executable_observer_construction.tex

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  pdftitle={An Executable Observer for Observer Patch Holography: Specification, Executed Components and Ordered Joins},
  pdfauthor={Bernhard Mueller; Mario Poneder; Jinwook Kim; Daniel Ari Friedman}}

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\title{An Executable Observer for Observer Patch Holography:\\
Specification, Executed Components and Ordered Joins}
\author{Bernhard Mueller\textsuperscript{1,2,*}\qquad Mario Poneder\qquad
Jinwook Kim\textsuperscript{3}\qquad
Daniel Ari Friedman\textsuperscript{4,5}}
\date{\OPHPaperReleaseDate}
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\begin{document}
\maketitle
\OPHPaperReleaseBanner

\begin{center}
\footnotesize
\textsuperscript{1}Pragma Research Inc.\quad
\textsuperscript{2}Information Physics Institute\quad
\textsuperscript{3}Oraclizer Labs\quad
\textsuperscript{4}Active Inference Institute\quad
\textsuperscript{5}FractiAI\\
\textsuperscript{*}Corresponding author:
\href{mailto:bernhard@floatingpragma.ai}{\nolinkurl{bernhard@floatingpragma.ai}}
\end{center}

\begin{abstract}
Observer Patch Holography (OPH) is made testable by its executable-model
hypothesis: a fully specified local repair and memory law on the proposed
carrier architecture admits continuing observers whose records have the
observed quantum-field and gravitational descriptions. This paper is the base
of that hypothesis. It specifies the twelve-port icosahedral carrier, the
native C/E seam repair with its exactly classified family of quantum record
laws, the finite observer algebra, the declared services and the scheduler,
and reports the finite components executed on that specification. Uniform
seam attempts have exact quantum gap $(5-\sqrt5)/60$; a record-steered
observer moves a final occupation from $1/6$ to $35/264$; nine
source-executed measurements identify every coordinate of the record-law
family. Overlapping observers read fields and event clocks from one
continuing history, and one register joins quantum electric flux, charged
fermions and a scalar with a feedback response certified nonzero at infinite
occupation. On the refining spherical source one repair history defines a
continuum operator of spectral dimension two governing Gaussian density
histories read by native incident clocks; a declared radial extension gives
a smooth $1+3$ free-scalar description of every fixed finite observer menu.
A specified coherent memory protects exchange-history sectors; complete
source factors further join fermionic propagation, Higgs-driven particle
creation, native repair and one calibrated clock. Their full outcome and
reference records are retained. Physical clock/energy validation, localized
particle identity, an interacting chiral $3+1$ limit and realized Einstein
premises are not established. Four negative controls remain: the native double-load defect decays with
survival $(5/6)^n$, the static radial product fails scalar null balance,
passive repair strictly attenuates coherence and has no wave sector, and the
tested mass laws miss the charged hierarchy.
\end{abstract}

\input{tex_fragments/EXECUTABLE_COMPLETION_HYPOTHESIS.tex}
\input{tex_fragments/FINITE_COMPLETION_CAMPAIGN.tex}

\section{This paper's hypothesis and place in the tower}
\label{sec:intro}

This paper is the executed part of layers two to four of the tower: the
carrier and its repair law, the executable observer, and the fields and
clocks read from one process. Its hypothesis is the executable-model
hypothesis stated above, taken at the level a finite execution can test.
The carrier, the native C/E repair law and its memory, the finite observer
algebra, the declared services and the scheduler are fixed before testing,
and the observers, records, local fields and event clocks of every result are
read from a process run under that one specification. Fields, particles and
geometry enter as descriptions of what observers record and jointly
interpret; no fundamental field or spacetime is reconstructed first
\cite{mCONSTRUCTION,mINTERP,basis}.

The paper proves finite theorems and certifies finite computations on the
declared objects: the seam rule from four requirements, the classification
of admissible quantum record laws, the exact gap, memory and recovery of the
native channels, record capacity, finite observers and field detectors with
every history retained, the common interacting source and its occupation
limits, overlapping-observer correspondence, local event clocks and matched
screens. For one repair history on the refining spherical source it proves
source limit theorems: a common continuum operator of spectral dimension
two, Gaussian density paths, incident-clock parametrizations and their
fluctuations. Conditional analytic theorems hold on supplied objects: a
declared radial product with a chosen wave action, and the supplied
interacting actions with their occupation limits.

None of these components is a physical clock, field theory, particle or
gravitational response. The five conjunctive obligations of the executable
hypothesis are open joins of this construction \cite{execHypothesis}:
observer algebras with faithful references, executed modular clocks and a
physical energy normalization; a nontrivial interacting chiral 3+1 field
theory with controlled refinement and detector maps; native defects
identified through charge, spin and statistics, propagation, interactions
and energy, with the measured spectrum; the Einstein reconstruction premises
realized on the same matter, state, modular flow, clocks and refining
geometry; and joint validation of complete observation and intervention
laws against independent measurements. Table~\ref{tab:obligations} separates
what this paper executes from what each obligation requires beyond it, and
Section~\ref{sec:joins} orders the joins with their acceptance conditions
and stopping tests.

A replayable failure of the specified law, state, access or readout contract
rejects that candidate and its tested interpretation. A proof covering every
realization of a frozen admissible class stops that class, and it rejects
OPH only if the excluded choice is proved necessary. Section~\ref{sec:joins}
records four such failures of the tested law: the native double-load defect
decays, the static radial product cannot balance a scalar excitation,
passive repair has no wave sector, and the tested mass laws miss the charged
hierarchy.

{
\footnotesize
\begin{longtable}{@{}L{0.17\linewidth}L{0.42\linewidth}L{0.35\linewidth}@{}}
\caption{The five obligations of the executable hypothesis
\cite{execHypothesis}, the components this paper executes toward each, and
the open join.}
\label{tab:obligations}\\
\toprule
\textbf{Obligation} & \textbf{Executed here} & \textbf{Open join}\\
\midrule
\endfirsthead
\multicolumn{3}{@{}l}{\footnotesize Table~\thetable\ continued}\\
\toprule
\textbf{Obligation} & \textbf{Executed here} & \textbf{Open join}\\
\midrule
\endhead
Clocks and energy normalization &
Finite modular flow of actual observation states and the fixed-reference
relation $t=-\beta s$ (Section~\ref{sec:readouts}); local, joint and active
event clocks (Sections~\ref{sec:local-tick}, \ref{sec:event-clocks});
native incident clocks with source limits and fluctuation predictions, and
faithful encoded-field states sharing one active clock
(Section~\ref{sec:native-history}). &
A reference selected by the process, regional compatibility, calibrated
duration and the energy normalization. Passive repair is no nontrivial
modular automorphism of an invariant observer algebra, and the uncapped
native-return reference tends to the trace.\\
\addlinespace
Interacting chiral 3+1 field theory &
Free-scalar refinement transfer and two-site occupation limits
(Section~\ref{sec:fields}); an interacting feedback signal at infinite
occupation (Section~\ref{sec:common}); the native continuum operator with
density, clock and admitted-wave limits, a declared radial product with
Lorentz-signature density, smoothing theorem and a quartic projection with
an occupation limit at fixed band (Section~\ref{sec:native-history}). &
The fields are free or two-mode; radial product, filters, preparations and
inverse temperature are declared; no interacting refining family and no
chiral matter. The native decay spectrum excludes a complete linear
identification with a compact free scalar.\\
\addlinespace
Native defects, identity and spectrum &
A charged realization with branch charge covariance
(Section~\ref{sec:p-detector}); a flux-dressed fermion code with operator
Gauss constraints (Section~\ref{sec:common-construction}); the collision
candidate tested (Section~\ref{sec:negative-controls}). &
No localized protected native sector, maintenance-energy law or spectrum.
The collision decays and the tested mass laws miss the charged hierarchy.\\
\addlinespace
Realized Einstein premises &
A rank-three response, native causal cones, the product's area and volume
dictionary, one characteristic cone for scalar, spinor and Maxwell symbols,
and a necessary spherical focusing equation
(Sections~\ref{sec:native-history},
\ref{sec:common-matter-metric-response}). &
Responsive geometry, conserved stress of the native process,
fixed-proper-volume generalized-entropy stationarity and a positive Newton
normalization. The static radial product fails null balance.\\
\addlinespace
Joint validation &
A prospectively fixed count protocol for the record-law coherence and its
sequential design; retrospective Bell and optical-clock replays; a
retrospective coupling audit (Section~\ref{sec:prediction}). &
One frozen physical dictionary compared with independent field, particle
and gravity data.\\
\bottomrule
\end{longtable}
}

The theorem library holds conditional theorems on consensus and repair
\cite{ophconsensus}, observers and records \cite{ophobservers}, gauge
structure \cite{ophgauge}, spacetime and Einstein dynamics
\cite{ophspacetime}, and screen microphysics \cite{ophmicro}; the flagship
synthesis gives the informal account \cite{flagship}. The labels A1, A2 and
A3 used there name theorem premises: A1 is the carrier, federation and
response layer of the specification, A2 the agreement condition with its
separately adopted response-endogeneity clause, and A3 the adopted
least-informative state-selection principle on its declared reference and
constraints. Neither selection nor connected-response endogeneity follows
from the closure equation alone. A theorem that uses them states an implication, and
Section~\ref{sec:premises} records, route by route, which antecedents the
construction supplies on its own finite objects.

The sections follow the tower. Section~\ref{sec:spec} states the
specification (layer two). Sections~\ref{sec:observer}
and~\ref{sec:native-process} give the executable observer on one carrier and
overlapping observers, event clocks and matched screens on one native process
(layer three). Sections~\ref{sec:fields} to~\ref{sec:native-history} give the
fields and clocks (layer four): field detectors on supplied actions, the
common interacting source, and the geometry, continuum operator, clocks and
fields of one repair history, followed by the common matter metric with its
gravitational control (the interface to layer six).
Section~\ref{sec:prediction} gives the prediction protocols,
Section~\ref{sec:premises} the premise map and Section~\ref{sec:joins} the
ordered joins. Every number is copied from a
retained receipt or a model document of the construction \cite{mREADME},
cited at a pinned revision of the public repository \cite{ophrepo}.

Results carry the claim classes of Table~\ref{tab:classes}. Actions,
preparations, couplings, instruments, services and schedules are inputs of
the specification, and apart from the retrospective replays of
Section~\ref{sec:public-records} no result is a comparison with independent
observations. Numbered theorems and propositions are finite theorems unless
they state a limit; items that quote a certified enclosure, table or
replayed count are certified finite computations; the occupation limits of
Section~\ref{sec:common} are conditional analytic theorems on their supplied
actions; the native continuum statements of
Section~\ref{sec:native-history} are source limit theorems with their
domain, normalization and order of limits; Sections~\ref{sec:public-records}
and~\ref{sec:endpoint} are diagnostics, and the count protocol of
Section~\ref{sec:ce-protocol} is a prospectively fixed conditional test.

\begin{table}[htbp]
\centering
\small
\caption{Claim classes used in this paper, from the flagship synthesis \cite{flagship}.}
\label{tab:classes}
\begin{tabular}{@{}L{0.30\linewidth}L{0.64\linewidth}@{}}
\toprule
\textbf{Class} & \textbf{Meaning}\\
\midrule
Finite theorem & A deductive result proved from displayed premises
on a stated finite domain. An exhaustive computation counts as part of the
proof only with a specified domain and a soundness argument.\\
Certified finite computation & An executable witness whose inputs,
controls, and output are fixed by a reproducible certificate; execution alone
does not promote its interpretation to a theorem.\\
Conditional analytic theorem & A theorem whose hypotheses include a supplied continuum action, state space or limit; its conclusion holds on that supplied object.\\
Source limit theorem & A theorem about the limit of a declared family of
finite source executions along a stated subsequence, with its
normalizations and order of limits displayed; the limit is derived, and its
physical identification is a separate claim.\\
Reconstruction implication & An exact implication whose antecedents
include displayed geometric or physical premises.\\
Unsupplied realization map & A required map from a finite object to a
physical event, current, field, clock, scale, or continuum structure for which
no construction is asserted.\\
Diagnostic & A certified computation compared against measured
values that it consumes; never counted as a prediction.\\
Prospectively fixed conditional test & A diagnostic with a custody-bound
target definition, kill band, precision floor, and decision rule fixed
before comparison.\\
\bottomrule
\end{tabular}
\end{table}

Seven claim records of the repository \cite{ophrepo} state the scope of the
executed components (Table~\ref{tab:registry}). Further model records own
the published-oscillator and anharmonic-transfer comparisons and the matter,
common-field and static-product witnesses; they are cited where those
results appear.

\begin{table}[htbp]
\centering
\footnotesize
\caption{Executed components and the repository's claim records that state
their scope.}
\label{tab:registry}
\begin{tabular}{@{}L{0.42\linewidth}L{0.14\linewidth}L{0.38\linewidth}@{}}
\toprule
\textbf{Component} & \textbf{Sections} & \textbf{Claim record}\\
\midrule
Two-event native observer, classification of record laws, quantum mixing of
the repair channels, recovery from complete records, phase protocol and its
source-executed implementation &
\ref{sec:record-laws}, \ref{sec:observer-algebra}, \ref{sec:native-observer}, \ref{sec:mixing},
\ref{sec:recovery}, \ref{sec:ce-protocol}, \ref{sec:ce-source} &
\texttt{OPH-MODEL-NATIVE-OBSERVER-EXECUTION}\\
Correctable record capacity and compact controller &
\ref{sec:capacity} & \texttt{OPH-MODEL-OBSERVER-RECORD-CAPACITY}\\
$P$-dependent binary detector and retrospective coupling audit &
\ref{sec:p-detector}, \ref{sec:endpoint} &
\texttt{OPH-MODEL-P-RESPONSE-EXECUTION}\\
Memory certificate at observer-local ticks for the reciprocal exchange law &
\ref{sec:local-tick} & \texttt{OPH-MODEL-N4-LOCAL-TICK-CERTIFICATE}\\
Scalar detector, two-read continuation and two-site source apparatus &
\ref{sec:scalar}, \ref{sec:two-read}, \ref{sec:two-site} &
\texttt{OPH-MODEL-SCALAR-RESPONSE-EXECUTION}\\
Common interacting source, occupation and readout certificates, smooth
continuation, compiled native attachment and encoded memory, observer
readouts, history repair, geometric and field readouts &
\ref{sec:common}, \ref{sec:history-geometry} &
\texttt{OPH-MODEL-COMMON-SOURCE-EXECUTION}\\
Overlapping observers, local and active event clocks, record-law
identification, federations and matched screens, the native continuum
operator and its limits, the sequential protocol and public-record replays &
\ref{sec:continuum}--\ref{sec:product-field}, \ref{sec:native-process},
\ref{sec:sequential}, \ref{sec:public-records} &
\texttt{OPH-MODEL-NATIVE-PROCESS-RECOVERY}\\
\bottomrule
\end{tabular}
\end{table}

\section{The specification}
\label{sec:spec}

The specification is layer two of the tower. The carrier, the repair law and
its record laws, the observer algebra, the services and the time parameters
below are fixed before any test, and every later result runs on them.

\subsection{The carrier and its load configurations}
\label{sec:carrier}

\begin{figure}[!htb]
\centering
\input{tex_fragments/figures/FIG_CARRIER_SEAM_RULE.tex}
\caption{The twelve-port carrier and its native repair. Left: the icosahedral incidence of the twelve ports, with thirty seams and twenty faces, in an orthographic view with back edges dashed; the filled ports carry a binary load of total two, and the drawn seam $(a,b)$ has $d=1$. Right: the seam rule for every mismatch the total-two carrier admits.}
\label{fig:carrier}
\end{figure}

The specification has one carrier with twelve primitive ports. Its incidence
is the oriented boundary of an icosahedron, with thirty seams and twenty
faces; five seams meet at every port, the neighbours of each port form a
five-cycle, and every seam lies on two oppositely oriented faces
\cite{ophmicro,flagship}. Write $A$ for the adjacency matrix of the seam
graph and $L=5I-A$ for its Laplacian. The spectrum of $L$ is $0$,
$5-\sqrt5$ with multiplicity three, $6$ with multiplicity five and
$5+\sqrt5$ with multiplicity three, so its characteristic polynomial is
$x(x-6)^5(x^2-10x+20)^3$ \cite{oQM,rQM,csSLOW}.

A load configuration assigns a nonnegative integer $x_p$ to each port $p$,
with total load $\sum_p x_p=2$. There are 78 configurations. The 66 with two
distinct occupied ports form the binary shell; the other twelve are
collisions $\ket{2u}$ with load two at one port $u$. We write $\Hil_{78}=\C^{78}$
for the quantum load space with basis $\ket{x}$, $\Hil_{66}$ for the span of
the binary shell, and $M_{78}$, $M_{66}$ for the corresponding full matrix
algebras. For a binary configuration occupied at ports $u$ and $w$ we also write $\ket{uw}$ or $\ket{\{u,w\}}$.

Ports are numbered $0$ to $11$ by the frame vectors $u_p$: $(-1,\varphi,0)$, $(1,\varphi,0)$, $(-1,-\varphi,0)$, $(1,-\varphi,0)$, $(0,-1,\varphi)$, $(0,1,\varphi)$, $(0,-1,-\varphi)$, $(0,1,-\varphi)$, $(\varphi,0,-1)$, $(\varphi,0,1)$, $(-\varphi,0,-1)$, $(-\varphi,0,1)$, with $\varphi=(1+\sqrt5)/2$. The thirty seams are the pairs with $\lvert u_a-u_b\rvert^2=4$, each written $(a,b)$ with $a<b$, and that order is the orientation used by the repair law below \cite{csSLOW,csCAUS}.

Premise A1 of the theorem library is the layer of carriers, their federation
through typed seams and coherent triple overlaps, the oriented spherical
support and the complete reversible port response \cite{axref}. The
construction realizes one carrier of that layer: its ports, seams and faces,
its load configurations and the repair events on its seams. Coherent quantum
operations run through a six-level processor whose twelve source generators,
one per port, exponentiate to every executed gate on two- and
four-dimensional proper codes through declared load and export services
\cite{csDERIV,ssREADME}. Spherical-screen executions join many carriers under
explicit access and refinement contracts \cite{nativeContinuum}; they
identify no physical current for the complete response algebra and do not
discharge the full A1 attachment.

\subsection{Native repair and its quantum record laws}
\label{sec:repair}

Fix an oriented seam $e=(a,b)$ and a fair placement coin $c\in\{0,1\}$. The
native repair law is
\begin{equation}
\label{eq:native}
d=x_a-x_b,\qquad
j=\tfrac12\bigl[d-(2c-1)(\lvert d\rvert \bmod 2)\bigr],\qquad
y=x+j\,(\mathbf e_b-\mathbf e_a),
\end{equation}
where $\mathbf e_p$ is the unit load vector of port $p$ and $y$ is the
repaired configuration. The two endpoint loads become the floor and the
ceiling of their mean; the coin assigns the ceiling to $a$ when $c=1$; all
other loads are unchanged.

\begin{proposition}[The classical seam rule]
\label{prop:classical-rule}
Let a stochastic rule on one seam (i) conserve $x_a+x_b$, (ii) end at a
configuration minimizing $x_a^2+x_b^2$, (iii) break any remaining tie by a
random choice independent of the loads, and (iv) be invariant under exchange
of the two endpoints. Then it is \eqref{eq:native} with a fair coin.
\end{proposition}

\begin{proof}
By (i) and (ii) the outputs are the floor and the ceiling of the mean. An odd
total leaves two outputs, and by (iii) endpoint $a$ receives the ceiling with
a constant probability $q$. Exchange of the endpoints maps $q$ to $1-q$, so
(iv) gives $q=\tfrac12$.
\end{proof}

Without (iii), the rule that leaves the extra unit with the endpoint holding
more satisfies the other three requirements. It fixes every binary
configuration, so the binary shell carries no motion and the records carry
no geometry. Requirement (iii) is an independent restriction on the
admitted tie-breaker: it excludes retaining a preference for the endpoint
holding more at the input. Internal realization alone does not impose that restriction.

The record is $r=(c,j)$, the coin and the signed
current. Compatibility on a seam is specified independently of the update;
it requires the two endpoint loads to differ by at most one. On the quantum load space the
record defines the current instrument $\Cn$ with Kraus operators and
record-discarded channel
\begin{equation}
\label{eq:kraus}
L^{e}_{(c,j)}=\frac{1}{\sqrt2}\sum_{x:\,j(x,c)=j}\ket{y(x,c)}\bra{x},
\qquad
\Cn_e(\rho)=\sum_{r}L^{e}_{r}\,\rho\,L^{e\dagger}_{r}.
\end{equation}
Both maps are written in Kraus form \cite{kraus1983}. The contrast-record
variant $\En$ keeps $r=(c,d)$ instead, with Kraus operators $M^{e}_{(c,d)}=2^{-1/2}\sum_{x:\,x_a-x_b=d}\ket{y(x,c)}\bra{x}$ and
record-discarded channel $\En_e$. Uniform independent seam attempts form the
baseline schedule; on diagonal binary states the repair is then the lazy symmetric exclusion process with two particles, with generator $\frac1{60}\sum_e(S_e-I)$ per attempt, $S_e$ exchanging the endpoint loads \cite{mCANDIDATES,oQM}.

\begin{proposition}[Native repair instrument]
\label{prop:native}
Let $e=(a,b)$ be an oriented seam of the twelve-port carrier, let the
placement coin be fair, and let $\Cn_e$ and $\En_e$ act on $\Hil_{78}$. Then:
\begin{enumerate}[(i)]
\item For every record value the branch map $x\mapsto y(x,c)$ is an injective
translation of its support. Consequently $\sum_rL^{e\dagger}_rL^e_r=I$, and
$\Cn_e$ is completely positive and trace preserving (CPTP), also when
tensored with an untouched reference system. The same holds for $\En_e$.
\item Every branch conserves the total load, leaves spectator ports
unchanged, never increases $\sum_px_p^2$, and maps the binary shell into
itself.
\item The current labels satisfy $c\in\{0,1\}$ and $j\in\{-1,0,1\}$; all six
labels occur on $\Hil_{78}$ and four occur on the binary shell.
\item $\Cn$ and $\En$ induce the same classical transition on every
configuration, and
\[
\En_e^2=\En_e,\qquad \En_e\Cn_e=\Cn_e\En_e=\En_e,\qquad
\Cn_e^n=\En_e+2^{1-n}(\Cn_e-\En_e)\quad(n\ge1).
\]
On the binary shell $\En_e$ is a trace-preserving conditional expectation.
\item With the seam-total parity $p=(x_a+x_b)\bmod 2$ one has
$d=2j+(2c-1)p$, and $\En$ equals $\Cn$ followed by a binary record of $p$
and this relabelling, as an identity of complete unnormalized branches.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
The inverse of a fixed-current branch adds $j$ at $a$ and subtracts it at
$b$; every basis vector has one branch for each coin, so each column of the
joint record/source isometry has two entries $2^{-1/2}$ with disjoint
supports \cite{oDERIV}. For fixed contrast, the unchanged spectators and the
endpoint sum reconstruct the input, which gives (i) for $\En$. Items (ii) and
(iv) are the single-seam relations of the candidate register
\cite{mCANDIDATES}. Item (v) follows from $j=\lceil d/2\rceil$ for $c=0$ and
$j=\lfloor d/2\rfloor$ for $c=1$; the parity-refined Kraus operator
$\Pi_pL_{(c,j)}$ has exactly the contrast group as support
\cite{sdDERIV}. The executable checks all 312 outputs of the observer's two
seams against an independent floor and ceiling rule \cite{oDERIV}.
\end{proof}

The current instrument is no classical coarse-graining of the contrast
instrument: grouping the labels of $\En$ sums density operators and cannot
restore the coherent Kraus sums of $\Cn$. Both belong to the family of quantum
record laws classified in Section~\ref{sec:record-laws}. One further law of the
candidate register, the reciprocal exchange N4 that couples the native
current to private six-level response factors, enters only the local-tick
certificate of Section~\ref{sec:local-tick} \cite{mCANDIDATES}.

\subsection{Admissible record laws}
\label{sec:record-laws}

The classical rule fixes the diagonal of a record law and leaves its
coherences free. For the seam $e=(a,b)$ write $\kappa(x)=(x_a,x_b)$ for the
seam key and $D=\mathbf e_b-\mathbf e_a$. A channel $\Phi$ on $\Hil_{78}$ is
an admissible record law for $e$ if it acts as the identity on the ten
spectator ports, reproduces the classical repair distribution on every
$\proj x$, commutes with $U_t=\exp(i\sum_pt_pN_p)$ for every $t\in\mathbb
R^{12}$, and is invariant under the carrier rotation that exchanges $a$ and
$b$ with the current reversed. The third condition makes every local load
phase a symmetry of the unread channel. It is a premise of the
classification, a local charge-superselection rule for the repair channel,
and it is what excludes a recorder of a pre-repair endpoint load. It guarantees that a
definite-current Kraus representation exists; it does not force every
recorded implementation of the channel to commit a definite current, since
a change of record basis before commitment mixes the current labels and
leaves the channel unchanged \cite{mINSTRUMENT}. The classification below
concerns unread channels, and statements about current histories use the
compatible definite-current instrument.

\begin{theorem}[Coherence tables]
\label{thm:record-laws}
A channel $\Phi$ is an admissible record law if and only if
\[
\Phi(\ket x\bra{x'})=\sum_j G_j\bigl(\kappa(x),\kappa(x')\bigr)\,
\ket{x+jD}\bra{x'+jD}
\]
for endpoint-symmetric positive semidefinite matrices $G_j$ on the seam keys
that can carry the current $j$, with $G_j(\kappa,\kappa)=\Pr(j\mid\kappa)$.
On $\Hil_{78}$ the free entries are five multipliers: $k$ between the empty
and the full seam, $g$ between the empty seam and a waiting single load, $h$
between the full seam and a waiting single load, a real $m$ between the two
waiting orientations, and $n$ between a moving single load and a moving
collision. Positivity gives $\lvert g\rvert,\lvert h\rvert,\lvert
n\rvert\le2^{-1/2}$ and $\lvert m\rvert\le\frac12$, and $k=1$ forces
$g=h$. Repairing one seam with $(k,g,h,m,n)$ and then with
$(k',g',h',m',n')$ gives $(kk',gg',hh',mm',n\bar h')$. Exactly two admissible
laws are idempotent, those with $g=h=m=n=0$ and $k\in\{0,1\}$, and every law
with $k=1$ converges under repetition to the idempotent law with $k=1$.
\end{theorem}

\begin{proof}[Proof sketch]
Covariance places $\Phi(\ket x\bra{x'})$ in the span of units $\ket
y\bra{y'}$ with $y-y'=x-x'$. Complete positivity confines nonzero Choi
entries to pairs with nonzero diagonal, which the classical law makes
$(x,x+jD)$, so both sides carry the same current and the Choi matrix is the
direct sum of the $G_j$. Trace preservation and the classical law fix the
diagonals, and spectator invariance makes each $G_j$ a function of the seam
keys. Under repetition every coherence keeps its seam key, so each multiplier
applies again; idempotence requires $k,g,h,m\in\{0,1\}$ and $n=n\bar h$, and
the bounds leave $g=h=m=n=0$. Exact arithmetic in $\mathbb Q(\sqrt2)(i)$
checks the composition law on every pair of seam keys for 64 ordered pairs
of tables and on all 6,084 matrix units for two pairs \cite{mRECORDLAWS}.
\end{proof}

\begin{figure}[!htb]
\centering
\input{tex_fragments/figures/FIG_RECORD_LAW_LADDER.tex}
\caption{The record-law ladder of Theorem~\ref{thm:record-laws} and Table~\ref{tab:record-laws}, and the one-event interference readout $\tfrac38+\tfrac12\operatorname{Re}g$ at phase zero that separates its members.}
\label{fig:record-law-ladder}
\end{figure}

Four members form a ladder (Table~\ref{tab:record-laws} and
Figure~\ref{fig:record-law-ladder}): $\Cn$ is $\Czero$ followed by a diagonal
readout of the coin, $\En$ is $\Cn$ followed by a readout of the seam parity
as in Proposition~\ref{prop:native}(v), and $\Tn$ is $\Cn$ followed by a
readout of the seam total. The completion of $\Cn$ in
Proposition~\ref{prop:native}(iv) is the case $g=h=n=\frac12$, $m=0$, and
$\En$ is the idempotent law with $k=1$. A recorder of an endpoint's pre-repair
load keeps the classical law and fails covariance, because one of its
branches carries two currents. $\Czero$ records only what conservation
requires, $\Cn$ records the tie-break as an event, $\En$ is the coarsest law
for which one repair completes the seam, and $\Tn$ records the full seam
occupation; none of these principles follows from the theorem premises. The
construction executes $\Cn$ as its convention and compares it with $\En$.
Under a common definite-current record interface all members assign the same
probabilities to every classical current observation;
Section~\ref{sec:prediction} reads $g$ in one coherent event, and
Section~\ref{sec:identification} identifies all nine real coordinates. Every admissible table, including complex phases and different
laws on different seams, has the uniform binary density as its unique
stationary source state and the same spectral gap $(5-\sqrt5)/60$, so
equilibrium selects none of the coordinates; a finite unread warm-up of
$221$ attempts on the binary shell, or $318$ on the full carrier, brings an
arbitrary initial density within $1/1000$ of that state, uniformly over
untouched references \cite{mRECORDLAWS}.

\begin{table}[htbp]
\centering
\small
\caption{Four distinguished admissible record laws \cite{mRECORDLAWS}.}
\label{tab:record-laws}
\begin{tabular}{@{}lL{0.36\linewidth}cccc@{}}
\toprule
Law & Each event records & Values per seam & $k$ & $g=h$ & Idempotent\\
\midrule
$\Czero$ & the current & 3 & 1 & $2^{-1/2}$ & no\\
$\Cn$ & coin and current & 6 & 1 & $1/2$ & no\\
$\En$ & coin and pre-repair contrast & 10 & 1 & 0 & yes\\
$\Tn$ & coin and both endpoint loads & 12 & 0 & 0 & yes\\
\bottomrule
\end{tabular}
\end{table}

\subsection{The finite observer algebra}
\label{sec:observer-algebra}

An observer of length $n$ on the carrier consists of a preparation on $\Hil_{78}$, a record tape $R^{\otimes n}$, seam policies $s_k$ that map committed records and classical randomness to seams or to supplied detector outcomes, the coherent writes $W_{s_k}$, optional supplied instruments whose committed bits may enter $s_k$, and a halting rule after $n$ writes; a policy reads only committed records. Definition~\ref{def:observer} is the instance $n=2$, and Sections~\ref{sec:p-detector}, \ref{sec:fields} and \ref{sec:common} add declared instruments to it.

\begin{definition}[Two-event native observer]
\label{def:observer}
Let $R=\C^6$ carry one basis vector per current label $(c,j)$. The observer
carrier is
\[
\Hil_{\rm cap}=\Hil_{78}\oplus(R\otimes\Hil_{78})\oplus(R\otimes R\otimes\Hil_{78}),
\]
with sectors of dimensions 78, 468 and 2808, total 3354; the sector index is
the phase, the number of filled record slots.
\begin{enumerate}[(a)]
\item \emph{Preparation.} The scored preparation is uniform on the 66 binary
configurations; the full input family uses all 78.
\item \emph{Measurement and record.} At phase zero the coherent write
$W_{e}=\sum_r\ket{r}\otimes L^{e}_r$ acts on the seam $e=(0,1)$.
\item \emph{Record-only feedback.} At phase one the seam is $s(r)=(0,5)$ if
the stored current is positive and $(0,1)$ otherwise, and
$V_1(\ket r\otimes\psi)=\ket r\otimes W_{s(r)}\psi$.
\item \emph{Forecast.} Before the second event the controller reads its
stored letter and issues a supplied forecast $q(r)$ of a positive second
current.
\item \emph{Halting.} At phase two the observer acts as the identity.
\end{enumerate}
The controller reads only the committed first record; it has no access to
source loads or future outcomes.
\end{definition}

\begin{proposition}[One CPTP map with classical-prefix preservation]
\label{prop:cptp}
Let $V_0=W_{(0,1)}$ and $V_1$ be extended by zero to the other sectors, giving
$A_0$ and $A_1$, and let $P_2$ project onto the full sector. Then:
\begin{enumerate}[(i)]
\item $V_0$ and $V_1$ are isometries on every nonfull sector, including
unreachable combinations of letters and loads, and
$T(\rho)=A_0\rho A_0^\dagger+A_1\rho A_1^\dagger+P_2\rho P_2$ is CPTP;
\item on the inhabited subspace $K=\Hil_1\oplus\Hil_2$ every first-record
projector $Q_r$ obeys $T^*(Q_r)=Q_r$, so a committed first record is reread
with probability one;
\item for the blank input with occupations $(1,1,0,\dots,0)$, the
first-record projector of the letter $(0,0)$, extended by zero to phase zero,
has expectation $0$ before the tick and $1/2$ after it, so writing a record is
a transition distinct from preserving it;
\item for the equal superposition of $\{0,2\}$ and $\{1,2\}$, the
reduced-density entry of the first-record pair $((0,1),(0,0))$ equals $1/4$
after the first write and $1/8$ after the second while the classical
diagonals are unchanged, so coherence between different record values need
not survive feedback.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
Orthogonal retained labels and $W_s^\dagger W_s=I$ give
$V_0^\dagger V_0=V_1^\dagger V_1=I$, hence $A_0^\dagger A_0+A_1^\dagger
A_1+P_2=I$. Every subsequent map intertwines $Q_r$ on $K$. In (iv) the
working state attached to the entry has occupations $\{1,2\}$, on which the
isometries of $(0,5)$ and $(0,1)$ overlap by $1/2$ \cite{oDERIV}. The
checkpoint adapter verifies all $78+468+2808$ columns and both the
preservation identity on the 3276-dimensional checkpoint space and the
blank-sector control \cite{csDERIV,rCS}.
\end{proof}

\begin{lemma}[Feedback requires classical record preservation]
\label{lem:multiplicative}
Let $\Lambda$ be a CPTP map from $R\otimes W$ to $R\otimes W'$ with
$\Lambda^*(B\otimes I)=B\otimes I$ for every $B$ in the full matrix algebra
of $R$. Then $\Lambda^*(I\otimes O)=I\otimes\Phi^*(O)$ for a unital
completely positive map $\Phi^*$, so changing only the record preparation of
a product input changes no output statistic on $W'$.
\end{lemma}

\begin{proof}[Proof sketch]
The operators $B\otimes I$ lie in the multiplicative domain of the unital
completely positive map $\Lambda^*$, because the Schwarz inequality is an
equality for them \cite{choi1974}. Then $\Lambda^*(I\otimes O)$ commutes with
every $B\otimes I$ and has the stated form \cite{oDERIV}.
\end{proof}

Lemma~\ref{lem:multiplicative} fixes the shape of the observer contract. A
record that steers the working state is preserved as a classical record,
with its projectors fixed, while coherence between different record values
may change.

\subsection{Declared services}
\label{sec:services}

Each component names the services it assumes; they belong to the
specification and enter the resource counts where a component counts
resources. The services are coherent proper-code load and export, with
complementary failure outcomes retained by a complete channel definition;
blank initialization, reset, storage and routing of quantum registers and
classical records; Born readout of pointers and fair coins; and analogue
precision of source pulses. Particular components add a calibrated external
flux and probe (Section~\ref{sec:p-detector}), incident-event recognition
and bare-occupation readout (Section~\ref{sec:local-tick}), coherent
phase preparation and readout (Section~\ref{sec:prediction}), and, for the
native process results, independent uniform seam scheduling, uniform
fixed-number preparation, full initial and current readback, coherent wave
processing, the return-length sampler and the response controller
(Sections~\ref{sec:native-history}, \ref{sec:native-process} and
\ref{sec:prediction}). The coherent
services are not products of the native grammar: committed C/E repair with
diagonal accessible inputs and fresh independent diagonal ancillas cannot
prepare the declared coherent reference to arbitrary accuracy, even with
adaptive heralding. This coherence floor of the native interface makes the
coherent services a necessary resource of the specification. The sharp joint
instrument of Section~\ref{sec:common-construction} uses native integer
repair as a declared primitive, and Section~\ref{sec:continued-native}
attaches the compiled 78-dimensional C/E channels, routing and encoded
recovery to the continuing work history. The processor algebra is $M_6$; its twelve generators are the exact source generators of the compiler, one per port \cite{ssREADME,csDERIV}; a proper code is a two- or four-dimensional subspace of $\C^6$ with its load and export channels and the complementary failure outcome; and the service error model is the compositional contract of Section~\ref{sec:ce-source}. Exact certificates exclude numerical execution error
and physical service error unless a component states otherwise.

\subsection{The closure inputs \texorpdfstring{$P$}{P} and \texorpdfstring{$N$}{N}}
\label{sec:closure-inputs}

The pixel ratio $P$ is the area of one screen cell in units of
$\ell_\star^2$, and $N$ is a record capacity in nats
\cite{ophobservers,pACCESS}. The construction accepts both as closure-derived
inputs and fits neither to an output. $P$ enters through a declared coupling map, the source-only structured-running branch of the fine-structure closure (one-loop running and matching conventions with the tree-level $m_Z$ closure), defined in the P-derivation code of the repository \cite{ophrepo} and consumed unchanged; the map is a declared selection, not a result of this construction. At the pinned closure value $P=1.630972095858897\ldots$, the midpoint of the pinned forward-closure interval, the map gives $\alpha^{-1}(P)\approx136.9948351774$ and $g(P)=\sqrt{4\pi\alpha(P)}\approx0.302867613117$ in Heaviside--Lorentz natural units, numerical evaluations of the executed map \cite{pREADME,rP}. $N$ enters only as a bound, and no component consumes its closure value. For a declared capacity carrier of dimension $D$, the record capacity $N_{\rm record}=\ln D$ bounds every correctable classical record code by $\ln M_0\le N_{\rm record}$ \cite{mCONSTRUCTION,oCAP}. The
cosmological relation $N=3\pi/(\Lambda\ell_\star^2)$ additionally requires a
physical identification of capacity with horizon area, which the
construction does not make.

\subsection{Time parameters}
\label{sec:time}

The construction keeps the time parameters of Table~\ref{tab:time} distinct
\cite{mINTERP}. Conversions between them need explicit maps. The
construction uses three: the fixed-reference relation $t=-\beta s$ of
Section~\ref{sec:readouts} between a model coordinate and a modular
parameter, the instrument-valued conversion between two local event clocks
of Section~\ref{sec:event-clocks}, which keeps their whole count law, and
the rescaling of native attempt count along the refining screens of
Section~\ref{sec:native-history}. A calibration of native calls against a
monotonic host counter is an engineered software correspondence. No
preferred universal physical clock is required.

\begin{table}[htbp]
\centering
\small
\caption{Time parameters and where they occur.}
\label{tab:time}
\begin{tabular}{@{}L{0.20\linewidth}L{0.38\linewidth}L{0.36\linewidth}@{}}
\toprule
\textbf{Parameter} & \textbf{Meaning} & \textbf{Occurrence}\\
\midrule
Host time & Runtime of a software execution; a resource cost & A software calibration of native calls; never a physical quantity\\
Solver counter & Position in a search or repair schedule; the attempt count of a declared schedule & History repair (Section~\ref{sec:history-repair}); attempt counts in the observer theorems\\
Causal order & Precedence of recorded events; counts of native operations & Observers, detectors, phase protocol, local ticks\\
Local event count & Attempted repairs incident to an observer region, or occupation changes at a port & Overlapping observers, local and active event clocks, matched screens (Section~\ref{sec:native-process})\\
Scaled attempt count & Native attempt count $n$ at level $l$ rescaled to $\tau=n/(1260\cdot16^l)$ on the refining screens & Native density limits and the controlled repair response (Sections~\ref{sec:density}, \ref{sec:prediction})\\
Coherent wave parameter & Parameter of the admitted coherent continuation $\cos(t\sqrt A)$ & Native waves and the return reference (Sections~\ref{sec:continuum}, \ref{sec:product-interacting})\\
Model coordinate & Parameter of a declared Hamiltonian or action & Scalar detectors, two-site apparatus, common source\\
Modular parameter & Parameter of the modular automorphisms of an algebra-state pair & Modular clock and fixed-reference relation (Section~\ref{sec:readouts})\\
Operational clock & Observer-accessible reading with calibrated duration & Supplied by no component\\
\bottomrule
\end{tabular}
\end{table}

\section{The finite observer}
\label{sec:observer}

This section is layer three on one carrier: an observer that prepares,
records, retains and acts on native records, with its capacity, memory and
recovery limits.

\subsection{The native observer}
\label{sec:native-observer}

The native observer executes Definition~\ref{def:observer} on the scored
preparation \cite{oREADME,oDERIV}. A trial is one fresh preparation followed
by two repair events and a halt; running many trials repeats the preparation
and does not extend one observer's memory. Event order is the time
parameter. The tape, prior, forecast table, seam selector and halting rule
are designed apparatus, and the forecast is a fixed table without learning.

\begin{proposition}[Statistics of the native observer]
\label{prop:native-observer}
Prepare the 66 binary configurations uniformly, use fair coins and the
controller of Definition~\ref{def:observer}, and let $Y$ indicate a positive
second current. Then:
\begin{enumerate}[(i)]
\item the 264 histories, two coins for each of the 66 configurations at each
of the two events, each have weight $1/264$;
\item the first record has masses $14/33$, $14/33$, $5/66$, $5/66$ for
$(c,j)=(0,0),(1,0),(0,1),(1,-1)$, the conditional probabilities
$q(r)=\Pr(Y=1\mid r)$ are $0$, $5/56$, $0$, $1/2$ in the same order, and
$\Pr(Y=1)=5/66$;
\item the Brier risk of forecasting $Y$ is $395/7392$ with $q(r)$,
$140/2013$ with the best forecast based on the chosen seam alone, and
$12875/150304$ when $q$ is evaluated on an independent record resampled
within the same seam;
\item the mean final load at port $0$ is $35/264$ under record feedback and
$1/6$ when the selector is disconnected or replaced by an independent
selector with the same seam frequencies; the feedback effect is $-3/88$.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
For (iii), $\Ex[(Y-q(R))^2]=\Ex[\operatorname{Var}(Y\mid R)]$. The chosen
seam $S$ is a function of $R$; the best $S$-only forecast has excess risk
$\Ex[\operatorname{Var}(q(R)\mid S)]$, and an independent resample within the
seam adds twice that amount. For (iv), a positive first current occurs on ten
binary inputs with coin zero, mass $5/66$. On that branch $x_0=0$, $x_1=1$
and $\Ex[x_5]=1/10$; a second repair on $(0,1)$ gives port-0 mean $1/2$ and
one on $(0,5)$ gives $1/20$. Hence the change is
$(5/66)(1/20-1/2)=-3/88$ and the mean is $1/6-3/88=35/264$. An independent
selector leaves the uniform shell uniform under either seam
\cite{oDERIV}.
\end{proof}

The other two current letters occur only on the full carrier, where the
forecast abstains. The exact receipt retains all 264 histories
\cite{rNATIVE}, and a seeded execution of 20,000 fresh trials matches every
trial against its exact path reference, with sample final port-0 loads
$0.1327$ with feedback, $0.16645$ disconnected and $0.1668$ with the
independent selector \cite{rRUN}. Since the selector is a design choice and
the forecast table is the exact conditional law, (iii) and (iv) are
consistency checks: the Bayes forecast beats coarser forecasts and the
record-wired selector moves the occupation.

\subsection{Continuing memory and the actual stationary reference}
\label{sec:finite-campaign}

The finite campaign~\cite{ophrepo} extends the
two-event observer with an explicitly declared memory law. It keeps the
same source and complete C/E instruments. A classical tag chooses C with
probability $\lambda$ and E otherwise; a repair occurs with probability $r$,
with an identity hold otherwise. A controller chooses a preferred seam from
its retained records. The actual seam distribution is
\[
 p(e\mid m)=\frac{\epsilon}{30}+(1-\epsilon)\pi(e\mid m),
 \qquad r,\epsilon\ge\frac1{16},\quad 0\le\lambda\le1.
\]
The rational parameter bound and permitted controller tables are part of the
campaign, not measured physical constants. The executed controller proposes
$(0,5)$ after a positive current and $(0,1)$ otherwise. It does not read the
hidden source load. A proposal intervention preserves the uniform exploration
floor. Holds are recorded with current zero.

For each classical memory input, every source branch is
$K_b=\sqrt{w_b}M_b$, with $M_b$ a native partial injection or identity and
$\sum_b K_b^*K_b=I_{78}$ exactly. Tensoring with the identity preserves an
arbitrary untouched reference. The tests independently reconstruct the branch
supports by endpoint floor/ceiling arithmetic, including their off-diagonal
action. Composition therefore defines every finite history with its
unnormalized quantum output and actual probability. Exact prefix replay
and retained six-attempt source/reference/last-record marginals verify C, E
and the delayed tagged mixture; the complete history tree remains implicit
in that channel kernel.

One to eight complete records occupy the declared buffer. Under the executed
one-slot export policy, $n$ attempts leave one record and export $n-1$ records
for $n\ge1$; expected repair and hold counts are $nr$ and $n(1-r)$.
Concatenating exports and the remaining buffer reconstructs the history.
The halt-on-full control instead preserves its source and records with an
explicit timeout outcome. These are different resource contracts. Bounded
working memory with growing exports is not closed finite-resource operation
for unlimited duration. No new native source-word implementation of the
supplied initialization, selection, write, export or delivery services is
claimed.

The stationary reference must include this controller. For $r=1$ and
$\epsilon=1/2$, a uniform binary source with blank memory becomes nonuniform
after two attempts: $30$ source probabilities change, with maximum deviation
$1/960$ from $1/66$. Exact rational solution of the actual source/current-sign
kernel instead gives one stationary law on a $129$-state recurrent subset of
the $132$ source/bit states. Applying the complete C/E observer instrument to
that law constructs the full-record reference. Its projection recovers the
bit law, and one further actual observer step preserves the entire full-record
density exactly. The resulting $128$ C or $192$ E record blocks have $4{,}218$
positive diagonal entries overall. Positivity establishes faithfulness only
on the explicitly compressed support of each block. The growing exported
tape is not a stationary object, and the delayed member's hold updates require
a separate stationary solve.

This reference also makes a clock obstruction precise. In an actual
nontracial block choose positive weights with ratio $q=p_i/p_j>1$ and the
dimensionless modular parameter $s_* = \pi/(2\log q)$. The state-dual modular
action takes the real pair probe $(|i\rangle+|j\rangle)/\sqrt2$ to a density
with off-diagonal entry $-i/2$. Every admitted history, with the declared real
joint ancillas and classical memory, preserves real density matrices.
A norm-one imaginary pair observable separates the target from every such
normalized output, giving trace-distance error at least $1/2$. This witness
is not a native detector, and the probe is an all-input map test, not an
asserted native preparation. On the accessible commutative record algebra
the modular action is trivial. Other algebras, encodings or coherent services
have different contracts; no general physical-clock exclusion follows.
The exact reference, implementation bound and full qualifications are
retained with the executable model~\cite{ophrepo}.

\subsection{Record capacity}
\label{sec:capacity}

The retained records are evaluated with the correctable-public-record
construction of the theorem library \cite{ophobservers,oCAP}. There $M_0$ is
the largest correctable set of reachable public records, the independence
number of the compound checkpoint confusability graph; the record capacity is
$N_{\rm record}=\ln M_0$ nats; and $M_0\le D$ for a declared carrier of
dimension $D$. The counts below describe the reachable record interface;
program, source, probe and service memory are additional.

\begin{proposition}[Record capacity of the native observer]
\label{prop:capacity}
Take halt and exact classical reread as the declared terminal continuations
of the observer of Definition~\ref{def:observer}. Then the compound graph
has no edges, and:
\begin{enumerate}[(i)]
\item $M_0=11$ on the binary preparation (four reachable first records, 264
histories) and $M_0=17$ on the full family of 78 inputs (six first records,
312 histories);
\item each terminal word projector has rank 78 in the 3354-dimensional
carrier, distinct projectors are orthogonal, and the eleven reachable ones
have total rank 858, so the bound is unsaturated with slack $\ln(3354/11)$;
\item the binary preparation reaches $1+4+11=16$ classical prefixes, so four
bits encode the complete controller history; a compact controller that reads
only this code reproduces the phase, action, forecast, source loads, record
and halt of all 264 histories; the full family reaches 24 prefixes and needs
five bits;
\item coarser retained readouts give correctable codes of size 4 for the
first letter alone, 4 for the second letter alone, 3 for the forecast alone,
2 for the chosen seam alone, 4 for forecast and seam, and 1 after complete
erasure.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
Halt and reread leave the tape unchanged, so the joint kernel is the
identity and the existing evaluator returns the number of reachable terminal
words. The terminal projector of word $w$ occupies the basis coordinates
$546+78w$ to $546+78w+77$ of the carrier \cite{oCAP,rCAP}.
\end{proof}

The terminal capacity is $\ln 11$ nats, about 3.459 bits, on the binary
preparation and $\ln 17$ nats on the full family; the Shannon entropy of the
benchmark record law, about 2.023 nats, is a different quantity. The
library's global proposal adds closure across the complete terminal fiber,
$N_\star=\ln M_0(U_{N_\star})$, and a de Sitter horizon identification giving
$\Lambda\ell_\star^2=3\pi/N_\star$ \cite{ophobservers,oCAP}; the native records
supply an actual finite input to that evaluator, and the two further steps
are not taken here.

\subsection{The \texorpdfstring{$P$}{P}-dependent detector}
\label{sec:p-detector}

The detector carries the closure input $P$ through the electromagnetic
coupling into a recorded binary observation that controls the next repair
\cite{pREADME,pDERIV}. The charge assignment, probe, calibrated flux, readout
and controller are specified apparatus (Section~\ref{sec:endpoint}).

Choose the charged realization $Q_v=g(P)N_v$ of the native occupation $N_v$.
A current record $j$ on the seam $(a,b)$ then transfers the charge
$\Delta Q=g(P)\,j\,(\mathbf e_b-\mathbf e_a)$, and total charge is conserved.
For port gauge parameters $\lambda_v$ and
$U_\lambda\ket{x}=e^{ig\sum_v\lambda_vx_v}\ket{x}$, every current-grouped
Kraus operator obeys
$U_\lambda L_{(c,j)}U_\lambda^\dagger=e^{igj(\lambda_b-\lambda_a)}L_{(c,j)}$.
The probe is a unit-charge path qubit prepared in $\ket{+X}$. The first
native current routes it around a calibrated external loop with signed
exposure $j\chi$, where $\chi=\oint A\cdot dx$ is prepared independently of
$g$, so the relative phase is $\varphi_j=g(P)\,j\,\chi$, the Aharonov-Bohm
phase \cite{aharonov1959}. A $Y$ measurement commits $y=\pm1$ with amplitudes
$m_y(j)=\frac12[e^{-i\varphi_j/2}-iy\,e^{i\varphi_j/2}]$ and effects
$E_y=\sum_r\frac12[1+y\sin\varphi_{j(r)}]\proj{r}$ on the old current
register. The actuator receives only $y$. It repairs $(0,5)$ for $y=+1$ and
$(0,1)$ for $y=-1$, keeps $y$ unchanged, and the protocol halts.

\begin{proposition}[$P$-dependent response]
\label{prop:p-response}
Prepare the 66 binary configurations uniformly, run the protocol above, and
put $s=\sin(g(P)\chi)$. Then:
\begin{enumerate}[(i)]
\item $\{E_y\}$ is a binary instrument on every input, including coherent
records; each $M_y=\sum_rm_y(j(r))\proj r$ commutes with the old record
projectors, and the composed experiment is one CPTP map that preserves the
committed bit;
\item $\Pr(y\mid j_1,P,\chi)=\frac12[1+y\sin(g(P)j_1\chi)]$;
\item at $\chi=1$ all 528 state, coin, detector and coin histories have
positive probability, and
\[
\Ex[y]=0,\quad \Ex[yj_1]=\tfrac{5}{33}s,\quad
\Ex[j_2]=\tfrac{5}{264}s,\quad \Ex[g(P)j_2]=\tfrac{5g(P)}{264}s;
\]
\item for weak exposure, $\Ex[g\,j_2]=\frac{5\pi}{66}\alpha(P)\chi+O(\chi^3)$;
\item disconnecting the probe, or replacing $y$ by an independent fair bit,
gives $\Ex[j_2]=0$ with the same detector and seam frequencies.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
The eigenvalues of $E_y$ lie in $[0,1]$ and sum to one, which gives (i). For
(iii), after the first repair let the source be $z$; averaging the second
coin gives $\Ex[j_2\mid z,j]=\frac14(2z_0-z_1-z_5)+\frac{sj}{4}(z_1-z_5)$.
The first term averages to zero because the first repair preserves the
uniform shell. Currents $\pm1$ each have mass $5/66$, and on either branch
the other load is uniform over ten ports, so $\Ex[j(z_1-z_5)]=5/66$. An
independent floor and ceiling enumerator reproduces every coefficient over
all $66\cdot2\cdot2\cdot2=528$ paths \cite{pDERIV,pVALID}.
\end{proof}

At the computed $P$ and $\chi=1$ the plus probability is $0.649129261647$ for
$j=+1$, $0.350870738353$ for $j=-1$ and $1/2$ for $j=0$; the mean second
current is $0.00564883566843$, the mean transferred charge
$0.00171084937579$ in natural units, and the final port-0 load
$0.161017830998235$ against $1/6$ for the controls. Off-closure controls at
$0.999P$ and $1.001P$ change the detector and feedback probabilities at fixed
$\chi$, and a seeded run of 200,000 trials gives mean second current
$0.006390$ with standard error $0.00085715$ \cite{pREADME,rP}. The audit
record $(r_1,y,r_2)$ has 24 terminal words and $M_0=24$ \cite{pACCESS}. If a
cut enclosing a retained record obeys the area-capacity law with area $K$
chart cells, then $N_{\rm cut}=KP/4$ and $\ln M_0\le N_{\rm cut}$ give
$K\ge4\ln M_0/P$; with the pinned $P$ one detector bit needs $K\ge2$
($4\ln2/P\approx1.6999608573$) and the full audit $K\ge8$
($4\ln24/P\approx7.7942567832$).

\subsection{Quantum mixing and memory}
\label{sec:mixing}

Discarding every seam, coin and repair record after each event turns the
repairs into channels $\Phi_\Cn$ and $\Phi_\En$ on the source. Under
independent seam attempts these channels have an exact spectral theory
\cite{oQM}. Theorems~\ref{thm:gap} and~\ref{thm:memory} concern this
record-discarded channel under independent uniform attempts; they do not
describe the retained tape or the feedback controller.

\begin{theorem}[Signed-difference reduction and quantum gap]
\label{thm:gap}
Let $G$ be a finite connected graph with $n\ge2$ vertices, positive edge
weights $w_{ab}$ and $W=\sum w_{ab}$. Let each event choose the edge $ab$ with
probability $w_{ab}/W$ and a fresh fair coin, and discard its record. On the
binary space $\Hil_k$ of load $k$, $1\le k\le n-1$, with $D=\binom nk$:
\begin{enumerate}[(i)]
\item for a matrix unit $\ket x\bra y$ the signed difference $z=x-y$ is
invariant; if $z$ has $r$ positive entries and $Z=\{v:z_v=0\}$, then on the
sector of $z$
\[
\Phi_{\mathsf L}\big|_z=I-\frac{L_{\rm excl}(G[Z],k-r)+K_{\mathsf L}(z)\,I}{2W},
\qquad \mathsf L\in\{\Cn,\En\},
\]
with the exclusion Laplacian $L_{\rm excl}$ of the induced graph,
$K_\Cn(z)=\sum w_{ab}\lvert z_a-z_b\rvert$ and
$K_\En(z)=2\sum w_{ab}\mathbf 1[z_a\neq z_b]$;
\item both channels are unital and Hilbert-Schmidt self-adjoint with spectrum
in $[0,1]$, their fixed algebra consists of the scalars, and both have gap
$\gamma=\lambda_1(G)/(2W)$, where $\lambda_1(G)$ is the first positive
Laplacian eigenvalue; and
$\lVert\Phi^t_{\mathsf L}-{\rm Dep}_D\rVert_{2\to2}=(1-\gamma)^t$;
\item for every input and every untouched reference $R$,
$\dist\bigl((\Phi^t_{\mathsf L}\otimes{\rm id}_R)(\rho),\tau\otimes\rho_R\bigr)
\le\min\{1,\tfrac{D}{2}(1-\gamma)^t\}$, with $\tau=I_D/D$ and
$\dist(\rho,\sigma)=\frac12\lVert\rho-\sigma\rVert_1$.
\end{enumerate}
For the twelve-port carrier at load two ($n=12$, $W=30$,
$\lambda_1=5-\sqrt5$, $D=66$) the gap is $\gamma=(5-\sqrt5)/60\approx0.046066$,
and the slowest off-diagonal sectors contract by exactly $r_\Cn=5/6$ and
$r_\En=7/10$ per attempt. On the full 78-state carrier the gap is the same,
the full channel is block triangular with a diagonal collision block whose
entries lie in $[0,5/6]$, and the unique stationary density is the uniform binary density, of rank 66 on
dimension 78.
\end{theorem}

\begin{proof}[Proof sketch]
A matrix unit survives a coin branch exactly when its two records agree, and
then both configurations translate by the same current; comparing the coin
branches gives (i). The diagonal sector is the symmetric $k$-particle
exclusion chain, whose gap equals $\lambda_1(G)$ by the theorem of Caputo,
Liggett and Richthammer \cite{caputo2010}. For a nonzero sector,
$K_\En\ge K_\Cn\ge\frac12\sum w_{ab}(z_a-z_b)^2\ge r\lambda_1\ge\lambda_1$, so
no off-diagonal mode is slower. Item (iii) uses Schmidt rank at most $D$ and
$\lVert Y\rVert_1\le D\lVert Y\rVert_2$. On the carrier, $r=1$ gives
$K_\Cn=10$ and $K_\En=2(10-\mathbf 1[uv\text{ edge}])$, while $r=2$ gives
$K_\Cn\ge16$ and $K_\En\ge24$. The collision block has largest entry
$1-d_{\min}/W=5/6\le1-\gamma$ \cite{oQM}. The executable checks
the reduction against the native primitive on all 4,356 binary matrix units
and the transient structure on all 6,084 full matrix units \cite{rQM}.
\end{proof}

Choose adjacent ports $u,v$ and the two mixtures
$\rho_\pm=\frac1{20}\sum_{w\notin\{u,v\}}(\ket{uw}\pm\ket{vw})(\bra{uw}\pm\bra{vw})$.
They have identical load populations and orthogonal supports, and
\[
\dist\bigl(\Phi^t_{\mathsf L}(\rho_+),\Phi^t_{\mathsf L}(\rho_-)\bigr)=r_{\mathsf L}^t,\qquad
\dist\bigl(\Phi^t_\Cn(\rho_+),\Phi^t_\En(\rho_+)\bigr)=\tfrac12(r_\Cn^t-r_\En^t).
\]
The second distance is $1/15$ after one attempt and peaks at four attempts at
$12259/101250$; the binary effect $\frac12(I+A_{uv}+A_{uv}^\dagger)$ with
$A_{uv}=\sum_{w\notin\{u,v\}}\ket{uw}\bra{vw}$ attains it. Load populations,
classical load histories and diagonal geometric observables obey the same
exclusion process under both laws, so only coherent preparations and readouts
distinguish them under this schedule.

\begin{theorem}[Finite quantum memory]
\label{thm:memory}
Under the schedule of Theorem~\ref{thm:gap} on the twelve-port carrier:
\begin{enumerate}[(i)]
\item on the binary shell both record-discarded channels are entanglement
breaking after at most 134 attempts;
\item on the full carrier let $u,v$ be adjacent and
$\ket\Psi=(\ket{2u}\ket0+\ket{uv}\ket1)/\sqrt2$ with a reference qubit; after
$t$ attempts the coefficient of $\ket{2u}\bra{uv}$ is $a_{\mathsf L}^t$, with
$a_\Cn=23/30$ and $a_\En=7/10$, and the partial transpose on the reference
has a principal $2\times2$ minor equal to $-a_{\mathsf L}^{2t}/4<0$; no finite
power of either full channel is entanglement breaking;
\item every input converges to $\tau_P\otimes\rho_R$, where $\tau_P$ is the
uniform binary density, with
$\dist\le\min\{1,\mu^{s/2}+\mu^s+\tfrac{D}{2}(1-\gamma)^{t-s}\}$ for all
$0\le s\le t$, $\mu=5/6$ and $D=66$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
For the normalized Choi state $J_t$,
$\lVert J_t-I/D^2\rVert_2^2=D^{-2}\sum_{i>0}\lambda_i^{2t}$. The separable
ball of Gurvits and Barnum \cite{gurvits2002} and the equivalence of
separable Choi states with entanglement-breaking channels \cite{horodecki2003}
reduce (i) to $65\,r_{\rm mix}^{2t}+4290\,r_{\mathsf L}^{2t}\le1/4355$, where
$r_{\rm mix}=1-\gamma$. The rational enclosure
$559/250<\sqrt5<2237/1000$ certifies this at $t=134$ and its failure at
$t=133$; treating all modes with $r_{\rm mix}$ would certify only 178. For
(ii), 25 edges avoid $u$ and four of them touch $v$; $\Cn$ keeps one coin on
each of those four and $\En$ keeps neither, and the triangular structure
prevents any return from the binary space. Separable states have positive
partial transpose \cite{peres1996}; the complete argument is in
\cite{oQM,rQM}.
\end{proof}

Replacing the full carrier by its attracting binary shell therefore changes
an exact finite-time quantum property, although both converge to the same
stationary density. The number 134 is a sufficient certificate time; the
exact index of entanglement breaking on the shell is not determined.

\subsection{Recovery from complete records}
\label{sec:recovery}

Preserving committed records and recovering the unknown input from them are
different questions \cite{oRR}. Here a seam policy may depend on committed
records and classical randomness and may not inspect hidden loads or insert
quantum corrections; a final decoder may act on the full load register and
tape, keep the tape, use blank ancillas and return a state on the original
carrier. That decoder access is stronger than the record-only access of the
observer's controller.

\begin{theorem}[Optimal recovery from complete native records]
\label{thm:recovery}
Let $X$ be the 78 configurations or the 66-state shell and $D=\lvert X\rvert$.
For a complete admitted history $h$ let $X_h$ be its compatible inputs, $P_h$
the projector onto them, $d_h=\rk P_h$, and $w_h$ the product of its seam and
coin factors. Then:
\begin{enumerate}[(i)]
\item the history Kraus operator is $K_h=\sqrt{w_h}\,U_hP_h$ with
$U_h^\dagger U_h=P_h$ and $\sum_hw_hP_h=I$; a tape-controlled inverse outputs
$\sum_hw_h\proj h\otimes P_h\rho P_h$, so every initial load population is
recovered;
\item among all final CPTP decoders the optimal entanglement fidelity for the
maximally mixed input is
$F_{\rm opt}=D^{-2}\sum_hw_hd_h^2=D^{-2}\sum_{x,y}H_{xy}$, with
$H_{xy}=\sum_{h:\,x,y\in X_h}w_h$;
\item for the two-event observer, $F_{\rm opt}$ equals $5543/8712$ (binary) and
$1301/2028$ (full) under $\Cn$, and $769/1452$ and $3329/6084$ under $\En$
with the same current-sign selector applied to its decoded current;
\item the maximal exactly correctable algebra on the whole input is the
commutant of $\{P_h\}$, and one event with full-support random seam choice
reduces it to the diagonal load algebra; nevertheless the qubit code
$\ket{0_L}=(\ket{\{0,2\}}+\ket{\{1,3\}})/\sqrt2$,
$\ket{1_L}=(\ket{\{0,3\}}+\ket{\{1,2\}})/\sqrt2$ satisfies the
Knill-Laflamme condition $QP_hQ=\alpha_hQ$ for every seam, coin and record of
either law and is recovered exactly after one event;
\item under independent seam attempts $H_{xy}(t)=q_{\mathsf L}(x-y)^t$, with
largest off-diagonal rates $5/6$ for $\Cn$ and $7/10$ for $\En$ on both
carriers; $F_{\rm opt}(t)\to1/D$, and neither retained instrument is
entanglement breaking at any finite horizon;
\item a fixed $\En$ seam word identifies every full-carrier input exactly
when at most one visited component has size one or two, and every binary
input exactly when at most one component is a singleton and at most one has
size two; the minimal such words have eight and seven distinct edges;
\item no bounded record-only adaptive policy makes $\Cn$ entanglement
breaking; a $\Cn$ word that doubles a minimal forest, 16 or 14 events,
excludes every nontrivial exact code; the policy that repeats each forest edge until its
coin first differs identifies the input almost surely, with stopping time
$T=m+\sum_{i=1}^mG_i$ for independent geometric $G_i$ of success probability
one half, $\Pr(T=t)=\binom{t-m-1}{m-1}2^{-(t-m)}$ for $t\ge2m$, and
$\Ex[T]=3m$, namely 24 on the full carrier and 21 on the binary shell.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Every retained current determines a translation, so reversing them gives the
input uniquely from the terminal configuration and $h$. Optimality follows
from Cauchy-Schwarz on each classical branch:
$\sum_a\lvert\Tr(R_{ha}U_hP_h)\rvert^2\le d_h^2$. For (iv), commutation is
sufficient by (i), and necessity follows from equality in the Schwarz
inequalities in a Stinespring dilation; the code has nonadjacent occupied
pairs, equal one-port means and vanishing products $N_aN_b$ on every edge,
which gives the Knill-Laflamme identity \cite{knill1997}. For (vi), matching
$\En$ histories are those for which $z$ is constant on every visited
component. For (vii), the inputs $\{u,w\}$ and $\{v,w\}$ with $u,v$
nonadjacent keep equal currents for at least one coin on every edge, so a
history of length at most $T$ shares a rank-two fiber with probability at
least $2^{-T}$; repeating an edge with the opposite coin reveals the initial
parity through $d=2j_1+j_*$ \cite{oRR,rRR}.
\end{proof}

The minimal fixed $\En$ words are the forests $(0,1),(0,5),(2,3),(2,4),
(6,8),(8,9),(7,10),(10,11)$ on the full carrier and
$(1,5),(2,3),(2,4),(6,8),(8,9),(7,10),(10,11)$ on the binary shell, with
component sizes $(3,3,3,3)$ and $(1,2,3,3,3)$. The full-input correctable
blocks of the two-event observer have sizes $46,10,9,1$ on the binary shell
and $56,10,9,1,1,1$ on the full carrier, for both laws. Existing source
inverse words implement the single-seam branch reversal of (i) while keeping
the classical archive \cite{oRR,rRR}. Native current
records thus suffice to reverse classical load transport, without restoring
arbitrary coherence lost across record groups. The source encoder and active
recovery of Section~\ref{sec:continued-native} instead preserve the stated
logical qubit between events; they do not invert the whole bare-load channel.

\subsection{A coherent continuation and its selection boundary}
\label{sec:coherent-continuation}

An exact finite experiment attaches a reflection walk to the binary native
branches, with coherent event access in place of a committed tape
\cite{oCOHERENT}. For each binary configuration $x$, seam $e$ and coin $c$,
let $(y,j)$ be the native output and current. On the $66\cdot60=3960$ arc
basis vectors define
\[
 S\ket{x,e,c}=\ket{y,e,c'},\qquad
 c'=\begin{cases}c,&j=0,\\1-c,&j\ne0,\end{cases}
 \qquad A\ket{x}=\frac1{\sqrt{60}}\sum_{e,c}\ket{x,e,c}.
\]
Every arc is checked: $S^2=I$, with 300 exchanged pairs and 3360 fixed arcs,
and the outgoing native record $(y,e,c,j)$ is recovered bijectively from the
reversed arc. Thus $A^\dagger A=I$ and $A^\dagger SA=K$, the actual symmetric
native transition kernel, and $W=S(2AA^\dagger-I)$ satisfies the hypotheses
of the quantum-walk spectral mapping theorem with discriminant $K$
\cite{pwhiguchi2016spectral,pwszegedy2004spectra}. The identity
$A^\dagger W^nA=T_n(K)$, with $T_n$ the Chebyshev polynomial, gives for the
centered loads $X_{x,p}=x_p-1/6$
\[
 KX=X(I-L/60),\qquad
 U_{n+1}-2U_n+U_{n-1}=-U_nL/30,
 \qquad U_n=A^\dagger W^nAX.
\]
These are compressed amplitudes, and $n$ counts the supplied coherent
continuation with no identification as duration. The binary domain is
essential: on the full carrier $(2,0,\ldots)\mapsto(1,1,\ldots)$ has no native
reverse event, so its asymmetric kernel equals $A^\dagger SA$ for no
involutive unitary on the same domain. Twelve port addresses and sixty
seam/coin events give a 720-arc source-compiled walk with discriminant
$K_{12}=I-L/60$; the map $F_{x,p}=(x_p-1/6)/\sqrt{10}$ satisfies
$F^*F=I-\mathbf1\mathbf1^T/12$ and $K_{66}F=FK_{12}$, so it realizes the
shared mean-zero density sector. Among the sixty proper carrier rotations a
unique inverse-symmetric order-five class $\mathcal C$ of size twelve gives
$L=6I-\frac12\sum_{g\in\mathcal C}U_g$, with multiplicity-free density
representation $1\oplus3\oplus5\oplus3$. The other order-five class is
isospectral but assigns probability zero to a native seam whose actual
one-attempt probability is $1/60$, so the seam records select the class.

The emitted walk word uses ten address qubits, five blank scratch qubits and
114,910 source pulses per walk with 229,820 load/export events; uniform event
preparation costs a separate 36,288 pulses. The reversible circuit is checked
on all 32,768 basis inputs, every clean-scratch address returns clean
scratch, including 304 padding addresses, and the fifteen-qubit execution
reuses raw pulse factors without target-matrix insertion or output
normalization. Floating routine guards sum to $s=829/(5\cdot10^9)$, giving
complete output trace-norm error at most $2\delta+\delta^2$ with
$\delta\le s/(1-s)$; these are numerical guards, not physical precision
certificates.

\paragraph{A finite reference and its modular clock.}
Biased native event amplitudes, the compiled reversible seam shift, inverse
event preparation and a herald realize $F_p=I-pL/60$ on the protected site
code. From the supplied $I_{12}/12$, $n$ successive successes give
$F_p^{2n}/\Tr F_p^{2n}$ with success probability at least $1/12$, every
failure recorded; for $p=30t/n\le1$ this tends to $e^{-tL}/\Tr e^{-tL}$
with relative-entropy and total-variation errors at most
$16t^2/[n(1-4t/n)]$. The execution at $p=1/2$, $n=2$, $t=1/30$ tests all
twelve input columns with success $4382551/5184000$ and bound $1/105$, and a
source interference readout has contrast $260343/43825510$ against zero after
dephasing. This state is central on the 44-dimensional represented group
algebra, where its modular flow is trivial; nontrivial modular gaps require
the full protected matrix algebra \cite{nativePredictions}.

Positive subordination gives square-root weights on the same register. With
$F=I-hL/2$, $h=p/30$, a Poisson number of Sibuya jumps gives positive
weights with
\[
 \sum_n w_nF^{2n}=e^{-R\sqrt{I-F^2}}=e^{-b\sqrt{L-hL^2/4}},\qquad b=R\sqrt h .
\]
Finite jump and count caps and dyadic rounding retain missing mass as sampler
rejection. The source instance $p=R=1/2$ uses 64 native coin records and at
most four raw filters, retains each first failure and all sixteen successful
interference outcomes, and has Gibbs trace-distance bound below $0.217$ and
spectral diagnostic $0.01344$. Its finite polynomial $M=\sum_nw_nF^{2n}$
fixes $H=-\log(M/m_0)$, $m_0=\sum_nw_n$. Raw source words execute that clock,
a current-controlled intersector kick and complete spectral-pair detectors on
the same register, retaining all 3,600 two-attempt native tapes, sampler and
herald failures, sixteen detector addresses and twelve reference columns.
The modular identity $\sigma_s=\alpha_{-s}$ holds for the ideal successful
marginal $M/\Tr M$ on $\mathcal B(\mathbb C^{12})$; it does not hold for
every retained request block.

This density supplies eleven pairs $Q_j=X_{0j}/\sqrt{2\Delta_j}$,
$P_j=Y_{0j}/\sqrt{2\Delta_j}$, $\Delta_j=p_0-p_j>0$, with
$\langle[Q_j,P_k]\rangle=i\delta_{jk}$, symmetric covariance
$\delta_{jk}\coth(g_j/2)/2$, $g_j=\log(p_0/p_j)$, and frequency $g_j$ under
$H$. For chronological factors $U_N=e^{iA_m/\sqrt N}\cdots e^{iA_1/\sqrt N}$
built from these pair observables on independent identical copies, a
parity commuting with the state and anticommuting with every $A_k$ gives
\[
 \left|z_N^N-e^{-C}\right|\le\frac{D^4/24+|C|^2/2}{N},\qquad
 z_N=\Tr(\rho U_N),\ D=\sum_k\|A_k\|,
\]
with $C=\tfrac12\sum_k\langle A_k^2\rangle+\sum_{k>j}\langle A_kA_j\rangle$.
Source pointer reads execute the one-copy factors for three band
representatives with all 32 work/pointer addresses retained; a growing copy
register is not executed. With physical normalization
$q_j=Q_j/\sqrt{\omega_j}$, $\pi_j=\sqrt{\omega_j}P_j$ and $g_j=b\omega_j$,
the clock $H/b$ gives thermal covariance
$\coth(b\omega_j/2)\cos(\omega_jt)/(2\omega_j)-i\sin(\omega_jt)/(2\omega_j)$.
For $m$ smeared factors with $R=\sum_kR_k$,
$R_k^2=\sum_j(f_{kj}^2/\omega_j+\omega_jh_{kj}^2)$, gaps
$g_j,b\omega_j\ge\gamma>0$, $\vartheta=\gamma/(1+\gamma)$ and $Z=1/p_0$,
\[
 |z_N^N-e^{-C}|\le
 \frac{R^4}{\vartheta^2N}\left(\frac{(m+1)Z}{96}+\frac18\right),
\]
so $Z/N\to0$ suffices on a fixed positive band; the family
$\rho=\diag(2,1,\ldots,1)/(d+1)$ with $N=d$ has instead the limit
$e^{3(\cos(1/\sqrt2)-1)}\ne e^{-3/4}$. On the native family $D_l=4^lL_l$ one
has $\lambda_n(D_l)\ge n/81940$, hence $\Tr e^{-b\sqrt{D_l}}\le1+163880/b^2$,
and with the lazy filter $p=630h16^l\le1$,
\[
 Z\le\frac{1+327760/b^2}{m_0},\qquad
 P_{\rm success}\ge\frac{m_0}{240\,4^l};
\]
both prepared and target gaps exceed $b/182$ when $(1-m_0)/m_0\le b/182$.
Preparation error $E$ with success floor $p>2E$ gives conditional trace
error at most $4E/p$, and with one-copy word error $\epsilon$ the $N$-copy
amplification is at most $N\zeta/(1-N\zeta)$, $\zeta=4E/p+\epsilon$.
On a fixed band $0\le D_0\le\Lambda$ with $h\Lambda<2$, retained positive
weights with missing mass $\delta<a\le e^{-b\sqrt\Lambda}$ give
\[
 \|\log(m_0)I-\log M-b\sqrt{D_0}\|
 \le \frac{\delta}{a-\delta}
 +\frac{bh\Lambda^{3/2}}{4(1+\sqrt{1-h\Lambda/4})}.
\]
A constructive growing word family sets $k=2l+c$ with
$4^c\ge b^2\max(630,\Lambda)$, $R=2^k$, $h=b^24^{-k}$, $a=4^{-k}$, jump cap
$2^{6k}$, count cap $6R+2k$ and Taylor order $4k+3$; the zero-request branch
keeps weight at least $f=2^{-2R}$. For $d=240\,4^l$, request cap $K$ and word
length $P$, primitive and load/export error at most $fa/(64dKP)$ gives
complete stopped preparation error $E\le fa/(32d)$ and
\[
 \|dS_{\rm raw}-M\|\le2dE,\qquad
 \|\log(dS_{\rm raw})-\log M\|\le\frac{a}{16-a}
\]
for the actual successful marginal $S_{\rm raw}$, without a reducing-band
premise. The derived partition bound permits $N=2^l$ copies with copy,
preparation and detector errors tending to zero on a fixed positive band.
Growing execution, spectral tails, accurate source services and a zero-energy
anchor for the radial product remain obligations, and fixed caps with
$N_ch\to0$ force the bounded-band generator to zero; the uncapped sampler has
infinite expected work because the constant sector survives every filter
\cite{nativePredictions}.

\paragraph{Selection controls.}
Two exact controls bound what the committed records select. For $|z|=1$,
$W_z=S[AA^\dagger+z(I-AA^\dagger)]$ has the same first event $W_zA=SA$ but
$A^\dagger W_z^2A=K^2+z(I-K^2)$; a complete added Ramsey read gives two-step
plus probability $1$ for $z=1$ and $113/150$ for $z=-1$. Multiplying each
coherent record branch on seam $(0,1)$ with current $+1$ by $-1$ preserves
every committed C and E branch on arbitrary references, yet the twisted
embedding gives $\Tr[(A_\eta^\dagger SA_\eta)^3]-\Tr(K^3)=-1/900$. Both real
continuations conserve load, respect all sixty proper rotations, fix the
uniform coherent state and admit the time reversal
$\Theta=S\,\mathrm{conjugation}$; under full event-permutation covariance,
real orthogonal coins fixing the uniform vector are identity or reflection.
A sign change on every moving arc preserves proper covariance and the
committed instruments but gives $\Tr(D^3)-\Tr(K^3)=-1/90$ for
$D=2\operatorname{diag}(K)-K$. Scalar record phases cancel throughout
committed-record histories. A source-executed interface test that reads the
low unread record bit in the $X$ basis gives complete $X$-plus probabilities
$3/4$ and $1/4$ for one preparation, against $1/2$ for both after commitment,
using 5380 C or 5383 E source pulses per attempt \cite{oSOURCESELECT}. These
are restrictions on selection: coherent access is a supplied interface, and
the test selects no phase.

\subsection{The local event clock}
\label{sec:local-tick}

N4 is a reciprocal extension of the binary repair law in the construction's
candidate register \cite{mCANDIDATES}. Each port carries a private six-level
response factor, and one coherent exchange
$W_e=\exp[\theta(Y\otimes\Gamma_e+X\otimes\Delta_e)]$, with
$X=S_e(N_a-N_b)^2$, $Y=iS_e(N_a-N_b)$ and response generators
$\Gamma_e,\Delta_e$ built from the endpoint response matrices, conjugates the
binary repair, $K_r=W_e(L_r\otimes I)W_e^\dagger$, with supplied coupling
$\theta$. Private preparations change bare load probabilities: at fixed bare
load and $\theta=1/8$ an endpoint occupation probability moves from
$0.482020673$ to $0.517979327$. An interval certificate proves an edge-Gram
margin $\mathcal G\ge mI$ at $\theta=1/16$ and $\theta=1/8$
\cite{mCANDIDATES,ltREADME,ophrepo}.

Fix an observer port $v$. A tick is any attempted seam incident to $v$,
including attempts that leave its load unchanged. Under independent uniform
seam selection a tick has probability $p=1/6$ per attempt, and remote attempts
continue between ticks. In independent trials the full tracial state is
prepared, the binary occupation $N_v$ is measured, the unmodified source runs
until one or two ticks have occurred, and $N_v$ is measured again, with no
intermediate measurement, reset or postselection. With $f=N_v-1/6$ and
$\lVert f\rVert^2_\tau=5/36$, write
$c_k=\Ex[(X_0-\frac16)(X_k-\frac16)]/(5/36)$ for the endpoint correlation
after $k$ ticks \cite{ltREADME,ltSPEC}. The numbers of
Theorem~\ref{thm:tick}(iii) are certified lower bounds; the full N4 two-tick
correlation is not evaluated, and preparation, incident-event detection and
bare-occupation readout are supplied operations that the detector of
Section~\ref{sec:native-observer} does not implement.

\begin{theorem}[Memory at observer-local ticks]
\label{thm:tick}
Let $\mathcal A$ and $\mathcal B$ be the averages over the thirty seams of the
remote and incident seam channels, so that the first-hit channel is
$\mathcal K=\mathcal B(I-\mathcal A)^{-1}$. Then:
\begin{enumerate}[(i)]
\item $c_k=\langle f,(\mathcal K^*)^kf\rangle/\lVert f\rVert^2$, and
$c_1=1-6\alpha_{\rm occ}/11$ for the occupation compression factor
$\alpha_{\rm occ}$;
\item the statistic
\[
b_{\rm tick}=c_2-c_1^2-\frac{r^2(209r+180)}{1331r^2+3300r+1800},\qquad r=1-c_1,
\]
vanishes for the native law and for the one-term exchange controls, which
have $c_1=5/11$ and $c_2=93/407$;
\item for both N4 completions, built on $\Cn$ and on $\En$,
$b_{\rm tick}\ge3m/50$, and the certified lower bounds are
\[
\begin{aligned}
b_{\rm tick}&\ge\frac{139996257}{12500000000000}\approx0.00001119970056
&&(\theta=1/16),\\
b_{\rm tick}&\ge\frac{11335290321}{50000000000000}\approx0.00022670580642
&&(\theta=1/8);
\end{aligned}
\]
\item $c_{2,\Cn}\ge c_{2,\En}$ and $b_{{\rm tick},\Cn}\ge b_{{\rm tick},\En}$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Each seam channel is a positive semidefinite contraction of Hilbert-Schmidt
space, with spectrum in $\{0,1/2,1\}$ for $\Cn$ and $\{0,1\}$ for $\En$; N4
conjugation preserves this. Hence $0\le\mathcal A\le(1-p)I$, the first-hit
series converges, and the stopped maps are unital and trace preserving.
Remote seams fix $f$, which gives (i). The closed comparator with seam mean
maps $I-\frac{\alpha_{\rm occ}}2w_ew_e^{\top}$ yields the correction term
in (ii) after substituting $\alpha_{\rm occ}=11(1-c_1)/6$. With the Schur complement $S$ of
$I-\mathcal A$ on the centered occupations, $S\le I$ and
$b_{\rm tick}=\langle z,S^{-1}z\rangle/(p\lVert f\rVert^2)\ge
\lVert z\rVert^2/(p\lVert f\rVert^2)$. The hidden defect is
$z=-\frac1{60}\sum_et_eS_e$ with five incident coefficients $\pm1$, so
$\lVert z\rVert^2\ge5m/3600$ and $b_{\rm tick}\ge3m/50$. On each seam
$\Cn-\En$ is positive semidefinite, which gives (iv) \cite{ltREADME}.
\end{proof}

The correction used for global attempts, $(1-c_1)^2/10$, assigns $-16/2035$
to the native null at local ticks; the correction of
Theorem~\ref{thm:tick}(ii) is the one adapted to this stopping instrument.
The receipt and eight independent checks bind the result, including a
comparison with the full 66-state native stopping kernel and a complete
dimension-six quantum model of the resolvent and Schur identities
\cite{ltREADME,rLT}.

\section{Observers, clocks and screens on one process}
\label{sec:native-process}

This section completes layer three on one native process. Its components
run the native C or E instrument itself, with the uniform independent seam
schedule and fair placement coin as declared scheduling inputs. Their time parameters are attempt counts and
local event counts. Each result states its preparation, observer access
and services; none identifies a count with laboratory duration
\cite{nativeRecovery}.

\subsection{Two overlapping observers}
\label{sec:correspondence}

On the $66$ binary total-two configurations, prepare the uniform state and
disclose the bits at ports $(0,1,5)$; all eight labels stay in the output
menu. Observer A accesses $(0,1)$ and observer B accesses $(0,5)$, so the
regions overlap at port $0$. Each receives its initial bits and every
incident local current, including zero, and ticks when an attempted seam
meets its region. The protocol requests two incidents on the union with at
most two native attempts per interval; after the first it applies one
native repair on $(0,1)$ if the reconstructed $n_0$ is one and on $(0,5)$
otherwise, and that repair ticks both observers.

\begin{proposition}[Exact overlapping-observer correspondence]
\label{prop:correspondence}
Normalize the native remote and incident branch counts by $60$, write
$P=A+B$ with $B=\sum_mB_m$ over incident marks $m$, and put $D=I-A$,
$p=2/5$ and $G=D/p$. With $K_m=B_mD^{-1}$ and $F_m=D^{-1}B_m$,
\begin{enumerate}[(i)]
\item $K_m=GF_mG^{-1}$ and $\mathbf 1^{\mathsf T}G=\mathbf 1^{\mathsf T}$; remote
projectors, the selective feedback maps and the disclosed uniform
preparation commute with or are fixed by $G$;
\item for attempt cap $h$ the marked and timeout maps are
$K_{m,h}=B_m(I+A+\dots+A^{h-1})$ and $Q_h=A^h$, their effective conjugates
are $F_{m,h}=G^{-1}K_{m,h}G$ and $A^h$, and the terminal decoder $G$
returns the complete unnormalized source state on every output word of
every finite adaptive protocol in this commuting intervention family;
\item the direct endpoint execution and the effective recursion agree
exactly on all $5{,}768$ output words, $689$ of positive probability,
including all terminal populations, and the timeout probability is
$369/625$; and
\item the unread completed transfer $F=D^{-1}B$ is self-adjoint in the
metric $G$ with $1/20\le F\le1$, so $H=-\log F$ is an effective Euclidean
generator per completed union incident tick.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
Remote repairs leave the union fields fixed and every remote seam is
disjoint from the feedback ports, so the conjugation by $G$ cancels between
successive reads and allowed feedback maps; induction over the protocol
gives (ii). Every native seam count equals $I+\mathrm{SWAP}$, which makes
$GF=B/p$ symmetric positive semidefinite, and the native triangle $(0,1,5)$
contributes at least $3I/60$ to $B$ \cite{nativeCorrespondence}.
\end{proof}

At every positive-weight prefix each observer's decoder, which sees only
its local initial pair and incident increments, returns fields that obey
native continuity, and the two port-zero values agree. The transformed
finite-cap maps preserve the transported cone $G^{-1}\mathbb R_+^{66}$; they
are not entrywise-positive stochastic maps, and the comparison is of
decoded full states. The quantum extension checks the native remote
resolvent on all $4{,}356$ matrix units with coherent preparations,
untouched references and complete local clock records, and an ordinary
completely positive effective instrument for local outputs is a positive
mixture of native repairs; its one-particle coherent experiment retains all
$81$ outputs \cite{nativeQuantumCorrespondence}. The compiled clock history
replaces the ideal preparation, intervention and pointer of that experiment
by words of the six-level source compiler. Exact rational propagation of
the binary source arrays keeps all $512$ code addresses and $1{,}656$ clock,
timeout and complement-abort outputs within $10^{-11}$ of the ideal
history, at most $51$ pulses ($54$ with the compiled Bell preparation), two
code services per pulse and five native repair attempts
\cite{nativeCompiledClock}. The native instrument, the scheduler and the
code services are specified inputs.

\subsection{Joint and active event clocks}
\label{sec:event-clocks}

A local event clock ticks on actual occupation changes at an observer port,
or on attempted repairs incident to a region, while remote repairs
continue; its stopped instrument sums the unresolved wait and retains finite
timeouts and conditional quantum outputs. Two such clocks on ports $0$ and
$72$ of joined carriers read one continuing history with an
observer-chosen field algebra. Exact elimination of the hidden factor gives
a record-conditioned quantum memory recurrence valid for arbitrary untouched
references. At completed events the experiment admits a four-mark
measure-and-prepare description, while unresolved waits retain coherence
that the completed instrument erases. C and E share the completed clock law
and predict field-plus probabilities $35/54$ and $14/27$ after three native
attempts. Without feedback each two-tick cycle of clock A carries B-count
mean two and variance $16$; instrument-valued clock conversion preserves
that count law, and direct native elimination and elimination through the
joint event instrument agree, including B-triggered field feedback
\cite{nativeJointClock}.

An active event clock lets each native event incident to port $0$,
including a zero-current one, trigger one fixed compiled modular word
$U(1)=\exp(iH)$ generated by the native three-state attempt transfer; the
carrier pauses during probe services, which are charged separately. For
caps $c\in\{1,2\}$ the execution retains all $3{,}661$ joint output labels,
the first timeout has probability $(5/6)^c$ and both ticks occur with
probability $[1-(5/6)^c]^2$. Closed native histories derive a faithful
thermal reference for that generator, and passive repair itself cannot act
as a nontrivial modular automorphism on an invariant observer algebra
\cite{nativeEventClock}. On the complete movement-bit tape
$B_t=|j_t|$, including zero currents, four consecutive covariances
$c_n=\mathbb E[B_tB_{t+n+1}]-p^2$ invert to the two transfer roots
$\lambda_\pm=(25\pm\sqrt3)/30$ with $p=5/33$, and the same roots predict the
coherent quadratures $q_X=(1+\cos(sE_j))/2$ and $q_Y=(1+\sin(sE_j))/2$,
$E_\pm=-\log\lambda_\pm$, with no frequency fit. The one-exponential
approximation matching $c_0,c_1$ misses $c_2$ by exactly $1/2673000$. The
inference does not distinguish C from E, whose classical tapes agree
\cite{nativePredictions}.

\subsection{Identification of the record law}
\label{sec:identification}

Nine source-executed scalar measurements identify all nine real coordinates
$(k,g,h,m,n)$ of the record-law table of Theorem~\ref{thm:record-laws}, and
with them the coarse definite-current instrument on all $78$ total-two
states with arbitrary spectator and reference coherences. The execution
comprises $36$ compiled probes and ten complete two-event histories, five
with untouched quantum references, run through the canonical observer
runtime with serialized checkpoint validation and continuation. An
admissible collision law agrees with E on the whole binary shell, and this
experiment separates it. A reference-uniform bound carries measurement
uncertainty through the finite feedback histories, so narrowing the
admissible probability tubes narrows the complete-history bound
\cite{nativeIdentification}. The probes are synthetic; they identify an
instrument within the classified family and select no law for nature.

\subsection{Federations and matched screens}
\label{sec:screens}

Federations join $20$, $80$ or $320$ identical twelve-port carriers through
the declared seam wiring, and native Kraus maps preserve
signed-difference operator blocks with individual outcomes and reference
correlations. The endpoint instruments fix the local Liouville action
$H_{\rm native}=\sum_e(I-\Phi_e)$, with $h_e=P_1+\tfrac12P_{1/2}$ for C and
$h_e=P_1$ for E on orthogonal projectors; the discrete uniform transfer is
$I-H_{\rm native}/M$ for $M$ seams. For an observed region $R$, maximum
degree $\Delta$ and retained radius $r$, truncating distant ports changes
the complete local record, quantum and reference output by at most
$\min(1,|R|(\Delta s)^{r+1}/(r+1)!)$ at auxiliary per-seam Poisson parameter
$s$ \cite{nativeAction}. Actual source neighborhoods are identified at levels
$1/3$, $2/4$ and $3/5$. On the $15{,}360$- and $245{,}760$-port screens
matched radius-nine initial restrictions give complete root-observer
output distance below $0.000382$ at $s=1/4$; after four local incident
ticks, with no auxiliary time coordinate, the pair allowance is below
$6.656\times10^{-9}$, and a density-$1/3$ preparation keeps a certified
two-time field covariance above $0.15332581621$. Full-screen executions
retain the actual seam scheduler, current records, feedback repair costs,
timeouts and serialized continuation. Conditional on a conserved total,
native connected exchange derives the unique diagonal equilibrium, so the
screens at totals $5{,}120$ and $81{,}920$ keep density $1/3$, the feedback
reversal of the connected four-field statistic and their finite-number
correlations without product preparation \cite{nativeTimeFields}.

A coherent observer runs on the same matched screens. A logical qubit on
ports $0$ and $1$, possibly entangled with an untouched reference, passes
through two native incident ticks and a local phase $Z_a$ after first mark
$1$, while every other port is occupied with probability $1/3$. A
degree-two operator dual evaluates the common $222$-port ball with $561$
seams and all $21$ capped record labels, and each full-screen completed
output lies within $7.779\times10^{-10}$ of it, uniformly over references.
C and E give identical classical records in this protocol and different
reference coherences \cite{nativeScreenObserver}. The native continuum
operator of the refining screens and its density, wave and response limits
are stated in Section~\ref{sec:native-history}.

\section{Field detectors on supplied actions}
\label{sec:fields}

Layer four begins with detectors that couple native records to fields whose
actions are supplied. They provide complete-output and refinement machinery
that a source-derived field description must reuse; they derive no field.

\subsection{The scalar detector and its refinement transfer}
\label{sec:scalar}

The scalar detector joins the native observer to the free-scalar continuum
family of the theorem library \cite{sREADME,sDERIV}. The scalar action,
quantization, vacuum, preparation, Born readout and model clock are
supplied. The field is the massive scalar on the golden tensor grids
$q\in\{13,34,89,233\}$, with Dirichlet boundary and mass one; after mass
whitening its one-particle operator is
$\mathsf A=I+B\otimes I\otimes I+I\otimes B\otimes I+I\otimes I\otimes B$,
with $\Omega=\sqrt{\mathsf A}$ and $H=d\Gamma(\Omega)$. A preparation profile
$f$ and a detector profile $g$ lie in compact left and right slabs.

One trial runs as follows. (1) Prepare one of the 66 binary configurations
uniformly, repair $(0,1)$ and retain $(c_1,j)$. (2) Prepare the field vacuum
and apply the momentum displacement of amplitude $\gamma jf$, $\gamma=4$.
(3) Evolve to model time $t$ and read the centered binary Weyl instrument
\[
K_y=\tfrac12\bigl[e^{-iX/2}-iy\,e^{iX/2}\bigr],\qquad
X=a^\dagger(v)+a(v),\qquad v=\Omega^{-1/2}g/\sqrt2,
\]
with $K_y^\dagger K_y=\frac12[I+y\sin X]$. (4) Commit $y$, repair $(0,5)$ for
$y=+1$ and $(0,1)$ for $y=-1$, and halt. Model times lie in $[19/20,1]$,
with primary setting $39/40$, and carry no laboratory calibration.

\begin{proposition}[Scalar feedback and refinement transfer]
\label{prop:scalar}
For the process above at model time $t$, let
$r(t)=\Pr(y=+1\mid j=1)-\frac12$. Then:
\begin{enumerate}[(i)]
\item the branch operators
$M_{(r,y,s)}=(L_sL_r)\otimes(K_ye^{-iHt}V_j)$, with $V_j$ the controlled
displacement, form one CPTP instrument on arbitrary joint source and field
inputs, and each branch retains its conditional field state as a
superposition of two coherent components with their relative phase;
\item $p(y\mid j)=\frac12+yj\,r(t)$ with
$r(t)=\frac12e^{-\lVert v\rVert^2/2}\sin\mu(t)$ and
$\mu(t)=\gamma\langle g,\Omega^{-1}\sin(\Omega t)f\rangle$;
\item over the 528 histories, $\Ex[j_2]=5r/132$,
$\Ex[N_{0,\rm final}]=\frac16-5r/132$, $\Ex[yj]=10r/33$ and
$\Pr(y=\pm1)=\frac12$; disconnecting the preparation or replacing $y$ by an
independent fair bit gives $\Ex[j_2]=0$ with the same seam frequencies, and
reversing the kick reverses the sign;
\item if $\lvert r_q-r_\infty\rvert\le\varepsilon_q$ on the window, then the
complete labelled histories satisfy the total-variation distance $\TV=\frac5{33}\lvert r_q-r_\infty\rvert\le5\varepsilon_q/33$, and
$\lvert\Delta\Ex[j_2]\rvert\le5\varepsilon_q/132$.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
The Weyl operators $e^{\pm iX/2}$ are displacements by $\pm iv/2$, which gives
the branch states and (ii). Conditioned on the initial configuration and first
coin, the two detector laws differ in total variation by
$\lvert j\rvert\,\lvert r_q-r_\infty\rvert$, and the subsequent native update
is common \cite{sDERIV}.
\end{proof}

At $q=233$ the parent certifies, uniformly for $19/20\le t\le1$,
\[
\varepsilon\le\frac{12513633763}{250000000000},\qquad
-\frac{111322580247}{500000000000}\le r\le-\frac{34438518047}{250000000000},
\]
so the finite-grid mean second current satisfies
\[
-\frac{37107526749}{4400000000000}\le\Ex[j_2]\le-\frac{34438518047}{6600000000000},
\]
negative throughout the window, with the continuum mean within
$\frac5{132}\varepsilon\approx0.001896$ \cite{rSCALAR,sPRED}. At $t=39/40$ the
plus probabilities for $j=+1$ are about $0.334515$, $0.319421$, $0.317093$ and
$0.316771$ on $q=13,34,89,233$, and the finest-grid prediction is
$\Ex[j_2]\approx-0.00694050$. A seeded execution with 200,000 fresh trials
per grid tests Born outcomes against predictions computed before sampling,
with reversal, donor and fixed-seam controls; at $q=233$ all 12,487,168 field
modes contribute. An independent diagonalization gives largest relative
frequency errors $2.137\%$, $0.2875\%$ and $0.04116\%$ at $q=13$, $34$ and
$89$ against the supplied dispersion $\omega^2=1+\pi^2(n_x^2+n_y^2+n_z^2)$
\cite{csCALIB}. A local kick-drift-kick extension through the six-neighbour
stencil runs at $q=13$ and $34$ and transfers the combined spatial and
temporal bound through the same factors $5/33$ and $5/132$; at $q=233$ the
temporal term is at most $0.000008071512$ and the combined probability error
at most $0.050062606564$ \cite{sREADME}.

\subsection{The retained two-read continuation}
\label{sec:two-read}

The continued observer keeps the disturbed field after its first read
\cite{sCONT}. The field is read at model times $19/20$ and $1$; the first bit
$y_1$ selects the second native seam, the conditional field evolves to time
$1$, the second bit $y_2$ selects the third seam, and the observer halts with
five records readable. This continuation uses exact continuous spectral
evolution and carries its own error bounds.

\begin{proposition}[Two reads of one field]
\label{prop:two-read}
In the continued protocol:
\begin{enumerate}[(i)]
\item all 2,112 labelled histories are retained; the record-prefix counts at
the six phases are $1,4,8,24,48,124$, so the 209 reached prefixes have an
eight-bit encoding and the 124 terminal words need seven bits;
\item with the Gram kernel $G_{ab}$ of the two detector vectors,
$V=G_{11}=G_{22}$, $C=\operatorname{Re}G_{12}$, $c=2\operatorname{Im}G_{12}$
(so that $[X_1,X_2]=ic$), $S_a(j)=e^{-V/2}\sin(j\mu_a)$, $D=\cos(c/2)$ and
$R_j=\frac12[e^{-(V-C)}\cos(j(\mu_1-\mu_2))-e^{-(V+C)}\cos(j(\mu_1+\mu_2))]$,
the joint law is
\[
p_j(y_1,y_2)=\tfrac14\bigl[1+y_1S_1(j)+y_2D\,S_2(j)+y_1y_2R_j\bigr];
\]
\item on the positive-current preparation, performing the first read changes
the second plus probability by $(-r(t_2))[1-\cos(c/2)]$, certified greater
than $0.000022281162$ at $q=233$ and greater than $0.0000252366$ in the
declared compact continuum; a commuting Gaussian control with the same means
and real covariance has $D=1$ and no such disturbance; the zero-current
vacuum correlation exceeds $0.045619388905$ at $q=233$;
\item the total variation distance of the complete labelled history from the
declared continuum is at most $1$, $0.451806$, $0.052189606637$ and
$0.025709913643$ on $q=13,34,89,233$, including the zero-current histories.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
A state is stored as $\sum_nc_nD(i\sum_ah_{na}v_a)\ket\alpha$ with exact
half-integer shifts, its Gram matrix is explicit in $G$, and both terms of
each Kraus operator are applied before the branch norm is taken. For four
atoms with Walsh-moment differences $a,b,d$,
$\TV=\frac12\max(\lvert a\rvert,\lvert b\rvert,\lvert d\rvert,
\frac12(\lvert a\rvert+\lvert b\rvert+\lvert d\rvert))$, bounded in directed
rational arithmetic \cite{sCONT}.
\end{proof}

At $q=233$ the execution gives $\Ex[j_3]\approx-0.00130126$ and
$\Ex[y_1y_2\mid j=0]\approx0.0514838$; deleting the first field record from
the forecast raises its Brier risk by about $0.00064677$. The 50,000-trial
runs per grid do not resolve the small disturbance statistically.

\subsection{The source-compiled two-site apparatus}
\label{sec:two-site}

The two-site apparatus executes preparation, evolution and pointer as
emitted twelve-port control words \cite{ssREADME,ssDERIV}. Its action lies
outside the golden refinement family, so its occupation limit carries no
spatial refinement. The Hamiltonian is
\[
H=\tfrac12(P_0^2+P_1^2+Q_0^2+Q_1^2)+\tfrac\kappa2(Q_0-Q_1)^2,\qquad
\kappa=\tfrac9{32},
\]
with normal-mode frequencies $1$ and $5/4$ and $d\in\{4,8,16\}$ number states
per site. The executed Hamiltonian uses the compressed squares
$Q_d^2+\frac d2\proj{d-1}$ and $P_d^2+\frac d2\proj{d-1}$, so that
$H_d=\Pi_dH\Pi_d$. The prepared state is the normalized projection of the
coupled vacuum onto total occupation at most $d-2$; at $d=16$ its binary
amplitude tree costs 5,379 native pulses. The current impulse is
$e^{ijQ_{0,d}}$, 64 or 128 symmetric split steps reach $t=1$, and the pointer
coupled by $e^{-iZQ_{1,d}/2}$ and read in the $Y$ basis gives
$\Pr(y=+1\mid j)=\frac12+jr_d$ and $\Ex[j_2]=5r_d/132$.

\begin{theorem}[Complete ideal error budget of the two-site instrument]
\label{thm:two-site}
For the declared two-site action, model time $0\le t\le1$ and native current
$j\in\{-1,0,1\}$:
\begin{enumerate}[(i)]
\item with $A$ the onsite and $B$ the bond part of $H_d$ and
$S(h)=e^{-ihA/2}e^{-ihB}e^{-ihA/2}$,
\[
\bigl\lVert S(t/n)^n-e^{-it(A+B)}\bigr\rVert\le\frac{t^3}{n^2}
\Bigl(\frac{\lVert[A,[A,B]]\rVert}{24}+\frac{\lVert[B,[A,B]]\rVert}{12}\Bigr),
\]
uniformly over inputs and references;
\item the prepared state differs from the infinite coupled vacuum in trace
distance by less than $0.007576$, $0.000079899$ and $1.0324\times10^{-8}$ at
$d=4,8,16$;
\item the trace distance between the complete ideal finite instrument,
including its classical outcome and unnormalized terminal field branches, and
the infinite two-oscillator instrument is below $0.799447$, $0.077171$ and
$0.000196$ at $d=4,8,16$, and the bound tends to zero as $d$ grows;
\item at $d=16$ and $n=128$ the combined occupation and splitting error is
below $0.001464436$, and the ideal source-word instrument satisfies
$0.514957066204\le\Pr(y=+1\mid j=1)\le0.517885937293$, an interval that
excludes $1/2$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Item (i) is a defect bound for the symmetric product formula
\cite{strang1968} with nested commutators bounded by exact rational Frobenius
norms. The infinite state is a displaced squeezed vacuum throughout; the
Gaussian generating function bounds the number tails along impulse,
evolution and both pointer arms, and projected Duhamel comparison with the
orthogonal leakage gives (iii) without dividing by a Born probability. Item
(iv) adds a rational enclosure of the infinite reference probability
\cite{ssDERIV,rSS}.
\end{proof}

The executed plus probabilities for a positive unit current, with their
rational splitting bounds, are $0.516680958$ ($0.000103957$) and
$0.516679563$ ($0.000025990$) at $d=4$ with 64 and 128 steps, $0.516424174$
($0.000693013$) and $0.516422803$ ($0.000173254$) at $d=8$, and $0.516423330$
($0.005076875$) and $0.516421959$ ($0.001269219$) at $d=16$. The infinite
reference has detector variance $9/20$ and mean
$\mu(t)=\frac12(\sin t-\frac45\sin(5t/4))$, so
$\Pr(y=+1\mid j)=\frac12(1+e^{-9/40}\sin(j\mu(t)))$, equal to
$0.516421501748$ at $t=1$ and $j=1$ \cite{ssREADME}. The $d=16$, 128-step
protocol uses nine work qubits, 18,612,125 native pulses and 37,224,250
proper-code load and export events, and retained 200,000-trial runs record
the reversal, disconnected and donor controls.

\section{The common interacting source}
\label{sec:common}

\subsection{Construction}
\label{sec:common-construction}

The common source places quantum electric flux, a charged fermion pair and a
neutral scalar on one evolving work register. The sharp protocol measures it
twice and feeds its records into the register and the declared native
integer primitive \cite{csREADME,csDERIV}; its four-read smooth extension
joins compiled native quantum channels and encoded recovery
(Section~\ref{sec:continued-native}). The action, preparation, coupling and
model-coordinate schedule are construction inputs.

Order four fermion modes as $L_0,R_0,L_1,R_1$. The two channels carry charges
$q_0=+1$ and $q_1=-1$ and one fermion each; $b_j=0$ means left and $b_j=1$
right occupation of channel $j$. The electric link has basis $\ket\ell$,
$\ell\in\mathbb Z$, with $\Eflux\ket\ell=\ell\ket\ell$ and
$U\ket\ell=\ket{\ell+1}$. The isometry
\[
W\ket{b_0,b_1}=\ket{\text{modes }b_0\text{ and }2+b_1\text{ occupied}}\otimes\ket{-b_0+b_1}
\]
maps onto four orthonormal occupation and flux vectors.

\begin{proposition}[Flux-dressed matter code]
\label{prop:gauss}
On the image of $W$:
\begin{enumerate}[(i)]
\item both operator Gauss constraints hold, $G_LW=G_RW=0$ with
$G_L=\Eflux-Q_L$ and $G_R=-\Eflux-Q_R$, the supported flux set is
$\{-1,0,1\}$ inside the infinite link, and fermion parity is even;
\item the dressed hopping terms $h_j=c_{Lj}^\dagger U^{q_j}c_{Rj}+c_{Rj}^\dagger
U^{-q_j}c_{Lj}$ preserve both ambient Gauss values and the code, with
$W^*(h_0+h_1)W=X_0+X_1=H_{\rm hop}$ and
$W^*\Eflux^2W=\frac12(I-Z_0Z_1)$, and
$\lVert[H_{\rm hop},\Eflux^2]\rVert=2$;
\item with $\theta=\operatorname{atan2}(4,3)$, $u=(3I-4iX)/5$,
$U_\theta=u\otimes u=e^{-i\theta H_{\rm hop}}$ and
$V_\theta=e^{-i\theta\Eflux^2}=\diag(1,z,z,1)$, $z=(3-4i)/5$, the echo
$U_\theta^*V_\theta U_\theta$ applied to $\ket{00}$ gives right occupation of
channel $0$ with probability exactly $576/3125$, and without $V_\theta$ the
probability is zero.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
On every image column $Q_L=-b_0+b_1=\Eflux$ and $Q_R=-\Eflux$. The four-mode
algebra of canonical anticommutation relations (CAR) is checked on all 16
occupation states with independent Jordan-Wigner signs, and all entries of
the echo unitaries are Gaussian rationals \cite{csDERIV}.
\end{proof}

The echo runs on the source controls with 22 native pulses and 44 coherent
service events. Two neutral scalar sites join the same register, and with
onsite quadratics compressed from the infinite oscillator the declared
dimensionless action is
\begin{equation}
\label{eq:common-H}
\begin{aligned}
H&=H_{\rm hop}+\Eflux^2+H_{\rm scalar}+H_{\rm int},\\
H_{\rm scalar}&=\sum_{j=0,1}\tfrac12\bigl[P_j^2+(1+\kappa)Q_j^2\bigr]-\kappa Q_0Q_1,
\qquad H_{\rm int}=g\sqrt2\,Q_0(n_{L0}-n_{R0}),
\end{aligned}
\end{equation}
with $\kappa=9/32$ and $g=1/4$. The interaction commutes with both Gauss
generators and with fermion parity and couples the sectors,
$[H_{\rm int},H_{\rm hop}]=2ig\sqrt2\,Q_0Y_0$. The experiment proceeds in four
steps. (1) Native repair on $(0,1)$ records a current $j$, which controls the
gauge pulse $V_j=e^{-ij\theta X_0}$. (2) The register evolves for interval
$1$ ($U_1$), and a source pointer commits the right occupation $a$ of the
first charged species ($P_a$). (3) Only $a$ selects the hopping pulse
$F_1=(3I-4iX_0)/5$, with $F_0=I$, and the register evolves for interval $1/2$
($U_2$). (4) A second pointer commits the sign $b$ of $Q_0$ ($R_b$), and only
$b$ selects the second native seam, $(0,5)$ for $b=1$ and $(0,1)$ otherwise.

\begin{proposition}[One joint instrument with retained histories]
\label{prop:joint}
The operators $K_{(a,b)}=R_bU_2F_aP_aU_1$ satisfy
$\sum_{a,b}K_{(a,b)}^\dagger K_{(a,b)}=I$, and
$M_{(r,a,b,s)}=(L_s^{e(b)}L_r^{(0,1)})\otimes(K_{(a,b)}V_j)$ form one CPTP
instrument on arbitrary joint native and quantum inputs. Under the uniform
binary preparation and the declared quantum vector $\psi$, the
$66\cdot2\cdot4\cdot2=1{,}056$ histories have probabilities
$p(a,b\mid j)/264$ with $p(a,b\mid j)=\lVert K_{(a,b)}V_j\psi\rVert^2$, each
bound to its terminal density $\proj{v_{j,a,b}}/264$, and the first native
record keeps the masses $14/33$, $14/33$, $5/66$, $5/66$. For a complete
record-controlled instrument $B_y=\sum_r\proj r\otimes K_{(r,y)}$ with
channel $\Lambda(X)=\sum_yB_yXB_y^\dagger$, every old record projector $Q_r$
satisfies $\Lambda^*(Q_r)=Q_r$, and the block readback
$E_{\rm block}(X)=\sum_rQ_rXQ_r$ and the public readback
$E_{\rm public}(X)=\sum_r\Tr(Q_rXQ_r)Q_r/\dim W$ are distinct conditional
expectations.
\end{proposition}

\begin{proof}[Proof sketch]
Summing $M^\dagger M$ over $s$, then $(a,b)$, then $r$ gives the identity.
Each $B_y$ commutes with $Q_r$, so $\Lambda^*(Q_r)=Q_r\sum_yB_y^\dagger
B_y=Q_r$; every reported probability is checked against the norm of the
retained unnormalized vector \cite{csDERIV,rCS}.
\end{proof}

At 512 steps per interval the emitted words use 4 working qubits, 176,182
native pulses and 352,364 coherent load and export events at $d=2$, and 6
qubits, 1,879,628 pulses and 3,759,256 events at $d=4$. Disconnected,
feedback-disabled, fixed-seam and independent-bit controls isolate the
causal effect of the records \cite{csREADME}.

\subsection{The feedback contrast at finite and infinite occupation}
\label{sec:finite-contrast}
\label{sec:infinite}

\begin{theorem}[Finite feedback contrast]
\label{thm:finite-contrast}
For the common source with intervals $1$ and $1/2$:
\begin{enumerate}[(i)]
\item every palindromic product $F(h)$ of the component exponentials whose
generators sum to $H$ satisfies
$\lVert F(t/n)^n-e^{-itH}\rVert\le S^3\lvert t\rvert^3/(3n^2)$ with
$S\le3+(1+2\kappa)(2d-1)+\lvert g\rvert\sqrt{4d-2}$, on all inputs and
references; at $n=512$ and $t=1$ the half-diamond bounds are below
$0.000728$ for $d=2$ and $0.004184$ for $d=4$;
\item at $d=2$ and zero native preparation current, exact rational Taylor
arithmetic of degree 64 with $\lVert H\rVert\le109/16$ encloses the
unconditional probability of a positive scalar coordinate and the feedback
contrast for the continuous finite Hamiltonian:
\[
\begin{aligned}
&\text{feedback enabled:} && [0.370697726608684,\,0.370697726608685],\\
&\text{feedback disabled:} && [0.402359206973491,\,0.402359206973492],\\
&\text{contrast:} && [-0.031661480364808,\,-0.031661480364807];
\end{aligned}
\]
\item for the ideal source word at 512 steps per interval the contrast lies in
\[
[-0.033297300667839,\,-0.030025660061775],
\]
which excludes zero.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
The multinomial Leibniz expansion bounds the third derivative of a product of
unitaries by $S^3$, palindromic symmetry gives second-order agreement, and
unitary telescoping gives (i). For (ii) the integral Taylor remainder gives
$\lVert T_m(t)-e^{-itH}\rVert\le(St)^{m+1}/(m+1)!$, and at $m=64$ the
probability radius is below $7.091\times10^{-37}$; measurements and feedback
contract trace distance, which gives (iii) \cite{csDERIV}.
\end{proof}

With $g=0$ the gauge-only feedback cannot change the scalar marginal, and a
coarse splitting or inadequate Taylor degree yields an interval containing
zero, which the evaluator reports as no resolved signal. The finite pointer
measures the sign projector of the truncated coordinate, which differs from
the infinite half-line effect; at $d=2$ the two differ in norm by
$\frac12-1/\sqrt{2\pi}\approx0.10105772$ \cite{csREAD}, so the
infinite-occupation statement needs the infinite detector.

\begin{theorem}[Infinite-occupation feedback contrast]
\label{thm:infinite}
Take the action~\eqref{eq:common-H} on two infinite oscillator Fock spaces,
the infinite free coupled vacuum, proportional to
$\exp[-(a_0^\dagger-a_1^\dagger)^2/36]\ket{0,0}$ with squared normalization
$4\sqrt5/9$, the blank gauge code, zero native preparation current, the
matter record and feedback of Section~\ref{sec:common-construction}, and the
half-line readout of $Q_0$. Then:
\begin{enumerate}[(i)]
\item the unconditional positive-coordinate probability with feedback minus
that without lies in $[-0.024045670143,\,-0.024043450955]$, and without the
interaction the contrast is exactly zero;
\item both matter records and both half-line outcomes retain their quantum
states, including the post-read infinite occupation tail;
\item along $d=2^k$, with exact palindromic factors and $n=d^2$ steps per
interval, the complete finite sign instruments converge in trace distance to
this infinite instrument, terminal states included, and the finite-family
contrast is strictly negative for all sufficiently large $d$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
The linear force is relatively bounded by the coupled oscillator Hamiltonian,
so $H$ is self-adjoint and finite Hermite polynomials lie in every
$\mathcal D(H^k)$. Each of 24 steps of length $1/16$ applies a degree-16
Taylor polynomial of $H$, charges its remainder exactly, truncates above total
occupation 16 and rounds to a $2^{-112}$ grid, charging both differences.
Half-line probabilities are $r+s/\sqrt{2\pi}$ with exact rationals, and $\pi$
is enclosed by Machin's formula. For (iii), $P_dQ_0P_d\to Q_0$ in the strong
resolvent sense and the spectral measure of $Q_0$ has no atom at zero
\cite{csINF,teschl2014,reedsimon1}.
\end{proof}

The $d=2$ finite experiment has contrast about $-0.03166148$; replacing only
its detector by the half-line effect gives about $-0.02526221$, and removing
the cutoff gives the interval of Theorem~\ref{thm:infinite}. The signal
survives both limits, which change its size; the interval is specific to zero
native preparation current and the declared two-site action \cite{csINF}.

\subsection{Occupation refinement and complete-output comparisons}
\label{sec:occupation}

Let $P_d$ project both oscillators onto occupations below $d$ and let
$N=n_0+n_1$ be the total oscillator occupation, a different object from the
closure input $N$.

\begin{theorem}[Uniform pre-readout occupation bound]
\label{thm:occupation}
Let the matter start in an arbitrary state entangled with an untouched
reference, let the scalar start in the fixed coupled vacuum, and let the
finite preparation retain total occupations $0,2,\dots,d-2$. Then:
\begin{enumerate}[(i)]
\item $M(s,t)=\langle e^{sN}\rangle$ satisfies
$\partial_tM\le\frac{11}{16}\sinh(s)\,\partial_sM+\frac{25}{32}\sinh(s)\,M$
through the matter measurement and feedback;
\item at total model time $T=3/2$ the resulting moment bound holds with base
$\varrho=113/64\ge7/4$ and $M<13$, where $\varrho\le e^{s(T)}$;
\item $\lVert P_dH(1-P_d)\rVert\le C_d=\frac\kappa2\sqrt{d(d+1)}+\frac{3\kappa}2d+\frac14\sqrt d$,
and the retained-history trace distance up to the final scalar read is at
most $\bigl[(\sqrt{2v_d}+TC_d\sqrt{\tau_d})^2+\tau_d\bigr]^{1/2}$ with
$\tau_d=M/\varrho^d$ and $v_d$ the omitted preparation probability, an
$O(d\,\varrho^{-d/2})$ rate;
\item this bound is below $0.011408$, $0.000002498$ and
$6.175\times10^{-14}$ at $d=32,64,128$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
In common and difference modes only the squeezing term and the force change
the number; weighted commutator, Cauchy and Young inequalities that hold for
matter-entangled vectors give (i), and integration along
$\tanh(s(t)/2)=e^{-11t/16}\tanh(s_0/2)$ from the initial value
$\sqrt{80/17}$ at $\varrho_0=8$ gives (ii). The record dilation and feedback
commute with $N$, and projected Duhamel comparison gives (iii) \cite{csOCC}.
\end{proof}

\begin{theorem}[Fourth moment and strong convergence of the sharp instrument]
\label{thm:fourth}
Let $w(N)=(N+2)^4$ and let $M(t)=\langle w(N)\rangle$ be summed over the
unnormalized record branches. Then:
\begin{enumerate}[(i)]
\item $\lvert\dot M\rvert\le CM$ with $C=8\kappa+4\lvert g\rvert=13/4$, and
every finite preparation has $M(0)\le M_0=143748821/8192000$;
\item the record-dilation vectors immediately before the final read satisfy
\[
\lVert\Psi_d-\Psi\rVert\le\Bigl[(e_0+I_TL_d)^2+\frac{M_0e^{CT}}{(d+2)^4}\Bigr]^{1/2}=O(1/d),
\]
with $I_T=\sqrt{M_0}\,(e^{CT/2}-1)/(C/2)$,
$L_d=\kappa/(d+2)+\lvert g\rvert/(d+2)^{3/2}$ and $e_0$ the preparation
error;
\item the sign projectors of $Q_{0,d}$ converge strongly to those of $Q_0$,
and the complete unnormalized classical and quantum history, together with
its native continuation, converges in trace norm for all three native
currents and both feedback settings.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
On occupation at most $m$, $\lVert H\psi\rVert\le4(m+2)\lVert\psi\rVert$ and
$H$ raises occupation by at most two, so finite-particle vectors are analytic
and core convergence gives strong resolvent convergence
\cite{reedsimon2,teschl2014}. The moment coefficients are bounded by $8\kappa$
and, through a positive polynomial certificate, $4\lvert g\rvert$; the
gauge measurement and rotations preserve $w(N)$ summed over branches
\cite{csOCCREF}.
\end{proof}

Strong convergence supplies no finite-$d$ rate for the sharp readout. The
complete-output certificate supplies fixed-state enclosures at
$d\in\{2,32,64,128\}$ for the zero-current preparation and the named
rectangular occupation codes \cite{csREAD,rSHARP}.

\begin{theorem}[Complete outputs of the sharp detector]
\label{thm:sharp}
For zero native preparation current, with both matter records and both sign
outcomes retained:
\begin{enumerate}[(i)]
\item for two complete post-read outputs with rank-one branches, the half
trace norm of their difference is
$\frac12\sum_b\sqrt{(p_b+q_b)^2-4\lvert z_b\rvert^2}$, summed over earlier
records, where $p_b$, $q_b$ are the branch weights and $z_b$ their overlap;
\item the complete continuous-output distances between the finite and the
infinite sharp histories lie in the intervals of Table~\ref{tab:sharp}; at
$d=2$ the ideal 512-step source budget widens them to
$[0.454336,\,0.455973]$ with feedback and $[0.456824,\,0.458461]$ without;
\item every output state supported in the rectangular occupation code of
$d=256$ lies at distance above $0.122$ from the true output of either
prepared history;
\item for the vacuum control, whose two sign probabilities are both $1/2$,
the best approximation by a state supported in the code with $d=2m$ has
distance between $\sqrt{3/(8\pi)}\,m^{-1/4}$ and
$\sqrt{4/(3\pi\sqrt3)}\,m^{-1/4}$, hence of order $d^{-1/4}$, and distance at
most $\varepsilon$ requires $d\ge9/(32\pi^2\varepsilon^4)$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Item (i) is the trace norm of a difference of rank-one operators. For (iii),
the operator exchanging the normalized inside and outside directions of each
branch is a norm-one witness, giving distance at least
$\sum_{a,b}\sqrt{a_{ab}t_{ab}}-\epsilon_{\rm poly}$; half-line Hermite matrix
elements are rational up to $1/\sqrt{2\pi}$ \cite{dlmf}. For (iv) the omitted
vacuum mass per sign branch obeys
$1/(4\pi\sqrt m)\le T_{2m}\le2/(3\pi\sqrt{3m})$ \cite{csREAD}.
\end{proof}

\begin{table}[htbp]
\centering
\small
\caption{Certified complete continuous-output distances between the finite
sharp history and the infinite sharp history, zero native preparation
current, rounded outward \cite{csREADME,rSHARP}.}
\label{tab:sharp}
\begin{tabular}{@{}ccc@{}}
\toprule
Onsite dimension $d$ & Feedback enabled & Feedback disabled\\
\midrule
2 & $[0.455154,\,0.455155]$ & $[0.457642,\,0.457643]$\\
32 & $[0.203713,\,0.226530]$ & $[0.203935,\,0.226752]$\\
64 & $[0.180917,\,0.180924]$ & $[0.181090,\,0.181097]$\\
128 & $[0.152201,\,0.152204]$ & $[0.152341,\,0.152344]$\\
\bottomrule
\end{tabular}
\end{table}

At $d=64$ the pre-readout error is below $0.000002498$, so the sharp detector
dominates the final error: accurate sign probabilities do not certify an
accurate post-measurement state.

\subsection{The smooth Ramsey instrument}
\label{sec:smooth}

A detector variant keeps the preparation, evolution, first record and
feedback and replaces the final sharp read by a source Ramsey pointer with
work-space Kraus maps
\[
M_0=e^{-i\pi/4}\cos\bigl(\tfrac{Q_0}2-\tfrac\pi4\bigr),\qquad
M_1=e^{i\pi/4}\cos\bigl(\tfrac{Q_0}2+\tfrac\pi4\bigr),
\]
and effects $\frac12(I\pm\sin Q_0)$ \cite{csSMOOTH}. Bit zero denotes the plus
effect, a detector meaning different from the sharp sign record, so the
sharp-sign certificates do not carry over.

\begin{theorem}[Smooth instrument]
\label{thm:smooth}
For the declared initial vacuum family, uniformly over the three native
current kicks and both feedback settings, with $T=3/2$ and the constants of
Theorem~\ref{thm:fourth}:
\begin{enumerate}[(i)]
\item the complete recorded instrument, including the pointer interaction and
both unnormalized outcomes, lies within trace distance $\min(1,E_d)$ of its
infinite counterpart, where $A_T=\sqrt{M_0}e^{CT/2}$ and
\[
E_d=\Bigl[\Bigl(e_0+I_TL_d+\frac{A_T(e-1)}{2(d+2)^{3/2}}\Bigr)^2
+\frac{M_0e^{CT+2}}{(d+2)^4}\Bigr]^{1/2}=O(1/d);
\]
\item with $n(d)=d^2$ symmetric steps per interval the combined ideal temporal
and occupation error is $O(1/d)$;
\item on normalized inputs in the form domain of the infinite Hamiltonian the
unconditional increment of the full energy is exactly $1/8$ in the declared
dimensionless convention.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Along the pointer parameter the generator $\frac12Z_{\rm anc}Q_0$ obeys
$dM/ds\le2M$, which gives the pointer term. For (iii), $f_0(q)=\cos(q/2-\pi/4)$
and $f_1(q)=\cos(q/2+\pi/4)$ satisfy $\sum_bf_b^2=1$, $\sum_bf_bf_b'=0$ and
$\sum_b(f_b')^2=\frac14$, so the kinetic term gains exactly
$\frac18\lVert\psi\rVert^2$ \cite{csSMOOTH}.
\end{proof}

The bound is about $0.2063$ at $d=64$, $0.04096$ at $d=256$ and $0.008840$ at
$d=1024$, and vacuous at the executed $d=2$ and $d=4$ \cite{csREADME}; the
energy identity belongs to the infinite oscillator and fails at a finite
cutoff \cite{csSMOOTH,csCALIB}.

\subsection{Continued instruments, compiled native repair and encoded memory}
\label{sec:continued-native}

One execution engine carries the sharp and smooth protocols and the
extension
\[
 U(1),\ P\to r_0,\ F(r_0),\ U(1/2),\ M\to r_1,\
 U(1/2),\ M\to r_2,\ F(r_2),\ U(1/2),\ P\to r_3,
\]
with $P$ the complete gauge instrument, $M$ the smooth instrument and $F$
the gauge feedback, model time $5/2$ and all sixteen unnormalized
work/reference branches retained, with no intermediate work preparation
\cite{csCOMPOSE}. Complete Kraus families preserve every old-prefix effect,
and imported checkpoints are validated by prefix reexecution; this supplies
neither physical quantum serialization nor cloning. Carrying a moment bound
$A$ and an error $e$ from stage to stage, with
$L_H=9/[32(d+2)]+1/[4(d+2)^{3/2}]$ and $L_S=1/[2(d+2)^{3/2}]$, each unitary
interval of length $t$ adds $A(e^{at}-1)L_H/a$ ($a=13/8$), each smooth read
$A(\mathrm e-1)L_S$, and the final tail gives
$E_d=[e^2+A^2/(d+2)^4]^{1/2}$; with $n(d)=d^2$ the complete output distance
is $O(1/d)$ for the scalar vacuum family and vacuous at the executed $d=2,4$;
the classical native attachment enumerates $4{,}224$ joint outcomes at each
executed cutoff. A separate state-specific
certificate retains all sixteen outputs for the blank-gauge and coupled
vacuum preparation and each $j\in\{-1,0,1\}$ \cite{csSMOOTHHISTORY}; the
rank-one trace-norm identity, complete-channel contractivity and directed
rational evaluation at 512 splits per interval give
\[
 \max_{j\in\{-1,0,1\}}D_{d,j}^{\mathrm{ideal\ product},\infty}
 <\begin{cases}0.264092,&d=2,\\0.053956,&d=4.\end{cases}
\]
This bound transfers through the native and protected-memory attachments for
arbitrary admitted native and reference inputs with this independently
prepared work state; it is not an arbitrary work-input bound and excludes
floating execution and physical service errors.

\paragraph{Coherent native attachment.}
Let $L_h^{A,e}$ be the compiled native Kraus operator for law $A\in\{\Cn,\En\}$,
including its fair coin, and $K_r^{j(h)}$ the work branch including the first
record's current-controlled kick. Source routing implements seam $(0,5)$ by
exchanging port labels 1 and 5. With $e(0)=(0,5)$ and $e(1)=(0,1)$,
\[
 M_{h,r,s}=(L_s^{A,e(r_2)}L_h^{A,(0,1)})\otimes K_r^{j(h)},\qquad
 \sum_{h,r,s}M_{h,r,s}^\dagger M_{h,r,s}=I_{78}\otimes I_W,
\]
complete on arbitrary entangled native, work and reference inputs
\cite{csQNATIVE}. The execution keeps all 512 native amplitudes, padding and
abort outcomes, with $295{,}106$ terminal basis outcomes for C and
$820{,}166$ for E. For input $(\ket{\{2,3\}}+e^{i\theta}\ket{\{1,2\}})/\sqrt2$
and independent work, the native interference effect gives
$p_C(\theta)=B+\frac{1+q}{8}\cos\theta$ and $p_E(\theta)=B$, with
$q=\Pr(r_2=0\mid j=0)$; the C phase contrast is about $0.349153278$ at $d=2$
and $0.351351028$ at $d=4$, against zero for E. Cyclic Gray ordering of the
shared nine-qubit controlled-flip macro reduces its source pulse count from
$11{,}782$ to $2{,}044$ while preserving the full 512-dimensional operator,
including global phase, inverse and padding
\cite{pwwelch2014efficient,csQNATIVE,csMEMORY}.

\paragraph{Active encoded memory.}
The code
$V\ket0=(\ket{\{0,2\}}+\ket{\{1,3\}})/\sqrt2$,
$V\ket1=(\ket{\{0,3\}}+\ket{\{1,2\}})/\sqrt2$ has the flagged
Knill--Laflamme identity $V^\dagger L_h^\dagger L_hV=p_hI_2$, $p_h=k_h/4$,
$k_h\in\{0,1,2\}$. A source encoder and record-controlled full-register
unitary $R_h$ realize $R_hL_hV=\sqrt{p_h}V$ \cite{csMEMORY}. Correcting after
each native event, with final seam $(2,3)$ for $r_2=0$ and $(0,1)$ otherwise,
\[
 M^{\rm corrected}_{h,r,s}(V\otimes I_W)
 =\sqrt{p_hp_s^{e(r_2)}}(V\otimes K_r^{j(h)}),
\]
which preserves the logical and reference marginal after summing all work and
record outputs, including initially entangled inputs. The passive control
without recovery has complete plus/minus output distance $1-\bar q$, with
$q_j=\Pr(r_2=0\mid j)$ and $\bar q=(q_{-1}+2q_0+q_1)/4$, so every later
decoder errs by at least $\bar q/2$ on one input, while the corrected
distance is one; for fixed seams $(0,1),(2,3)$ passive Bell fidelity is $1/2$
and corrected fidelity one. Encoding, blank registers, source precision,
storage and record services are supplied. This is active protection of the
stated qubit before readout, not arbitrary-carrier recovery, fault tolerance
or an autonomous observer.

\subsection{Observer readouts and the fixed-reference clock}
\label{sec:readouts}

The readouts below extract observer representations, modular flow and
regional structure from the same executed history \cite{csOBS}. The quantum
algebra, state pairing and access choices are premises; the Born rule is not
derived from native repair.

\begin{proposition}[Observer representations of the executed history]
\label{prop:gns}
Let $v_{ab}$ be the four unnormalized terminal vectors of the history at
$d\in\{2,4\}$ and $\rho_{\rm hist}=\bigoplus_{a,b}\proj{v_{ab}}$ on
$\bigoplus_{a,b}M_{4d^2}$.
\begin{enumerate}[(i)]
\item For a positive functional $\omega$ the
Gelfand-Naimark-Segal (GNS) construction $\langle[A],[B]\rangle=\omega(A^*B)$,
$\pi(X)[A]=[XA]$, $\Omega=[I]$ has dimension $dr$ on a block $M_d$ whose
density has rank $r$. The record and work algebra has GNS dimension
$4(4d^2)$, that is 64 and 256; the four committed record projectors give a
four-dimensional classical GNS space; the site-0 observer that forgets both
records has GNS dimension $d^2$, that is 4 and 16; and the regional maps
$J_R[A]=[{\rm embed}_R(A)]$ are isometric and intertwine the regional
actions.
\item For faithful $\rho_0$ the modular group
$\sigma_s(A)=\rho_0^{is}A\rho_0^{-is}$ moves number observables; the
commutative record algebra has identity flow; states conditioned on the sharp
terminal bit are nonfaithful on the full scalar algebra, with rank at most
$d/2$.
\item At $d=2$, $\rho_0=\frac12(I-\delta X)$ with $\delta=P(b=0)-P(b=1)$ and
modular frequency $\nu=\log(P(b=0)/P(b=1))$. A source-compiled
continuation of every terminal branch by $Z(\pi/4)$, $X(s\nu/2)$ and a
fresh number pointer preserves the old two-bit record marginal and has
readout law $P(1\mid s)=\frac12(m+\delta\sin(\nu s))$, where $m$ is the
total branch mass, equal to one for the normalized history.
\item Scalar regional restrictions are completely positive conditional
expectations, normalized partial trace followed by identity embedding; they
are unital and idempotent and in general not multiplicative. The centered
components of the actual Hamiltonian give an interaction graph that is a
path through scalar site 1, scalar site 0 and the shared charged-matter and
link sector.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
For $\rho=V\diag(\lambda)V^*$ of rank $r$ the map
$[A]\mapsto AV\sqrt{\diag(\lambda)}$ is the GNS isometry \cite{bratteli1}; the
modular identities, including the finite Kubo-Martin-Schwinger (KMS)
condition, are checked with this sign convention \cite{bratteli2}, and at
$d=2$ the probe implements the Schr\"odinger dual of $\sigma_s$ \cite{csOBS}.
\end{proof}

In the executed history $\nu\approx0.5292248981159262$ and
$\delta\approx0.2586046272446501$ \cite{rBRIDGE}; ensemble frequencies read a
modular interval on a declared monotone window, and full-cycle aliases are
indistinguishable. The post-measurement site-0 state taken as its own
modular reference fails to reproduce the interacting dynamics: its modular
flow preserves it while its density changes under $H$ with derivative norms
about $0.2086$ at $d=2$ and $0.6603$ at $d=4$ \cite{csCALIB}. A fixed
reference on the full work algebra removes this obstruction \cite{csTHERM}.

\begin{proposition}[Fixed-reference modular relation]
\label{prop:thermal}
Let $H$ be the finite interacting Hamiltonian on the work space of dimension
$4d^2$, $\beta>0$ and $\sigma_\beta=e^{-\beta H}/\Tr e^{-\beta H}$. Then
\[
\sigma_\beta^{is}A\,\sigma_\beta^{-is}=e^{-i\beta sH}Ae^{i\beta sH}
=\alpha_{-\beta s}(A),\qquad \alpha_t(A)=e^{itH}Ae^{-itH},
\]
so continuous Hamiltonian segments correspond to modular flow at
$t=-\beta s$. For the full matrix algebra this identification of the two
automorphism groups forces the Gibbs form of the reference. With record
blocks $\Sigma_\beta=\bigoplus_rq_r\sigma_\beta$, $q_r>0$, the weights cancel
and every record projector is fixed. The actual state
$\omega(t)=e^{-itH}\omega e^{itH}$ can change while $\sigma_\beta$ is
stationary, and $\Tr(\omega(t)E)=\Tr(\omega\,\sigma_{-t/\beta}(E))$ for every
effect $E$.
\end{proposition}

\begin{proof}[Proof sketch]
$\log\sigma_\beta=-\beta H-\log Z_\beta\,I$, and equality of the generators
requires $[\log\sigma+\beta H,A]=0$ for every matrix $A$, whose commutant is
scalar \cite{csTHERM,connesrovelli1994}.
\end{proof}

The reference, $\beta$ and full-work access are declared, and measurements
and feedback act outside this one-parameter group. The receipt uses
$\beta=1$ and reference weights $(1/10,2/10,3/10,4/10)$, with source
continuations at model times $0$, $1/2$ and $1$ appending a gauge-number
pointer to every post-sharp branch and retaining all eight outcomes
\cite{csTHERM}.

\subsection{History repair of the completed record}
\label{sec:history-repair}

\begin{theorem}[Local repair of the completed history]
\label{thm:history}
Fix $d\in\{2,4\}$, the native current $j$ and the feedback setting. Let twelve
event nodes carry unnormalized work vectors subject to the compatibility
equations $x_{\rm root}={\rm prepared}(j)$, $x_1=U_1x_{\rm root}$,
$x_{(a)}=P_ax_1$, $y_{(a)}=F_ax_{(a)}$, $z_{(a)}=U_2y_{(a)}$ and
$w_{(a,b)}=B_bz_{(a)}$, of depths $0$ to $5$. Then:
\begin{enumerate}[(i)]
\item exactly one assignment satisfies all equations;
\item the common fixed set of the local updates
$x_v\leftarrow A_vx_{{\rm parent}(v)}$ equals that compatibility set;
\item every node-fair schedule reaches the compatible assignment from
arbitrary initial vectors after finitely many updates and stays there; five
covering rounds suffice, a forward topological pass needs eleven updates, and
a reverse order may need 55;
\item with subtree weights $w_v$, equal to $11$, $5$, $4$, $3$ and $1$ at
depths one to five, the potential
$\Phi=\sum_{v\neq{\rm root}}w_v\lVert x_v-A_vx_{{\rm parent}(v)}\rVert$
decreases by at least the repaired residual at every active ideal repair;
\item every deterministic extraction from the completed assignment, including
branch probabilities, terminal densities, GNS representations and modular
parameters, is independent of the schedule.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
Agreement at depth zero forces agreement at each further depth. A node whose
ancestors are correct becomes correct permanently at its next selection, and
after the $k$th covering round every node of depth at most $k$ is correct.
Updating $v$ zeroes its residual and changes each child's residual by at
most $\lVert r_v\rVert$, since all maps are contractions \cite{csHIST}.
\end{proof}

The solved object is the complete branching instrument law for given
apparatus inputs; its uniqueness selects no realized record pair, and the
auxiliary solver is a host computation distinct from a native hardware
process.

\input{tex_fragments/SAME_ACTION_FINITE_INTERFACES.tex}

\section{One repair history: geometry, continuum, clocks and fields}
\label{sec:native-history}
\label{sec:repair-geometry}

This section derives what one native repair history supplies on the carrier
and on the refining spherical source. Its statements are dimensionless
carrier displacements, attempt counts and event counts; physical position,
length scale and propagation speed need their own operational
identification.

\subsection{Native geometry, causal cones and field relations}
\label{sec:history-geometry}

\begin{proposition}[Native geometric readout]
\label{prop:geometry}
\begin{enumerate}[(i)]
\item Exact enumeration of all 4,680 state, seam and coin branches gives the
one-attempt mean $T=I-L/60$ under uniform seam selection.
\item With $Q$ removing the constant mode and $C_n=QT^{2n}Q$, the normalized
response $12C_n/\Tr C_n$ tends to $G=4P_{\rm slow}$, where $P_{\rm slow}$
projects onto the rank-three band $5-\sqrt5$; all eleven nonconstant modes
are present at every finite attempt count.
\item For the readout $x(n)=\sum_pn_pu_p$ with frame directions $u_p$, a
committed current $j$ on the seam $(a,b)$ gives the relative displacement
$x(n')-x(n)=j(u_b-u_a)$, computable from the record without hidden loads.
\item The two-event controller spans two displacement directions. A declared
continuation that sends the final scalar bit to $(0,5)$ or $(0,7)$ after the
first event re-executes all 1,056 second events, and its three displacement
directions $(2,0,0)$, $(1,1-\varphi,\varphi)$, $(1,1-\varphi,-\varphi)$, with
$\varphi=(1+\sqrt5)/2$, have determinant $4$.
\end{enumerate}
\end{proposition}

\begin{proposition}[Native causal cones]
\label{prop:causal}
For a fixed seam schedule and shared coins, the load at a port $v$ after the
schedule is a function of the initial loads and coins in its backward
dependency set. An initial difference supported on $D$ influences $v$ only
along a time-ordered path, so
$\dist(D,v)\le\text{edge crossings on that path}\le\text{elapsed seam events}$,
and at most one edge is crossed per causal layer of disjoint simultaneous
events.
\end{proposition}

Exhaustive checks cover all 78 states for 27 schedule and coin pairs, and two
executed examples show influence at graph distance two arriving only after
two properly ordered seams \cite{csCAUS}. One edge per causal layer is a
native signal speed of at most one seam per ordered repair event; remote
repairs can carry a difference to a receiver before its first local tick
\cite{nativeRecovery}. Units are observer conventions \cite{csCALIB}: if a
common operational propagation law fixed a dimensionless speed
$c_{\rm model}$, choosing the SI speed of light would impose only
$L/T=c_{\rm SI}/c_{\rm model}$ on the model's length and time units, an
operational action constant would add $ET=\hbar_{\rm SI}/\hbar_{\rm model}$,
and one overall scale would stay free. The construction supplies no
$c_{\rm model}$ relating its native distance readout to its modular clock.

\begin{proposition}[Native field relations]
\label{prop:fields}
\begin{enumerate}[(i)]
\item Every event satisfies the discrete continuity law $x'-x=-b_ej$ with
$b_e=\mathbf e_a-\mathbf e_b$.
\item Conditional on the past and the selected seam,
$\Ex[j]=d/2$ and $\operatorname{Var}(j)=(\lvert d\rvert\bmod2)/4$.
\item Under the uniform calibration the mean load obeys
$\mu_{k+1}-\mu_k=-L\mu_k/60$, exactly, by $\mathcal KX=XT$ for the 78-state
kernel $\mathcal K$ and load matrix $X$; the deviation
$V=\frac12\lVert\mu-\frac16\mathbf 1\rVert^2$ decreases strictly unless the
mean is constant, while every binary trajectory keeps
$\lVert x-\frac16\mathbf 1\rVert^2=5/3$.
\item The slow band has mean multiplier $q=(55+\sqrt5)/60\approx0.953934466291663$,
every nonconstant multiplier lies in $(0,1)$, and the second-difference
coefficient $q+1/q-2>0$ has the opposite sign to the coefficient
$-4\sin^2(\omega\Delta t/2)\le0$ of an undamped oscillator; no positive
rescaling of the attempt count turns this closed mean law into a wave law.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch of Propositions~\ref{prop:geometry}
to~\ref{prop:fields}]
The fair coin averages a seam to its endpoint mean, which gives the mean
operator and the conditional current moments; the other two positive response
eigenvalues have strictly smaller magnitude than $1-(5-\sqrt5)/60$, which
gives the limit in Proposition~\ref{prop:geometry}(ii). Conservation gives
the displacement identity. For the causal cones, scanning the schedule
backwards and adding both endpoints of every seam that touches the
dependency set proves the dependency claim by induction
\cite{csSLOW,csCAUS,csFIELDS}.
\end{proof}

The recovered frame also supplies an unbounded three-component \emph{axial
circulation}. With oriented incidence matrix $B$, seam current $J_n$ and
columns $W_{(a,b)}=\hat u_a\times\hat u_b$ of the unit frame vectors, exact
source identities give $WB^T=0$, $WW^T=8I_3$, $\Ex[J_n\mid x_n]=-B^Tx_n/60$
and $\Ex[J_nJ_n^T]=I_{30}/198$ in the uniform binary equilibrium. Hence
$Z_N=\sum_{n<N}WJ_n$ is a record-readable martingale with
$\operatorname{Cov}(Z_N)=(4N/99)I_3$, and
$N^{-1/2}Z_{\lfloor Nt\rfloor}$ converges to three-dimensional Brownian
motion with covariance $(4/99)\min(t,s)I_3$ and Cameron--Martin form
$(99/8)\int|\dot z(t)|^2\,dt$ in a rescaled attempt count. A three-event face
loop restores the carrier state while adding a nonzero counter increment.
The counter's memory grows with the observation horizon; it supplies neither
a radial field nor a physical trajectory \cite{nativeContinuum}.

\subsection{The common continuum operator}
\label{sec:continuum}

On the declared spherical source family, the native cell Green operators
converge in norm along one common subsequence to a positive compact
injective operator $G$ on the mean-zero observation space. Its inverse
$A=G^{-1}$ is densely defined and self-adjoint, and restoring the constant
zero mode gives a conservative strongly local regular Dirichlet form with
domain $H^1$ on the supplied flat-polyhedron comparison surface, energy
bounded above and below by positive multiples of the gradient energy, and
heat-trace spectral dimension two. A boundary-action test excludes a
boundedly invertible $L^2$ energy identification of $A$ with the static
boundary action of a smooth compact three-dimensional Dirichlet bulk with
smooth boundary, whose Steklov eigenvalues grow like $\sqrt n$;
bounded-energy internal rank-three modes vanish at this scaling
\cite{nativeContinuum}. An exact terminal reduction gives local dual-edge
conductances in $[5/8,15/19]$ with Green-operator error at most $182\,2^{-l}$
relative to native cell means, so this weighted graph has the same continuum
subsequences.

A positive native lag readout realizes the Poisson extension. For
$K_l=I-L_l/(2M_l)$, Sibuya jump probabilities
$p_n=\binom{2n}{n}/[4^n(2n-1)]$ and a Poisson number of jumps with mean $R$
give $\Ex K_l^N=e^{-R\sqrt{I-K_l}}$. Restricting jumps to $b$ and their count
to $j$, with overflow retained, has maximum native lag $bj$ and operator
error at most $\min(1,R\binom{2b}{b}4^{-b}+R^{j+1}/(j+1)!)$; at $R=1/2$,
$b=64$, $j=8$ the full 66-state source gives $0.034581$ within the bound
$0.036$. Taking $R_l=r\sqrt{2M_l4^l}$ yields the compressed limit
$e^{-r\sqrt A}$; here $r$ labels a positive mixture of native lags, not a
physical radius.

Full native density correlations converge to $e^{-\tau A}$, and the admitted
coherent continuation converges strongly to $\cos(t\sqrt A)$. The full-space
extension $\bar A=A\oplus0$ satisfies
$\|\mathbf1_F e^{-\tau\bar A}\mathbf1_E\|\le e^{-d(E,F)^2/(2\tau)}$, and its
cosine propagator vanishes between these supports when $|t|<\sqrt2\,d(E,F)$
in the declared flat comparison distance. Two consecutive long observation
blocks of one history supply a continuous classical Gaussian wave readout
with covariance $G\cos((t-u)\sqrt A)$ in $C([0,T],H^{-2}(S^2))$, with
observation count taken to infinity before spatial refinement
\cite{nativeContinuum}. A declared static filter $A_l^{1/4}$ changes it to
$A_l^{-1/2}\cos((t-u)\sqrt{A_l})$, twice the real part of the free-field
vacuum covariance; the stationary increments obey
$2A_l^{-1/2}(I-\cos(\Delta t\sqrt{A_l}))\le(\Delta t)^2\sqrt{A_l}$ and, with
$\sum_m\langle h_{jm,l},\sqrt{A_l}h_{jm,l}\rangle\le12j^2$ for centered cell
averages of spherical harmonics, the geometric $H^{-s}$ increment moment is
at most $12[1+1/(2s-3)](\Delta t)^2$ for $s>3/2$, giving convergence in
$C([0,T],H^{-s}(S^2))$. The wave readout is declared classical processing,
distinct from passive density waiting and from a quantum vacuum.

\subsection{Native density histories and their limits}
\label{sec:density}

At level $l$, let $V_l=240\,4^l$, $q_l=V_l/3$ and
$\chi_l=q_l(V_l-q_l)/[V_l(V_l-1)]$. In the uniform canonical equilibrium,
with independent uniform seams and fair repair, the actual snapshot readout
of a mean-zero test function repeated on its twelve native ports,
\[
 Y_l(\tau;f)=\frac{\sum_p f_l(p)
 [n_{\lfloor1260\,16^l\tau\rfloor}(p)-1/3]}{\sqrt{V_l\chi_l}},
\]
converges in joint distributions at finitely many times as space and the
native attempt count are refined together. The limiting family is stationary
and Gaussian, with covariance $\langle f,e^{-|\tau-u|A}g\rangle$.
Representing each unread repair as a fair
identity-or-swap, the conditional profile overlaps concentrate with variance
at most $4\|g\|_\infty^2\|f\|_{\ell^2}^2/V_l^2$ uniformly in attempt count,
and a conditional finite-population central limit theorem applies. The
declared filter $G_l^{1/2}$ gives covariance $Ge^{-|\tau-u|A}$. The law is
reflection positive with one-particle semigroup $e^{-\tau A}$, generator $A$
and not $\sqrt A$. The interpolated density converges in
$C([0,T],H^{-3}(S^2))$ and its Green-filtered readout in
$C([0,T],H^{-2}(S^2))$, by a forward--backward martingale decomposition with
$g=2(I+P_l)^{-1}f$ and harmonic tail bounds; the step paths have the same
Skorohod limits \cite{nativeContinuum}. For $\phi=\Phi/\sqrt2$ a mode of
eigenvalue $\lambda$ has covariance $e^{-\lambda|\tau-u|}/(2\lambda)$, so the
Gaussian precision is the Euclidean form
\[
 S_E(\phi)=\frac12\int_{\mathbb R}
       \bigl(\|\partial_\tau\phi\|^2+\|A\phi\|^2\bigr)\,d\tau,
\]
with reconstructed Hamiltonian $d\Gamma(A)$ and spatial precision $A^2$. The
action follows from the native readout law and has diffusive dispersion.

Two restrictions bar identifying this sector with a relativistic field.
On the shorter attempt scale $b_l=1260\,8^l$ both the density and
Green-filtered paths freeze to their initial field, with expected squared
supremum increments at most $50(T2^{-l}+a_l^{-1})$, $a_l=1260\,16^l$. For a
complete linear readout with bounded injective dense-range composite map
$B=TC^{1/2}$, $C=I$ or $G$, the left polar decomposition $B=D_0^{1/2}U$ gives
\[
 D_0^{-1/2}D_\tau D_0^{-1/2}=Ue^{-\tau A}U^*,
\]
so normalization preserves the ordered decay rates $\lambda_n(A)\asymp n$.
No common positive clock factor turns these into the $n^{1/3}$ Euclidean
decay rates of a complete stationary free scalar on a smooth compact
three-dimensional manifold without boundary, and the same argument excludes
every spatial dimension $d\ge2$, whose rates grow like $n^{1/d}$; a
continuous nonconstant time change preserving the semigroup is a scalar
factor. The excluded object is the complete linear Gaussian identification;
restricted boundary observations, nonlinear dictionaries and interacting
fields need their own analysis, and a causal microscopic theory may have
this diffusive sector as an effective description \cite{nativeContinuum}.

\subsection{Incident clocks and active waves}
\label{sec:incident-clocks}

Let $a_l=1260\,16^l$, $M_l=630\,4^l$ and let a fixed event set contain
$d_l>0$ seams. Its incident counter $N_l(n)$ includes zero-current repairs,
with $p_l=d_l/M_l$; write $T_l(k)$ for the attempt of its $k$th tick. For
$n_*=\lceil a_l(T+\delta)\rceil$,
\[
 \Pr\!\left[\max_{n\le n_*}|N_l(n)-np_l|>a_lp_l\delta\right]
 \le \frac{n_*(1-p_l)}{a_l^2p_l\delta^2},
\]
and on the complementary event the normalized count
$C_l(t)=N_l(\lfloor a_lt\rfloor)/(a_lp_l)$ and the inverse clock
$U_l(t)=T_l(\lfloor a_lp_lt\rfloor)/a_l$ differ from $t$ by at most
$\delta+1/a_l$ and $\delta+1/(a_lp_l)$ on $[0,T]$. A primitive port has
$d_l=5$ or $6$ and a whole carrier $d_l=33$. The density path theorem
transfers to these local count parametrizations without independence
between clock marks and fields \cite{nativeClockLimit}. Passive port access
chosen before each seam draw from earlier records also works: integer
credits $R_n=\sum_{j\le n}(30/d_j)I_j$ have conditional mean $30/M_l$ per
attempt, deviation probability at most $n_*(M_l-5)/(5a_l^2\delta^2)$, and
inverse clock error at most $\delta+M_l/(5a_l)$.

For fixed incidence counts $d_i$ and shared-seam counts $c_{ij}$, centered
clock counts divided by $\sqrt{a_l/M_l}$ converge to Gaussian increments with
covariance $\min(t,u)c_{ij}$, asymptotically independent of the density
snapshots. Adjacent degree-five port clocks have limiting correlation $1/5$,
the endpoints of an external seam $1/6$, and nonadjacent clocks zero. These
are native model predictions under the stated schedule and preparation,
without an atomic-clock or proper-time dictionary \cite{nativeClockLimit}.

The admitted coherent walk can use the same tick. A protected wave register
receives one fixed step $W_l$ per incident event at a degree-$d$ port,
including zero current, while the timing source pauses during coherent
services. With $Q_l=630\,16^l$, $D_l=4^lL_l$, $\beta=2d/\sqrt{630}$,
$\eta=\delta+1/a_l$ and $e_l=\Lambda/[6(4Q_l-\Lambda)]$, on the band
$0\le D_l\le\Lambda<4Q_l$ and the count good event,
\[
 \sup_{\tau\le T}\left\|
 \cos\!\left(N_l(\lfloor a_l\tau\rfloor)\arccos K_l\right)
       -\cos(\beta\tau\sqrt{D_l})\right\|
 \le\beta\sqrt\Lambda\,[\eta+(T+\eta)e_l],
\]
so the compressed wave converges to $\cos(\beta\tau\sqrt A)$. Step error
$\epsilon_l$ accumulates at most $N_{\max}\epsilon_l$ with
$N_{\max}=O(\sqrt{Q_l})$, so $\sqrt{Q_l}\epsilon_l\to0$ suffices. An exact
control retains 237,600 two-attempt clock histories; the continuing
Ramsey-plus probability is $1297/1320$ against $7901/7920$ when
zero-current triggers are dropped. The wave step is compiled apparatus;
passive repair does not supply it \cite{nativeClockLimit}.

\subsection{An encoded quantum field and its modular clock}
\label{sec:encoded-field}

The classical covariance does not determine commutators. $N$ independent
proper-code registers in $\ket0$ give $Q_N=\sum_jX_j/\sqrt{2N}$ and
$P_N=\sum_jY_j/\sqrt{2N}$, with commutator $i\sum_jZ_j/N$,
$\Ex Q_NP_N=i/2$ and $\Ex Q_N^4=3/4-1/(2N)$. For Weyl factors
$W_N(v)=e^{i(sQ_N+tP_N)}$ the quantum central limit theorem \cite{jppIID}
gives
\[
 \lim_{N\to\infty}\Ex\prod_jW_N(v_j)
 =\exp\!\left[-\frac14\left|\sum_jv_j\right|^2
       -\frac i2\sum_{j<k}\sigma(v_j,v_k)\right],
 \qquad \sigma((s,t),(s',t'))=st'-ts',
\]
the CCR vacuum with its ordered-product phase. Source pulses execute four
codes and one Ramsey pointer with all 32 outcomes retained; the pointer
quadratures give $0.7452257133-0.3013609949i$ for the declared three-factor
word, reversal conjugates the phase and the tracial preparation removes it.
The native density bands fix the mode rotation $e^{i\theta_lZ/2}$ per
incident trigger, $\theta_l=\arccos(1-\lambda/(2Q_l))$, with encoded
frequency $\Omega_l=\sqrt{Q_l}\theta_l\to\sqrt\lambda$, ordered covariance
$e^{-i\Omega_l(t-u)}/(2\Omega_l)$ and kick response
$\sin(\Omega_l(t-u))/\Omega_l$, executed for both kick signs at three native
bands \cite{nativeContinuum}. Polarized preparation, mode encoding, storage,
pulses and readout are specified services. The sine kernel
$A^{-1/2}\sin(t\sqrt A)$ vanishes outside the comparison cone for mean-zero
tests; unrestricted smearings add $tP_{\rm const}$, and restoring the compact
massless zero mode gives a free particle with no normalizable ground state,
so the mean-zero field is no complete local net.

For $\rho_r=(I+rZ)/2$, $0<r<1$, and $H=\Omega(I-Z)/2$, the finite Gibbs and
Kubo--Martin--Schwinger identity gives $\sigma_s=\alpha_{-bs}$ for every real
$s$ exactly when $\log\frac{1+r}{1-r}=b\Omega$, that is
$r=\tanh(b\Omega/2)$; equal polarization at unequal frequencies fails a
common $b$, the pure endpoint is not faithful and the tracial endpoint has
trivial flow. With $Q_N=\sum X_j/\sqrt{2Nr}$ and $P_N=\sum Y_j/\sqrt{2Nr}$
the ordered copy limit has thermal covariance
$C(\delta)=[\cos(\Omega\delta)-ir\sin(\Omega\delta)]/(2r\Omega)$ with
$C(t-ib)=C(-t)$, executed on two codes with retained purifying references.
The native bound $\lambda_n(A)\ge n/49920$ gives $\Tr e^{-b\sqrt A}\le99840/b^2$,
so the mean-zero Fock space has the faithful normal state
$Z_b^{-1}e^{-b\,d\Gamma(\sqrt A)}$ with the same modular flow, and under
$t=\kappa\tau$, $\kappa=2d/\sqrt{630}$, its modular parameter is
$s=-\kappa\tau/b$. The common $b$ is a measurable preparation condition,
not a repair-selected temperature.

\subsection{The product field, its clock and its Lorentzian density}
\label{sec:product-field}

Independent long blocks of one native history supply radial innovations.
Their mean-zero covariance is $B_l=G_l-P/(2520\,16^l)$, and a second block
adds a normalized constant-direction readout, giving $B_l+P_{\rm const}$. On
a declared interval $(0,L)$ with Dirichlet ends, a mode with $h=\sqrt\lambda$
has covariance
\[
 C_\lambda(r,s)=
 \frac{\sinh(h\min(r,s))\sinh(h(L-\max(r,s)))}{h\sinh(hL)},
 \qquad C_0(r,s)=\frac{\min(r,s)(L-\max(r,s))}{L},
\]
realized by applying $Q_{rs}^{1/2}(B_l+P_{\rm const})^{-1/2}$ to successive
blocks, with $Q_{rs}$ the conditional innovation. The whitening means the
radial direction carries no native information beyond the spectrum of $A$.
Complete 96-event current tapes check the finite decoder. The spatial
subsequence gives the product operator $\mathcal B=-\partial_r^2+\bar A$ with
action $\frac12\int_0^L(\|\partial_r\phi\|^2+\mathcal E(\phi))\,dr$; for
$L=\pi$ its count satisfies
$\Lambda^{3/2}/(12\sqrt2)\le N_{\mathcal B}(\Lambda)\le49921\Lambda^{3/2}$
for $\Lambda\ge3$, and its tensor form gives comparison wave speed one in
$d_{\rm product}^2=|r-s|^2+2d_{\rm screen}^2$. On a radial grid the sampled
bridge has an explicit sine-mode covariance, and a spectral filter realizes
the thermal covariance $\coth(b\sqrt{\mathcal B}/2)/(2\sqrt{\mathcal B})$
from the same records; the full-port generator and the cell Schur generator
have compressed resolvents within $1/(8\,4^l)$ at each positive shift, so
both give the same product target. Radial labels, filters, boundaries and
block access are specified operations.

For a full-port angular mode set $a_l=\sqrt{Q_l}\arccos(1-\lambda/(2Q_l))$
and $c=\sqrt{\rho_{k,h}}$. Each incident trigger applies the source $Z$ pulse
of angle $a_l/(2\sqrt{Q_l})$ and the radial $X$ pulse of angle
$c/(2\sqrt{Q_l})$; anticommutation gives $\omega_{l,k}=\sqrt{a_l^2+c^2}$
with no fitted product frequency, and the split word differs from the joint
Hamiltonian by at most $na_lc/(4Q_l)$ after $n$ triggers. The faithful
reference is polarized along $(cX+a_lZ)/\omega_{l,k}$ with
$r=\tanh(b\omega_{l,k}/2)$; ordered two-time measurements retain all 32
outputs. A deterministic packet $v\propto(12I-L)^{1024}(\mathbf1_{\mathrm{cell}\,0}
-\mathbf1_{\mathrm{cell}\,1})$ on the 240-port carrier has Rayleigh quotient
$\lambda\simeq0.044442251696$, residual $\|(L-\lambda)v\|<0.000304$ and
squared cell compression above $0.996458773594$; Householder encoders and a
Bell reference read it after incident-triggered phase, with 25-trigger
Chebyshev discrepancy about $0.000149026$ at $M=630$, at a cost of 1,747,068
source pulses. Copy and spectral limits followed by radial refinement give
\[
 C_b(t)=\frac{\coth(b\sqrt{\mathcal B}/2)\cos(t\sqrt{\mathcal B})
                     -i\sin(t\sqrt{\mathcal B})}{2\sqrt{\mathcal B}},
 \qquad
 D(t)=\frac{\sin(t\sqrt{\mathcal B})}{\sqrt{\mathcal B}},
 \qquad \sigma_s=\alpha_{-bs},
\]
and with $C_L=L(49920+L^2/9)/3$, $\Tr e^{-b\sqrt{\mathcal B}}\le6C_L/b^3$,
so the encoded Fock algebra has the faithful normal state
$Z_b^{-1}e^{-b\,d\Gamma(\sqrt{\mathcal B})}$ with $s=-\kappa\tau/b$. Vanishing
count tolerances require $4^l\eta_l^2\to\infty$ and complete-step errors
$\sqrt{Q_l}\epsilon_l\to0$. Radial design, common $b$, preparation, mode
access and source precision are declared services \cite{nativeContinuum}.

\begin{figure}[!htb]
\centering
\input{tex_fragments/figures/FIG_FIELD_CHAIN.tex}
\caption{What the effective $(1+3)$ scalar description derives from native records and what it declares. Blue: derived from one native repair history by a limit theorem. Grey: constructed from derived and declared parts. Orange: declared or compiled inputs. Red: the proved obstruction that rules out a wave sector from passive native repair. The smoothing theorem is an existence result for fixed finite readout menus.}
\label{fig:field-chain}
\end{figure}

\paragraph{Lorentzian density.}
The regular strongly local angular form with its $H^1$ domain admits a
measurable uniformly elliptic tensor in each chart
\cite[Theorem 7.2 and Corollary 7.3]{pwcipriani2006dirichlet}, recoverable
from complete native records: with the disclosed total-one kernel
$K_l=I-L_l/(2M_l)$, including every zero-current hold, the ordered limits
\[
 \lim_{t\downarrow0}\lim_l\int\phi\,
 \frac{\bar P_l(t)(fg)-f\bar P_l(t)g-g\bar P_l(t)f+fg}{2t}\,d\mu
 =\int\phi\,a^{ij}\partial_i f\partial_j g\,d\mu
\]
recover weighted tensor averages. In barycentric face coordinates the
one-attempt increment tensor obeys
$\tfrac37a_{\rm bare}\le a_{\rm screen}\le\tfrac{15}{19}a_{\rm bare}$ with
$a_{\rm bare}=\begin{pmatrix}1/36&-1/72\\-1/72&1/36\end{pmatrix}$, so the
one-attempt-first limit reads a different tensor. The radial product adds
$a^{rr}=1$. With coordinate measure $\mu(x)\,d^3x$ the chosen wave action
has coefficient density $C=\mu\diag(-1,a)$, and
\[
 g=\sqrt{-\det C}\,C^{-1}
   =\mu\sqrt{\det a}\,\diag(-1,a^{-1}),
 \qquad \sqrt{-\det g}\,g^{-1}=C
\]
has signature $(-,+,+,+)$ almost everywhere, with weak wave equation
$\partial_\alpha(C^{\alpha\beta}\partial_\beta\phi)=0$
\cite[Equations (23)--(26)]{pwvisser1998acoustic}. The signature is that of
the chosen action; the derived content is the measurable tensor $a$. The
native measure is the radial pushforward of flat face area, with density
$\rho_F(u)=c/(n_F\cdot u)^3$, $c=\pi r^2/(5a_0)$, on a face of unit normal
$n_F$ and plane distance $r$. Adjacent faces agree on their common edge but
their spherical gradients differ by $-3c(n_F-n_G)/(n_F\cdot u)^4\ne0$, so the
density is not $C^1$, and since $C^{tt}=-\mu$ the exact dictionary in this
atlas fails the smooth-density premise. An alternative smooth dictionary
would need a separate argument.

The density is globally Lipschitz relative to smooth sphere volume. Smoothing
only the angular variables and the tensor, with the unitaries
$U_\varepsilon f=\sqrt{\rho_\varepsilon/\rho}\,f$ between the changing $L^2$
spaces, gives transported Mosco convergence and then norm-resolvent
convergence with gap $(\pi/L)^2$. Thermal covariance and sine response
converge in operator norm, and ordered Weyl laws, finite adaptive Ramsey
histories and finite detector outputs converge. Hence for any fixed finite
admissible menu on bounded wave times and any positive tolerance one smooth
static $1+3$ metric approximates the joint readouts. Both sides are free
fields on the same supplied codes; the theorem removes the roughness of the
native coefficients, not the declared radial product, clock or state, and
gives no computable rate. In the smooth comparison $g=N^2\diag(-1,a^{-1})$,
$N^2=\mu\sqrt{\det a}$, the canonical momentum is $\mu\partial_t\phi$ and
the thermal covariance and response have the standard static CCR/KMS
normalization with a reflecting radial boundary.

Angular observations do not select the radial law. For the 240-port $L$,
three 720-port products $B_R=I_3\otimes L+R\otimes I$ with the unit path,
fourfold path and unit triangle Laplacians satisfy $J^*F(B_R)J=F(L)$ for the
radial average $J$ and every spectral function, while a radial-contrast probe
has cosine second derivatives $-5/2,-10,-3$.

For positive finite-measure regions $E,F$ of the product, let
$d(E,F)=\sup_u(\mathop{\mathrm{ess\,inf}}_F u-\mathop{\mathrm{ess\,sup}}_E u)$
over bounded local-form functions with $\Gamma_{\mathcal B}(u)\le1$.
Davies--Gaffney estimates and the short-time theorem of Ariyoshi and Hino
\cite[Theorem 2.7]{pwariyoshi2005smalltime},
$\lim_{\tau\downarrow0}-4\tau\log\langle\mathbf1_F,e^{-\tau\mathcal B}\mathbf1_E\rangle=d(E,F)^2$,
combined with heat-cosine transmutation give
\[
 \inf\{t>0:\|\mathbf1_F\cos(t\sqrt{\mathcal B})\mathbf1_E\|>0\}
 =\inf\{t>0:\|\mathbf1_F\mathcal B^{-1/2}
       \sin(t\sqrt{\mathcal B})\mathbf1_E\|>0\}=d(E,F).
\]
This is the sharp finite-propagation front of the chosen wave equation,
fixed by its intrinsic metric; it supplies no independent propagation speed,
and same-field modular time has onset $|s|=d/b$.

\subsection{Interacting readouts on the product}
\label{sec:product-interacting}

Retain the angular constant and radial profiles $\sqrt2\sin r$ and
$\sqrt2\sin(2r)$ on $(0,\pi)$. The projected local quartic polynomial is
\[
 W=\tfrac32(q_1^4+q_2^4)+6q_1^2q_2^2,\qquad
 H=\tfrac12(p_1^2+q_1^2+p_2^2+4q_2^2)+\frac{g_{\rm eff}}{24}W,
\]
with $g_{\rm eff}=(\lambda/\pi)\int\mu_{\rm screen}^2\sqrt{\det a_{\rm screen}}\,dx$
for a constant covariant coupling; $g_{\rm eff}=1$ is an input. At occupation
cutoff $4\times4$, native source pulses prepare the purification of
$e^{-\beta H}/\Tr e^{-\beta H}$ and execute two reads with feedback,
retaining all work, reference and record outputs. For $\beta=1/4$, intervals
$1/16$ and 64 split steps, the complete-output distance from an independent
exponential is below $1.97\times10^{-8}$; deleting the interaction or
feedback changes the record law by about $7.31\times10^{-4}$ or
$1.95\times10^{-3}$. The worst branch uses 217,110 source pulses, 434,220
load/export events and two native repairs. A separate protocol advances this
Hamiltonian on every incident event of two native timing attempts,
retaining all 3,600 tapes and 4,900 field-prefix outputs, with timeout
probability $35/36$ and complete error below $3.44\times10^{-9}$. On the
refined clock family, incident rate $R_l=2d4^l$ gives a controlled
inverse-count comparison; each metric readout uses its own clock pullback,
and no universal physical conversion is inferred.

For $g\ge0$ the positive quartic form has a Friedrichs Hamiltonian
$H\ge H_{\rm harm}$, Hermite compressions form a core, and
\[
 Z_d\longrightarrow Z,\qquad
 0<Z\le\frac{e^{-3\beta/2}}{(1-e^{-\beta})(1-e^{-2\beta})},
 \qquad \rho_d\longrightarrow\rho\quad\hbox{in trace norm},
\]
so every fixed finite complete adaptive history converges and the limiting
Gibbs state has $\sigma_s=\alpha_{-\beta s}$. This proves occupation existence
for the two-mode theory, without a small-code certificate: the complete
$d=4$ versus $d=8$ output distance is about $0.56045$, and $d=8$ versus
$d=12$ about $0.24501$. An inverse control recovers
$(g_{\rm eff},\beta,c)=(1,1/4,1)$ within $10^{-4}$ from six noiseless source
probabilities and predicts a withheld interval within $10^{-8}$, with local
Jacobian singular values about $0.264713,0.001489,0.000110$. For fixed
covariant $\lambda$ the smoothed metrics give convergent quartic projections.
The quartic excites omitted modes; no invariant two-mode band, interacting
ultraviolet limit or finite-cutoff microcausality follows
\cite{mCONSTRUCTION}.

\paragraph{The reference-state boundary.}
Increasing native-return caps match modular phase magnitudes to the cosine
limit, and a fixed cap freezes the low-band phase. On the full configuration
algebra the uncapped reference approaches normalized trace with half-trace
error at most $3/(8\,2^l)$, its density-wave sector has vanishing weight, and
fixed-parameter modular evolution becomes trivial in normalized-trace $L^2$.
If the relevant fluctuation moments converge, any limiting scalar canonical
commutator in this bounded-overlap regime is zero. Refinement-rescaled time,
other states and other fluctuation regimes are not excluded
\cite{nativeContinuum}.

\input{tex_fragments/COMMON_MATTER_METRIC_AND_RESPONSE.tex}

\section{Prediction protocols}
\label{sec:prediction}

\paragraph{A response without a fitted coefficient.}
In the stationary binary number sector, disclose local occupations and
apply one extra fair native repair on seam $(a,b)$ exactly when $n_a=1$.
Detailed balance gives, for any fixed future record event $F$ whose
continuation does not consult the earlier pulse mark,
\[
 \Pr_{\rm feedback}(F)-\Pr_{\rm passive}(F)
 =\tfrac12\Ex_{\rm passive}[(n_b-n_a)\mathbf1_F].
\]
Complete capped clock and field forecasts and the native spatial Green
response satisfy this identity exactly, with every trigger, coin and current
label retained \cite{nativeContinuum}. No matched public intervention dataset
exists for these forecasts.

\paragraph{A controlled continuum response.}
For a disclosed port profile $h$, a uniform seam $(a,b)$ executes its
unchanged fair repair with probability
$[1+\epsilon(h_b-h_a)(n_a-n_b)]/2$, and the clock counts executed repairs,
including zero currents. The exact linear occupation response is
$2\chi K(I-K^r)h$, and on the refining source
\[
 \left.\partial_\epsilon\Ex\eta_l(g)\right|_0
 \longrightarrow 2\langle g,(I-e^{-\tau A})f\rangle.
\]
At $\tau=1$ with four selection slots per target repair, stopped likelihood
bounds control the nonlinear mean error by $36\epsilon^2\|g\|$, and for a
unit-axis flat coordinate at $\epsilon=10^{-6}$,
$\liminf_l\Ex_\epsilon\eta_l(f)\ge\epsilon\,9/32792-36\epsilon^2>0$
\cite{nativeContinuum}. In the selection-slot clock the derivative converges
to $\langle g,R_\tau f\rangle$ with $R_\tau=2(I-e^{-\tau\bar A})$, and positivity
improvement of the heat semigroup gives, for nonzero nonnegative Lipschitz
profiles at distance $r>0$,
\[
 0<-\langle g,R_\tau f\rangle
   \le2e^{-r^2/(2\tau)}\|f\|\|g\|,\qquad \tau>0.
\]
This derivative has no exact finite support cone. Reading its control,
support and slot parameter as physical local interventions and time would
contradict an exact relativistic cone; that identification is not derived,
and a causal microscopic theory with diffusive effective observables is
consistent with it.

\subsection{The phase experiment}
\label{sec:ce-protocol}

The classical repair law fixes the diagonal of every admissible record law
and none of its coherences (Theorem~\ref{thm:record-laws}). A fixed
interference experiment reads the multiplier $g$ that separates the members,
and its count protocol, comparing the executed laws $\Cn$ and $\En$, is fixed
before any data \cite{prREADME,prDERIV}. All members are ordinary quantum
instruments, so the test separates record laws and makes no claim to
separate OPH from quantum mechanics.

Write $x=\{2,3\}$, $z=\{1,2\}$ and $w=\{0,2\}$. Prepare
$\ket{\phi_\theta}=(\ket x+e^{i\theta}\ket z)/\sqrt2$ for
$\theta\in\{0,\pi,\pi/2\}$, or the incoherent mixture
$\frac12(\proj x+\proj z)$, apply one repair on the seam $(0,1)$, retain its
record, and measure $P_+=\proj{\phi_0}$ and $P_-=I-P_+$ inside each committed
record block.

\begin{proposition}[Phase law of the two record instruments]
\label{prop:ce}
For the preparations above:
\begin{enumerate}[(i)]
\item the nonzero recorded source densities of $\Cn$ are
$\frac12\proj{\phi_\theta}$ for $(c,j)=(0,0)$, $\frac14\proj x$ for $(1,0)$
and $\frac14\proj w$ for $(1,-1)$; those of $\En$ are $\frac14\proj x$ for
$(c,d)=(0,0)$ and $(1,0)$, $\frac14\proj z$ for $(0,-1)$ and $\frac14\proj w$
for $(1,-1)$;
\item the plus probabilities are $p_\Cn(\theta)=\frac38+\frac14\cos\theta$
and $p_\En(\theta)=\frac38$, and the incoherent preparation gives $3/8$ for
both; the phase-reversal contrast is $1/2$ for $\Cn$ and zero for $\En$;
\item all four preparations have identical classical load probabilities,
the readout preserves the probability of every old record, and no record is
erased and no outcome postselected.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
On the seam $(0,1)$ the configuration $x$ has contrast zero for both coins,
and $z$ has contrast $-1$, current $0$ for coin zero and current $-1$ for
coin one, which moves it to $w$. The coherent sum survives only in the
$\Cn$ branch $(0,0)$, where both currents vanish; $\En$ separates $x$ from $z$
by their contrasts \cite{prDERIV}.
\end{proof}

\begin{table}[htbp]
\centering
\small
\caption{Plus probabilities of the four fixed settings \cite{prREADME,mRECORDLAWS}.}
\label{tab:ce}
\begin{tabular}{@{}lcccc@{}}
\toprule
Preparation & $\Czero$ & $\Cn$ & $\En$ & $\Tn$\\
\midrule
Phase $0$ & $\frac38+\frac{\sqrt2}{4}$ & $5/8$ & $3/8$ & $3/8$\\
Phase $\pi$ & $\frac38-\frac{\sqrt2}{4}$ & $1/8$ & $3/8$ & $3/8$\\
Phase $\pi/2$ & $3/8$ & $3/8$ & $3/8$ & $3/8$\\
Incoherent mixture of $x$ and $z$ & $3/8$ & $3/8$ & $3/8$ & $3/8$\\
\bottomrule
\end{tabular}
\end{table}

For every admissible record law the plus probability is
$p_\Phi(\theta)=\tfrac38+\tfrac12\operatorname{Re}(g\,e^{-i\theta})$, since the
empty seam of $x$ stays in place and $z$ waits with probability $\frac12$
while keeping coherence $g$ with $x$. The settings $0$, $\pi$ and $\pi/2$
therefore estimate $\operatorname{Re}g$ and $\operatorname{Im}g$ for whichever
law is realized; laws with equal $g$, such as $\En$ and $\Tn$, are separated
by the same construction with the full seam $\{0,1\}$ in place of $z$, which
gives $\frac12+\frac12\operatorname{Re}(k\,e^{-i\theta})$ \cite{mRECORDLAWS}.

The count protocol fixes four settings with 8,192 fresh independent attempted
trials each and no optional stopping. Each attempt falls in one of twelve
common bins of coin, current and outcome, or aborts. At each setting it
scores the twelve bin indicators, the plus indicator and the abort indicator,
$m=4(12+2)=56$ statistics, with sampling half-width $b=1/32$, independently
established apparatus probability budget $\eta=1/64$ and familywise sampling
requirement $\delta=0.001$.

\begin{proposition}[Sampling certificate]
\label{prop:sampling}
If the attempted trials of each setting are independent, then with $n=8192$
and $b=1/32$ the probability that any of the 56 observation intervals
$[\hat p-b,\hat p+b]$ misses its actual probability is at most
$112e^{-16}<112/S_{32}<0.000012606<\delta$, where
$S_{32}=\sum_{k=0}^{32}16^k/k!$.
\end{proposition}

\begin{proof}
Hoeffding's inequality \cite{hoeffding1963} gives
$2e^{-2nb^2}=2e^{-16}$ for each statistic, a union bound over 56 statistics
gives $112e^{-16}$, and $S_{32}<e^{16}$; the final comparison is exact
rational arithmetic \cite{prDERIV}.
\end{proof}

A candidate is rejected when some observation interval is disjoint from its
ideal probability widened by $\eta$. Under the apparatus contract, rejection
occurs only when an observation interval fails, and both candidates use the
same intervals; if the calibration guarantee fails with probability at most
$\beta$, the combined bound is $112/S_{32}+\beta$. An abort has ideal
probability zero, and a resolved excess returns the verdict that the
apparatus budget is exceeded. Without supplied counts the verdict is ``no
data'', and software demonstrations are labelled synthetic \cite{prREADME}.
One survivor discriminates the conditional instruments, two survivors do
not separate them, and two rejections mean neither specified apparatus model
explains the counts; in every case the same counts estimate $g$.

\subsection{The source-executed implementation}
\label{sec:ce-source}

Both complete recorded instruments, the four preparations and the readout
also run as emitted twelve-port control words \cite{sdREADME,sdDERIV}. A
configuration is written as the seam pair and the ten spectators: the six
endpoint pairs use three bits and the spectators six, placing the 78
configurations injectively in a nine-qubit register with 434 unused
addresses. A local permutation with blank record bits realizes each repair
from phase-exact transpositions, and the fair coin is prepared by a source
pulse and recorded. The encoding is a mathematical isometry, not a physical
interface from occupation states.

\begin{proposition}[Exact source realization of the two instruments]
\label{prop:ce-source}
In the nine-qubit encoding $J$:
\begin{enumerate}[(i)]
\item the encoded recorded channels intertwine the native ones,
\[
\mathcal E_{\rm code}(J\rho J^*)=(I_{\rm record}\otimes J)\,
\mathcal E_{\rm native}(\rho)\,(I_{\rm record}\otimes J^*),
\]
on all 6,084 matrix units of the 78-state carrier and therefore with every
untouched reference; the complete Gram identity has 156 nonzero Kraus
entries;
\item $\En$ is $\Cn$ followed by a three-pulse source controlled-NOT (CNOT) that copies the
seam-total parity $p$, with contrast decoded as $d=2j+(2c-1)p$; relative to
the retained coin and current labels one additional binary record is
necessary and sufficient, and forgetting it leaves parity dephasing without
restoring the coherence of $\Cn$;
\item the readout is the L\"uders instrument \cite{luders1950} of the
rank-one plus projector and its complement, realized by coupling the plus
projector to a blank pointer and measuring only the pointer; the minus
outcome is one rank-77 projector whose internal coherences survive, and
reading the full computational basis would implement a different
instrument.
\end{enumerate}
\end{proposition}

\begin{proof}[Proof sketch]
For fixed current the endpoint map is an injective translation, and for fixed
contrast the spectators and endpoint sum reconstruct the input, so every
input column of the joint coin, record and source isometry has two entries
$2^{-1/2}$ with disjoint supports; an independent enumeration checks all
6,084 matrix units. For (ii), $\{2,3\}$ and $\{1,2\}$ share the $\Cn$ record
$(0,0)$ and have contrasts $0$ and $-1$ \cite{sdDERIV}.
\end{proof}

The complete attempted trial costs 32,006 to 32,010 native pulse requests,
with one proper-code load and one export per pulse; a readout costs 26,632
pulses, a local $\Cn$ repair 5,370 and a local $\En$ repair 5,373, with peak
live width 11 qubits for $\Cn$ and 12 for $\En$ \cite{sdREADME}. Executed
terminal states reproduce Table~\ref{tab:ce}, and on synthetic counts drawn
from executed branch norms the evaluator keeps the generating candidate and
rejects the other. The compositional error contract
\[
N_pu+N_l\,l+N_e\,e+p+z+c+a+r+s+m+g+n\le\frac1{64},
\]
with pulse, load and export counts and their channel errors and whole-trial
allowances for preparation, initialization, coin, randomizer, records,
storage, measurement, coefficient compilation and numerical execution, gives
sufficient conditions for $\eta=1/64$; an equal allocation at the largest
schedule is $1/6146496$ per term. The same source alphabet, code services and
classical transitions admit two quantum instruments whose phase-zero
observable differs by $1/4$, so these ingredients alone do not select the
record law.

\subsection{Retrospective audit of the closure coupling output}
\label{sec:endpoint}

The detector of Section~\ref{sec:p-detector} evaluates the structured-running
branch of the closure map; the flagship synthesis reports a second,
self-consistent closure map with a much smaller residual \cite{flagship}, and
this construction consumes the bare source map only. Its output,
approximately $136.994835177413$, falls short of the 2022 CODATA value
$\alpha^{-1}=137.035999177(21)$ \cite{codata2022} by about $300.388218$ parts
per million, and exact rational arithmetic shows its narrow numerical
interval disjoint from the band of five reported standard uncertainties
\cite{prREADME,prDERIV}. The band is a declared comparison allowance, not a
confidence interval. The audit rejects the additional hypothesis that this
output is the complete physical endpoint with zero missing correction; no fit
is performed and $P$ is unchanged.

\subsection{A sequential design on one carrier}
\label{sec:sequential}

The phase test above assumes fresh preparations. A sequential design runs it
on one continuing carrier in its binary total-two sector without resets or
independent trials. Each trial applies $379$ unread native attempts to the
state the carrier holds, performs the fixed-seam native current herald, whose
useful yield is $5/33$, and on either useful branch executes the source
mixer, three unread native attempts, a second mixer and the native coin and
current pointer; a failed herald counts as a trial. After exactly $3{,}927$
trials the two locked likelihoods are compared on all six counts, and an
impossible record returns a verdict outside both models. Conditional-law
bounds on the dependent transcript give a wrong-candidate probability of at
most $3/400$ under the exact-source model; a uniform service allowance of
$1/1570800$ per complete trial would suffice for a one-percent bound, and
that allowance is not authenticated \cite{nativeProtocol}. Two consecutive
trials per candidate and their reference controls are executed.

\subsection{Native records against public data}
\label{sec:public-records}

Published records test precise attachments of native records without a
laboratory realization of the carrier; each test is retrospective and states
its readout hypotheses.

The binary endpoint map is an identity or a local exchange, so diagonal
preparations stay diagonal under recorded operations and local classical
feedback. Under independent settings and local factorized responses the
diagonal branch obeys $P(\mathrm{win})\le3/4$ in the CHSH game, that is
$S\le2$; a control that lets one side read both settings wins every trial,
and the native example on the 66-state carrier, with all $1{,}056$ histories,
gives $S=26/33$. Replaying the deposited loophole-free records of Hensen et
al.\ \cite{hensen2015} reproduces $245$ eligible trials, $196$ wins and
$S=16036094/6619647\approx2.42250$, with a published fixed-count tail bound
near $0.03917$ \cite{nativeBell}. This rejects the local classical reading of
the diagonal branch under that experimental contract; it does not test the
quantum C/E instruments.

Two native ports, or two regions of one carrier, counting incident repairs
under the uniform scheduler have independent interval increments with exact
count laws. Attaching the published $300$-second blocks of the
Ar$^{13+}$/Yb$^+$ frequency ratios of King et al.\ \cite{king2022} to the
normalized count ratio, through a gain of at least one, a declared bracket on
the Yb$^+$ frequency, an absolute error of $10^{-12}$ per block and direct
phase-count readback, gives conditional probability bounds below $10^{-104}$
for two ports and below $10^{-22}$ over all ordered pairs of the $3{,}433$
regional event sets, after charging every choice of subwindow, gain and
retained blocks \cite{nativePublicData}. This rejects the direct attachment
with its readout assumptions; the native event clocks are unaffected as
clocks of the model, and a clock calibration needs an explicit model of the
measuring apparatus.

Published trapped-ion records test a supplied experimental model executed
through the same source interface: two calibrated preparation parameters
predict $100$ reserved complex readouts with per-spin-mean root-mean-square
error $0.06647$, and a changed-potential transfer predicts $125$ derived
observations with root-mean-square error $0.2876427868$ in two-point
position-estimator units \cite{mREADME}. They test the supplied oscillator
model, whose action and apparatus do not follow from native repair.

\section{The premise map}
\label{sec:premises}

The premise map relates the construction to the conditional theorems of the
library \cite{mPREMISES,mMAP} and is the starting ledger of the joins of
Section~\ref{sec:joins}. It applies one rule: a result ``$H$ implies $C$''
discharges an input $I$ only when $C$ establishes $I$ on the same domain with
the required errors and $H$ has independent support; a change of regulator,
state or code, or a calibration relabelled as a derivation, discharges
nothing. An antecedent is either discharged for the finite objects of the
construction by execution or proof, or supplied, and discharge on the finite
domain differs from discharge of the physical premise on a physical
refinement tower. Six of the seven claim records of Table~\ref{tab:registry}
consume no canonical physical premise; the detector record consumes the
accepted closure inputs $P$ and $N$ and the selected coupling map, and lists
the same-quantity bridge for $P$ and the physical Thomson readback contract as
unsupplied \cite{ophrepo}. The crosswalk represents all 84 canonical premise
identifiers in 27 groups and inventories 343 registered claims and 661 Lean
modules, without individually instantiating every listed claim
\cite{mPREMISES}. Table~\ref{tab:premises} summarizes seven theorem routes.

\paragraph{Finite entropy and the gravitational premise.}
For positive sector counts $d_i$, normalized probabilities $p_i$ and fixed
$M=\sum_i d_i$, the counting entropy obeys
\[
 S_{\rm count}=H(p)+\sum_i p_i\log d_i
 =\log M-D(p\|d/M)\le\log M.
\]
The maximizer is $p_i=d_i/M$. In the positive-real relaxation, joint
fixed-$M$ variations have zero first derivative there and second derivative
$-\sum_i(p_i'-d_i'/M)^2/p_i$. This is no operation varying integer
dimensions, and with bulk factors full saturation also requires maximally
mixed bulk blocks. Uniform native preparation realizes counting weights.
Fixed count is not identified with fixed physical volume, nor the
logarithmic sector term with a geometric area term of universal coefficient
\cite{nativeEntropy}.
For every cap of the wave reference's return-length law, an absolute bound
holds on the full
$D=\binom{V}{V/3}$ configuration space:
\[
 \Delta=\log D-S(\sigma)\le\frac{2025}{64V},\qquad
 I_\sigma(R:R^c)\le\log J+\Delta,
\]
where $J$ counts the allowed regional particle numbers. A complete base-face
patch has $V/20$ ports and $3\,2^l$ crossing seams. Its stationary reduced
entropy is extensive in port count, while the return reference's mutual
information divided by crossing-seam count vanishes. Neither observable
therefore has a nonzero universal coefficient times that boundary count.
This does not exclude interpreting a whole spherical screen as a bulk
boundary; that physical region map is a premise \cite{nativeEntropy}.

{
\footnotesize
\begin{longtable}{@{}L{0.14\linewidth}L{0.42\linewidth}L{0.38\linewidth}@{}}
\caption{Theorem routes, antecedents discharged on the declared construction,
and supplied antecedents \cite{mPREMISES,mCONSTRUCTION}.}
\label{tab:premises}\\
\toprule
\textbf{Route} & \textbf{Derived from native records (N) or constructed on declared inputs (D)} & \textbf{Supplied}\\
\midrule
\endfirsthead
\multicolumn{3}{@{}l}{\footnotesize Table~\thetable\ continued}\\
\toprule
\textbf{Route} & \textbf{Derived from native records (N) or constructed on declared inputs (D)} & \textbf{Supplied}\\
\midrule
\endhead
Consensus and repair \cite{ophconsensus} &
Committed-record preservation; the exact gap of the record-discarded
channels; schedule-independent repair of the completed common-source
history; the exact law of two overlapping observers; federation output
bounds below $0.000382$ at $s=1/4$ and $6.656\times10^{-9}$ after four
incident ticks. &
Repair completeness, local diamonds and transactional descent for an
unrestricted observer menu; conflicting updates; global history repair by
native hardware; all-level federation routing.\\
\addlinespace
Quantum records \cite{ophobservers} &
(D) All-input CPTP maps with coherent write isometries and complete binary
instruments; the exact correctable algebra, optimal recovery and active
encoded memory; the compiled joint native/work channel; GNS representations
of the actual observation state; the quantum correspondence on $4{,}356$
matrix units and the coherent screen observer within $7.779\times10^{-10}$. &
Hilbert representation, subsystem split, Born readout, coherent preparation
and readout services (not available from diagonal native resources,
Section~\ref{sec:services}), a physical encoding of loads in the code.\\
\addlinespace
Record capacity \cite{ophobservers} &
$M_0$ for each declared record interface with a carrier projector
certificate; $\ln M_0\le N$ on a declared carrier. &
Complete-fiber closure, a physical selector and the horizon identification
behind $\Lambda\ell_\star^2=3\pi/N$.\\
\addlinespace
Gauge structure \cite{ophgauge} &
Branch charge covariance of $Q=g(P)N$; a four-state CAR and link code with
operator Gauss constraints and even parity. &
Physical current and action for the complete response algebra, coupling
attachment, nonabelian and chiral matter.\\
\addlinespace
Scalar continuum \cite{ophobservers,flagship} &
Refinement transfer and occupation limits with complete-output bounds; the
infinite-occupation interacting signal. (N) The native $H^1$ form of
spectral dimension two, Gaussian density paths, incident-clock
parametrizations, the precision $-\partial_\tau^2+A^2$ and active
incident-clock waves \cite{nativeContinuum}. (D) A thermal CCR field on the
declared radial product with a shared active clock; a quartic projection with
interacting Gibbs readouts and a fixed-band occupation limit. &
The physical scalar action, quantization and vacuum; inertia; a calibrated
clock; a spatially refining interacting family.\\
\addlinespace
Modular clocks \cite{ophspacetime} &
Finite modular flow of actual observation states; $t=-\beta s$ for a fixed
reference; a source-prepared finite reference with its positive-log clock;
an active native-transfer clock whose quadratures four movement-bit
covariances predict \cite{nativeEventClock,nativePredictions}. (D) Faithful
encoded product-field states sharing the active clock. &
A physical reference and its complete modular dynamics;
Bisognano-Wichmann identification; common null translations; clock
calibration; the relation to causal depth.\\
\addlinespace
Gravity \cite{ophspacetime} &
A rank-three response and axial circulation from committed records; native
causal cones; the common spherical continuum with density and incident-clock
limits \cite{nativeContinuum}; exclusion of a bounded energy identification
with a smooth compact three-dimensional Dirichlet bulk. (D) A measurable
Lorentzian density of the product wave with its recovered angular tensor;
the exact measure fails $C^1$; radial-average observations do not select the
radial architecture. &
Physical region, metric and area attachment; geometric meaning of
counting-entropy stationarity; small-ball geometry; conserved stress;
universal coupling and normalization; a responsive metric.\\
\bottomrule
\end{longtable}
}

Four retained audit controls fix exact boundaries of the construction
\cite{mCONSTRUCTION}. Each rejects a named identification or an overly broad
inference; none rejects the theorem premises.

\paragraph{Complete-carrier center.} The centers of $M_{66}$ and $M_{78}$
consist of scalars, so the unextended working algebra cannot supply twelve
nonzero primitive central ports. Binary occupations are noncentral, overlap
and sum to $2I$, and total-two occupations also admit the eigenvalue two. The
native matrix algebra is kept as a sector; a separate twelve-block
central-address extension supplies a central algebra with port-controlled
operations, and occupation measurement is a separate contract.

\paragraph{Marked triplets.} The slow Gram matrix satisfies $G^2=4G$,
$\Tr G=12$ and $(5I-A)G=(5-\sqrt5)G$. One fixed order-five symmetry has trace
$(1-\sqrt5)/2$ on the slow frame and $(1+\sqrt5)/2$ on the lower response
triplet, which excludes an invertible intertwiner for the same action, so
spatial and internal response domains are kept distinct. The comparison
separates these two specified triplets only; it neither shows every internal
triplet inequivalent to the frame nor identifies carrier rechartings with
physical rotations. The local rank-three carrier band and the refining global
low band are distinct. The conditional Spin--matter construction has
independent commuting factors, and its locked diagonal action fails to give a
uniform fermionic central sign. A physical separation requires the spatial
and internal actions to be identified on the same clocked interacting
response and preserved by its dynamics.

\paragraph{Feedback class.} With the two endpoint loads in the uniform
mixture of their one-particle configurations, a rational number-preserving
exchange gives endpoint probabilities $769/1250$ and $481/1250$ for the two
private preparations, a difference of $144/625$. Four Kraus operators certify
that it is CPTP, and its private commutant has dimension one, which excludes
every single-product exponential representation, alternate logarithm
branches included. The no-feedback theorems for reduced loads and single
products hold within their stated classes, and this channel marks the class
boundary.

\paragraph{Configuration equilibrium.} For $A=I+L_{\rm path3}$ the controls
$\varepsilon=\pm1/16$ keep the normalized configuration Gibbs marginal and
change the full reference and the vacuum. For a stiffness response
$Y=\gamma(A+\varepsilon A^2)$ one has $Y_{13}=\gamma\varepsilon$ and a
third-mode fit residual $6\gamma\varepsilon$, while the actual acceleration is
$-Yq$. The theorem identifying inertia with the same capacity keeps that
premise, and equal equilibrium configurations do not select the kinetic law.

\section{Ordered joins, acceptance conditions and stopping tests}
\label{sec:joins}

The completion target is one native repair and memory process whose
observers' compatible records and interventions admit the observed
3+1-dimensional quantum-field description, with its matter and interactions,
and a common classical-gravity limit, each as a controlled effective
description in its regime of validity \cite{execHypothesis,mCONSTRUCTION}.
Observer clocks, the defect and maintenance-energy mechanism, the
interacting field and particle description and the realized Einstein
premises must come from that same process. Component benchmarks keep their
own contracts and are not pooled into that realization. Hardware, protocol
and implementation choices are design variables, and no uniqueness theorem
for a chosen rule is a prerequisite; closure-derived $P$ and $N$ are accepted
inputs that discharge no other interface.

Every comparison of observable laws uses one complete-output criterion. For
every protocol $p$ of a declared menu,
\[
 \sup_p\tfrac12\bigl\lVert D_{\rm native}(\rho_{\rm native}(p))-D_{\rm eff}(\rho_{\rm eff}(p))\bigr\rVert_1\le\epsilon ,
\]
with failures, unresolved waits and admitted references retained in both
outputs and decoders that keep the claimed observations. An acceptance names
the protocol menu, decoders, refinement family and error budget before the
comparison \cite{mCONSTRUCTION}. Exact equality of limiting observable laws
is the mathematical target; finite simulations and experiments are assessed
with declared approximation and measurement errors, and agreement must be
constrained by measurement, since a consistent world model can describe
physics different from that of nature.

\paragraph{Finite constructive joins.}
A separately versioned campaign applies existing finite theorems to four
executed component joins \cite{finiteTheoremJoins}. Its clock inventory
covers all 66 transported two-port cores at the declared modular parameter,
with supplied coherent services and complete failure accounting. A joint
instrument advances the transported modular detector and a protected field
walk on the same native incident history; a control preserving both separate
marginals changes their joint law. This operational comparison does not
require equal generators. The positive native-return wave reference remains
available even though the tested three-charge reference fails direct
substitution for it.

On the binary two-load configuration complex, 300 movement edges and 495
declared triangle/square cells give integral first homology
$\mathbb Z/2$. Actual C/E records realize an odd exchange loop, but its
native branch amplitude is positive: a history class alone supplies no
fermionic phase or protected particle. Complete native currents also give
the positive form $D(v)=v^TLv/396$. In the existing declared product-field
symbol, rest gap and independent radial curvature agree in the ideal limit;
the executed split words have rest/inertia ratio $\chi\cot\chi$ with a
controlled subdivision error. This does not assign mass to the history
class. Reference-like localization, the physical momentum and clock maps,
and one defect action supplying rest energy, inertia and stress remain open.
These finite joins add no physical comparison or integrated acceptance.

\paragraph{A compiled Ramsey source service.}
On the finite density representation of the existing native coherent walk,
one additional Ramsey arm gives a 16-qubit controlled source program. Its
walk uses 123{,}646 source pulses; the full-register reversible shift is
checked on all 65{,}536 basis addresses. The 720-arc mean-zero density
sector has an exact isometry into the original 3{,}960-arc binary-source
walk, but the full binary-arc unitary is not compiled. Source arrays execute
the Ramsey preparation, controlled walk and terminal analysis, and complete
two-attempt $\Cn/\En$ joint laws retain clock-export aborts, both field
outcomes and untouched references. One selected four-attempt continuation
keeps the wave register and three FIFO exports without fresh field
preparation. It is a selected unnormalized history, not an exhaustive
four-attempt distribution. Numerical routine guards and counted proper-code
loads/exports are separate from physical apparatus accuracy and calibrated
seconds \cite{ramseyService}.

\paragraph{A bounded repair/reference alternative.}
A separately frozen finite family replaces the fair placement coin on a seam
$(a,b)$ by probability $\theta_a/(\theta_a+\theta_b)$ of placing the one
endpoint load at $a$, for declared positive port weights $\theta$ and a
fixed source-independent seam selector. Complete $\Cn$ and $\En$ record
instruments then fix the faithful binary-shell
product-pair reference $\pi(x)\propto\prod_i\theta_i^{x_i}$ and covary with
its modular action. This is a changed local law, with the inherited native
branch support; its full branch algebra has no nontrivial reducing
source charge. The earlier transported three-charge reference cannot be
made reversible by this endpoint-only rule: its per-configuration weights
obey $w_1^2/(w_0w_2)=125/64$, whereas this product family requires one
\cite{localRepairFamily}. Neither its weights nor a physical clock are
selected by these finite statements.

\paragraph{Reference balance and relational charge.}
For the unchanged uniform-seam $\Cn/\En$ law, the complete binary-shell
population kernel is symmetric on 300 movement edges and has the unique
uniform reversible measure. Holding the transported clock reference fixed
produces a one-attempt total-variation drift of $17/750$ or $17/600$ for
the two tested cores. A recorded two-port core that follows every nonzero
current instead gives a reversible joint source/core measure on $66^2$
states. Its overlap charge $q=|x\cap c|$ is preserved by every complete
record block; the three sectors are connected separately and have stationary
weights $2/5$, $5/9$, $2/45$. The charge's native gradient form is zero.
These are source-plus-controller sectors, not a localized intrinsic particle
or an energy splitting. The reduced diagonal distribution converges to the
joint stationary measure from the stated fixed-core preparation under
independent uniform seams; adaptive feedback and the full archived record
are outside that stationarity result \cite{finiteTheoremJoins}.

\paragraph{Sharp sectors and active memory.}
An exact test of the same binary source/core charge finds that every state
in a sharp $q=0,1,2$ sector differs from the declared transported reference
on the full ten-port exterior by trace distance at least $3/5$, $4/9$,
$43/45$, respectively; conditioned references attain these bounds. For an
uninterrupted complete native $\Cn/\En$ history, every positive closed odd
exchange flag has a rank-one source input domain. The result prevents that
flag from carrying an arbitrary unknown coherent qubit through a final
decoder, but intermediate active recovery is a different operation grammar.
The existing source-executed $q=1$ code attains the all-seam exact-correction
dimension bound two. Its recovery leaves a moved core with $q=1$ probability
$7/8$ unless an explicit classical core reset is appended. The code has
native movement probability $1/6$ per uniform-seam attempt, an excess of
$1/66$ over the uniform-source baseline, while coherent recovery work is
outside that statistic. Neither the code nor this activity identifies a
localized reference-preserving particle or its rest energy
\cite{exchangeSector}.

\paragraph{A joint readout of the incident count.}
The compiled modular detector and density-wave Ramsey service also support
an operational comparison when the native tape is hidden from the readout.
Conditional on a successful preparation and terminal read in the declared
two-attempt policy, let $N\in\{1,2\}$ count port-0 incidents. Rational
enclosures give a reduction in optimal squared-error risk from the joint
detector/field output, relative to the detector alone, exceeding $0.0027325$
for $\Cn$ and $0.0027329$ for $\En$. Independent resampling of the field
bit with both separate marginals preserved removes the gain. The finite
preparation and complete source-abort allowances retain a gain above
$0.00272$ under the declared numerical routine guards. This is inference of
one dimensionless shared count. The stochastic readout does not instantiate
a strictly monotone reading, a two-event affine calibration or its distinct
third-event check from the finite clock interfaces, and supplies no calibration in seconds
\cite{clockCountInference}.

\paragraph{Optimal small-cap screening for the corrected qubit.}
For every exterior cap size $1\le k\le10$, hypergeometric averaging gives
the minimum over sharp-$q$ states of the worst trace distance over all
$k$-port exterior caps to the transported reference. The conditional
reference attains the bound.
At $q=1$, the unrestricted one-port value is $8/225$. Exact all-seam
correction imposes a stronger restriction: the finite flagged-code theorem
forces nonedge source support and scalar port-occupation compressions.
An exterior port adjacent to either core endpoint can then carry no code
occupation. Edge cores have four safe exterior ports, distance-two cores
have two, and opposite cores have none. A crossed four-port qubit attains
worst one-port distance $61/450$, optimal among exact all-seam sharp-$q=1$
qubits on an edge core and improved from the earlier code's $82/225$.
All complete raw $\Cn/\En$ branches satisfy its correction conditions;
the compiled recovery described below supplies its source pulse service. Its full
exterior differs by at least $4/9$. Separately, a sharp-$q=2$ pair
starting on any native edge exits that edge set with hazard $1/15$ per
uniform-seam attempt, excluding an invariant edge bond in this finite
candidate \cite{particleScreening}.

\paragraph{Same-code activity and unselected sector action.}
The improved code and its predecessor both have native moving-current
probability $1/6$ per uniform-seam attempt. The stationary co-moving
$q=1$ ensemble has rate $5/33$, but averages over core labels, so their
difference $1/66$ is a specified diagnostic comparison. On the joint
source/core graph, each connected $q$ sector has a uniform stationary
measure. Any positive reweighting of those three sectors remains
stationary for the same reduced diagonal source/core population law under
uniform seams. Doubling the $q=1$ weight leaves its
conditional record law unchanged while shifting a separately declared
Gibbs modular contrast by $-\log2$. Thus native stationarity and the
current form do not select the rest gap. The earlier code's mean repair
plus recovery costs, $439409$ and $863483/2$ source pulses for $\Cn/\En$,
belong to the distinct fixed-seam $(0,5)$ protocol; they are not a measured
uniform-seam energy. The finite action-measure uniqueness theorem applies
within each connected sector, and the proper-time/inertia and Ward theorems
require the same candidate's action, kinetic and stress maps
\cite{activeCodeEnergy}.

\paragraph{Compiled screened recovery and the same-code transport test.}
The screened encoder and all 208 positive complete $\Cn/\En$ recovery
branches have a source execution on the 512-row work register. Their ideal
identity $R_hK_hV=\sqrt{p_h}V$ preserves arbitrary logical/reference states
through any finite corrected history, with immediate supplied recovery and
an explicit archived core-label reset. Initial encoding uses 65,484 source
pulses. Uniform-seam mean native-plus-recovery costs are $37499681/60$ for
$\Cn$ and $36870437/60$ for $\En$ per attempt; retained records use 23 or
24 classical bits. Numerical source composition has a stated 100-attempt
bound under its routine guards, separate from exact ideal composition and
physical apparatus error \cite{screenedRecovery}.

On this same code, the native current second moment is a rank-eleven
weighted port Laplacian with trace $1/3$. For the earlier source eigenprofile,
its quadratic form is $13/120$, while the two logical basis states have
variances $1/2$ and $5/2$. The ratios $13/60$ and $13/300$ therefore do not
supply one wave rate for this code. Summing every corrected branch gives
identity logical transfer, with four unit superoperator eigenvalues, and
the core resets to $(0,1)$ at each attempt boundary. Transient native
current is present, but continuing coherent dispersion and a rest gap are
not identified. A common global phase is invisible to the channel and is
not a measured zero rest energy \cite{screenedTransport}. The finite
conserved-sector theorem additionally proves stationary reweighting,
conditional cancellation and zero sector-potential gradient for every
positive sector factor. Exact enumeration binds its premises to the 4,356
native source/core states; the general implication is checked in Lean,
while the actual finite kernel is checked in Python \cite{conservedSectorAction}.

\paragraph{Four shared clock events.}
A continuation compiles four transported-core modular words at parameter
one and source pair preparation/analyzers. Four terminal horizons on copies
of the same C/E protocol retain all stopping outcomes and source aborts,
with up to three FIFO exports. Conditional clock and Ramsey ensemble means
are strictly increasing at the four shared checkpoint types. A raw affine
fit on events one and two fails the reserved fourth event by approximately
$0.0105927$ in field probability; the source-error allowance leaves a
positive cross-product margin. These are ensemble readings with nonlinear
detector response. Model-conditioned response inversion recovers a common
count but does not establish a physical clock rate, individual-shot monotone
reading or timelike atlas \cite{sharedEventClock}.

\paragraph{Conditional SI attachment.}
The unit utility takes an explicit $T$ seconds per model-clock unit and
signal speed $v$ in model lengths per clock unit. Identifying that signal
with $c$ fixes $L=cT/v$. If the same candidate supplies a coherent rest
angular frequency $\omega_0$ and the physical identification
$E=\hbar\omega$, its conditional rest mass is
$m=\hbar\omega_0/(Tc^2)$. The SI defining values of $c$ and $h$ do not
supply $T$ or $\omega_0$ \cite{siConstants}. The independent small-momentum
check is $\omega_0\omega''(0)/v^2=1$ for a massive relativistic dispersion.
The corrected screened code supplies no such dispersion or rest gap, so
no particle-mass or GeV prediction is assigned by this conversion.

\subsection{The joins in dependency order}
\label{sec:join-order}

The joins below are ordered by acceptance dependency \cite{completionProgram}.
Each states what must be shown on the same process, the results of this paper
it starts from, and the test that would stop it. A theorem premise assigned
to a join must be proved on the actual model and retained domain, with its
state, algebra, operation, clock and refinement maps bound in the premise map
of Section~\ref{sec:premises}, before the theorem's conclusion is used.

\paragraph{The admissible class.} A versioned class of carrier and
refinement, repair and memory laws, record instruments, resources,
preparations, clocks and interpretation maps is fixed before validation,
together with the observable domains, Standard Model, neutrino and gravity
targets, allowed calibrations, input ancestry, uncertainties, budgets and
decision thresholds. It distinguishes the fixed C/E law, the permitted
quantum record implementations of Theorem~\ref{thm:record-laws}, and any
bounded alternative-law family. \emph{Stopping test:} if the declared class is
empty or jointly inconsistent, that exclusion is proved. An unbounded
unspecified escape class leaves a whole-program exclusion unarmed, and an
unfinished freeze is neither confirmation nor falsification.

\paragraph{One continuing native observer.} One replayable native repair and
memory process prepares, acts, reads, retains records and continues on
overlapping accessible regions, with coherent resources, ancillas,
initialization, read and write, retention, feedback, timeout and recovery
services each constructed or counted, and complete classical and quantum
outputs including untouched references, failures and continuation. It
starts from the finite observers of Sections~\ref{sec:observer}
and~\ref{sec:common}, the compiled native attachment and active encoded
memory of Section~\ref{sec:continued-native}, and the overlapping observers
and matched screens of Section~\ref{sec:native-process}; several of these run
on separate schedules with separate error budgets, and their joins need
their own implementation and error analysis. In parallel the theorem
library is mapped onto the model: every theorem the chain needs receives a
native interface and an owning join, and results that no join needs leave
the active chain with their provenance. \emph{Stopping test:} a
source-access, coherence, retention or resource impossibility rejects the
tested implementation class. The coherence floor of
Section~\ref{sec:services} is such a result for diagonal native resources: it
makes coherent services a necessary resource of that class.

\paragraph{Continuing regime, faithful reference and local modular clocks.}
The same process prepares a reference with support and faithfulness on the
accessible observer algebras, keeps a continuing regime, executes the
modular actions and clock readouts, and derives compatible local clock
comparisons and the common normalization used by energy and propagation.
Passive repair, dimensionless modular flow, event counts and physical
duration stay distinct, and tracial and compressed-algebra controls are
retained. It starts from the fixed-reference relation and modular
representations of Section~\ref{sec:readouts}, the finite reference and
positive-log clock of Section~\ref{sec:coherent-continuation}, the event
clocks of Sections~\ref{sec:local-tick} and~\ref{sec:event-clocks}, and the
incident clocks and faithful encoded-field states of
Sections~\ref{sec:incident-clocks} to~\ref{sec:product-field}. Two retained
limits bound it: passive repair cannot act as a nontrivial modular
automorphism of an invariant observer algebra, and the uncapped native-return
reference tends to the trace. \emph{Stopping test:} a proof that the frozen
preparation and access grammar forces trivial modular action, incompatible
regional flows or no admissible physical clock map; a failed chosen reference
alone is a candidate obstruction.

\paragraph{A localized native defect and its protecting invariant.} An actual
native state or observable sector has a localized core, a reference-like
exterior and a boundary charge invariant under the full admitted repair law,
defined independently of a target mass, with autonomous protection or a
controlled lifetime, every seam and branch retained, and quantum and record
sectors checked separately. \emph{Stopping test:} an exhaustive invariant and
localization no-go over every sector and allowed refinement of the frozen
defect class stops this mechanism. The classical and collision exclusions of
Section~\ref{sec:negative-controls} do not exhaust arbitrary quantum or
history sectors.

\paragraph{Maintenance energy, kinetics and conserved stress.} One intrinsic
action and energy, and one kinetic response, are derived for the same
defect, with reference subtraction, clock normalization and an independently
justified stress and Ward identity. Action, energy, entropy production, event
counts and host work stay distinct; moving and rest responses share one
signal speed with no species-dependent conversion, and interaction energy and
reference or cutoff dependence are accounted. \emph{Stopping test:} a proved
inability to define a common positive energy, kinetic and stress dictionary
in the frozen class, or a mandatory incompatibility between maintenance and
propagation. A scalar code cost or a failed ansatz has narrower scope.

\paragraph{Transport, particle identity and currents.} The same excitations
carry charges, spin and statistics, family structure, gauge-current
couplings and allowed creation, fusion, annihilation and decay, with
transport and charge conservation shown on the native process, leakage and
recovery bounded, extra modes and chirality retained, and stable defects,
metastable excitations and massless collective modes distinguished. It
starts from the charged realization of Section~\ref{sec:p-detector} and the
flux-dressed code of Section~\ref{sec:common-construction}. \emph{Stopping
test:} incompatible required charge, spin, statistics, current or transport
structures exclude the covered defect realization.

\paragraph{An interacting chiral 3+1 field description.} A refining
same-process observation law has a nontrivial interacting chiral
3+1-dimensional effective description, compared through the complete-output
criterion with locality, quantum correlations, positivity and controlled
errors. Every taste, chirality, dark mode and residue is retained until a
derived decoupling map removes it, kinetic, Higgs and current responses are
obtained jointly, and the photon-containing theory stays gapless, with
massive thresholds, confinement observables and any pure Yang--Mills gap
stated separately. It starts from the occupation limits of
Section~\ref{sec:common}, the continuum operator and its density, clock and
wave limits of Section~\ref{sec:native-history}, the declared radial product
with its smoothing theorem and the fixed-band quartic projection.
\emph{Stopping test:} a covered chirality, locality, positivity, continuum or
interaction no-go stops the frozen realization class. A supplied scalar
product, finite free symbol or pole list does not pass this join.

\paragraph{The physical dictionary and the matched spectrum.} A
source-selected spectrum, mixing and interaction packet in compatible
physical conventions uses the same defects, energy, kinetics and clocks. The
charged benchmark of nine running Yukawa couplings and four
Cabibbo--Kobayashi--Maskawa parameters is derived together, with source and
decoder revisions, empirical ancestry, matching scheme and scale, inputs,
covariance and theory uncertainty bound before the comparison; neutrino,
unstable and confined sectors keep their own obligations. \emph{Stopping
test:} a wrong unavoidable matched spectrum with an independently justified
error budget rejects the fixed candidate, and the class only if every
permitted nuisance and source choice is excluded. Calibration on known data
is not independent validation.

\paragraph{Responsive geometry and causal clocks.} A source-derived event,
causal and geometry dictionary has an effective Lorentzian domain, compatible
observer directions, local clock comparisons and count, volume and length
maps with controlled errors, and responds to admitted matter changes. It
starts from the rank-three response and causal cones of
Section~\ref{sec:history-geometry} and the common matter metric of
Section~\ref{sec:common-matter-metric-response}, and it keeps the finite
rank-three carrier geometry distinct from physical 3+1 event geometry.
\emph{Stopping test:} a mandatory incompatibility between source geometry,
clocks and nonzero matter stress excludes the covered common realization. The
static radial-product failure is a scoped control, not a universal gravity
no-go.

\paragraph{Realized Einstein premises.} A premise-by-premise same-tower
witness supplies geometrically normalized local modular flow whose
directional charges define the matter stress used in propagation, its Ward
identity, the edge-center entropy split and area normalization, the modular
first law, generalized-entropy stationarity at fixed proper volume with
controlled small-ball remainders and enough null directions, universal
coupling and one positive Newton constant \cite{ophspacetime}. A stationary
Markov law, a Gibbs reference or a stationary conditional record entropy is
not this variation; the counting-entropy and boundary-proxy controls of
Section~\ref{sec:premises} and the cavity comparison of
Section~\ref{sec:common-matter-metric-response} are retained. \emph{Stopping
test:} a proof that a necessary entropy sign, modular or stress
normalization, or stationarity variation is incompatible across every
allowed same-process realization. An unmet premise is an open join, not a
theorem failure.

\paragraph{The composed gravitational response.} The conditional Einstein
theorem is applied with a checked map for every premise, giving the Einstein
equation up to its cosmological term with one positive Newton constant and
universal coupling of the same defect energy and stress, and finite and
refinement errors are propagated into gravitational observations. A quantum
spin-two graviton lies outside the classical theorem. \emph{Stopping test:} a
counterexample satisfying every theorem premise refutes the implication; a
necessary failure of joint gravitational response excludes the frozen
program only with coverage of the whole admissible class.

\paragraph{Joint independent tests.} One frozen physical dictionary is
compared with independent field, particle and gravity observations through
complete joint and conditional laws and intervention responses, with
predeclared rejection rules, eligible data, nuisance ranges, covariance and
theory errors. It starts from the prospectively fixed count protocol, the
sequential design and the retrospective replays of
Section~\ref{sec:prediction}; the phase test needs a realized preparation and
observable and an independently established service budget, and it measures
where the realized record law lies in the classified family. \emph{Stopping
test:} the fixed formulation is rejected when an unavoidable prediction fails
a valid frozen comparison; a whole-class empirical exclusion requires every
admitted realization and nuisance choice to fail.

Across all joins an obstruction review records, for every exclusion, the
target version, mandatory premises, quantified class, evidence, independently
checked coverage, surviving escape routes and consequence. A mathematical
error, a candidate failure, a class no-go, an empirical rejection, an
inconclusive search and a budget stop are kept distinct. Whole-program
falsification requires a verified contradiction in indispensable premises, a
proof excluding every frozen realization of a required join, or an
unavoidable independently excluded prediction; a finite unsuccessful search,
an unconstructed map or an exhausted budget proves no impossibility.

\subsection{Negative controls on the tested law}
\label{sec:negative-controls}

Four results reject specified mechanisms of the tested law. Each is a fact
about that law and its stated class.

\paragraph{The native double-load defect is transient.} On the twelve-port
carrier, the only configuration functions invariant under every admitted
seam and coin are constants, and the binary shell is the unique closed
sector. A collision state carries one double load above that shell. Under
the uniform seam probe, the complete C and E instruments both satisfy
$\sum_hK_h^\dagger P_{\rm coll}K_h=\frac56P_{\rm coll}$ for the projector onto
the twelve collisions, so after $n$ attempts the collision probability is
$(5/6)^n\Tr[(P_{\rm coll}\otimes I)\rho]$ for arbitrary quantum inputs and
untouched references. The mean number of attempts to split is six, and the
double load never hops before splitting. Load differences telescope around
all twenty faces for all 78 states, so this gradient map carries no
holonomy. Holding one port fixed creates sectors only by disabling its five
repairs, which changes the law. The record costs of the corrected memory code
have expectations independent of the logical state, so they cannot split
logical-state masses \cite{nativeDefect}. No persistent quantum subspace
supported on the collisions exists under this evolution; other quantum and
record-encoded sectors are untested.

\paragraph{The static radial product fails null balance.} For the smooth
static product metrics of the conditional scalar description, both radial
null Einstein contractions vanish, while a coherent scalar excitation in the
restored angular-constant Dirichlet sector has positive relative null
stress. That geometry cannot satisfy the Einstein balances for the reference
and the excitation at positive coupling with other sources fixed
(Proposition~\ref{prop:static-product}). A responsive geometry is required.

\paragraph{Passive repair has no wave sector.} The record-discarded native
channels are Hilbert--Schmidt self-adjoint with spectrum in $[0,1]$ and
scalar fixed algebra (Theorem~\ref{thm:gap}). The executed quantum channel
strictly attenuates a coherent mode, which rules out unitary conjugation on
this carrier, and the half-step map of its principal logarithm is not
positive, so it generates no completely positive semigroup \cite{mREADME}.
The closed mean law is damped (Proposition~\ref{prop:fields}(iv)), ballistic
rescaling freezes the density paths, and the native decay rates grow like
Laplacian eigenvalues, so no common clock factor gives a wave dispersion
(Section~\ref{sec:density}). Every wave in this paper is declared record
processing or a compiled apparatus step on supplied codes; other dilations
or generators are not excluded.

\paragraph{The tested mass laws miss the charged hierarchy.} The declared
linear Weyl--Higgs candidate gives charged-lepton coefficient ratios
$1:1.1903:1.4982$ against the observed $1:206.77:3477.23$, and the whole
complex affine class bounds the largest-to-smallest mass ratio by $543.55$
against an observed lower bound of $3470$
\cite{physicalAudit,massResponse}. These are conditional diagnostics of
unmatched coefficients under a direct pole-ratio identification; no
maintenance-energy map underlies them.

\section{Reproducibility}
\label{sec:repro}

Every executable result has a replayable receipt. Package manifests bind
source, document and receipt artifacts by SHA-256. The repository's
verification entry point recomputes these hashes, replays retained receipts
and compares canonical bytes. Exact certificates must match exactly, and
floating diagnostics carry declared replay tolerances. Independent test
suites check the executable statements against separately written oracles,
including adversarial mutations. Analytic limit proofs state their domains,
normalizations and order of limits; a finite receipt does not replace those
proofs. Model sources and the premise crosswalk are available at the pinned
revisions cited with their results \cite{nativeContinuum,mPREMISES}; the
flagship synthesis and basis document are cited on the main branch
\cite{flagship,basis}.

\section*{Statements and declarations}

\noindent\textbf{Funding.}
The authors declare that no external grants or dedicated third-party research
funding were received for the preparation of this manuscript.

\noindent\textbf{Competing interests.}
The authors participate in OPH-related research, software, simulation, or
prototype-development programs and may receive professional or reputational
benefit from this work. These relationships are disclosed. The authors declare
no other competing interests.

\noindent\textbf{Data availability.}
The model documents, executable sources, retained receipts and claim records
supporting this study are available in the public OPH repository at the
baseline revision \cite{ophrepo} and the revisions cited with individual results.

\noindent\textbf{Corresponding author.}
Correspondence should be addressed to Bernhard Mueller,
\href{mailto:bernhard@floatingpragma.ai}{\nolinkurl{bernhard@floatingpragma.ai}}.

{\footnotesize
\InputIfFileExists{tex_fragments/PRIOR_WORK_EXECUTABLE_OBSERVER_CONSTRUCTION.tex}{}{}
\begin{thebibliography}{99}
\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}

\bibitem{jppIID}
Jak\v{s}i\'c, V., Pautrat, Y., and Pillet, C.-A. (2009).
\emph{A quantum central limit theorem for sums of IID random variables}.
Author manuscript, Sections 3--4.
\url{https://jaksic.xyz/papers_pdf/qcltiid.pdf}

\bibitem{ophrepo}
FloatingPragma (2026). \emph{Observer Patch Holography}: papers, Lean library,
executable model, receipts and claim registries. Public repository; model
documents, receipts, papers and the basis document are cited at the release
tag of this paper, and frozen custody records at their registration
revisions.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/r2053}

\bibitem{flagship}
Mueller, B., Poneder, M., Osika, A., Xue, K., Cassie, B., Nguyen, P., Kim, J.,
Matscheko, D., Friedman, D.\,A., Hill, J., Glynn, W.\,T., Visser, M.\,A., Anirudha, K.\,A., and de La Fourni\`ere, B. (2026). \emph{Observers Who Read Their Universe into Existence: Finite Consensus, an Executable Observer, and Standard Physics}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/flagship/from_observer_consensus_to_standard_physics.pdf}

\bibitem{basis}
FloatingPragma (2026). \emph{The OPH basis}. Basis document: the closure hypothesis, the no-outside rule, the agreement condition and the executable specification.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/BASIS.md}

\bibitem{axref}
FloatingPragma (2026). \emph{OPH premise reference}. Formal statements of the theorem-library premises A1, A2 and A3.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/AXIOM_REFERENCE.md}

\bibitem{ophconsensus}
Mueller, B., Xue, K., Kim, J., Anirudha, K.\,A., Matscheko, D., and Hill, J.
(2026). \emph{Reality as a Consensus Protocol: The Fixed-Point Computation That
Implements Physics}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/paper/reality_as_consensus_protocol.pdf}

\bibitem{ophobservers}
Mueller, B., Poneder, M., Osika, A., Xue, K., Cassie, B., Nguyen, P., and Friedman, D.\,A.
(2026). \emph{Observers Are All You Need}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/paper/observers_are_all_you_need.pdf}

\bibitem{ophgauge}
Mueller, B., Poneder, M., Osika, A., Xue, K., Nguyen, P., Matscheko, D., and Friedman, D.\,A.
(2026). \emph{Standard Model Gauge Structure under Complete-Response and Observer-Transport Premises}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/paper/deriving_standard_model_gauge_structure_from_observer_overlap_consistency.pdf}

\bibitem{ophspacetime}
Mueller, B., Poneder, M., Osika, A., Xue, K., Nguyen, P., Visser, M.\,A.,
Matscheko, D., and de La Fourni\`ere, B. (2026). \emph{Recovering Observer
Spacetime and Einstein Dynamics from Overlap Consistency}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/paper/recovering_observer_spacetime_and_einstein_dynamics_from_overlap_consistency.pdf}

\bibitem{ophmicro}
Mueller, B., Osika, A., Xue, K., Cassie, B., de La Fourni\`ere, B., and Friedman, D.\,A. (2026).
\emph{Twelve-Port Icosahedral Carriers and Native Seam Repair: The Carrier Layer of Observer-Patch Holography}. Public manuscript.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/paper/screen_microphysics_and_observer_synchronization.pdf}

\bibitem{mREADME}
FloatingPragma (2026). \emph{The runnable OPH model}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/README.md}

\bibitem{nativeContinuum}
FloatingPragma (2026). \emph{A common native continuum operator, wave
observations, modular phases and repair response}. Executable model,
analytic bounds and retained source receipts.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/README.md}

\bibitem{nativeEntropy}
FloatingPragma (2026). \emph{Joint counting stationarity and its geometric
boundary}. Exact entropy identity and finite source controls.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/code/geometry/ENTROPY_FIRST_LAW_AUDIT.md}

\bibitem{nativeClockLimit}
FloatingPragma (2026). \emph{Native incident clocks, predictable access and
active waves}. Analytic count and inverse-clock bounds, correlated
clock fluctuations and exact native controls.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/relative_observer_geometry/README.md}

\bibitem{nativeRecovery}
FloatingPragma (2026). \emph{Observer-relative time, fields and shared
records}. Recovery assessment of the native process results.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/OBSERVER_RELATIVE_RECOVERY.md}

\bibitem{nativeCorrespondence}
FloatingPragma (2026). \emph{One process, overlapping observer records and
an effective transfer}. Executable model and exact receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer_correspondence/README.md}

\bibitem{nativeQuantumCorrespondence}
FloatingPragma (2026). \emph{Coherent quantum histories on overlapping
native clocks}. Executable model and receipts.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/quantum_observer_correspondence/README.md}

\bibitem{nativeCompiledClock}
FloatingPragma (2026). \emph{Source-compiled apparatus in a complete
observer-clock history}. Executable model and exact receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/compiled_clock_history/README.md}

\bibitem{nativeJointClock}
FloatingPragma (2026). \emph{Joint local clocks and quantum fields with
explicit memory}. Executable model and exact receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/joint_clock_process/README.md}

\bibitem{nativeEventClock}
FloatingPragma (2026). \emph{An executable modular clock for the native
record-transfer description}. Executable model and exact receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/transfer_modular_probe/README.md}

\bibitem{nativePredictions}
FloatingPragma (2026). \emph{Fixed observer predictions and complete-count
comparison}. Modular clock recovery and native repair response.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/predictions/README.md}

\bibitem{nativeIdentification}
FloatingPragma (2026). \emph{Identifying the complete current-resolving
repair law}. Executable model and receipts.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/repair_identification/README.md}

\bibitem{nativeAction}
FloatingPragma (2026). \emph{A local action for native quantum
observations}. Executable model, locality bounds and screen comparisons.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/native_action/README.md}

\bibitem{nativeTimeFields}
FloatingPragma (2026). \emph{Joint time and field laws on matched source
screens}. Executable model and receipts.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/observer_time_fields/REFINEMENT.md}

\bibitem{nativeScreenObserver}
FloatingPragma (2026). \emph{Coherent observer outputs on matched source
screens}. Executable model and receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/coherent_screen_observer/README.md}

\bibitem{nativeProtocol}
FloatingPragma (2026). \emph{A finite sequential native-law prediction
protocol}. Executable model and receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/closed_prediction_protocol/README.md}

\bibitem{nativeBell}
FloatingPragma (2026). \emph{Classical record responses and published Bell
measurements}. Replay of the deposited records.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/predictions/BELL_RECORDS.md}

\bibitem{nativePublicData}
FloatingPragma (2026). \emph{OPH predictions and public-data comparisons}.
Retrospective audit, including the direct optical-clock attachments.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/docs/research/PUBLIC_DATA_PREDICTION_AUDIT.md}

\bibitem{hensen2015}
Hensen, B. et al. (2015). Loophole-free Bell inequality violation using
electron spins separated by 1.3 kilometres. \emph{Nature} 526, 682--686.
\url{https://doi.org/10.1038/nature15759}

\bibitem{king2022}
King, S. A. et al. (2022). An optical atomic clock based on a highly
charged ion. \emph{Nature} 611, 43--47.
\url{https://doi.org/10.1038/s41586-022-05245-4}

\bibitem{execHypothesis}
FloatingPragma (2026). \emph{OPH and the executable-model hypothesis}.
Statement of the hypothesis, its five conjunctive obligations and their
acceptance conditions.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/OPH_EXECUTABLE_HYPOTHESIS.md}

\bibitem{completionProgram}
FloatingPragma (2026). \emph{Executable OPH completion program}. Ordered
joins, retained foundations, acceptance conditions and stopping rules.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/COMPLETION_PROGRAM.md}

\bibitem{finiteTheoremJoins}
FloatingPragma (2026). \emph{Finite clock, defect and energy constructions}.
Versioned source contracts, complete clock services and joint readouts,
native exchange homology, and independent mode-inertia checks.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/clock_defect_energy}

\bibitem{localRepairFamily}
FloatingPragma (2026). \emph{Bounded local repair family and faithful reference}.
Exact source laws, retained records, theorem applications and finite
class boundary.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/local_repair_family}

\bibitem{exchangeSector}
FloatingPragma (2026). \emph{Exchange history, sharp sectors and active memory}.
Exact full-exterior bounds, complete native record fibers and source-executed
recovery/readout controls.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/exchange_sector}

\bibitem{ramseyService}
FloatingPragma (2026). \emph{Source-pulse Ramsey control of the native density wave}.
Complete C/E clock/field outputs, retained source services and finite
continuation controls.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/main/model/clock_defect_energy/RAMSEY_SERVICE.md}

\bibitem{clockCountInference}
FloatingPragma (2026). \emph{A shared-event count read from the compiled clock and field}.
Complete source records, rational risk bounds and finite theorem interfaces.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/main/model/clock_defect_energy/CLOCK_COUNT_INFERENCE.md}

\bibitem{particleScreening}
FloatingPragma (2026). \emph{Local screening and native-edge binding tests}.
Exact cap minimax, all-seam code geometry and native first-exit bounds.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/particle_screening}

\bibitem{activeCodeEnergy}
FloatingPragma (2026). \emph{Same-code current and sector-action ambiguity}.
Common candidate definition, counted correction work and sectorwise action limits.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/active_code_energy}

\bibitem{screenedRecovery}
FloatingPragma (2026). \emph{Compiled continuing recovery of the screened qubit}.
Complete C/E source recovery, finite composition and retained service costs.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/screened_recovery}

\bibitem{screenedTransport}
FloatingPragma (2026). \emph{Transient current and corrected transport of the screened code}.
Same-code current form, logical transfer and core-reset countercontrols.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/screened_transport}

\bibitem{conservedSectorAction}
FloatingPragma (2026). \emph{Conserved-sector action theorem on the native kernel}.
Quantified finite reference freedom and explicit model-premise bindings.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/conserved_sector_action}

\bibitem{sharedEventClock}
FloatingPragma (2026). \emph{Four shared source events for a clock comparison}.
Source-executed conditional ensemble readouts and an affine consistency test.
\url{https://github.com/FloatingPragma/observer-patch-holography/tree/main/model/clock_defect_energy/shared_event_clock}

\bibitem{siConstants}
BIPM. \emph{SI defining constants}.
\url{https://www.bipm.org/en/measurement-units/si-defining-constants}

\bibitem{nativeDefect}
FloatingPragma (2026). \emph{Native defect decision}. Exact sector,
collision-survival, load-gradient and memory-cost controls on the twelve-port
carrier.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/native_defect_decision_results.json}

\bibitem{physicalAudit}
FloatingPragma (2026). \emph{Physical interpretation of the executable
model}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/PHYSICAL_AUDIT.md}

\bibitem{massResponse}
FloatingPragma (2026). \emph{Particle construction: results, assumptions and
research tasks}. Mass-response guide with the hierarchy diagnosis.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/research/OPH_PARTICLE_MASS_RESPONSE.md}

\bibitem{mCONSTRUCTION}
FloatingPragma (2026). \emph{OPH model construction from local repair}.
Construction guide with completion target, acceptance criteria and audit
controls.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/CONSTRUCTION.md}

\bibitem{mRECORDLAWS}
FloatingPragma (2026). \emph{Admissible record laws for native repair}. Model
document and exact receipt.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/record_laws/README.md}

\bibitem{mINSTRUMENT}
FloatingPragma (2026). \emph{The coherence table and the recorded
instrument}. Exact control on the actual carrier.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/record_laws/INSTRUMENT_CONTRACT.md}

\bibitem{mCANDIDATES}
FloatingPragma (2026). \emph{Concrete repair-law candidates}. Candidate
register.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/CANDIDATES.md}

\bibitem{mPREMISES}
FloatingPragma (2026). \emph{OPH reverse-engineering table: model and theorem
premises}. Generated premise map and concrete theorem bindings.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/PREMISES.md}

\bibitem{mMAP}
FloatingPragma (2026). \emph{Machine-readable premise crosswalk}. Source of
the generated premise map.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/tracking/premise_register.json}

\bibitem{mINTERP}
FloatingPragma (2026). \emph{OPH interpretation and working terms}.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/docs/INTERPRETATION_AND_TERMS.md}

\bibitem{oREADME}
FloatingPragma (2026). \emph{Runnable finite observers}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/README.md}

\bibitem{oDERIV}
FloatingPragma (2026). \emph{Finite current-record controller: mathematical
account}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/DERIVATION.md}

\bibitem{oCAP}
FloatingPragma (2026). \emph{Native observer records reuse the existing capacity
construction}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/CAPACITY.md}

\bibitem{oQM}
FloatingPragma (2026). \emph{Quantum mixing of the native C/E repairs}. Model
document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/QUANTUM_MIXING.md}

\bibitem{oRR}
FloatingPragma (2026). \emph{Recovery with complete native records}. Model
document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/RECORD_RECOVERY.md}

\bibitem{oCOHERENT}
FloatingPragma (2026). \emph{Coherent continuation: native attachment and
nonselection}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/observer/COHERENT_CONTINUATION.md}

\bibitem{oSOURCESELECT}
FloatingPragma (2026). \emph{Selecting one source, interface and repair law}.
Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/SOURCE_SELECTION.md}

\bibitem{pREADME}
FloatingPragma (2026). \emph{P-dependent electromagnetic observation and native
feedback}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/p_response/README.md}

\bibitem{pDERIV}
FloatingPragma (2026). \emph{Charged realization, binary instrument and
feedback}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/p_response/DERIVATION.md}

\bibitem{pACCESS}
FloatingPragma (2026). \emph{Detector access and the conditional capacity
budget}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/p_response/ACCESS_CAPACITY.md}

\bibitem{pVALID}
FloatingPragma (2026). \emph{Validation of the native-record electromagnetic
adapter}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/p_response/VALIDATION.md}

\bibitem{sREADME}
FloatingPragma (2026). \emph{A scalar detector inside the native observer
protocol}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/scalar_response/README.md}

\bibitem{sDERIV}
FloatingPragma (2026). \emph{A native record coupled to a scalar field
instrument}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/scalar_response/DERIVATION.md}

\bibitem{sPRED}
FloatingPragma (2026). \emph{Conditional detector prediction}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/scalar_response/PREDICTION.md}

\bibitem{sCONT}
FloatingPragma (2026). \emph{Two reads of the retained field}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/scalar_response/CONTINUATION.md}

\bibitem{ssREADME}
FloatingPragma (2026). \emph{Scalar preparation, evolution and detection through
source controls}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/source_scalar/README.md}

\bibitem{ssDERIV}
FloatingPragma (2026). \emph{A finite source implementation of the two-site
scalar instrument}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/source_scalar/DERIVATION.md}

\bibitem{sdREADME}
FloatingPragma (2026). \emph{Execute the native C/E discrimination experiment
from source controls}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/source_discrimination/README.md}

\bibitem{sdDERIV}
FloatingPragma (2026). \emph{Source implementation of the complete C/E
discriminator}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/source_discrimination/DERIVATION.md}

\bibitem{prREADME}
FloatingPragma (2026). \emph{Fixed observer predictions and complete-count
comparison}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/d85db928b240224c1306ada12961f18d9df9d199/model/predictions/README.md}

\bibitem{prDERIV}
FloatingPragma (2026). \emph{Exact record-instrument discrimination}. Model
document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/predictions/DERIVATION.md}

\bibitem{ltREADME}
FloatingPragma (2026). \emph{N4 memory survives the observer's local tick
clock}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/local_tick/README.md}

\bibitem{ltSPEC}
FloatingPragma (2026). \emph{N4 occupation memory at observer-local ticks}.
Protocol specification.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/local_tick/SPEC.md}

\bibitem{csREADME}
FloatingPragma (2026). \emph{A common interacting observer process}. Model
document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/README.md}

\bibitem{csDERIV}
FloatingPragma (2026). \emph{One finite interacting source: derivation and
theorem attachments}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/DERIVATION.md}

\bibitem{csINF}
FloatingPragma (2026). \emph{The interacting feedback signal survives infinite
occupation}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/INFINITE_HISTORY.md}

\bibitem{csOCC}
FloatingPragma (2026). \emph{Occupation refinement of the interacting retained
history}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/OCCUPATION.md}

\bibitem{csOCCREF}
FloatingPragma (2026). \emph{Occupation limit of the same interacting observer
history}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/OCCUPATION_REFINEMENT.md}

\bibitem{csREAD}
FloatingPragma (2026). \emph{Infinite coordinate-sign readout}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/READOUT.md}

\bibitem{csSMOOTH}
FloatingPragma (2026). \emph{A smooth binary observer with a complete occupation
error bound}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/SMOOTH_REFINEMENT.md}

\bibitem{csCOMPOSE}
FloatingPragma (2026). The finite executable observer specification.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/COMPOSITION.md}

\bibitem{csSMOOTHHISTORY}
FloatingPragma (2026). Complete-output certificate for the continued smooth history.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/SMOOTH_HISTORY.md}

\bibitem{csQNATIVE}
FloatingPragma (2026). Coherent native repairs around one continuing quantum history.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/QUANTUM_NATIVE.md}

\bibitem{csMEMORY}
FloatingPragma (2026). Active encoded memory in the continuing observer.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/ENCODED_MEMORY.md}

\bibitem{csOBS}
FloatingPragma (2026). \emph{Hilbert space, modular clocks and operational
regions}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/OBSERVER_READOUTS.md}

\bibitem{csTHERM}
FloatingPragma (2026). \emph{A fixed thermal reference for the changing common
history}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/THERMAL_CLOCK.md}

\bibitem{csHIST}
FloatingPragma (2026). \emph{One complete history from local constraint
repair}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/HISTORY_REPAIR.md}

\bibitem{csSLOW}
FloatingPragma (2026). \emph{The existing three-dimensional response carrier on
actual observer records}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/SLOW_GEOMETRY.md}

\bibitem{csCAUS}
FloatingPragma (2026). \emph{Exact native causal cones}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/CAUSALITY.md}

\bibitem{csFIELDS}
FloatingPragma (2026). \emph{Field relations of the existing native repair}.
Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/NATIVE_FIELDS.md}

\bibitem{csCALIB}
FloatingPragma (2026). \emph{Testing the readouts before assigning physical
units}. Model document.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/CALIBRATION.md}

\bibitem{rNATIVE}
FloatingPragma (2026). Retained receipt of the exact native observer histories
and calculation checks.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/results/native.json}

\bibitem{rRUN}
FloatingPragma (2026). Retained receipt of the 20,000-trial sequential native
execution.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/results/run.json}

\bibitem{rCAP}
FloatingPragma (2026). Retained receipt of the record capacity and compact
controller.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/results/capacity.json}

\bibitem{rQM}
FloatingPragma (2026). Retained receipt of the quantum mixing and memory
certificates.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/results/quantum_mixing.json}

\bibitem{rRR}
FloatingPragma (2026). Retained receipt of complete-record recovery and
identifying schedules.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/results/record_recovery.json}

\bibitem{rP}
FloatingPragma (2026). Retained receipt of the $P$-dependent detector.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/p_response/results.json}

\bibitem{rLT}
FloatingPragma (2026). Retained receipt of the local-tick memory certificate.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/local_tick/results.json}

\bibitem{rSCALAR}
FloatingPragma (2026). Retained receipt of the scalar detector on four grids.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/scalar_response/results.json}

\bibitem{rSS}
FloatingPragma (2026). Retained receipt of the two-site source apparatus.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/observer/source_scalar/results.json}

\bibitem{rCS}
FloatingPragma (2026). Retained receipt of the common interacting source.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/results.json}

\bibitem{rSHARP}
FloatingPragma (2026). Retained receipt of the complete sharp-output
comparisons.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/sharp_output_results.json}

\bibitem{rBRIDGE}
FloatingPragma (2026). Retained receipt of the observer readouts and the
modular clock.
\url{https://github.com/FloatingPragma/observer-patch-holography/blob/r2056/model/common_source/bridge_results.json}

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\section*{AI Assistance Disclosure}
This research project used research-grade commercial models, including
Anthropic's Fable and OpenAI's GPT-6 Astra, for research support, software
development, editing, and synthesis. The authors are responsible for the
paper's claims, methods, and final text.

\end{document}