proof} The kernel \(1/\abs x\) is locally integrable in three dimensions, with scaling degree one. Two extensions with scaling degree at most one differ by a point-supported distribution, whose nonzero terms have scaling degree at least three. Hence its extension is unique in that class. VII-H's four-dimensional Wick-square contact freedom is not imported \cite{CheungVIIH}. Euclidean translations and rotations preserve the energy. A general coordinate change must transform both mass measures and the distance kernel; retaining the Euclidean kernel in arbitrary new coordinates changes the prescription. Finally, if a constraint root \(z(\theta)\) satisfies \[ \partial_zH\ge c>0,\qquad \sup_z\abs{H(z,\theta)-H(z,\theta')}\le L\abs{\theta-\theta'}, \] the mean-value theorem gives \(\abs{z(\theta)-z(\theta')}\le L\abs{\theta-\theta'}/c\). Without nondegeneracy, \(z=\sqrt\theta\) from \(z^2-\theta=0\) is a counterexample to Lipschitz boundary movement. \section{Track A5: complete-process identification} \label{sec:process} For a finite instrument \(\{\mathcal I_r\}\), each map is completely positive and trace nonincreasing, and their sum is trace preserving \cite{DaviesLewis}. A Markov history law is \[ P(r_{1:m})= \Tr\!\left[ \mathcal I_{r_m}^{(m)}\Lambda_{m:m-1}\cdots \mathcal I_{r_1}^{(1)}\Lambda_{1:0}(\rho)\right]. \] Adaptive settings are functions of previous records. For a memory-bearing bath, propagate the joint carrier--bath state between interventions rather than resetting it implicitly. This distinction is central to multitime process theory \cite{PollockProcess,PollockMarkov}. \begin{theorem}[Complete-process equality and testing obstruction] \label{thm:process} Two Markov models with identical initial accessible states, interval channels after every allowed setting and history, and complete instruments have identical record laws for all permitted adaptive experiments, including untouched references. For equal laws every randomized test has \[ \alpha+\beta=1. \] For arbitrary laws, \[ \alpha+\beta\ge1-\TV(P_0,P_1). \] \end{theorem} \begin{proof} Induction gives equality of every unnormalized conditional state. For \(\varphi\in[0,1]\), \[ \alpha+\beta=1+\E_0\varphi-\E_1\varphi. \] Use total variation. \end{proof} \begin{example}[One-time maps do not determine the process] Process one SWAPs the input with an independent maximally mixed bath qubit and later SWAPs back. Process two applies a uniformly random Pauli, retains its classical label, and later applies the inverse. Without interventions both have a replacement channel at the first time and identity at the second. Replacing the carrier by \(\ketbra00\) between the steps makes the first process return the original input and the second return \(I_2/2\). \end{example} For reduction-only dynamics on the two-dimensional code, let \(L=\sqrt{R/2}\,Z\). Both \[ U_t=e^{-\ii\sqrt{R/2}\,ZW_t} \] and \[ \dd\psi_t= -\tfrac12(L-\langle L\rangle_t)^2\psi_t\,\dd t +(L-\langle L\rangle_t)\psi_t\,\dd W_t \] have ensemble generator \(\Diss L\). The second process preserves norm by It\^o's rule. Its branch populations are bounded martingales, and for \(V=\Var_\psi(Z)\), \[ \frac{\dd}{\dd t}\E V=-2R\E V^2. \] For \(R>0\), population convergence and finite \(\E\int_0^\infty V^2dt\) force \(V\to0\); the selected code vertex has its initial population as probability. The random-unitary process instead preserves those populations trajectory by trajectory. They nevertheless have identical complete system-only records if each conditional interval is specified by the same Markov channel and the hidden noises are inaccessible. This conclusion uses Theorem~\ref{thm:process}, not one-time equality alone. \begin{theorem}[Shared-amplitude nuisance overlap] \label{thm:overlap} Let \(b_0,b_1\in[b_-,b_+]\), \(a\ge0\), and suppose at every stage, setting, and history \[ \mathcal L_0=\mathcal L_{\mathrm{known}}+b_0\mathcal D,\qquad \mathcal L_1=\mathcal L_{\mathrm{known}}+(b_1+a)\mathcal D, \] with common apparatus. If \(a\le b_+-b_-\), a null--alternative pair has identical complete records. No sample size distinguishes that pair. \end{theorem} \begin{proof} Choose \(b_1=b_-\), \(b_0=b_-+a\). All conditional generators and instruments agree. Apply Theorem~\ref{thm:process}. \end{proof} This is the carrier-specific analogue of En01-B's calibrated overlap logic \cite{CheungEn01B}. The same nuisance coefficient must match all histories; independent refitting at every setting is a different null family. \section{Track A6--A7: forward instrument and robust viability} \label{sec:detection} At time \(t\), phase compensation gives projectors \[ \Pi_\pm=\tfrac12(P\pm X_{\omega t}),\qquad \Pi_\Omega=\ketbra{\Omega}{\Omega}. \] Their selective maps are \(\rho\mapsto\Pi_x\rho\Pi_x\). A stochastic assignment matrix flips \(+\leftrightarrow-\) with probability \(\varepsilon<1/2\), and independently erases with probability \(1-\eta\). Preparation failure, loss, and erasure are recorded. Use the score \[ f(-)=1,\qquad f(+)=0,\qquad f(\Omega)=f(e)=f(p_{\mathrm{fail}})=1/2. \] \begin{theorem}[Complete forward expectation and robust gap] \label{thm:gap} Let \(v=1-2\varepsilon\), \(A=p\eta v/2\), and \(r=\kappa+\gamma\). Then \[ \E_R f=\tfrac12-Ae^{-(r+R)t}. \] For rectangular calibrated families \[ A\in[A_-,A_+],\quad r\in[r_-,r_+],\quad R\in[R_-,R_+],\quad A_->0,\ R_->0, \] the null \(R=0\) and alternative have exact ideal oriented gap \[ g_{\mathrm{ideal}}(t) =A_-e^{-r_+t}-A_+e^{-(r_-+R_-)t}. \] For nonrectangular families enclosed by these intervals, the same expression is a lower bound, not necessarily the exact gap. Uniform complete-law errors \(e_0,e_1\) give \[ g_{\mathrm{robust}}=g_{\mathrm{ideal}}-e_0-e_1. \] For \(t>0\), the displayed ideal lower bound is positive exactly when \[ R_->r_+-r_-+t^{-1}\log(A_+/A_-). \] \end{theorem} \begin{proof} The ideal signed outcome mean is \(qc\). Assignment, observation, and preparation multiply it by \(v,\eta,p\). Neutral outcomes have zero centered score. For a rectangle, the null supremum is \(1/2-A_-e^{-r_+t}\), and the alternative infimum is \(1/2-A_+e^{-(r_-+R_-)t}\). Subtract and apply total-variation control for a bounded score. \end{proof} The projective instrument has the mathematical isometry \(\psi\mapsto\sum_x\ket x\otimes\Pi_x\psi\). No conformational Ramsey apparatus is thereby engineered. For an independently justified additive-noise measurement, \[ F_\nu(S)=\int\nu(S-x)E_A(\dd x),\qquad p_{\mathrm{meas}}=p_{\mathrm{int}}*\nu. \] This is a forward POVM. On an \(L^2\) frequency band, stable inversion requires \(\inf\abs{\widehat\nu}>0\). A zero-free multiplier gives uniqueness on classes whose Fourier differences it does not annihilate, but unrestricted stability does not follow. Gaussian multipliers decay at high frequencies. At \(\varepsilon=1/2\), the present detector destroys the score signal completely. For a calibrated response \(y_h=Ba_h+hs+r_h\), weights with \(B^{\mathsf T}w=0\) give \[ w^{\mathsf T}y_h=hw^{\mathsf T}s+w^{\mathsf T}r_h,\qquad \abs{w^{\mathsf T}r_h}\le\sum_i\abs{w_i}\delta_{h,i}. \] A retained invariant signal exists exactly when \(s\notin\operatorname{range}B\). Thus a contrast can cancel the proposed gravitational response as well as the nuisance, as emphasized by En01-C \cite{CheungEn01C}. For unbounded setting-dependent records, randomize setting \(i\) with probability \(p_i>0\), clip at \(L>0\), and use \[ C_w=\max_i\abs{w_i}/p_i,\qquad f(i,z)=\tfrac12+\frac{w_ic_L(z)}{2LC_wp_i}. \] If \(\E_hZ_i^2\le M_{h,i}\), clipping reduces an oriented mean-contrast gap \(G_T\) by at most \[ B_T/L,\qquad B_T=\sum_i\abs{w_i}(M_{0,i}+M_{1,i}), \] since \(\abs{z-c_L(z)}\le z^2/L\). Independent common erasure multiplies the nonconstant gap by its observation probability; arbitrary missingness must be included in the complete law. \subsection{Survival and independent viability conditions} For \(R>0\), \(\tau_{\mathrm{OR}}=1/R\), the Markov survival test is \[ D_R(\tau_{\mathrm{OR}})\ge D_{\min} \quad\Longleftrightarrow\quad \frac{\kappa+\gamma}{R} \le\log(D_0/D_{\min})-1. \] For a separately specified colored endpoint factor it becomes \[ \kappa/R+\chi(1/R)\le\log(D_0/D_{\min})-1. \] Neither is an identification theorem. Define a viability set by the independent conditions \[ T\ge1/R,\quad D_R(1/R)\ge D_{\min},\quad \Var(F_k)\le V_{k,\max},\quad R_-\le R\le R_+,\quad g_{\mathrm{robust}}>0. \] \begin{theorem}[Nonempty dimensionless pilot region] \label{thm:pilotexample} For a reference unit \(t_0>0\), choose \[ p=1,\quad \eta=0.9,\quad \varepsilon=0.02,\quad \kappa=0.05/t_0,\quad 0\le\gamma\le0.05/t_0, \] \[ 1/t_0\le R\le1.1/t_0,\qquad e_0=e_1=0.01. \] Sense at \(t_0\), take \(T=t_0\), \(D_{\min}=0.1\), and a white-force pilot \(B(t)=\sqrt{\gamma/2}\,\dot W_t\) with filter \(\mathbf1_{[0,t_0]}\) and variance allowance \(0.025\). Then the robust score gap exceeds \(0.2\), survival holds at every allowed reduction time, and the filter allowance holds. These are design parameters, not biological measurements. \end{theorem} \begin{proof} Here \(A=0.432\), and \[ g_{\mathrm{robust}} =0.432(e^{-0.10}-e^{-1.05})-0.02 >0.432(0.9-3/8)-0.02=0.2068. \] Also \(D_R(t_0)\ge e^{-1.2}/2>0.1\); every \(1/R\le t_0\), and the witness decreases. The filtered force variance is \(\gamma t_0/2\le0.025\). A nonzero localized Gaussian displacement has energy proportional to \(m^2\), so a declared mass scale can realize any selected positive pilot rate. This is not tubulin calibration. \end{proof} \begin{theorem}[Sampling with explicit dependence assumptions] \label{thm:sampling} For a score in \([0,1]\), actual null mean at most \(a\), alternative mean at least \(b\), and \(g=b-a>0\), a midpoint test on \(N\) independent attempted runs has \[ \alpha,\beta\le e^{-Ng^2/2}. \] Thus \(N\ge2g^{-2}\log(1/\alpha_*)\) bounds each error by \(\alpha_*\). The same result holds under dependence if these mean bounds hold conditional on every past record. Unconditional mean bounds alone do not suffice. \end{theorem} \begin{proof} Hoeffding's bound gives the independent result \cite{Hoeffding}. For dependence, the conditional Hoeffding lemma gives \[ \E[e^{\lambda(f_j-\E[f_j\mid\mathcal F_{j-1}])} \mid\mathcal F_{j-1}]\le e^{\lambda^2/8}. \] Iterate and optimize the exponential bound. Repeating one common Bernoulli variable supplies a counterexample under arbitrary dependence. \end{proof} Bounding the sum of both errors uses \(\log(2/\alpha_*)\). Calibration failure probability \(\beta_{\mathrm{cal}}\) adds that probability when the test assumptions hold conditional on the successful calibration. \section{Track A8: causal and interpretive boundaries} \label{sec:causal} For a finite endpoint channel \(\Phi_B\), absence of influence from all local trace-preserving interventions on \(A\), for every input, is equivalent to \[ \Phi_B=\Xi\circ\Tr_A \] and to \[ \Phi_B^*(L(\mathcal H_B)) \subseteq\One_A\otimes L(\mathcal H_{A^c}). \] To prove the nontrivial implication, compare identity with replacement of \(A\) by \(I_A/\dim\mathcal H_A\), and define \(\Xi(Z)=\Phi_B(I_A/\dim\mathcal H_A\otimes Z)\). Factorization preserves equality with every untouched reference. Backwards support in a complete block-channel schedule bounds actual influence by scheduled ancestors. Unital adjoints on disjoint blocks preserve identity factors. Persistent baths and controllers must be included \cite{CheungVIIJ,CheungVIIIC}. Positive influence instead requires a lower certificate: an ideal bounded-record contrast \(\Delta\), with errors \(e_a,e_b\), has actual contrast at least \(\Delta-e_a-e_b\). For the coherent-versus-dephased preparation, the present score contrast has magnitude \[ p\eta v e^{-(\kappa+\gamma+R)t}/2. \] It concerns the apparatus record, not a constructed neuronal channel. Apply scheduled support, actual intervention contrasts, and physical causal interpretation separately to each structural, quantum-carrier, and gravity-related hypothesis. The epothilone-B study's loss-of-righting endpoint \cite{Khan} does not supply these branch, bath, or instrument certificates. If physical axioms do not constrain an added consciousness predicate on a nonempty event set, two expansions with different predicates have identical physical predictions. This relative non-entailment does not claim that every physical account of consciousness is impossible. \section{Track B: source-aligned definitions and their exact scope} \label{sec:Bdefinition} \subsection{Three different types of statement} Track B uses: \begin{enumerate} \item \textbf{Definitions and proofs here:} explicit retarded distributions, parameterized local-source and response interfaces, tree composition, reconstruction descent, contact obstructions, normal jets, and conditional admission. \item \textbf{Source records:} mathematical statements reported in the supplied Cheung excerpts, with exact source labels where available. They are not independently re-proved merely by being restated. \item \textbf{Missing inherited data:} exact density components, some response coefficients, the full selected characteristic locus, and the exact full-shell ideal and ordering conventions. No replacement choice is asserted to be the unchanged inherited model. \end{enumerate} The frozen documentary data are the XII-C cubic representative, inverse coefficient \(-1\), the two completion legs, XII-E insertion coefficient \(-1\), the scalar-sourced slot, XII-A full-shell carrier, and XIII-C partial-diagonal ambiguity class \cite{CheungXIIA,CheungXIIC,CheungXIID,CheungXIIE,CheungXIIIC}. The additional excerpts establish filenames for \texttt{XII-A-v2.tex}, \texttt{XII-C.tex}, \texttt{XII-D.tex}, \texttt{XII-E.tex}, and \texttt{XIII-C.tex}. Their full contents are still not included in the supplied package. \subsection{Spacetime, tests, and propagator conventions} \label{sec:Bspacetime} Take \[ M=\R^{1+3},\qquad \eta=\diag(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta_{\bm x}. \] All distributions act on compact smooth test densities; the fixed Minkowski volume identifies them with test functions. Set \(\mathcal D(U)=C_c^\infty(U)\), \(\mathcal D'(U)=\mathcal D(U)'\), and \(\mathcal E'(U)\) for compactly supported distributions. For a finite species space, use componentwise distribution spaces. The massless retarded fundamental solution is \begin{equation} \Rker(t,\bm x)= \frac1{2\pi}\Theta(t)\delta(t^2-\abs{\bm x}^2). \label{eq:Rret} \end{equation} The formula is understood as the standard cone measure, including its defined extension at the vertex, with \(\Box\Rker=\delta_0\). Use \(E_0^{\mathrm R}\) for convolution with \(\Rker\) on compact or suitably past-compact inputs, and \(E_0^{\mathrm A}\) for its transpose. Similarly \(E_m^{\mathrm R/A}\) are the Green operators of \(\Box+m^2\). Existence, causal support, and Cauchy uniqueness are standard \cite{BGP}. The fixed smooth observation \(f\) and interaction switch \(g\) are compactly supported. The fixed smooth ramp \(\sigma_\varepsilon(t)\) vanishes for \(t\le0\), with fixed \(\varepsilon>0\); no sharp-ramp limit is used. Its precise transition profile is an input. For a symmetric covariant tensor define trace reversal \[ (\Pmet h)_{\mu\nu} =h_{\mu\nu}-\tfrac12\eta_{\mu\nu}\eta^{\alpha\beta}h_{\alpha\beta}. \] In four dimensions \(\Pmet^2=\id\). The metric response operator and prescribed inverse are \[ L_{\mathrm{gr}}=-\Box\Pmet,\qquad G_{\mathrm{gr}}^{\mathrm R}=-\Pmet E_0^{\mathrm R}. \] They must not be confused with the coordinate-response kernel \(\Lcoord\) below, or with Track A's rate \(R\). \begin{lemma}[Retarded convolution and fixed Fourier pole] \label{lem:retarded} The causal convolution is defined and \[ \Bker:=\Rker*\Rker =\frac1{8\pi}\Theta(t)\Theta(t^2-\abs{\bm x}^2). \] For \(t>0\) away from the vertex, \[ \partial_\alpha\Bker(x)=\frac{x_\alpha}{4\pi}\delta(x^2). \] With temporal Fourier convention \[ \widehat u(\omega)=\int e^{-\ii\omega t}u(t)\,\dd t,\qquad u(t)=\frac1{2\pi}\int e^{\ii\omega t}\widehat u(\omega)\,\dd\omega, \] and \(\rho=\abs\xi>0\), the retarded multiplier is \[ M_{\mathrm R}= -\frac1{2\rho}\left[ (\omega-\rho-\ii0)^{-1} -(\omega+\rho-\ii0)^{-1}\right]. \] On \(\lambda_s=\omega-s\rho\), \(\abs{\lambda_s}a-c-1\). The Taylor subtraction \[ W_p\phi= \phi-\chi(q)\sum_{|\alpha|\le p} \frac{q^\alpha}{\alpha!}\partial_q^\alpha\phi(s,0) \] gives the extension \[ U(\phi)=\int u(s,q)W_p\phi(s,q)\,\dd s\,\dd q. \] The remainder is \(O(\abs q^{p+1})\), so radial integrability is \(r^{p+c-a}\), with exponent greater than \(-1\). This proves extension for that printed class. It does not prove a full characteristic extension from frequency homogeneity alone. For general distributional transverse extensions, scaling and wavefront hypotheses must be checked against an applicable theorem \cite{Dang,BrunettiFredenhagen}. One cannot cite such a theorem without establishing its domain, scaling, conormal-landing, and support conditions. \begin{proposition}[Smoothness off a locus does not ensure extension] \label{prop:noextension} A smooth distribution on a punctured domain need not extend across the puncture. \end{proposition} \begin{proof} Take \(e^{1/x^2}\) on \(0