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\title{
\textbf{Psy02-A}\\[0.4em]
Conditional Scalar-Carrier Bounds and Operational Separation:\\
A Mathematical Framework for Evaluating Orch-OR Hypotheses
}

\author{
Ngo Cheung, MBBS, FHKAM(Psychiatry)\\
Independent Researcher, Hong Kong SAR, China\\
\texttt{info@cheungngomedical.com}
}

\date{}

\begin{document}
\maketitle

\begin{abstract}
A mathematical assessment of orchestrated objective reduction
requires separating the construction of a quantum carrier from
observable concentration, environmental coherence loss, measurement,
causal influence, and interpretation as conscious experience.
We develop a modular conditional framework motivated by Cheung's
scalar-carrier, response, and causal-support manuscripts, while
retaining their distinct models and proof boundaries.
For displaced Fock-normal states, we prove finite-family
concentration, ordered Weyl bounds, and exact residual-state
fluctuation identities with explicit number-moment constants.
An environmental moment inequality gives a time-uniform
concentration certificate, but an independently specified
trace-distance witness is required for coherence survival.
Explicit attenuation and dephasing examples show that concentration,
occupation, and a single accessible marginal do not determine
quantum-information preservation.
We construct a separate massless Wick-square response benchmark,
classify its declared contact ambiguity, and evaluate an invariant
nonzero test.
Finite channel schedules yield reference-preserving ancestor
bounds, whereas mechanism identification requires separated
complete-record laws.
A regularized finite-dimensional mass-density model supplies a
well-defined reduction semigroup and stochastic unraveling,
including the rate \(E_G/\hbar\); its mass assignments and physical
interpretation remain additional inputs.
The complete work plan and a claim-status ledger identify the
remaining carrier, biological, continuum, detector, causal, and
interpretive obligations.
No derivation of Orch-OR, neuronal quantum viability, quantum
gravity, or conscious experience is asserted.
\end{abstract}

\noindent\textbf{Keywords:}
Orch-OR; scalar carrier; Fock space; concentration;
quantum coherence; instruments; response normalization;
causal support; collapse models; identifiability.

\section{Introduction and statement repair}
\label{sec:introduction}

The Orch-OR proposal associates biologically orchestrated quantum
processes with a proposed gravity-related reduction mechanism and
an interpretation in terms of conscious experience
\cite{HameroffPenrose,Penrose,Diosi}.
These are distinguishable commitments.
A calculation concerning a free scalar observable does not, without
additional constructions, identify its carrier with a neuronal
degree of freedom, establish the relevant environmental dynamics,
or assign an experiential meaning to an event.

Cheung's Em06-A and Em06-B manuscripts supply the immediate
mathematical motivation for distinguishing scalar-only classical
targets, residual-state concentration, and normalized fluctuations
\cite{CheungEm06A,CheungEm06B}.
VII-H motivates normalization-independent response diagnostics,
while VII-J and VIII-C motivate a separation between scheduled
support and actual influence
\cite{CheungVIIH,CheungVIIJ,CheungVIIIC}.
En01-A, En01-B, En01-C, and Em05-A motivate the additional separation
of accessible observations from coherence, recovery, and
mechanism identification
\cite{CheungEn01A,CheungEn01B,CheungEn01C,CheungEm05A}.

\paragraph{Source boundary.}
The supplied landmark excerpts are documentary source records.
They are not treated as an independently audited collection of
complete proofs.
The results below are proved for newly specified objects.
When a bridge to a landmark's classical input map or to a biological
model is not constructed here, it is explicitly an assumption or an
unresolved obligation.
Similar notation does not identify different Hilbert spaces,
test classes, states, channels, prescriptions, or physical models.

\paragraph{The repaired claims.}
The following corrections are part of the mathematical framework,
not qualifications added after its conclusion.

\begin{table}[ht]
\centering
\small
\caption{Required repairs to the proposed landmark incorporation.}
\label{tab:repairs}
\begin{tabularx}{\linewidth}{@{}p{0.34\linewidth}X@{}}
\toprule
Proposed identification & Correct statement\\
\midrule
Concentration replaces decoherence-time analysis
&
Concentration controls specified observable laws.
A coherence or recovery witness needs its own dynamics and bound.\\
Residual number counts surviving degrees of freedom
&
\(n_\epsilon=\Tr(\rho_\epsilon N)\) is an excitation-number
expectation in a declared residual frame.\\
Decoherence is input variation or detector noise
&
These are particular perturbation models.
General environmental dynamics require a reduced channel or
system--environment evolution.\\
A scalar field replaces a spin or dipole carrier
&
This is a surrogate choice unless an operational correspondence
is proved for named states and observations.\\
Ordered Weyl expectations are sequential probabilities
&
Sequential probabilities require instruments.
An ordered expectation need not be a joint characteristic function.\\
Contact ambiguity is residual-state variance
&
These concern different objects:
response-distribution normalization and state-dependent moments.\\
A collapse timescale specifies a collapse model
&
A stochastic law, unconditional evolution, regularization,
outcomes, and compatibility conditions must also be supplied.\\
Scheduled dependence identifies the mechanism
&
Scheduled ancestors bound possible influence.
Mechanism identification requires separated accessible record laws.\\
\bottomrule
\end{tabularx}
\end{table}

\paragraph{Contribution and novelty boundary.}
The Fock estimates and channel inequalities use standard methods.
The contribution is their explicit assembly into a two-budget
certification framework, together with counterexamples and
interfaces that prevent invalid transfers between its modules.
The finite reduction model is a declared regularized comparison
model, not a derivation of a new physical collapse law.

\section{Carrier specification and finite realizations}
\label{sec:carrier}

\subsection{Fock conventions, domains, and number moments}

Let \(\mathfrak h\) be a separable complex Hilbert space whose inner
product is linear in the second argument.
Let
\[
 \mathcal H=\Fock(\mathfrak h)
 =\bigoplus_{n=0}^{\infty}\mathfrak h^{\otimes_{\mathrm s}n},
 \qquad
 N=\dd\Gamma(\One).
\]
On the finite-particle domain \(\Dfin\), use
\[
 [a(v),a^\dagger(w)]=\ip vw\,\One,
 \qquad
 \Phi(v)=a^\dagger(v)+a(v).
\]
The map \(v\mapsto\Phi(v)\) is real linear.
Define
\[
 A(v,w)=\Impart\ip vw,
 \qquad
 W(v)=e^{\ii\Phi(v)}.
\]
These conventions give
\begin{align}
 [\Phi(v),\Phi(w)]&=2\ii A(v,w)\One,
 \label{eq:CCR}\\
 W(v)W(w)&=e^{-\ii A(v,w)}W(v+w).
 \label{eq:Weyl}
\end{align}
Standard Fock and Weyl constructions are discussed in
\cite{Derezinski}; the domain argument used below invokes
the analytic-vector theorem \cite{Nelson}.

A normal state is represented by a positive trace-class operator
\(\rho\) with \(\Tr\rho=1\).
Its residual number moment is
\[
 n(\rho)=\Tr(\rho N)\in[0,\infty].
\]
For an unbounded self-adjoint \(B\),
\(\Tr(\rho B^2)\) means the spectral quadratic moment, which may be
finite even when an operator product is not trace class in a naive
sense.

\begin{lemma}[Field domains and the number estimate]
\label{lem:number}
For \(v\in\mathfrak h\), \(\Phi(v)\) is essentially self-adjoint
on \(\Dfin\).
Its closure satisfies, as a quadratic-form inequality on
\(D(N^{1/2})\),
\[
 \norm{\Phi(v)\psi}^2
 \le
 2\norm v^2\left(
      \norm{N^{1/2}\psi}^2+
      \norm{(N+\One)^{1/2}\psi}^2
            \right).
\]
Consequently, whenever \(n(\rho)<\infty\),
\begin{equation}
 \Tr\!\left(\rho\Phi(v)^2\right)
 \le (4n(\rho)+2)\norm v^2.
 \label{eq:number-bound}
\end{equation}
\end{lemma}

\begin{proof}
The creation and annihilation estimates are
\[
 \norm{a(v)\psi}\le\norm v\,\norm{N^{1/2}\psi},
 \qquad
 \norm{a^\dagger(v)\psi}
 \le\norm v\,\norm{(N+\One)^{1/2}\psi}.
\]
Apply \(\norm{x+y}^2\le2\norm x^2+2\norm y^2\).

If \(\psi\) has particle support at most \(m\), repeated application
of these estimates bounds
\[
 \norm{\Phi(v)^k\psi}
 \le
 (2\norm v)^k
 \sqrt{\frac{(m+k)!}{m!}}\norm\psi.
\]
The corresponding analytic-vector series converges for every
fixed argument. Thus \(\Dfin\) is a dense invariant domain of
analytic vectors, and the analytic-vector theorem establishes
essential self-adjointness.

The form estimate extends to \(D(N^{1/2})\) by graph-norm
approximation.
Expand a positive trace-class \(\rho\) in an orthonormal
eigenbasis and sum the form estimate.
Finite \(n(\rho)\) ensures that the relevant eigenvectors with
positive eigenvalue lie in \(D(N^{1/2})\).
The resulting bound is \eqref{eq:number-bound}.
\end{proof}

\subsection{Spacetime tests are not one-particle vectors}

A compact spacetime test \(f\) and its associated one-particle vector
\(v=\kappa(f)\) are different objects.
Support restrictions apply initially to \(f\), not automatically
to a position representation of \(v\).

For an explicit flat-space example, put
\[
 \omega_m(\bm k)=\sqrt{\abs{\bm k}^2+m^2},
 \qquad m\ge0,
\]
and, for real \(f\in C_c^\infty(\R^4)\), define
\[
 \widehat f(\omega,\bm k)
 =
 \int_{\R^4}
 e^{\ii(\omega t-\bm k\cdot\bm x)}
 f(t,\bm x)\,\dd t\,\dd^3\bm x,
\]
\[
 \kappa_m(f)(\bm k)
 =
 \frac{\widehat f(\omega_m(\bm k),\bm k)}
 {(2\pi)^{3/2}\sqrt{2\omega_m(\bm k)}}.
 \label{eq:test-map}
\]
The value at the single point \(\bm k=0\) is irrelevant to the
\(L^2\) class.
Coordinates in the following estimate are expressed in a fixed
reference unit.

\begin{proposition}[A concrete compact-test allowance]
\label{prop:compact-test}
The map \(\kappa_m:C_c^\infty(\R^4;\R)\to L^2(\R^3)\)
is well defined and satisfies
\begin{equation}
 \norm{\kappa_m(f)}
 \le
 \frac{\norm f_{L^1}+\norm{\Delta_{\bm x}f}_{L^1}}
      {2\sqrt2\,\pi}.
 \label{eq:test-bound}
\end{equation}
The same inequality bounds
\(\norm{\kappa_m(f)-\kappa_m(g)}\) with \(f-g\) on the right.

If \(f=(\partial_t^2-\Delta_{\bm x}+m^2)q\) for compact smooth
\(q\), then \(\kappa_m(f)=0\).
\end{proposition}

\begin{proof}
Integration by parts gives
\[
 (1+\abs{\bm k}^2)\abs{\widehat f(\omega,\bm k)}
 \le
 \norm f_{L^1}+\norm{\Delta_{\bm x}f}_{L^1}.
\]
Since \(\omega_m(\bm k)\ge\abs{\bm k}\),
\[
 \norm{\kappa_m(f)}^2
 \le
 \frac{(\norm f_{L^1}+\norm{\Delta_{\bm x}f}_{L^1})^2}
      {2(2\pi)^3}
 \int_{\R^3}
 \frac{\dd^3\bm k}
 {\abs{\bm k}(1+\abs{\bm k}^2)^2}.
\]
The integral is \(2\pi\), proving \eqref{eq:test-bound}.
Linearity gives the difference estimate.
The Fourier multiplier of the displayed wave operator vanishes
on \(\omega=\omega_m(\bm k)\), proving the last assertion.
\end{proof}

For a different background, classical metric input, switch, or
relational test prescription, the corresponding map \(\kappa_p\)
must be constructed separately.
The flat-space estimate does not prove a curved-background
or unrestricted gauge theorem.

\begin{definition}[Admissible carrier data]
\label{def:carrier}
An admissible carrier specification consists of:
\begin{enumerate}
\item a fixed \(\mathfrak h\), Fock representation, and field convention;
\item a named compact spacetime test class and a specified map
\(f\mapsto\kappa_p(f)\), or an explicit designation as an abstract
oscillator surrogate;
\item real offsets \(b_j\), vectors \(v_j\), and a displacement
direction \(d\), for \(j=1,\ldots,J\);
\item a normal residual state with finite number moment;
\item any common-support, plateau, and gauge restrictions used by
the input map.
\end{enumerate}
\end{definition}

\subsection{Carrier correspondence and the finite-dimensional obstruction}

\begin{definition}[Operational carrier correspondence]
A bounded-observation correspondence consists of a state class
\(\mathcal S_{\mathrm{bio}}\), an encoding channel
\(\mathcal E\), matched effects \(M_{\mathrm{bio}}\) and
\(M_{\mathrm{sc}}\), and a bound
\begin{equation}
 \sup_{\rho\in\mathcal S_{\mathrm{bio}}}
 \abs{
 \Tr(\rho M_{\mathrm{bio}})
 -
 \Tr(\mathcal E(\rho)M_{\mathrm{sc}})
 }
 \le\eta_{\mathrm{carrier}}.
 \label{eq:carrier-effect}
\end{equation}
The participating state and effect classes are part of the definition.
\end{definition}

This comparison does not automatically control an unbounded field
moment, an entire output channel, or reference-assisted recovery.
Those require stronger, separately named comparisons.

\begin{proposition}[Finite-dimensional CCR obstruction]
\label{prop:finite-CCR}
No finite-dimensional matrices \(Q,P\in M_d(\C)\), \(d\ge1\),
satisfy
\[
 [Q,P]=\ii c\,\One_d
\]
when \(c\ne0\).

For the oscillator truncated to
\(\operatorname{span}\{\ket0,\ldots,\ket{d-1}\}\), define
\[
 a_d=\sum_{n=1}^{d-1}\sqrt n\,\ketbra{n-1}{n},
 \quad
 Q_d=\frac{a_d+a_d^\dagger}{\sqrt2},
 \quad
 P_d=\frac{a_d-a_d^\dagger}{\ii\sqrt2}.
\]
Then
\begin{equation}
 [Q_d,P_d]
 =
 \ii\left(\One_d-d\ketbra{d-1}{d-1}\right).
 \label{eq:compressed-CCR}
\end{equation}
\end{proposition}

\begin{proof}
The trace of a commutator is zero, while
\(\Tr(\ii c\One_d)=\ii cd\ne0\).
Direct multiplication gives
\(
[a_d,a_d^\dagger]=\One_d-d\ketbra{d-1}{d-1}
\),
which implies \eqref{eq:compressed-CCR}.
\end{proof}

An approximate spin--scalar bridge must therefore specify its
restricted observations and its boundary or compression error.
It cannot assert an exact nonzero canonical commutator.

\subsection{Deterministic finite Fock approximations}

For \(\mathfrak h=\C^m\), let \(P_K\) project onto total number
at most \(K\), and let \(Q_K=\One-P_K\).
Introduce a visible failure state \(\ket{\perp}\).

\begin{theorem}[Flagged cutoff and its correct error regimes]
\label{thm:cutoff}
The map
\[
 \mathcal E_K(X)
 =
 P_KXP_K+\Tr(Q_KX)\ketbra{\perp}{\perp}
\]
is deterministic and CPTP.
The retained subspace has dimension
\[
 \dim P_K\mathcal H=\binom{m+K}{K}.
\]
For \(t_K=\Tr(\rho Q_K)\),
\[
 t_K\le\frac{n(\rho)}{K+1}.
\]
In the common output space containing the original Fock space
and the flag,
\begin{align}
 \dtr(\rho\oplus0,\mathcal E_K(\rho))
 &\le\sqrt{t_K}+\frac{t_K}{2},
 \label{eq:cutoff-general}\\
 \dtr(\rho\oplus0,\mathcal E_K(\rho))
 &=t_K
 \quad\text{if }[\rho,P_K]=0.
 \label{eq:cutoff-diagonal}
\end{align}
\end{theorem}

\begin{proof}
Use \(P_K\) as the low-block Kraus operator and
\(\ket{\perp}\bra{e_\alpha}\) for an orthonormal basis of the
high block. Their effects sum strongly to \(\One\).
The dimension follows from counting occupation vectors.

On \(Q_K\), \(N\ge K+1\), giving the tail bound.
Purify \(\rho\) to \(\ket\Psi\), and set
\(\ket{\Psi_P}=(P_K\otimes\One)\ket\Psi\).
A calculation in their two-dimensional span gives
\[
 \norm{
 \ketbra\Psi\Psi-\ketbra{\Psi_P}{\Psi_P}
 }_1
 =
 \sqrt{4t_K-3t_K^2}
 \le2\sqrt{t_K}.
\]
Partial trace contracts the trace norm.
The orthogonal flag contributes \(t_K/2\) to the half trace
distance, proving \eqref{eq:cutoff-general}.
When \([\rho,P_K]=0\), the difference is a positive high block
of trace \(t_K\) and a negative orthogonal flag block of the same
trace, proving \eqref{eq:cutoff-diagonal}.
\end{proof}

The exact linear tail error agrees with the number-diagonal
regime described in Cheung's En02-E
\cite{CheungEn02E}.
It must not be transferred unchanged to arbitrary coherent states.

\begin{proposition}[Independent ancillary transfer]
\label{prop:ancilla}
Let \(\rho,\sigma\) be ancillary states independent of a message
and its reference.
For a fixed channel \(\Lambda\), define
\[
 \mathcal N_\rho(X)=\Lambda(X\otimes\rho).
\]
Then
\[
 \ddiam(\mathcal N_\rho,\mathcal N_\sigma)
 \le\dtr(\rho,\sigma).
\]
Here and below \(\ddiam=\tfrac12\norm{\cdot}_\diamond\).
\end{proposition}

\begin{proof}
For every referenced input state \(\xi\), the input difference
is \(\xi\otimes(\rho-\sigma)\), of trace norm
\(\norm{\rho-\sigma}_1\).
Apply channel contraction and take the supremum.
\end{proof}

This is a common-interface ancillary statement.
It is not an automatic biological encoding or a dynamical
simulation theorem.

\section{Static concentration and normalized fluctuations}
\label{sec:static}

\subsection{Displacement-compatible targets}

For \(0<\epsilon\le1\), set
\[
 U_\epsilon=W(d/\sqrt\epsilon),
 \qquad
 \widetilde\rho_\epsilon
 =U_\epsilon\rho_\epsilon U_\epsilon^\dagger,
\]
\[
 \widehat O_{j,\epsilon}
 =b_j\One+\sqrt\epsilon\,\Phi(v_j),
 \qquad
 o_j=b_j+2A(d,v_j).
 \label{eq:observable}
\]
Thus the target is displacement compatible.
An arbitrary collection of numbers is not silently presumed
to arise from one common displacement.

By the Weyl convention,
\begin{equation}
 U_\epsilon^\dagger
 (\widehat O_{j,\epsilon}-o_j\One)
 U_\epsilon
 =
 \sqrt\epsilon\,\Phi(v_j).
 \label{eq:displacement}
\end{equation}
Write
\[
 n_\epsilon=\Tr(\rho_\epsilon N),
 \qquad
 C_\epsilon=4n_\epsilon+2.
\]

\begin{theorem}[Finite-family concentration]
\label{thm:concentration}
For each \(j\), let \(\mu_{j,\epsilon}\) be the spectral law
of \(\widehat O_{j,\epsilon}\) in
\(\widetilde\rho_\epsilon\).
Then
\begin{align}
 q_{j,\epsilon}^2
 &:=
 \Tr\!\left(
 \widetilde\rho_\epsilon
 (\widehat O_{j,\epsilon}-o_j\One)^2
       \right)
 =
 \epsilon\Tr(\rho_\epsilon\Phi(v_j)^2),
 \label{eq:exact-mse}\\
 q_{j,\epsilon}^2
 &\le\epsilon C_\epsilon\norm{v_j}^2,
 \label{eq:mse-bound}\\
 \abs{
 \Tr(\widetilde\rho_\epsilon\widehat O_{j,\epsilon})-o_j
 }
 &\le q_{j,\epsilon},
 \label{eq:mean-bound}\\
 \Var_{\widetilde\rho_\epsilon}
       (\widehat O_{j,\epsilon})
 &\le q_{j,\epsilon}^2,
 \label{eq:variance-bound}\\
 \Wtwo(\mu_{j,\epsilon},\delta_{o_j})
 &=q_{j,\epsilon}.
 \label{eq:W2-exact}
\end{align}
For \(r>0\),
\[
 \mu_{j,\epsilon}\{x:\abs{x-o_j}\ge r\}
 \le\frac{q_{j,\epsilon}^2}{r^2}.
\]

If \(A(v_i,v_j)=0\) for every pair, the family strongly commutes
and has a joint spectral law \(\mu_\epsilon\) on \(\R^J\), with
\begin{equation}
 \Wtwo(\mu_\epsilon,\delta_o)^2
 =
 \sum_{j=1}^J q_{j,\epsilon}^2
 \le
 \epsilon C_\epsilon
 \sum_{j=1}^J\norm{v_j}^2.
 \label{eq:joint-W2}
\end{equation}

The exact criterion for concentration of a selected coordinate
is
\[
 \epsilon\Tr(\rho_\epsilon\Phi(v_j)^2)\longrightarrow0.
\]
The condition \(\epsilon n_\epsilon\to0\) is sufficient for
fixed bounded vector families, but is not necessary.
\end{theorem}

\begin{proof}
Equation \eqref{eq:displacement} and unitary invariance of
spectral moments prove \eqref{eq:exact-mse}.
Lemma~\ref{lem:number} proves \eqref{eq:mse-bound}.
Cauchy--Schwarz gives the mean estimate.
Variance about the actual mean is at most the mean-square
error about a prescribed target.

There is only one coupling of a probability law with a point
mass, so its squared Wasserstein distance is its squared
distance moment.
Markov's inequality gives the tail bound.

If all symplectic pairings vanish, all one-parameter Weyl
groups commute by \eqref{eq:Weyl}; hence the self-adjoint
generators strongly commute.
The joint spectral theorem applies.
Its squared distance moment is the sum of the coordinate
moments, proving \eqref{eq:joint-W2}.
The final criterion is the exact identity
\eqref{eq:W2-exact}.
\end{proof}

\begin{corollary}[Uniform bounded-family version]
\label{cor:uniform}
For a fixed finite \(J\), suppose
\[
 \sup_{\epsilon,j}\norm{v_{j,\epsilon}}\le K,
 \qquad
 \epsilon n_\epsilon\to0.
\]
Then all coordinate mean errors, variances, mean-square errors,
and spectral distances to their displacement-compatible
targets vanish uniformly in \(j\).
For commuting families the joint Wasserstein distance also
vanishes.
\end{corollary}

\begin{proof}
The common upper bound is
\(\epsilon(4n_\epsilon+2)K^2\), which tends to zero.
\end{proof}

\subsection{Ordered expectations for noncommuting families}

Put
\[
 A_{ij}=A(v_i,v_j),\qquad
 V(t)=\sum_{j=1}^Jt_jv_j,\qquad
 a(t)=\sum_{i<j}t_it_jA_{ij}.
\]

\begin{theorem}[Ordered Weyl identity and bound]
\label{thm:ordered}
For arbitrary real \(t=(t_1,\ldots,t_J)\),
\begin{align}
 &\Tr\!\left(
 \widetilde\rho_\epsilon
 \prod_{j=1}^J e^{\ii t_j\widehat O_{j,\epsilon}}
       \right)
 \nonumber\\
 &\qquad=
 e^{\ii t\cdot o-\ii\epsilon a(t)}
 \Tr\!\left(
 \rho_\epsilon e^{\ii\sqrt\epsilon\,\Phi(V(t))}
      \right),
 \label{eq:ordered}\\
 &\abs{
 \Tr\!\left(
 \widetilde\rho_\epsilon
 \prod_j e^{\ii t_j\widehat O_{j,\epsilon}}
       \right)-e^{\ii t\cdot o}
 }
 \nonumber\\
 &\qquad\le
 \epsilon\abs{a(t)}
 +\sqrt{\epsilon C_\epsilon}\,\norm{V(t)}.
 \label{eq:ordered-bound}
\end{align}
For residual vacuum,
\begin{equation}
 \Tr\!\left(
 \widetilde\rho_\epsilon
 \prod_j e^{\ii t_j\widehat O_{j,\epsilon}}
       \right)
 =
 \exp\!\left(
 \ii t\cdot o-\frac{\epsilon}{2}\norm{V(t)}^2
                 -\ii\epsilon a(t)
       \right).
 \label{eq:ordered-vacuum}
\end{equation}
Under the bounded-family assumptions of
Corollary~\ref{cor:uniform}, convergence to \(e^{\ii t\cdot o}\)
is uniform on compact \(t\)-sets.
\end{theorem}

\begin{proof}
Repeated application of \eqref{eq:Weyl} produces the phase
\(-\ii\epsilon a(t)\) and the final direction
\(\sqrt\epsilon V(t)\).
Conjugation by \(U_\epsilon\) gives the target phase.

For self-adjoint \(B\), spectral calculus gives
\[
 \abs{\Tr(\rho(e^{\ii B}-\One))}
 \le
 \sqrt{\Tr(\rho B^2)}.
\]
Apply this with \(B=\sqrt\epsilon\,\Phi(V(t))\), use
Lemma~\ref{lem:number}, and combine with
\(\abs{e^{-\ii\epsilon a}-1}\le\epsilon\abs a\).
Vacuum normal ordering gives
\(\bra0W(v)\ket0=e^{-\norm v^2/2}\).
On compact \(t\)-sets, the vector norms and symplectic phase
have common bounds.
\end{proof}

This reproduces the mathematical form of the supplied Em06-B
ordered-correlation interface with the conventions fixed here
\cite{CheungEm06B}; it is not an independent audit of the
landmark's classical relational map.

\begin{proposition}[Ordered correlations need not be a joint law]
\label{prop:not-joint}
The ordered expression in \eqref{eq:ordered} need not be the
characteristic function of a classical joint probability law.
\end{proposition}

\begin{proof}
Take residual vacuum, zero targets, \(v_1=e\), \(v_2=\ii e\),
with \(\norm e=1\).
Then
\[
 f(t,s)
 =
 \exp\!\left(
 -\frac{\epsilon}{2}(t^2+s^2)-\ii\epsilon ts
       \right).
\]
A joint characteristic function must satisfy
\(f(-t,-s)=\overline{f(t,s)}\).
The displayed function violates this identity whenever
\(\sin(\epsilon ts)\ne0\).
\end{proof}

\subsection{Fluctuations and failure of a unique limit}

Define
\[
 \widehat F_{j,\epsilon}
 =
 \frac{\widehat O_{j,\epsilon}-o_j\One}{\sqrt\epsilon}.
\]

\begin{theorem}[Exact residual fluctuations]
\label{thm:fluctuations}
The fluctuation family has the exact residual representation
\[
 U_\epsilon^\dagger\widehat F_{j,\epsilon}U_\epsilon
 =\Phi(v_j).
\]
Its commutators, on the conjugated common core or as the
corresponding form identities, are
\[
 [\widehat F_{i,\epsilon},\widehat F_{j,\epsilon}]
 =2\ii A_{ij}\One.
\]
The unnormalized observables instead satisfy
\[
 [\widehat O_{i,\epsilon},\widehat O_{j,\epsilon}]
 =2\ii\epsilon A_{ij}\One.
\]

Concentration does not determine a unique fluctuation limit.
A sufficient additional condition for convergence of bounded
fluctuation expectations is trace-norm convergence of the
residual states in the fixed representation.
\end{theorem}

\begin{proof}
The representation follows directly from
\eqref{eq:displacement}; the commutators follow from
\eqref{eq:CCR}.
For every bounded operator \(B\),
\[
 \abs{\Tr((\rho_\epsilon-\rho_0)B)}
 \le\norm{\rho_\epsilon-\rho_0}_1\norm B.
\]
Apply this to fluctuation Weyl products or bounded spectral
functions.

For nonuniqueness, let \(\epsilon_k=1/k\), and alternate
\(\rho_{\epsilon_k}=\ketbra00\) and \(\ketbra11\)
in one mode.
Then \(n_{\epsilon_k}\le1\), so concentration holds.
For \(v=e\),
\[
 \bra0e^{\ii t\Phi(e)}\ket0=e^{-t^2/2},
 \qquad
 \bra1e^{\ii t\Phi(e)}\ket1=(1-t^2)e^{-t^2/2}.
\]
The two subsequences have distinct characteristic functions.
\end{proof}

\begin{remark}[Variance alone misses bias]
In one mode, a residual coherent state with real amplitude
\(c/(2\sqrt\epsilon)\) has
\[
 \E(\widehat O_\epsilon-o)=c,
 \qquad
 \Var(\widehat O_\epsilon)=\epsilon.
\]
Thus vanishing variance does not imply concentration about the
prescribed target.
A symmetric mixture of the two opposite amplitudes has zero
mean error but mean-square error \(c^2+\epsilon\).
\end{remark}

\begin{remark}[Residual and physical occupation]
The residual number is not the occupation of the displaced
state.
The displacement shifts annihilators by a scalar, giving the
safe estimate
\[
 \Tr(\widetilde\rho_\epsilon N)
 \le2n_\epsilon+\frac{2\norm d^2}{\epsilon}.
\]
Even displaced vacuum can therefore have occupation of order
\(\epsilon^{-1}\).
Neither quantity counts modes or surviving biological degrees
of freedom.
\end{remark}

\section{Environmental dynamics and the two-budget principle}
\label{sec:environment}

\subsection{A conditional time-uniform concentration budget}

Let \(s\in[0,T]\) be physical or model time.
Specify normal residual states \(\rho_{\epsilon,s}\), vectors
\(v_j(s)\), offsets, and displacement-compatible targets.
The actual state is
\[
 \widetilde\rho_{\epsilon,s}
 =
 U_{\epsilon,s}\rho_{\epsilon,s}U_{\epsilon,s}^\dagger.
\]

\begin{assumption}[Environmental moment certificate]
\label{ass:environment}
The chosen reduced dynamics preserves normality and finite
number moments.
The function
\(n_\epsilon(s)=\Tr(\rho_{\epsilon,s}N)\) is absolutely
continuous and satisfies, almost everywhere,
\[
 n_\epsilon'(s)
 \le\alpha(s)n_\epsilon(s)+\beta(s),
\]
where \(\alpha,\beta\) are integrable and \(\beta\ge0\).
These statements must be justified from the chosen dynamics;
they are not consequences of static concentration.
\end{assumption}

Define
\[
 A(s)=\exp\!\left(\int_0^s\alpha(r)\,\dd r\right),
 \qquad
 B(s)=A(s)\int_0^s\frac{\beta(r)}{A(r)}\,\dd r.
\]

\begin{theorem}[Time-uniform observable concentration]
\label{thm:time-concentration}
Under Assumption~\ref{ass:environment},
\[
 n_\epsilon(s)\le A(s)n_\epsilon(0)+B(s).
\]
If
\[
 K_T=\sup_{s\le T,j}\norm{v_j(s)}<\infty,
 \qquad
 \overline n_{\epsilon,T}
 =\sup_{s\le T}\{A(s)n_\epsilon(0)+B(s)\},
\]
then
\begin{equation}
 \sup_{s\le T,j}
 \Tr\!\left(
 \widetilde\rho_{\epsilon,s}
 (\widehat O_{j,\epsilon}(s)-o_j(s)\One)^2
       \right)
 \le
 \epsilon K_T^2(4\overline n_{\epsilon,T}+2).
 \label{eq:time-budget}
\end{equation}
The bound vanishes if
\(\epsilon\overline n_{\epsilon,T}\to0\).
\end{theorem}

\begin{proof}
Multiply the moment inequality by \(A(s)^{-1}\) and integrate.
Apply Theorem~\ref{thm:concentration} at each time.
\end{proof}

A bounded-generator theorem does not automatically justify this
assumption for an unbounded field generator.
Domain, moment, and conservativity arguments remain necessary
for such a model.

\subsection{A separately specified coherence witness}

Use the trace and channel distances
\[
 \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1,
 \qquad
 \ddiam(\Lambda,\Gamma)
 =\tfrac12\norm{\Lambda-\Gamma}_\diamond.
\]
For channels, the latter controls all referenced input states.
The relevant operational norm facts are summarized in
\cite{Watrous}.

Fix a dephasing channel \(\Delta\), an input state \(\rho\), and
an accessible channel \(\Lambda_s\).
Define
\begin{equation}
 D_s(\rho;\Lambda,\Delta)
 =
 \frac12\norm{\Lambda_s(\rho-\Delta\rho)}_1.
 \label{eq:coherence-witness}
\end{equation}
The basis, decomposition, or record algebra defining \(\Delta\)
is part of the specification.

\begin{theorem}[Operational meaning and stability of the witness]
\label{thm:witness}
For the witness \eqref{eq:coherence-witness},
\[
 D_s
 =
 \sup_{0\le M\le\One}
 \abs{
 \Tr(M\Lambda_s(\rho))
 -
 \Tr(M\Lambda_s(\Delta\rho))
 }.
\]
Further deterministic processing cannot increase \(D_s\).

Suppose
\[
 \ddiam(\Lambda_s,\Lambda_s^0)\le e_s,
 \qquad
 \dtr(\rho,\rho_0)\le p,
\]
with the same \(\Delta\).
Then
\begin{equation}
 \abs{D_s(\rho;\Lambda,\Delta)
       -D_s(\rho_0;\Lambda^0,\Delta)}
 \le2e_s+2p.
 \label{eq:witness-stability}
\end{equation}
Thus a survival certificate is
\[
 D_s(\rho_0;\Lambda^0,\Delta)-2e_s-2p
 \ge d_{\min}>0.
\]
\end{theorem}

\begin{proof}
The output difference is Hermitian and traceless.
Its positive and negative parts have equal trace, each equal
to half its trace norm.
The positive spectral projector attains the effect supremum.

Trace distance contracts under a channel.
For channel perturbations, each of the two normalized input
states contributes trace-norm error at most \(2e_s\).
After multiplying their sum by \(1/2\), the allowance is
\(2e_s\).
For preparation changes, contractivity of \(\Lambda_s\) and
\(\Delta\) gives
\[
 \tfrac12\norm{
 \Lambda_s[(\rho-\rho_0)-\Delta(\rho-\rho_0)]
 }_1
 \le\norm{\rho-\rho_0}_1=2p.
\]
The reverse triangle inequality proves
\eqref{eq:witness-stability}.
\end{proof}

This is the independently proved channel interface suggested
by Cheung's En01-A
\cite{CheungEn01A}.
It is not a statement about every possible definition of
coherence.

\begin{corollary}[A recovery certificate implies a witness allowance]
\label{cor:recovery}
On a declared finite code, suppose a decoder \(\mathcal R_s\)
satisfies
\[
 \ddiam(\mathcal R_s\Lambda_s,\id)\le r_s.
\]
For code states \(\rho,\Delta\rho\),
\[
 D_s(\rho;\Lambda,\Delta)
 \ge
 \dtr(\rho,\Delta\rho)-2r_s.
\]
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:witness} to
\(\mathcal R_s\Lambda_s\) and use contraction under the decoder.
\end{proof}

\subsection{An explicit attenuator and its separate budgets}

Let the system and bath have \(m\) oscillator modes.
A passive beam-splitter unitary has output annihilators
\[
 a_{j,\mathrm{out}}
 =
 \sqrt\eta\,a_j+\sqrt{1-\eta}\,b_j,
 \qquad 0\le\eta\le1.
\]
Prepare an independent bath state \(\tau\) with
\(\Tr(\tau b_j)=0\) and finite total number \(n_B\).
Tracing the bath defines a channel \(\mathcal A_\eta\).
This is a standard attenuation construction
\cite{HolevoWerner}; the needed identities are proved below.

\begin{proposition}[Attenuation moment and displacement identities]
\label{prop:attenuation}
For every finite-number input \(\rho\),
\[
 n(\mathcal A_\eta(\rho))
 =
 \eta n(\rho)+(1-\eta)n_B.
\]
Furthermore,
\[
 \mathcal A_\eta(U_\epsilon\rho U_\epsilon^\dagger)
 =
 W(\sqrt\eta\,d/\sqrt\epsilon)
 \mathcal A_\eta(\rho)
 W(\sqrt\eta\,d/\sqrt\epsilon)^\dagger.
\]
\end{proposition}

\begin{proof}
Expand \(a_{j,\mathrm{out}}^\dagger a_{j,\mathrm{out}}\).
The cross terms vanish because the bath is independent and
centered.
Sum over \(j\).
A displacement of the system transforms into a system
displacement of amplitude \(\sqrt\eta\) and a bath displacement.
The latter disappears under partial trace.
\end{proof}

For a vacuum bath and one mode, let
\[
 \rho_+=\ketbra{+}{+},
 \qquad
 \ket+=\frac{\ket0+\ket1}{\sqrt2},
\]
and let \(\Delta\) dephase in the number basis.

\begin{proposition}[Concentration and coherence have different trajectories]
\label{prop:attenuator-witness}
For the preceding input,
\[
 n(\mathcal A_\eta(\rho_+))=\frac{\eta}{2},
 \qquad
 D(\rho_+;\mathcal A_\eta,\Delta)=\frac{\sqrt\eta}{2}.
\]
With \(\eta(s)=e^{-\kappa s}\), \(\kappa>0\), the selected
observable concentration bound is
\[
 \epsilon(2e^{-\kappa s}+2)\norm v^2,
\]
whereas the witness is
\[
 D_s=\frac12e^{-\kappa s/2}.
\]
For \(0<d_{\min}\le1/2\), witness survival up to \(T\) is
equivalent to
\[
 T\le\frac2\kappa\log\frac1{2d_{\min}}.
\]
\end{proposition}

\begin{proof}
Vacuum attenuation maps
\[
 \ketbra01\longmapsto\sqrt\eta\,\ketbra01,
 \qquad
 \ketbra11\longmapsto
 \eta\ketbra11+(1-\eta)\ketbra00.
\]
The output difference between the coherent and dephased inputs
has off-diagonal entries \(\sqrt\eta/2\), hence eigenvalues
\(\pm\sqrt\eta/2\).
The remaining statements follow by substitution.
\end{proof}

Concentration can therefore improve while the independently
specified coherence witness decreases.
A concentration condition does not replace the survival
timescale.

\subsection{Counterexamples to occupation-only and marginal-only inference}

\begin{proposition}[Number-preserving complete dephasing]
\label{prop:dephasing}
In one mode,
\[
 n(\rho_+)=n(\Delta\rho_+)=\frac12,
\]
and both inputs satisfy the same number-based concentration
bound.
Yet
\[
 D(\rho_+;\id,\Delta)=\frac12,
 \qquad
 D(\rho_+;\Delta,\Delta)=0.
\]
Thus finite number and small scaled observable spread do not
imply preservation of this quantum witness.
\end{proposition}

\begin{proof}
Both states have number populations \(1/2,1/2\).
Their difference has eigenvalues \(\pm1/2\).
The dephasing channel annihilates that difference.
\end{proof}

\begin{proposition}[One accessible marginal does not determine recovery]
\label{prop:Bell}
The identity qubit channel and complete \(Z\)-dephasing produce
the same accessible output \(I_2/2\) for the system half of a
Bell input.
Their optimal deterministic referenced recovery errors are,
respectively,
\[
 0,\qquad\frac12
\]
in half-diamond distance to the identity.
\end{proposition}

\begin{proof}
Both channels fix \(I_2/2\).
Identity is recovered exactly.
Complete dephasing is entanglement breaking, so every decoder
leaves its Bell-input output separable across reference and
receiver.
A separable two-qubit state has Bell-projector expectation at
most \(1/2\), by Cauchy--Schwarz for pure product states and
convexity.
The Bell test therefore gives error at least \(1/2\).

For attainment,
\[
 \Delta_Z=\tfrac12(\id+\Ad_Z),
\]
so
\(
\ddiam(\Delta_Z,\id)\le1/2
\).
\end{proof}

This finite proof supplies the relevant distinction without
importing the separate extraction model of Em05-A
\cite{CheungEm05A}.

\section{Input, preparation, and measurement budgets}
\label{sec:budget}

\subsection{A declared input-control interface}

Let \(p\) collect classical metric data, switching data,
observation tests, and any other classical inputs.
A common representation and common admissible support class
must be fixed before comparing \(p\) and \(p'\).

\begin{assumption}[Input map certificate]
\label{ass:input}
For each selected observation, the constructed input map
provides targets \(o_j(p)\), vectors \(v_j(p)\), and bounds
\[
 \abs{o_j(p')-o_j(p)}\le\eta^o_j,
 \qquad
 \norm{v_j(p')-v_j(p)}\le\eta^v_j.
\]
Any exact gauge identity asserts equality only on its declared
common plateau.
An outside-plateau defect is included in \(\eta^o_j,\eta^v_j\)
or in a separately bounded operator remainder.
\end{assumption}

For the explicit flat test map, Proposition~\ref{prop:compact-test}
provides an instantiated \(\eta^v_j\).
For the full Em06-B relational input map, the corresponding
metric, switch, and plateau estimates require that manuscript's
complete definitions and constants
\cite{CheungEm06B}.
The present assumption does not pretend to reconstruct them
from a release summary.

\subsection{Preparation in a moment-compatible metric}

For a state \(\rho\), write
\[
 \norm B_{\rho,2}=\sqrt{\Tr(\rho B^\dagger B)}
\]
when finite, interpreted through the appropriate quadratic form.

\begin{definition}[Preparation allowance for a selected moment]
\label{def:preparation}
A preparation allowance \(p_j\) certifies
\begin{equation}
 \sqrt\epsilon\,\norm{\Phi(v_j)}_{\sigma,2}
 \le
 \sqrt\epsilon\,\norm{\Phi(v_j)}_{\rho,2}+p_j.
 \label{eq:prep-allowance}
\end{equation}
One sufficient construction uses common purifications
\(\Psi_\rho,\Psi_\sigma\) and
\[
 p_j
 =
 \sqrt\epsilon\,
 \norm{
 (\Phi(v_j)\otimes\One)
 (\Psi_\sigma-\Psi_\rho)
 }.
\]
\end{definition}

The sufficient construction follows from the triangle
inequality.
It controls a selected unbounded moment, unlike an
unqualified trace-distance preparation bound.

\begin{proposition}[Trace distance alone does not bound field moments]
\label{prop:unbounded-prep}
There are one-mode states \(\sigma_M\) with
\[
 \dtr(\sigma_M,\ketbra00)\to0
\]
but
\[
 \Tr(\sigma_M\Phi(e)^2)\to\infty.
\]
\end{proposition}

\begin{proof}
Take
\[
 \sigma_M
 =
 (1-M^{-1/2})\ketbra00+M^{-1/2}\ketbra MM.
\]
The trace distance is \(M^{-1/2}\).
Since
\(\bra M\Phi(e)^2\ket M=2M+1\),
the second moment is \(1+2\sqrt M\).
\end{proof}

For completeness, a trace-distance allowance becomes usable
when supplemented by a tail certificate.
Let
\[
 A_L=\min\{(B-o\One)^2,L^2\One\},
 \qquad
 T_\sigma(L)=\Tr\!\left(
 \sigma[(B-o\One)^2-A_L]
                       \right).
\]
If \(\dtr(\rho,\sigma)\le\delta\), then
\begin{equation}
 \Tr(\sigma(B-o\One)^2)
 \le
 \Tr(\rho(B-o\One)^2)+L^2\delta+T_\sigma(L).
 \label{eq:prep-tail}
\end{equation}
Indeed \(0\le A_L\le L^2\One\), and an effect expectation
changes by at most trace distance.

\subsection{Intrinsic and recorded observables}

Suppose the implemented self-adjoint observable, in the
declared residual frame, satisfies
\[
 B_j-o_j(p)\One
 =
 r_j\One+\sqrt\epsilon\,\Phi(v_j(p)+\delta v_j)+E_j,
 \label{eq:implementation}
\]
where
\[
 \abs{r_j}\le\eta^o_j,\qquad
 \norm{\delta v_j}\le\eta^v_j,\qquad
 E_j=E_j^\dagger,\qquad \norm{E_j}\le e_j.
\]
The bounded remainder preserves self-adjointness on the field
domain.
Let \(n_\sigma=\Tr(\sigma N)<\infty\).

Let \(X_j\) be its spectral outcome in \(\sigma\).
A recorded surrogate outcome \(Y_j\) has a specified coupling
\[
 Y_j=X_j+Z_j,
 \qquad
 \norm{Z_j}_{L^2}\le z_j.
\]
This representation may be independent additive detector noise
or a certified correlated readout error.

Finally, if a physical carrier is claimed, require a
same-quantity comparison between its recorded law
\(\nu_j^{\mathrm{bio}}\) and the surrogate recorded law:
\begin{equation}
 \Wtwo(\nu_j^{\mathrm{bio}},\nu_j^{\mathrm{sc}})
 \le c_j.
 \label{eq:carrier-W2}
\end{equation}
Both laws must have finite second moments.

\begin{theorem}[End-to-end scalar readout budget]
\label{thm:readout}
Under the preceding hypotheses, define
\begin{equation}
 \mathcal B_j
 =
 q_j+p_j+\eta^o_j
 +\sqrt{\epsilon(4n_\sigma+2)}\,\eta^v_j
 +e_j+z_j+c_j,
 \label{eq:complete-budget}
\end{equation}
where
\[
 q_j=\sqrt{\epsilon\Tr(\rho\Phi(v_j(p))^2)}.
\]
Then
\[
 \Wtwo(\nu_j^{\mathrm{bio}},\delta_{o_j(p)})
 \le\mathcal B_j.
\]
In the surrogate-only case set \(c_j=0\) and replace the
biological law by the surrogate law.

The mean error is at most \(\mathcal B_j\), and
\[
 \nu_j^{\mathrm{bio}}
 \{y:\abs{y-o_j(p)}\ge r\}
 \le\frac{\mathcal B_j^2}{r^2}.
\]
\end{theorem}

\begin{proof}
Use the triangle inequality for the state quadratic norm in
\eqref{eq:implementation}.
Definition~\ref{def:preparation} bounds the principal field
term; Lemma~\ref{lem:number} bounds the vector-change term.
The scalar and bounded-operator contributions are at most
\(\eta^o_j\) and \(e_j\).

The spectral distance to the target point mass is the
root mean-square error.
Minkowski gives
\[
 \norm{Y_j-o_j(p)}_{L^2}
 \le
 \norm{X_j-o_j(p)}_{L^2}+\norm{Z_j}_{L^2}.
\]
Finally apply the Wasserstein triangle inequality to
\eqref{eq:carrier-W2}.
Cauchy--Schwarz and Markov's inequality give the last claims.
\end{proof}

Every term in \eqref{eq:complete-budget} has the units of the
same scalar readout.
A dimensionless effect-probability error such as
\eqref{eq:carrier-effect} cannot simply be added to this
root mean-square budget.

\subsection{An additive-noise POVM and its exact cross term}

Let \(E_B\) be the spectral measure of a self-adjoint \(B\),
and let \(\lambda\) be a probability law for independent
detector noise.

\begin{proposition}[Classical additive-noise POVM]
\label{prop:noise}
The operator-valued measure
\[
 F_\lambda(S)
 =
 \int_{\R}\lambda(S-x)\,E_B(\dd x)
\]
is a POVM.
Its outcome law in a state \(\sigma\) is the convolution of
the spectral law of \(B\) with \(\lambda\).

For finite second moments and independent \(X,Z\),
\begin{equation}
 \E(X+Z-o)^2
 =
 \E(X-o)^2+\E Z^2
 +2(\E X-o)\E Z.
 \label{eq:noise-exact}
\end{equation}
If the noise is centered, the last term vanishes.
Without independence or a separately proved zero cross term,
only the Minkowski allowance is asserted.
\end{proposition}

\begin{proof}
Positivity, normalization, and countable additivity follow
from those of \(\lambda\) and the spectral measure, using weak
operator integration.
The convolution statement follows by taking state
expectations.
Expand the square and use independence for the cross term.
\end{proof}

This is a detector model.
It does not represent arbitrary environmental decoherence.

\subsection{Sequential measurement requires instruments}

An instrument with finite outcomes is a family of completely
positive trace-nonincreasing maps
\(\{\mathcal I_x\}\) whose sum is trace preserving
\cite{DaviesLewis}.

\begin{proposition}[Sequential operational law]
\label{prop:instrument}
For specified instruments
\(\mathcal I^{(1)},\ldots,\mathcal I^{(k)}\),
\[
 P(x_1,\ldots,x_k)
 =
 \Tr\!\left(
 \mathcal I^{(k)}_{x_k}\circ\cdots\circ
 \mathcal I^{(1)}_{x_1}(\rho)
       \right)
\]
is a probability law.
The law depends on the instruments, not merely on a list of
self-adjoint observables or their ordered Weyl expectations.
\end{proposition}

\begin{proof}
Complete positivity gives nonnegative outcome probabilities.
Summing the last outcome applies a trace-preserving map.
Repeat backwards through the sequence to obtain total
probability \(\Tr\rho=1\).
Different state-update maps with the same effects can give
different later probabilities.
\end{proof}

\section{A separate normalization-independent response benchmark}
\label{sec:response}

This section constructs a distinct free massless vacuum
benchmark in the spirit of VII-H
\cite{CheungVIIH}.
It is not a susceptibility of the displaced family in
Section~\ref{sec:static}, and its contact ambiguity is not
a residual-state second moment.
The Wick framework and distribution-extension principles are
standard \cite{BrunettiFredenhagen,HollandsWald}.

\subsection{Model and punctured response}

Work on four-dimensional Minkowski space, with \(c=1\), and write
\[
 z=(t,\bm x),\qquad s(z)=t^2-\abs{\bm x}^2.
\]
Use a free massless scalar vacuum with
\[
 \omega_0(\varphi(x)\varphi(y))=\hbar W(x-y),
 \qquad
 W(z)=-\frac1{4\pi^2}(s(z)-\ii0t)^{-1}.
\]
Let
\[
 Q(x)=:\!\varphi(x)^2\!:
\]
be vacuum-calibrated.
Fix the source sign so that the first retarded response is
\[
 R^\circ(z)
 =
 \frac{\ii}{\hbar}\Theta(t)
 \omega_0([Q(z),Q(0)]),
 \qquad z\ne0.
 \label{eq:response-definition}
\]

\begin{lemma}[Punctured Wick-square kernel]
\label{lem:response-kernel}
The declared first-order response satisfies
\begin{equation}
 R^\circ(z)
 =
 \gamma\,\Theta(t)\delta'(s(z)),
 \qquad
 \gamma=\frac{\hbar}{4\pi^3},
 \qquad z\ne0.
 \label{eq:cone-response}
\end{equation}
It is real, retarded, and homogeneous of degree \(-4\)
on the punctured space.
\end{lemma}

\begin{proof}
Vacuum Wick contraction has exactly two connecting pairings,
so
\[
 \omega_0(Q(z)Q(0))=2\hbar^2W(z)^2.
\]
These positive-frequency products are defined in the free
Wick framework.
On the nonzero future null cone, \(t>0\) and \(\dd s\ne0\).
The boundary-value identity
\[
 (u-\ii0)^{-2}-(u+\ii0)^{-2}
 =-2\ii\pi\delta'(u)
\]
therefore pulls back to \(s\).
It gives
\[
 W(z)^2-W(-z)^2
 =-\frac{\ii}{8\pi^3}\delta'(s(z))
 \quad(t>0).
\]
Substitute in \eqref{eq:response-definition}.
Away from the origin, the commutator is supported on the
nonzero null cones; the time cut creates no extra singular
product there.
Under positive dilation, \(s\) has degree two and
\(\delta'(s)\) degree \(-4\).
\end{proof}

\subsection{An explicit extension and complete ambiguity in its class}

Choose a smooth compactly supported function
\(\chi:[0,\infty)\to\R\), equal to one near zero.
For real \(F\in C_c^\infty(\R^4)\), define
\begin{equation}
\begin{split}
 \langle R_\chi,F\rangle
 =
 \frac{\gamma}{4}
 \int_0^\infty\int_{S^2}
 \bigg[
 &\frac{F(t,t\bm\omega)-\chi(t)F(0)}{t}\\
 &+\partial_rF(t,r\bm\omega)\big|_{r=t}
 \bigg]\dd\bm\omega\,\dd t.
\end{split}
 \label{eq:finite-part}
\end{equation}
The cutoff includes its chosen reference time unit.

\begin{lemma}[Existence and scaling of the reference extension]
\label{lem:extension}
Equation \eqref{eq:finite-part} defines a real retarded
distribution extending \eqref{eq:cone-response}.
Its scaling degree at the origin is four.
\end{lemma}

\begin{proof}
Near zero,
\(F(t,t\bm\omega)-F(0)=O(t)\) uniformly in \(\bm\omega\).
Both terms are therefore integrable; away from zero all
supports are bounded.
On a fixed compact support, the formula is bounded by a
finite smooth-test seminorm, so it is a distribution.

For a test supported away from zero, use radial variables
\(r=\abs{\bm x}\) and \(u=t^2-r^2\).
The identity
\[
 \int_0^\infty r^2\delta'(t^2-r^2)
 F(t,r\bm\omega)\,\dd r
 =
 \frac{F(t,t\bm\omega)}{4t}
 +
 \frac{\partial_rF(t,r\bm\omega)|_{r=t}}4
\]
shows agreement with the punctured kernel.
Support is contained in the future null cone and the origin.

For \(F_\lambda(z)=\lambda^{-4}F(z/\lambda)\),
a change of variables gives
\[
 \langle R_\chi,F_\lambda\rangle
 =
 \lambda^{-4}
 \left[
 \langle R_\chi,F\rangle
 +\gamma\pi F(0)\log\lambda
 \right].
\]
The logarithm does not increase scaling degree above four.
The nonzero homogeneous punctured restriction has degree
\(-4\), so the scaling degree is exactly four.
\end{proof}

\begin{definition}[Permitted response prescription class]
\label{def:response-class}
A permitted response is a real retarded distribution \(R\)
on \(\R^4\) that:
\begin{enumerate}
\item agrees with \eqref{eq:cone-response} away from the origin;
\item has scaling degree at most four at the origin;
\item is used as a translation-covariant relative-coordinate
kernel \(R(x-y)\).
\end{enumerate}
No larger prescription or state class is included.
\end{definition}

\begin{theorem}[Complete contact ambiguity]
\label{thm:contact}
Every permitted response has the form
\[
 R=R_\chi+c\hbar\delta_0,
 \qquad c\in\R.
\]
Consequently, for every real test \(F\) satisfying \(F(0)=0\),
\[
 \langle R,F\rangle=\langle R_\chi,F\rangle.
\]
\end{theorem}

\begin{proof}
The difference \(D=R-R_\chi\) is supported at the origin.
A finite-order distribution supported at one point is a
finite sum of derivatives of the delta distribution.
This follows by subtracting a sufficiently high Taylor jet
of a test at the point and using the finite-order continuity
estimate.

In four dimensions, \(\partial^\alpha\delta_0\) has scaling
degree \(4+\abs\alpha\).
The common scaling-degree restriction therefore permits
only \(\delta_0\).
Reality makes its coefficient real.
Since \(\hbar>0\), write that coefficient as \(c\hbar\).
Evaluation on \(F\) gives the last assertion.
\end{proof}

If the prescription class permitted higher scaling degree,
derivatives of \(\delta_0\) could appear.
The condition \(F(0)=0\) would then not annihilate every
ambiguity.

\subsection{An exactly evaluated invariant}

Choose \(t_0>a>0\), and define
\[
 \rho_{\mathrm b}(u)=
 \begin{cases}
 e^{-1/(1-u^2)},&\abs u<1,\\
 0,&\abs u\ge1,
 \end{cases}
\]
\[
 b(t)=
 \frac{\rho_{\mathrm b}((t-t_0)/a)}
 {\sqrt{a\int_{-1}^1\rho_{\mathrm b}(u)^2\,\dd u}}.
\]
Then
\[
 \int b(t)^2\,\dd t=1,\qquad
 \int t\,b(t)^2\,\dd t=t_0.
\]
Set
\begin{equation}
 F_*(t,\bm x)
 =
 (t^2-\abs{\bm x}^2)b(t)b(\abs{\bm x}).
 \label{eq:Fstar}
\end{equation}
The radial factor vanishes near \(\bm x=0\), so this is a
smooth compact test.

\begin{theorem}[Nonzero prescription-independent response]
\label{thm:response-invariant}
For every permitted response,
\begin{equation}
 \langle R,F_*\rangle
 =
 -\frac{\hbar t_0}{2\pi^2}\ne0.
 \label{eq:response-value}
\end{equation}
\end{theorem}

\begin{proof}
The test vanishes near the origin, so its response is
independent of \(c\).
On \(r=t\),
\[
 F_*(t,t\bm\omega)=0,
 \qquad
 \partial_rF_*(t,r\bm\omega)|_{r=t}
 =-2t\,b(t)^2.
\]
Equation \eqref{eq:finite-part} gives
\[
 \langle R,F_*\rangle
 =
 \frac{\gamma}{4}(4\pi)
 \int_0^\infty -2t\,b(t)^2\,\dd t
 =
 -2\pi\gamma t_0.
\]
Substitute \(\gamma=\hbar/(4\pi^3)\).
\end{proof}

The normalization and source sign are essential.
Changing them changes the displayed coefficient.
The value is a response pairing, not an outcome probability.

\subsection{Delayed product realization and robustness}

For source \(h\) and readout \(f\), define the relative test
\[
 F_{f,h}(z)
 =
 \int_{\R^4}f(y+z)h(y)\,\dd^4y.
\]
Let \(h_\delta\) be a nonnegative smooth mollifier of integral
one, supported in the Euclidean ball of radius \(\delta\)
around zero, and take \(f=F_*\).

\begin{proposition}[A delayed nonzero experiment]
\label{prop:delayed}
For sufficiently small \(\delta>0\), the source and readout
supports are strictly time separated and
\[
 F_{f,h_\delta}(0)=0,
 \qquad
 \langle R,F_{f,h_\delta}\rangle
 \longrightarrow
 -\frac{\hbar t_0}{2\pi^2}.
\]
Thus the product experiment is prescription independent and
nonzero for sufficiently small \(\delta\).
\end{proposition}

\begin{proof}
The support of \(f\) has time coordinate at least \(t_0-a>0\).
For \(\delta<(t_0-a)/2\), the source occurs strictly earlier,
and source and readout do not overlap.
Therefore
\[
 F_{f,h_\delta}(0)=\int f(y)h_\delta(y)\,\dd y=0.
\]
Mollification converges to \(f\) in every smooth-test seminorm
on one common compact support.
Distributional continuity and
Theorem~\ref{thm:response-invariant} prove the claim.
\end{proof}

For an explicit allowance, suppose the relative tests have
time support in \([d,T]\), \(0<d<T\).
Define a \(C^1\) seminorm using the supremum of the test and
its spatial gradient.
Equation \eqref{eq:finite-part} gives
\[
 \abs{\langle R,G\rangle}
 \le
 C_{d,T}\norm G_{C^1},
 \qquad
 C_{d,T}
 =
 \abs\gamma\pi\left(\log(T/d)+T-d\right),
\]
for tests with this time support.
Taylor's theorem gives
\[
 \norm{F_{f,h_\delta}-f}_{C^1}
 \le\delta\norm f_{C^2},
\]
where the \(C^2\) seminorm includes the gradient and Hessian
bounds needed for translations.

\begin{corollary}[Robust nonzero response certificate]
\label{cor:response-robust}
Suppose preparation, test implementation, and detector
calibration together contribute a certified response-unit
error \(\eta_{\mathrm{response}}\).
If
\[
 \frac{\hbar t_0}{2\pi^2}
 >
 C_{d,T}\delta\norm f_{C^2}
 +\eta_{\mathrm{response}},
\]
then the implemented response is strictly negative.
\end{corollary}

\begin{proof}
Subtract the complete error allowance from the absolute
reference value.
\end{proof}

A finite instrument for measuring this Wick-square response
has not been constructed merely by defining a response
distribution.
A transfer to a different state, mass, geometry, or biological
carrier requires a new comparison theorem.

\subsection{Finite-bank nuisance separation}

\begin{proposition}[Linear nuisance annihilation]
\label{prop:bank}
Let a response bank be \(r\in\R^J\), with complete permitted
nuisance class \(r\mapsto r+Ba\).
A linear readout \(w^{\mathsf T}r\) is nuisance invariant
exactly when
\[
 B^{\mathsf T}w=0.
\]
An invariant readout with nonzero response exists exactly
when
\[
 r\notin\operatorname{range}(B).
\]
If component errors are bounded by \(\eta_i\), the readout
error is at most
\[
 \sum_i\abs{w_i}\eta_i.
\]
\end{proposition}

\begin{proof}
Invariance is
\(w^{\mathsf T}Ba=0\) for every \(a\).
The orthogonal complement of \(\operatorname{range}(B)\)
is \(\ker B^{\mathsf T}\).
If \(r\) lies outside the range, its orthogonal projection
onto that kernel provides a nonzero invariant response.
The error allowance is the triangle inequality.
\end{proof}

An incomplete nuisance catalogue cannot certify invariance
under nuisances that were never included.

\section{Causal support and mechanism identification}
\label{sec:causal}

\subsection{A complete finite schedule}

Let \(X\) be a finite site set with finite-dimensional local
Hilbert spaces.
All persistent quantum memories, classical controllers,
environments retained for later use, and record registers
must be included in the declared schedule.

At tick \(r\), let \(\mathcal B_r\) be a partition of \(X\),
and specify a channel
\[
 \Phi_r=\bigotimes_{B\in\mathcal B_r}\Phi_{r,B}
\]
on its fixed block spaces.
Adaptive programs can be represented only after their
controllers and conditional operations have been included
in the specification.

For endpoint \(Y\subseteq X\), define backwards ancestor sets:
\begin{align}
 N_{t,t}(Y)&=Y,\\
 N_{u,t}(Y)
 &=
 \bigcup\{B\in\mathcal B_{u+1}:
 B\cap N_{u+1,t}(Y)\ne\varnothing\},
 \qquad u<t.
 \label{eq:ancestors}
\end{align}
Write \(\mathcal A(S)\) for operators supported on \(S\).

\begin{theorem}[Reference-preserving scheduled support]
\label{thm:support}
For every endpoint observable \(A\in\mathcal A(Y)\),
\[
 (\Phi_{u+1}^*\circ\cdots\circ\Phi_t^*)(A)
 \in\mathcal A(N_{u,t}(Y)).
\]
Consequently, the endpoint channel factors through the
input reduction on \(N_{u,t}(Y)\).
If two input states, possibly with a spectator reference,
have equal reductions on \(N_{u,t}(Y)\) together with that
reference, their endpoint-reference outputs are equal.

In particular, a local trace-preserving intervention supported
outside \(N_{u,t}(Y)\) cannot influence this endpoint.
\end{theorem}

\begin{proof}
A block disjoint from the current support acts on an identity
factor.
Its adjoint is unital and preserves that identity.
A block meeting the support can enlarge it only within the
same block.
Iterating proves the support assertion.

All endpoint observables therefore pull back to the ancestor
algebra.
Equal input reductions give equal expectations of every
pullback, hence equal endpoint states.
The argument tensors with the identity on a reference.

For explicit factorization, choose any fixed state on the
nonancestor sites, insert it after the input reduction, and
then apply the endpoint channel.
The equality of reductions proves that this constructed
channel agrees with the original one on every input.
A local trace-preserving intervention outside the ancestor
set preserves its reduction, including the reference.
\end{proof}

This independently proves the relevant finite support
statement associated with VIII-C and VII-J
\cite{CheungVIIIC,CheungVIIJ}.
It does not identify the ancestors with a geometric causal
past.

If \(X\) has a metric and every block has diameter at most
\(\Delta\), then
\[
 N_{u,t}(Y)
 \subseteq
 \{x:d_X(x,Y)\le(t-u)\Delta\}.
\]
The proof is the triangle inequality applied at each
backwards step.

\begin{proposition}[Approximate no-influence allowance]
\label{prop:signal-error}
Suppose an actual endpoint channel \(\Gamma\) is within
half-diamond distance \(e\) of a specified channel
\(\Gamma_0\).
If \(\Gamma_0\) gives identical referenced outputs for two
allowed inputs, then their actual outputs have trace
distance at most \(2e\).
\end{proposition}

\begin{proof}
Insert the common ideal output between the two actual
outputs and use the triangle inequality.
\end{proof}

\subsection{Scheduled ancestors need not be actual influence}

\begin{example}[A scheduled path that cancels]
Apply SWAP on registers \(A,B\), then apply the same SWAP
again.
The composite channel is the identity.
The backwards scheduled ancestor set for endpoint \(B\)
contains both registers, but the final \(B\) state is its
initial state and is unaffected by an initial intervention
on \(A\).
\end{example}

\begin{example}[Marginals do not certify a joint endpoint]
Let an input bit be \(b\), draw an independent uniform bit
\(r\), and output
\[
 (Y_1,Y_2)=(r,r\oplus b).
\]
Each individual output is uniform and input independent.
The joint output determines \(b\) exactly.
This classical stochastic map is also a quantum
measure-and-prepare channel.
Individual endpoint certificates do not replace a joint
endpoint certificate.
\end{example}

\begin{proposition}[Conditional physical causal containment]
\label{prop:physical-causal}
Suppose an independently supplied physical embedding assigns
events to the schedule and a transitive causal relation.
Assume every allowed dependency through every local channel
is certified to respect that relation.
Then scheduled ancestors, and hence actual influences, lie
in the physical causal past of the endpoint.
\end{proposition}

\begin{proof}
Every scheduled dependency path is a finite composition of
certified causal steps.
Transitivity gives containment.
Theorem~\ref{thm:support} places actual influence inside the
scheduled set.
\end{proof}

Neither the physical embedding nor its causal relation is
derived from the support theorem.

\subsection{Observational equivalence}

Let \(P_0,P_1\) be complete attempted-run record laws.
Failures, heralding, intervention settings, and missing
records are included rather than silently postselected away.

\begin{theorem}[Mechanism non-identifiability]
\label{thm:nonidentifiable}
If \(P_0=P_1\), every randomized test
\(\varphi\in[0,1]\), declaring model \(1\), satisfies
\[
 \alpha+\beta=1,
 \qquad
 \alpha=\E_0\varphi,\quad
 \beta=\E_1(1-\varphi).
\]
More generally,
\[
 \alpha+\beta\ge1-\TV(P_0,P_1).
\]
\end{theorem}

\begin{proof}
Write
\[
 \alpha+\beta
 =1+\E_0\varphi-\E_1\varphi.
\]
The difference of expectations of a \([0,1]\)-valued function
is at most total variation.
\end{proof}

\begin{example}[Different mechanisms with identical restricted records]
For each intervention value \(d\), a quantum model prepares
\[
 \ket{\psi_d}
 =
 \sqrt{1-p_d}\ket0+\sqrt{p_d}\ket1
\]
and measures in the \(Z\) basis.
An alternative model samples a classical Bernoulli variable
with parameter \(p_d\).
For the declared accessible records, the models have
identical laws for every \(d\), despite different internal
descriptions.
Additional phase-sensitive measurements could distinguish
some such models, but are not part of this experiment.
\end{example}

Thus an intervention effect on an endpoint need not be a
mechanism-specific quantum signature.
The obstruction is experiment relative, not a universal
classical simulation theorem.

\subsection{A robust identifying statistic}

\begin{theorem}[Oriented complete-record separation]
\label{thm:statistical}
Let a predeclared statistic \(f\) take values in \([0,1]\).
Suppose ideal null and alternative families satisfy
\[
 \sup_{P\in\mathcal P_0}\E_Pf\le a,
 \qquad
 \inf_{P\in\mathcal P_1}\E_Pf\ge b,
 \qquad b>a.
\]
Suppose actual complete laws are within total variation
\(e_0,e_1\) of their corresponding ideal families.
Define
\[
 g=b-a-e_0-e_1.
\]
If \(g>0\), the midpoint threshold between
\(a+e_0\) and \(b-e_1\) gives, after \(L\) independent
identically distributed attempted runs from a fixed family
member,
\[
 \alpha,\beta\le e^{-Lg^2/2}.
\]
Thus
\[
 L\ge\frac2{g^2}\log\frac1{\alpha_*}
\]
suffices for each error to be at most \(\alpha_*\).
A simultaneous calibration failure probability at most
\(\beta_{\mathrm{cal}}\) gives unconditional error at most
\(\alpha_*+\beta_{\mathrm{cal}}\).
\end{theorem}

\begin{proof}
Total variation changes the expectation of \(f\) by at most
\(e_h\).
The common threshold is therefore at least \(g/2\) from
each allowed expectation.
Hoeffding's one-sided inequality \cite{Hoeffding} gives
\[
 \exp[-2L(g/2)^2].
\]
A union bound includes calibration failure.
\end{proof}

A lower bound on an absolute signal magnitude does not
supply one common oriented threshold if the alternative
sign varies.
This is the same distinction emphasized in En01-B
\cite{CheungEn01B}.

\subsection{Unbounded records require clipping and tail control}

Suppose setting \(i\) is sampled with probability \(p_i>0\),
and a scalar record \(Z_i\) has a certified second moment.
Let \(w_i\) be fixed contrast weights and
\[
 C_w=\max_i\frac{\abs{w_i}}{p_i}>0,
 \qquad
 c_R(z)=\max\{-R,\min\{z,R\}\}.
\]
Define
\[
 f(i,z)=\frac12+\frac{w_i c_R(z)}{2RC_wp_i}.
\]

\begin{proposition}[Bounded statistic from a mean contrast]
\label{prop:clipping}
The statistic lies in \([0,1]\), and
\[
 \E_hf
 =
 \frac12+\frac1{2RC_w}
 \sum_iw_i\E_hc_R(Z_i).
\]
If the ideal oriented mean-contrast gap is \(G>0\), and
\(\E_hZ_i^2\le M_{h,i}\), put
\[
 B=\sum_i\abs{w_i}(M_{0,i}+M_{1,i}).
\]
Then its clipped expectation gap is at least
\[
 \frac1{2RC_w}\left(G-\frac BR\right).
\]
For independent setting-independent erasure with observation
probability \(\gamma\), score erased runs by \(1/2\).
The gap is multiplied by \(\gamma\).
Actual total-variation implementation errors then subtract
\(e_0+e_1\).
\end{proposition}

\begin{proof}
The definition of \(C_w\) proves boundedness.
Average over the randomized setting.
Pointwise,
\[
 \abs{z-c_R(z)}
 \le\frac{z^2}{R},
\]
so each mean changes by at most \(M_{h,i}/R\).
Apply the weighted triangle inequality.
Independent erasure multiplies the nonconstant expectation
by \(\gamma\).
Finally use total-variation control.
\end{proof}

This proves a general form of the complete-record design
principle in En01-C
\cite{CheungEn01C}.
Independence, second-moment bounds, and the attempted-run
record law are substantive hypotheses.

\section{A separately declared reduction model}
\label{sec:collapse}

The relation \(\tau\sim\hbar/E_G\) is not a complete
stochastic or operational dynamics.
We now construct one finite regularized model in which
that rate appears.
The construction is related to the mass-density structure
of Diósi-type models \cite{Diosi}, but is not derived from
the scalar-carrier theorems or from quantum gravity.

\subsection{Regularized mass data and units}

Let \(\mathcal H_{\mathrm c}=\C^d\), with basis
\(\ket a\), \(a=1,\ldots,d\).
Assign smooth compactly supported real mass-density profiles
\(\mu_a(\bm x)\).
They may be required to be nonnegative and to have a fixed
total mass; the mathematical positivity below does not
require equal total mass.

Define
\begin{align}
 K_{ab}
 &=
 \frac G\hbar
 \int_{\R^3\times\R^3}
 \frac{\mu_a(\bm x)\mu_b(\bm y)}
      {\abs{\bm x-\bm y}}
 \,\dd^3\bm x\,\dd^3\bm y,
 \label{eq:mass-Gram}\\
 E_{G,ab}
 &=
 \frac G2
 \int_{\R^3\times\R^3}
 \frac{
 [\mu_a(\bm x)-\mu_b(\bm x)]
 [\mu_a(\bm y)-\mu_b(\bm y)]
 }
 {\abs{\bm x-\bm y}}
 \,\dd^3\bm x\,\dd^3\bm y.
 \label{eq:EG}
\end{align}
Here \(G\) has its gravitational units,
\(E_{G,ab}\) has energy units, and \(K_{ab}\) has inverse-time
units.
The smoothing profiles are explicit regulators, not
parameter-free point masses.

\begin{lemma}[Positive mass-density Gram matrix]
\label{lem:mass-positive}
The integrals in \eqref{eq:mass-Gram} are finite, and \(K\)
is real symmetric positive semidefinite.
There are real coefficients \(\ell_{\alpha a}\) such that
\[
 K_{ab}=\sum_{\alpha=1}^r\ell_{\alpha a}\ell_{\alpha b},
 \qquad r\le d.
\]
Moreover,
\[
 \frac{E_{G,ab}}{\hbar}
 =
 \frac12\sum_\alpha
       (\ell_{\alpha a}-\ell_{\alpha b})^2.
 \label{eq:EG-rate}
\]
\end{lemma}

\begin{proof}
The Coulomb singularity is locally integrable in three
dimensions; compact smooth profiles make the integrals finite.
For \(q=\sum_ac_a\mu_a\), the Coulomb quadratic form is
nonnegative:
its Fourier representation has nonnegative multiplier
\(4\pi/\abs{\bm k}^2\).
The integral is finite at zero and infinity for a smooth
compact profile.
Thus \(c^{\mathsf T}Kc\ge0\).
A real spectral factorization gives the coefficients.
Expanding the squared profile difference proves
\eqref{eq:EG-rate}.
\end{proof}

\subsection{Unconditional dynamics}

Let
\[
 L_\alpha=\sum_a\ell_{\alpha a}\ketbra aa,
 \qquad
 H=\sum_aE_a\ketbra aa.
\]
Define the finite generator
\begin{equation}
 \mathcal L_{\mathrm c}(\rho)
 =
 -\frac{\ii}{\hbar}[H,\rho]
 -\frac12\sum_\alpha[L_\alpha,[L_\alpha,\rho]].
 \label{eq:collapse-generator}
\end{equation}
This is a bounded finite-dimensional generator of the
standard completely positive form
\cite{GoriniKossakowskiSudarshan,Lindblad}.

\begin{theorem}[Finite regularized reduction semigroup]
\label{thm:collapse-channel}
The unique evolution
\(\mathcal C_s=e^{s\mathcal L_{\mathrm c}}\) is CPTP and
satisfies
\begin{equation}
 [\mathcal C_s(\rho)]_{ab}
 =
 e^{-\ii(E_a-E_b)s/\hbar}
 e^{-E_{G,ab}s/\hbar}
 \rho_{ab}.
 \label{eq:collapse-solution}
\end{equation}
All populations are preserved.
For \(E_{G,ab}>0\), the \(ab\) coherence has e-folding time
\[
 \tau_{ab}=\frac{\hbar}{E_{G,ab}}.
\]
\end{theorem}

\begin{proof}
Each matrix unit is an eigenvector of the generator,
giving \eqref{eq:collapse-solution}.
For complete positivity, let \(W_\alpha(s)\) be independent
real Brownian variables of variance \(s\), and set
\[
 V_s=
 \exp\!\left(
 -\frac{\ii Hs}{\hbar}
 -\ii\sum_\alpha L_\alpha W_\alpha(s)
       \right).
\]
Then
\[
 \mathcal C_s(\rho)=\E(V_s\rho V_s^\dagger),
\]
because Gaussian averaging multiplies the \(ab\) matrix unit
by
\[
 \exp\!\left[
 -\frac s2\sum_\alpha
       (\ell_{\alpha a}-\ell_{\alpha b})^2
       \right].
\]
An average of unitary channels is CPTP.
Finite linear-ODE uniqueness gives the specified evolution.
\end{proof}

The random-unitary representation proves the unconditional
channel.
It does not by itself select a conditional collapse history.
Different unravelings of an ensemble law need not have the
same physical interpretation.

\subsection{A normalized stochastic reduction unraveling}

For a normalized initial vector, specify the stochastic law
\begin{equation}
\begin{split}
 \dd\psi_s={}&
 \left[
 -\frac{\ii H}{\hbar}\dd s
 -\frac12\sum_\alpha
     (L_\alpha-m_\alpha(s)\One)^2\dd s
 \right]\psi_s\\
 &+
 \sum_\alpha
 (L_\alpha-m_\alpha(s)\One)\psi_s\,\dd W_\alpha(s),
 \qquad
 m_\alpha(s)=\bra{\psi_s}L_\alpha\ket{\psi_s}.
\end{split}
 \label{eq:collapse-SDE}
\end{equation}

\begin{theorem}[Conditional reduction and Born weights]
\label{thm:collapse-unraveling}
Equation \eqref{eq:collapse-SDE} has a global norm-preserving
solution.
Its ensemble density solves
\eqref{eq:collapse-generator}.

Suppose the vectors
\[
 \bm\ell_a=(\ell_{1a},\ldots,\ell_{ra})
\]
are pairwise distinct.
Then the populations
\(p_a(s)=\abs{\ip a{\psi_s}}^2\) converge almost surely to a
vertex of the probability simplex.
The probability of the vertex \(a\) is \(p_a(0)\).

If some \(\bm\ell_a\) coincide, only reduction to their
indistinguishable groups is guaranteed.
\end{theorem}

\begin{proof}
It\^o's rule gives zero drift and zero stochastic increment
for \(\norm{\psi_s}^2\).
The coefficients are smooth and bounded on the unit sphere.
Local finite-dimensional existence, norm preservation, and
compactness of that sphere give a global solution.

For \(P_s=\ketbra{\psi_s}{\psi_s}\), It\^o's rule gives
\[
 \dd P_s
 =
 \mathcal L_{\mathrm c}(P_s)\dd s
 +
 \sum_\alpha
 (L_\alpha P_s+P_sL_\alpha-2m_\alpha P_s)\dd W_\alpha.
\]
The nonlinear mean terms cancel from the drift.
The stochastic integrands are bounded, so their means
vanish.
Taking expectations proves the ensemble equation.

The populations satisfy
\[
 \dd p_a
 =
 2p_a\sum_\alpha
 (\ell_{\alpha a}-m_\alpha)\dd W_\alpha.
\]
They are bounded martingales and hence converge almost surely.
Let
\[
 C_{\alpha\beta}
 =
 \sum_ap_a
 (\ell_{\alpha a}-m_\alpha)
 (\ell_{\beta a}-m_\beta),
\]
\[
 V=
 \sum_ap_a\norm{\bm\ell_a}^2-\norm{\bm m}^2
 =\Tr C.
\]
Since
\(
\dd m_\alpha=2\sum_\beta C_{\alpha\beta}\dd W_\beta
\),
It\^o's rule gives
\[
 \frac{\dd}{\dd s}\E V(s)
 =
 -4\E\Tr(C(s)^2).
\]
Therefore
\[
 \E\int_0^\infty\Tr(C(s)^2)\,\dd s
 \le\frac{V(0)}4.
\]
The integral is finite almost surely.

The convergent populations make \(C(s)\) converge.
If the limiting \(V\) were positive, the limiting
\(\Tr(C^2)\) would be positive, contradicting integrability.
Hence the limiting population law is supported on a set of
identical vectors \(\bm\ell_a\).
With pairwise distinct vectors, that set is a singleton.
Finally,
\(\E p_a(\infty)=p_a(0)\) by bounded martingale convergence,
which gives the stated weights.
\end{proof}

Convergence concerns populations.
An overall phase can continue to evolve and is not asserted
to converge.

\subsection{Compatibility with a finite scalar surrogate}

\begin{proposition}[Number-preserving Fock extension]
\label{prop:collapse-Fock}
Embed the code basis into a finite set of oscillator
occupation vectors.
Extend \(H,L_\alpha\) diagonally by zero on its orthogonal
complement.
The resulting bounded operators define a CPTP semigroup
on the full finite-mode Fock space.
It preserves the code, finite number moments, and total
number expectation:
\[
 \Tr(\widehat{\mathcal C}_s(\rho)N)=\Tr(\rho N).
\]
Consequently, residual states subjected to this extension
retain the number-based concentration bounds of
Section~\ref{sec:static}.
\end{proposition}

\begin{proof}
The extended operators are bounded, mutually commuting, and
commute with \(N\).
The same random-unitary construction defines the normal
channel on trace-class operators.
Each unitary preserves number spectral subspaces and number
moments.
Average the exact number identity.
The concentration theorem then applies to the resulting
residual states.
\end{proof}

The assignment of mass profiles to oscillator occupation
vectors is an extra model input.
It is not the mass-density operator of a neuronal carrier
derived from a free scalar field.

\begin{proposition}[Environmental composition and implementation errors]
\label{prop:composition}
Finite compositions of the declared environmental,
reduction, and nonselective measurement channels are CPTP.
If corresponding implemented channels have half-diamond
errors \(e_1,\ldots,e_k\), the composite error is at most
\[
 \sum_{j=1}^ke_j.
\]
In finite dimensions, sums of bounded environmental and
reduction generators can also be obtained as the norm limit
of their alternating short-time channel products.
\end{proposition}

\begin{proof}
Composition preserves complete positivity and trace.
For the error, expand the difference of two composites into
a telescoping sum and use channel contraction.

For bounded finite generators \(\mathcal L_1,\mathcal L_2\),
\[
 e^{h\mathcal L_1}e^{h\mathcal L_2}
 =
 \One+h(\mathcal L_1+\mathcal L_2)+O(h^2).
\]
At \(h=s/n\), telescoping against
\(e^{h(\mathcal L_1+\mathcal L_2)}\) gives an error
\(O(nh^2)=O(n^{-1})\).
\end{proof}

The finite argument does not establish a continuum
unbounded-generator limit.

\subsection{Channels are not algebra automorphisms}

\begin{proposition}[Why commutator preservation is the wrong requirement]
\label{prop:not-automorphism}
A dissipative channel adjoint need not preserve products or
commutators.
For qubit dephasing with parameter \(0\le\lambda<1\),
\[
 \Lambda^*(X)=\lambda X,\quad
 \Lambda^*(Y)=\lambda Y,\quad
 \Lambda^*(Z)=Z,
\]
so
\[
 [\Lambda^*(X),\Lambda^*(Y)]
 =2\ii\lambda^2Z
 \ne
 \Lambda^*([X,Y]).
\]
For oscillator attenuation, the enlarged output annihilator
still obeys its canonical commutator because the bath
contribution restores it.
\end{proposition}

\begin{proof}
The qubit calculation uses \([X,Y]=2\ii Z\).
For attenuation,
\[
 [a_{\mathrm{out}},a_{\mathrm{out}}^\dagger]
 =
 \eta[a,a^\dagger]+(1-\eta)[b,b^\dagger]
 =\One.
\]
Discarding the bath and applying a reduced adjoint is a
different operation from retaining this enlarged algebra.
\end{proof}

Thus compatibility must distinguish the underlying algebra,
the dual reduced channel, and a system--environment
realization.
It must not demand that every reduced channel act as an
automorphism.

\paragraph{Remaining physical obligations.}
The finite construction proves a conditional mathematical
compatibility statement.
It does not establish a microscopic mass operator for
microtubules, a regulator-removal theorem, relativistic
causality, a gravitational derivation, or the Orch-OR
interpretation.
Experimental exclusions of particular gravity-related
collapse models likewise retain their regulator and
dynamical assumptions \cite{Donadi}.
A detector or environmental error allowance is not itself
a translation of such experimental bounds to a different
collapse model.

\section{Interpretive non-entailment}
\label{sec:interpretation}

\begin{definition}[Physical theory and added assignment]
Let \(T_{\mathrm{phys}}\) be the axioms specifying the
physical objects and predictions of a declared model.
Let \(E\ne\varnothing\) be a named event set.
An added consciousness assignment is a predicate
\(\mathcal C\subseteq E\), or a function from \(E\) to a
declared set of experiential labels.
\end{definition}

\begin{theorem}[Relative non-entailment of an unconstrained assignment]
\label{thm:nonentailment}
Suppose \(T_{\mathrm{phys}}\) has a model and contains no
premise constraining an added predicate \(\mathcal C\).
Then the physical premises do not determine that predicate.
In particular, there are two expansions with identical
physical objects and predictions but different assignments.
\end{theorem}

\begin{proof}
Take one model of \(T_{\mathrm{phys}}\).
Expand it once with \(\mathcal C_1=\varnothing\) and once
with \(\mathcal C_2=E\).
The physical reduct is unchanged, so every physical premise
and prediction has the same interpretation.
The assignments differ because \(E\ne\varnothing\).
\end{proof}

This is a relative logical result.
It does not state that consciousness can never be explained
physically.
If an additional theory constrains the assignment through
report, content, or functional premises, those premises
must be supplied and assessed.
They are not consequences of concentration or a collapse
timescale.

\section{The modular conditional certification theorem}
\label{sec:synthesis}

The preceding modules can be assembled without identifying
their underlying models.

\begin{theorem}[Conditional scalar-carrier and operational certificate]
\label{thm:main}
Fix a time horizon \(T\) and the following declared data:
\begin{enumerate}
\item admissible scalar-carrier data from
Definition~\ref{def:carrier}, with the environmental
certificate of Assumption~\ref{ass:environment};
\item moment-compatible preparation and input certificates,
a implemented-observable representation
\eqref{eq:implementation}, and detector and carrier-law
allowances as in Theorem~\ref{thm:readout};
\item a separately specified dephasing operation and accessible
channel family, with ideal witnesses \(D_s^0\) and errors
\(e_s,p\);
\item, if a response claim is made, the separate response model
and prescription class of Section~\ref{sec:response},
together with response-unit implementation errors;
\item, if a causal claim is made, a complete finite schedule,
and, for a geometric claim, an independently certified
physical embedding;
\item, if a mechanism-identification claim is made, complete
record families, a common oriented statistic, implementation
allowances, and independent attempted runs.
\end{enumerate}

Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item intrinsic scalar mean-square errors obey
\eqref{eq:time-budget}, and complete scalar readout errors
obey \eqref{eq:complete-budget};
\item ordered scalar expectations obey
\eqref{eq:ordered-bound}, while normalized fluctuations
retain their exact residual-state representation and CCR;
\item the independently specified coherence witness obeys
\[
 D_s\ge D_s^0-2e_s-2p;
\]
\item the response claim is normalization independent within
its declared class and robustly nonzero when
Corollary~\ref{cor:response-robust} applies;
\item actual finite-model influence is contained in scheduled
ancestors, with geometric containment only under the
additional embedding certificate;
\item mechanism identification is certified by
Theorem~\ref{thm:statistical} when its oriented gap is positive;
identical complete record laws instead obey
Theorem~\ref{thm:nonidentifiable}.
\end{enumerate}

If the finite regularized reduction model of
Section~\ref{sec:collapse} is additionally chosen, it has
the proved channel and conditional reduction laws there.
That choice is not derived from items (i)--(vi).

None of these conclusions identifies a biological carrier
or determines an unconstrained consciousness assignment.
\end{theorem}

\begin{proof}
For (i), apply Theorem~\ref{thm:time-concentration} to the
intrinsic residual family and Theorem~\ref{thm:readout}
to its declared implementation.
For (ii), apply Theorems~\ref{thm:ordered} and
\ref{thm:fluctuations}.
For (iii), use Theorem~\ref{thm:witness}.
For (iv), apply Theorem~\ref{thm:contact},
Theorem~\ref{thm:response-invariant}, and
Corollary~\ref{cor:response-robust}.
For (v), use Theorem~\ref{thm:support} and, only when its
extra hypotheses hold, Proposition~\ref{prop:physical-causal}.
For (vi), apply the two complete-record theorems.

The optional finite reduction statements are
Theorems~\ref{thm:collapse-channel} and
\ref{thm:collapse-unraveling}.
The carrier correspondence is not among the constructed
biological data, and Theorem~\ref{thm:nonentailment}
establishes the relative interpretive boundary.
\end{proof}

\begin{remark}[What a conjunction does not do]
A conjunction of valid certificates is not a correspondence
theorem.
In particular, the response distribution is not a channel
schedule, the scalar fluctuation algebra is not an
instrument, and a finite reduction comparison is not a
derived neuronal mechanism.
\end{remark}

\section{Mapping the six Orch-OR difficulties}
\label{sec:six}

\begin{enumerate}
\item \textbf{Identifying the carrier.}
The framework supplies named Fock states, compact-test maps,
field domains, commutation matrices, and finite compression
errors.
Identification with spins, dipoles, or microtubules still
requires a correspondence such as \eqref{eq:carrier-effect}
or a stronger channel-level bridge.

\item \textbf{Maintaining relevant quantum resources.}
The concentration budget and the coherence or recovery
budget are reported separately.
The attenuation and dephasing examples prove that one does
not determine the other.
Biological viability requires an environmental model for
the actual participating degrees of freedom.
The contrasting calculations of Tegmark and of Hagan,
Hameroff, and Tuszy\'nski concern different modeling
assumptions and are not replaced by a static
\(\epsilon n_\epsilon\) condition
\cite{Tegmark,HaganHameroffTuszynski}.

\item \textbf{Unconfirmed reduction physics.}
The finite regularized model proves that a chosen
mass-density kernel can define a consistent channel and
conditional process.
Its rate formula does not derive the profiles, regulator,
gravity hypothesis, or continuum physical law.
The Penrose proposal remains an additional physical
commitment \cite{Penrose}.

\item \textbf{Nonunique microtubule evidence.}
An intervention can alter a behavioral endpoint without
identifying an Orch-OR-specific mechanism.
Khan and colleagues report an epothilone-B effect on the
latency to anesthetic loss of righting reflex in rats
\cite{Khan}.
That endpoint is a specified behavioral proxy; the reported
drug contrast does not by itself supply the carrier,
collapse, or mechanism-separation certificates defined here.

\item \textbf{Molecular events and conscious content.}
Instruments, channels, and causal ledgers can specify a
physical chain of observations.
They do not automatically provide a dictionary from a
molecular event to an experiential content.
Such a dictionary is a further model component.

\item \textbf{Why a physical event should be experiential.}
An assignment unconstrained by the physical premises is
not selected by those premises.
The relative non-entailment theorem states this boundary
without claiming a universal impossibility of physical
explanation.
\end{enumerate}

\section{The complete work plan and completion criteria}
\label{sec:plan}

The plan is retained in full as eight work packages.
The proved modules replace vague targets by explicit
certificates.
Open bridges remain open rather than being declared solved
by a change of terminology.

\subsection*{Work package A: carrier audit and correspondence}

\textbf{Required data.}
Specify \(\mathcal H,\mathcal A,\rho_\epsilon\), the spacetime
test class, its one-particle map, displacement convention,
domains, support restrictions, and the commutation matrix.

\textbf{Completed here.}
Lemma~\ref{lem:number},
Proposition~\ref{prop:compact-test},
Proposition~\ref{prop:finite-CCR}, and
Theorem~\ref{thm:cutoff} establish an explicit scalar-surrogate
carrier audit.

\textbf{Remaining target.}
For a biological carrier, construct matched state and
observation classes and prove a carrier error.
For coherent processing, construct a referenced channel
comparison rather than only an effect comparison.

\textbf{Completion criterion.}
Either a proved correspondence with a declared error, or
an explicit designation as a scalar surrogate without
biological identification.

\subsection*{Work package B: concentration and ordered correlations}

\textbf{Required data.}
Finite residual number moments, displacement-compatible
targets, vectors, and a fixed finite family.

\textbf{Completed here.}
Theorems~\ref{thm:concentration},
\ref{thm:ordered}, and \ref{thm:fluctuations} give all constants,
the joint commuting law, the noncommuting ordered bound,
and the exact fluctuation representation.

\textbf{Remaining target.}
A unique fluctuation limit requires additional residual-state
information.
A landmark-specific relational claim requires source alignment
of its classical test map and plateau conditions.

\textbf{Completion criterion.}
A stated input family with finite constants and verified
uniformity.
No inference of coherence preservation from these bounds.

\subsection*{Work package C: environmental dynamics and two budgets}

\textbf{Required data.}
A reduced evolution or system--environment model,
a residual-frame identification, a moment certificate,
and a separately declared quantum witness.

\textbf{Completed here.}
Theorem~\ref{thm:time-concentration} proves the conditional
time-uniform estimate.
Propositions~\ref{prop:attenuation} and
\ref{prop:attenuator-witness} instantiate both budgets
for attenuation.
Theorem~\ref{thm:witness} gives operational error allowances.

\textbf{Remaining target.}
Derive the actual neuronal moment and witness dynamics,
including bath preparation, coupling, temperature,
memory, and accessible observations.
For unbounded generators, prove domain and moment preservation.

\textbf{Completion criterion.}
Report both selected-observable concentration and survival
or loss of the named quantum witness.

\subsection*{Work package D: input, preparation, and readout errors}

\textbf{Required data.}
Common representations, admissible input changes,
plateau restrictions, preparation control, and a detector
model in a compatible metric.

\textbf{Completed here.}
Theorem~\ref{thm:readout} proves an end-to-end scalar budget.
Proposition~\ref{prop:noise} constructs the additive-noise
POVM and states the exact cross term.
Proposition~\ref{prop:instrument} supplies sequential
probabilities for specified instruments.

\textbf{Remaining target.}
Instantiate the classical metric and switch bounds from
complete source definitions.
Construct any proposed physical detector.
Replace a probability-only carrier allowance by a
moment-compatible law comparison when RMS claims are made.

\textbf{Completion criterion.}
Every term controls the same stated quantity and carries
the correct units.
Outside-plateau errors and nonzero noise cross terms are
retained.

\subsection*{Work package E: normalization-independent response}

\textbf{Required data.}
Field, state, source sign, Wick-square calibration, test
space, and the complete prescription class.

\textbf{Completed here.}
Section~\ref{sec:response} constructs a punctured kernel,
a finite-part extension, the complete contact ambiguity
in its stated class, and an invariant nonzero benchmark.
It also proves a delayed product realization and a robust
response margin.

\textbf{Remaining target.}
Construct a physical readout and any proposed transfer to
another state, geometry, mass, carrier, or biological system.

\textbf{Completion criterion.}
The response value exceeds the complete response-unit error.
Prescription independence remains restricted to the
classified nuisance class.

\subsection*{Work package F: causal support and identifiability}

\textbf{Required data.}
Complete finite schedules, local channels, references,
joint endpoints, complete record laws, and any independent
physical causal embedding.

\textbf{Completed here.}
Theorem~\ref{thm:support} gives backwards support.
Propositions~\ref{prop:signal-error} and
\ref{prop:physical-causal} give conditional stability and
geometric containment.
Theorems~\ref{thm:nonidentifiable} and
\ref{thm:statistical} give the two possible mechanism verdicts.

\textbf{Remaining target.}
Design an actual experiment whose nuisance-enlarged null
and alternative record families are separated by one
common oriented statistic.

\textbf{Completion criterion.}
Either a robust mechanism-specific separation certificate
or an explicit observational-equivalence obstruction.

\subsection*{Work package G: reduction dynamics and compatibility}

\textbf{Required data.}
Regularized mass profiles, units, a conditional stochastic
law, an unconditional channel, domain and moment statements,
environmental composition, and causal conditions.

\textbf{Completed here.}
Theorems~\ref{thm:collapse-channel} and
\ref{thm:collapse-unraveling} construct a finite regularized
model.
Proposition~\ref{prop:collapse-Fock} proves one scalar-surrogate
compatibility statement.
Proposition~\ref{prop:composition} proves finite composition
and error control.

\textbf{Remaining target.}
Derive or independently justify physical mass data;
control regulator and continuum limits; certify locality
or a physical causal embedding; and compare the resulting
model with experiments in its own parameter class.

\textbf{Completion criterion.}
A well-defined proposed physical reduction dynamics with
its compatibility certificate, or an obstruction within
a specified class.
A timescale formula alone does not meet the criterion.

\subsection*{Work package H: interpretive boundary}

\textbf{Required data.}
A named physical theory and a separately stated assignment
to experience or conscious content.

\textbf{Completed here.}
Theorem~\ref{thm:nonentailment} proves relative non-entailment
when that assignment is unconstrained.

\textbf{Remaining target.}
Supply and test any proposed additional interpretive,
functional, report-based, or content-level premises.

\textbf{Completion criterion.}
State precisely which premises select the assignment.
Do not identify the remaining interpretive step with a
previously proved physical bound.

\subsection*{Recommended execution order}

The principal path is
\[
 \boxed{
 \text{carrier audit}
 \longrightarrow
 \text{static bounds}
 \longrightarrow
 \text{environmental extension}
 \longrightarrow
 \text{operational readout}
 \longrightarrow
 \text{identifiability}
 }.
\]
The response construction, reduction model, and interpretive
boundary are separate branches.
Their outputs enter the principal path only through
explicit comparison maps.

The earlier proposed ``Further development'' steps that
merely restated input control are replaced by the
instantiation tasks in work packages C and D.
Black-hole evaporation, Page curves, and holographic
reconstruction are not Orch-OR proof targets of this
manuscript and are not included as successor obligations.

\section{Discussion and conclusion}
\label{sec:conclusion}

The strongest conclusion established here is a controlled
scalar-surrogate and operational framework.
It provides:
\begin{itemize}
\item explicit static and time-dependent concentration bounds;
\item exact residual fluctuation identities;
\item a separate operational coherence budget;
\item a same-quantity input, preparation, and readout ledger;
\item a normalization-independent response benchmark;
\item finite causal-support and complete-record tests;
\item one well-defined finite regularized reduction comparison;
\item a precise relative interpretive boundary.
\end{itemize}

The framework does not establish that microtubules realize
its Fock carrier, that relevant quantum resources survive
in neurons, that a gravity-related reduction law occurs
there, or that a reduction event is conscious.
Those claims need the separately listed bridge hypotheses
and evidence.

No clinical recommendation follows.
In particular, the mathematical model does not recommend
microtubule-active drugs, anesthetic changes, or treatments
for any psychiatric or neurological condition.

The landmark incorporation is therefore constructive but
conditional.
It sharpens the mathematical obligations of an Orch-OR
model instead of presenting those obligations as already
discharged.

\section*{Declarations}

\paragraph{Funding.}
No specific funding is declared.

\paragraph{Competing interests.}
None declared.

\paragraph{Ethics.}
The present work is a mathematical analysis and involves no
new human or animal experimentation.

\paragraph{Data availability.}
No new empirical dataset is used.
The supplied landmark excerpts are documentary materials;
their complete manuscripts and any physical correspondence
require separate source and empirical audits.

\paragraph{Verification provenance.}
The manuscript contains analytic proofs and declared
comparison models.
It does not report newly executed simulations, experimental
reproduction, machine-checked proofs, or an executed
\LaTeX{} compilation.
The source is designed for \texttt{pdflatex}; two compilation
passes resolve internal references.

\appendix

\section{Claim-status and source ledger}
\label{app:ledger}

The labels in the following ledger have distinct meanings:
\begin{itemize}
\item \textbf{Source record:} a claim reported in the supplied
landmark excerpts;
\item \textbf{Proved here:} a result proved for the objects
defined in this manuscript;
\item \textbf{Assumption:} an input needed by a conditional
result;
\item \textbf{Open bridge:} a construction not supplied by
the present proofs.
\end{itemize}

\small
\begin{longtable}{@{}p{0.25\linewidth}p{0.20\linewidth}p{0.48\linewidth}@{}}
\caption{Status of the incorporated components.}
\label{tab:status}\\
\toprule
Component & Status & Exact boundary\\
\midrule
\endfirsthead
\toprule
Component & Status & Exact boundary\\
\midrule
\endhead

Em06-B scalar concentration and ordered interface
&
Source record; Fock form proved here
&
The source's full relational input map, classical metric
estimates, and plateau constants are not independently
audited by their restatement.\\[0.5em]

Fock domains and number estimate
&
Proved here
&
Uses the stated field normalization, analytic-vector
theorem, and finite residual number expectation.\\[0.5em]

Compact-test map
&
Proved here
&
Explicit flat-space mass-shell map only; not a construction
on an arbitrary metric or microtubule geometry.\\[0.5em]

Scalar--biological correspondence
&
Open bridge
&
Requires participating states, observations, an encoding
or comparison map, and an error in the relevant metric.\\[0.5em]

Time-uniform concentration
&
Conditional theorem
&
The environmental number-moment inequality and its domain
justification are inputs.\\[0.5em]

Attenuation two-budget example
&
Proved here
&
A specified oscillator and independent bath model, not
neuronal environmental dynamics.\\[0.5em]

Coherence witness and recovery distinction
&
Proved here
&
Relative to the selected dephasing and accessible channel;
does not exhaust every quantum resource.\\[0.5em]

Input and preparation allowances
&
Conditional theorem
&
Input-map constants and moment-compatible preparation
control must be instantiated.\\[0.5em]

Detector noise
&
Proved model
&
Independent additive noise gives a convolution POVM;
general dynamics are not identified with this noise.\\[0.5em]

Sequential probabilities
&
Proved here
&
Require named instruments; ordered Weyl expectations
alone are insufficient.\\[0.5em]

VII-H-type response
&
Benchmark proved here
&
Free massless vacuum, fixed source sign and Wick-square
calibration, scaling degree at most four.\\[0.5em]

Response transfer or physical detector
&
Open bridge
&
A response distribution does not itself construct an
instrument or a biological susceptibility.\\[0.5em]

VII-J/VIII-C-type support
&
Finite theorem proved here
&
Complete block-channel schedule; physical causal
containment needs an independent embedding.\\[0.5em]

Mechanism identification
&
Conditional certificate or obstruction
&
Requires complete record laws and a common oriented
statistic after nuisance and implementation errors.\\[0.5em]

Finite regularized reduction
&
Proved comparison model
&
Mass profiles, regularization, and interpretation are
declared inputs; no continuum physical law is derived.\\[0.5em]

Continuum collapse and relativistic compatibility
&
Open bridge
&
Finite complete positivity does not solve unbounded
domains, regulator removal, or physical causal embedding.\\[0.5em]

Consciousness assignment
&
Relative non-entailment proved here
&
Only for an assignment unconstrained by the stated physical
premises; not a universal impossibility theorem.\\

\bottomrule
\end{longtable}
\normalsize

\section{Mathematical debugging ledger}
\label{app:debug}

\begin{enumerate}
\item \textbf{Field normalization.}
The choice \(\Phi=a^\dagger+a\) gives vacuum variance
\(\norm v^2\), CCR \(2\ii\Impart\ip vw\), and the
number bound \(4n+2\).
A different convention changes these constants.

\item \textbf{Displacement compatibility.}
The targets are explicitly
\(o_j=b_j+2A(d,v_j)\).
This avoids assuming that arbitrary targets come from one
common coherent displacement.

\item \textbf{Unbounded products.}
Second moments are spectral quadratic moments.
Commutator statements are made on a common core or as form
identities, not as unjustified products of arbitrary
trace-class and unbounded operators.

\item \textbf{Strong commutation.}
Vanishing symplectic pairings imply commutation of the
entire one-parameter Weyl groups.
This, rather than a formal core commutator alone, justifies
the joint spectral law.

\item \textbf{Concentration criterion.}
Vanishing variance alone is insufficient.
The exact condition is vanishing mean-square error about
the prescribed target.

\item \textbf{Cutoff error.}
The exact tail probability is the state error only in the
block-diagonal regime.
Arbitrary coherence requires the square-root allowance.
Failures are retained rather than renormalized away.

\item \textbf{Preparation metrics.}
Trace-distance closeness alone does not bound an unbounded
field moment.
Weighted preparation control or a tail certificate is
required.

\item \textbf{Noise cross term.}
The cross term in \eqref{eq:noise-exact} is retained.
Its removal requires centering with independence or a
separate zero-cross-term proof.

\item \textbf{Witness error constants.}
Changing a channel affects both the coherent and dephased
inputs; the resulting allowance is \(2e_s\), not \(e_s\).
Preparation changes similarly contribute \(2p\).

\item \textbf{Response ambiguity.}
The contact classification includes its scaling-degree
restriction.
Dropping that restriction can permit derivatives of
\(\delta_0\), which \(F(0)=0\) does not all annihilate.

\item \textbf{Response units and sign.}
The source sign and \(\hbar\) convention are fixed before
evaluating the benchmark.
The response pairing is not a probability.

\item \textbf{Environmental moment domains.}
A formal generator calculation does not by itself prove
absolute continuity of an unbounded number moment.
That property is part of the environmental certificate.

\item \textbf{Reduced algebra.}
A channel adjoint is not generally multiplicative.
Preservation of the underlying enlarged CCR is not
preservation of commutators by the reduced adjoint.

\item \textbf{Causal completeness.}
Unrecorded persistent bath memories or classical controls
can invalidate a proposed local schedule.
They must be included before computing ancestors.

\item \textbf{Joint endpoints.}
Input-independent marginals do not imply an
input-independent joint channel.

\item \textbf{Statistical independence.}
The sample bound assumes independent attempted runs from
a fixed family member.
Adaptive dependence or repeated reuse of one preparation
requires a different statistical argument.

\item \textbf{Collapse interpretation.}
The random-unitary ensemble representation and the
normalized reduction unraveling are distinct conditional
descriptions of one unconditional law.
Selecting the latter as physical is an additional choice.

\item \textbf{Finite versus continuum dynamics.}
Bounded finite generators justify the displayed
composition limit.
No unbounded continuum Trotter or regulator-removal result
is inferred from it.

\item \textbf{Proof and execution.}
An analytic derivation, a script, a compilation, a
regression, and a machine proof certificate are different
forms of evidence.
None is silently substituted for another.
\end{enumerate}

\section{Landmark incorporation without model identification}
\label{app:landmarks}

The central incorporated source records are
Em06-A, Em06-B, VII-H, VII-J, VIII-C, En01-A, En01-B,
En01-C, Em05-A, and En02-E.
Their roles are specified in the preceding sections.

Cheung's Em03-D supplies a useful documentary example of
a compact-field to bounded-observation comparison with
resource and state-class restrictions
\cite{CheungEm03D}.
Such a comparison illustrates the kind of bridge required
here; it does not identify its Schwarzschild packet with
a neuronal carrier.
Em04-A similarly illustrates the distinction between a
declared dissipative model, instrument construction, and
implementation certificates \cite{CheungEm04A}.

The remaining retrieved manuscripts mentioned in the supplied
corpus record concern other classical, field-theoretic,
operational, reconstruction, or gravitational programs.
They are not hidden premises of this manuscript.
In particular:
\begin{itemize}
\item a classical quotient or prescribed Einstein--scalar
background is not derived gravitational dynamics;
\item a response kernel is not a causal schedule or an
instrument;
\item a finite boundary readout is not a continuum carrier
correspondence;
\item a fixed-background emission calculation is not a
collapse mechanism or neuronal model;
\item a recovery ledger is not a consciousness assignment;
\item a source dependency, continuation, or shared vocabulary
does not establish a mathematical identification.
\end{itemize}

The present framework therefore neither imports unresolved
gravitational constructions nor opens their independent
proof obligations.
Its conditional theorem stands on the objects and hypotheses
printed in this source.

\begin{thebibliography}{99}

\bibitem{HameroffPenrose}
S.~Hameroff and R.~Penrose,
Consciousness in the universe: a review of the
``Orch OR'' theory.
\emph{Physics of Life Reviews}
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\url{https://doi.org/10.1016/j.plrev.2013.08.002}.

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R.~Penrose,
On gravity's role in quantum state reduction.
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\bibitem{Diosi}
L.~Di\'osi,
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M.~Tegmark,
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\textbf{61} (2000), 4194--4206.
\url{https://doi.org/10.1103/PhysRevE.61.4194}.

\bibitem{HaganHameroffTuszynski}
S.~Hagan, S.~R.~Hameroff, and J.~A.~Tuszy\'nski,
Quantum computation in brain microtubules:
decoherence and biological feasibility.
\emph{Physical Review E}
\textbf{65} (2002), 061901.
\url{https://doi.org/10.1103/PhysRevE.65.061901}.

\bibitem{Donadi}
S.~Donadi, K.~Piscicchia, C.~Curceanu, L.~Di\'osi,
M.~Laubenstein, and A.~Bassi,
Underground test of gravity-related wave function collapse.
\emph{Nature Physics}
\textbf{17} (2021), 74--78.
First published online 7 September 2020.
\url{https://doi.org/10.1038/s41567-020-1008-4}.

\bibitem{Khan}
S.~Khan et al.,
Microtubule-stabilizer epothilone B delays anesthetic-induced
unconsciousness in rats.
\emph{eNeuro}
\textbf{11}, no.~8 (2024), ENEURO.0291-24.2024.
\url{https://doi.org/10.1523/ENEURO.0291-24.2024}.

\bibitem{Derezinski}
J.~Derezi\'nski,
Introduction to representations of the canonical
commutation and anticommutation relations.
Lecture notes (2005).
\url{https://arxiv.org/abs/math-ph/0511030}.
Cited for the standard CCR, Weyl, and Fock framework.

\bibitem{Nelson}
E.~Nelson,
Analytic vectors.
\emph{Annals of Mathematics}
\textbf{70}, no.~3 (1959), 572--615.
\url{https://doi.org/10.2307/1970331}.

\bibitem{HolevoWerner}
A.~S.~Holevo and R.~F.~Werner,
Evaluating capacities of bosonic Gaussian channels.
\emph{Physical Review A}
\textbf{63} (2001), 032312.
\url{https://doi.org/10.1103/PhysRevA.63.032312}.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's manuscript and chapter resources:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{DaviesLewis}
E.~B.~Davies and J.~T.~Lewis,
An operational approach to quantum probability.
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\bibitem{BrunettiFredenhagen}
R.~Brunetti and K.~Fredenhagen,
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renormalization on physical backgrounds.
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\url{https://doi.org/10.1007/s002200050004}.
\url{https://arxiv.org/abs/math-ph/9903028}.

\bibitem{HollandsWald}
S.~Hollands and R.~M.~Wald,
Local Wick polynomials and time ordered products of
quantum fields in curved spacetime.
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\url{https://doi.org/10.1007/s002200100540}.
\url{https://arxiv.org/abs/gr-qc/0103074}.

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V.~Gorini, A.~Kossakowski, and E.~C.~G.~Sudarshan,
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\bibitem{CheungEm06A}
N.~Cheung,
\emph{Em06-A: a scalar-only coherent limit of the first
plateau coefficient}.
Manuscript \texttt{Em06-A-v1.tex}, version 1, undated.
Enter Calculus Companion.
Supplied release excerpt, lines 1374--1419,
\texttt{thm:release}.
\url{https://enter.grok.me/browse/Em06-A}.
Documentary source record; the full source construction is
not presumed verified by its release summary.

\bibitem{CheungEm06B}
N.~Cheung,
\emph{Manuscript Em06-B}.
\texttt{Em06-B-v1.tex}, version 1, undated.
Enter Calculus Companion.
Supplied excerpts:
``Ordered Weyl expectations and their classical limit,''
lines 991--1045,
\texttt{thm:ordered},
\texttt{eq:ordered-general},
\texttt{eq:ordered-bound}, and
\texttt{eq:ordered-vacuum};
and analytical release, lines 1798--1858,
\texttt{thm:release}.
\url{https://enter.grok.me/browse/Em06-B}.
The source's full input-stability constants and relational
construction are not reconstructed from these excerpts.

\bibitem{CheungVIIH}
N.~Cheung,
\emph{Landmark VII-H: normalization-independent connected
quantum responses}.
Manuscript \texttt{VII-H.tex}, version 1, undated.
Enter Calculus Companion.
Supplied release excerpt, lines 1748--1780,
\texttt{thm:release}.
\url{https://enter.grok.me/browse/VII-H}.
Cited for the free massless Wick-square benchmark,
complete contact-ambiguity boundary, and invariant response
design. Section~\ref{sec:response} prints and proves the
benchmark used in this manuscript.

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Manuscript VII-J}.
\texttt{VII-J.tex}, version 1, undated.
Enter Calculus Companion.
Supplied release excerpt, lines 2035--2078,
\texttt{thm:release}, and the supplied scheduled-path example.
\url{https://enter.grok.me/browse/VII-J}.
Cited for the distinction between scheduled ancestors,
actual influence, and independently certified physical
causal containment.

\bibitem{CheungVIIIC}
N.~Cheung,
\emph{Manuscript VIII-C}.
\texttt{VIII-C-v2.tex}, version 2, undated.
Enter Calculus Companion.
Supplied ``Backwards support for prescribed block channels,''
lines 1512--1566,
\texttt{thm:C5},
\texttt{eq:backwards-set},
\texttt{eq:support}, and
\texttt{eq:metric-support}.
\url{https://enter.grok.me/browse/VIII-C}.
The finite support theorem needed here is independently
proved in Section~\ref{sec:causal}.

\bibitem{CheungEn01A}
N.~Cheung,
\emph{Manuscript En01-A}.
\texttt{En01-A.tex}, version 1, undated.
Enter Calculus Companion.
Supplied ``Channel-interface theorem,''
lines 1063--1093,
\texttt{thm:channel} and \texttt{eq:channelbias}.
\url{https://enter.grok.me/browse/En01-A}.
Cited for operational separation of coherent and specified
dephased inputs.

\bibitem{CheungEn01B}
N.~Cheung,
\emph{Manuscript En01-B}.
\texttt{En01-B-v2.tex}, version 2, undated.
Enter Calculus Companion.
Supplied nuisance-cancellation and robust-margin theorem,
lines 1518--1562, \texttt{thm:contrast};
and calibrated quadratic obstruction,
lines 1215--1266, \texttt{thm:width}.
\url{https://enter.grok.me/browse/En01-B}.
Cited for common orientation, nuisance-enlarged families,
and observational-equivalence boundaries.

\bibitem{CheungEn01C}
N.~Cheung,
\emph{Manuscript En01-C}.
\texttt{En01-C.tex}, version 1, undated.
Enter Calculus Companion.
Supplied complete-record certificate,
lines 1658--1723,
\texttt{thm:certificate},
\texttt{eq:statistic}, and \texttt{eq:g}.
\url{https://enter.grok.me/browse/En01-C}.
Cited for bounded complete-record statistics, tail
allowances, and independent-trial requirements.

\bibitem{CheungEm05A}
N.~Cheung,
\emph{Manuscript Em05-A}.
\texttt{Em05-A-v1.tex}, version 1, undated.
Enter Calculus Companion.
Supplied ``Diagnostic separation,''
lines 1257--1298, \texttt{thm:release}.
\url{https://enter.grok.me/browse/Em05-A}.
Its finite extraction model is not identified with the
oscillator or Bell counterexamples used here.

\bibitem{CheungEn02E}
N.~Cheung,
\emph{Manuscript En02-E}.
\texttt{En02-E-v2.tex}, version 2, undated.
Enter Calculus Companion.
Supplied flagged-cutoff theorem, lines 572--602,
\texttt{thm:cutoff};
and independent ancillary interface,
lines 933--952, \texttt{thm:ancilla}.
\url{https://enter.grok.me/browse/En02-E}.
Cited with its number-diagonal and common-interface
restrictions retained.

\bibitem{CheungEm03D}
N.~Cheung,
\emph{Manuscript Em03-D}.
\texttt{Em03-D-v1.tex}, version 1, undated.
Enter Calculus Companion.
Supplied compact bulk-to-packet comparison,
lines 887--954, \texttt{thm:packetbridge};
induced Weyl effect, lines 1062--1124,
\texttt{thm:induced};
and analytical release, lines 2024--2085.
\url{https://enter.grok.me/browse/Em03-D}.
A separate fixed-background, state-class, and
resource-dependent comparison; not a neuronal carrier
correspondence.

\bibitem{CheungEm04A}
N.~Cheung,
\emph{Manuscript Em04-A}.
\texttt{Em04-A-v2.tex}, version 2, undated.
Enter Calculus Companion.
Supplied analytical release, lines 1658--1698,
\texttt{thm:release}.
\url{https://enter.grok.me/browse/Em04-A}.
Cited for separation of dissipative dynamics, instrument
construction, and implementation verification.

\end{thebibliography}

\end{document}