En02-E From Schwarzschild Radiation to Finite Recovery Channels Selected packet states, controlled truncation, and nonvacuum-calibrated memory benchmarks

We construct a controlled finite representation of selected radiation modes in a specified stationary Schwarzschild free-field model. A finite orthonormal packet family has a normal, zero-mean, gauge-invariant Gaussian state determined by its full packet covariance. We retain complex off-diagonal correlations, derive its thermal normal-mode representation and occupation generating function, and construct a deterministic flagged total-occupation cutoff. For this state, the cutoff trace-distance error is exactly the discarded probability.

We prove covariance-to-state continuity, reference-compatible cutoff bounds for general inputs, bandwidth-dependent energy estimates, and occupation-dependent scattering-channel bounds that retain complex amplitudes. A common ancillary interface supplies a precise condition under which a state approximation controls an encoded channel. Separately, a declared Clifford message channel is calibrated by the selected state's nonvacuum probability. Its unrestricted and atomic-delivery memory-limited recovery errors are evaluated exactly. This benchmark transfers one field statistic; its decoder does not access the selected field state's full correlations.

The implementation appendix specifies phase-sensitive covariance regressions, complete decoder extensions, basis dictionaries, event-wise resource ledgers, radial boundary bounds, and failure-preserving SDP residual checks. It is not reported as executed. The analytical results concern selected stationary radiation states and explicitly declared finite channels, not collapse, backreaction, complete evaporation, or a microscopic Hawking information-release mechanism.

The construction separates the following objects:

\begin{gathered}
\text{specified field state}
\longrightarrow
\text{selected packet state}
\longrightarrow
\text{finite flagged representation},\\
\text{declared encoding interface}
\longrightarrow
\text{message channel}
\longrightarrow
\text{complete recovery protocol}.
\end{gathered}
A field state does not determine the second line.
Connections between the lines are proved only for explicitly stated
interfaces.

\paragraph{Route A.}
We construct and approximate the full selected radiation state,
including multiparticle occupation and packet correlations.

\paragraph{Route B.}
We transfer its vacuum/nonvacuum probability to an independently
specified logical benchmark. This is a nonvacuum-calibrated recovery
model, not a derived microscopic evaporation channel.

The stationary field formulation is the one specified in the supplied
Em03-A manuscript \cite{Em03A}, with the external Unruh-state
construction retained \cite{DMP}. The operational benchmark is a
specialization of the supplied En02-D model \cite{En02D}; its needed
claims are proved here.

\begin{center}
\renewcommand{\arraystretch}{1.13}
\begin{tabular}{@{}p{0.26\textwidth}p{0.27\textwidth}p{0.38\textwidth}@{}}
\toprule
Ingredient & Dependency & Status in this paper\\
\midrule
Stationary boundary covariance
& Em03-A; standard Hawking model
& Explicitly stated analytical input\\
Bulk state identification
& Dappiaggi--Moretti--Pinamonti
& External construction and Hadamard theorem\\
Radial scattering
& Specified Schwarzschild ODE
& Flux-normalized construction; approximation bounds\\
Selected density operator
& Quasifree field restriction
& Proved finite-mode representation\\
Occupation cutoff
& This paper
& Deterministic CPTP map and explicit errors\\
Message encoding and core
& Operational specification
& Additional model inputs\\
Recovery performance
& Specified channel and interface
& Analytical optima; numerical checks unexecuted\\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Evidence status.}
No Python, solver, or pdfLaTeX execution is asserted.
The numerical appendix is an implementation specification, not a
results section. Analytical examples are explicitly identified.
An executed incorporation study requires actual archived runs and
inspection of their numerical and compilation records.

We use \(c=\hbar=k_B=1\), \(G>0\), and base-two entropy.
For states and channels,
\[
d_{\mathrm{tr}}(\rho,\sigma)=\frac12\left\lVert \rho-\sigma\right\rVert_1,\qquad
d_\diamond(\Phi,\Psi)=\frac12\left\lVert \Phi-\Psi\right\rVert_\diamond.

Channel comparisons include arbitrary references. Matrix ${\Vert \cdot \Vert }_{\infty }$ denotes operator norm and ${\Vert \cdot \Vert }_{\mathrm{F}}$ denotes Frobenius norm.

Fix $M_{\ast }>0$, $r_{\mathrm{h}}=2G{M}_{\ast }$, and one real angular harmonic of degree $\ell$. The exterior metric and tortoise coordinate are

The Killing field ${\partial }_t$ is normalized at infinity. For the massless minimally coupled scalar, the radial equation is

Surface gravity and inverse horizon temperature are

The state is the quasifree product prescription with the affine vacuum on the complete past horizon and the Killing-frequency vacuum on past null infinity. On the exterior half-horizon,

The bulk-to-boundary construction and the Hadamard property on the stated Schwarzschild domain are external inputs DMP. They are not a finite-core construction.

The current convention is

The affine vacuum current covariance pulls back to

These are the stationary boundary identities imported from Em03-A. The positive- and negative-frequency current coefficients are $\omega (1+n_{\mathrm{H}})/(4\pi )$ and $\omega n_{\mathrm{H}}/(4\pi )$, respectively; Fourier transformation contributes $2\pi$. Hence

In particular, a normalized packet covariance has no extra $1/(2\pi )$ factor.

Let $S_{\ell }(\omega )$ be the unitary, flux-normalized two-channel scattering matrix of (eq:radial). Write its horizon-to-infinity transmission amplitude as ${\tau }_{\ell }$, and put ${\Gamma }_{\ell }=|{\tau }_{\ell }{|}^2$. For $\omega >0$, $0<{\Gamma }_{\ell }\le 1$. The outgoing annihilator is

The incoming channels are independent and have zero means. Their number covariances give

Quasifreeness is part of the input. Number covariance alone would not determine a general non-Gaussian state.

The associated time–frequency mode-window convention is

It is distinct from a normalized packet occupation and from a specified detector instrument.

Hawking's fixed-background calculation motivates this model Hawking. A collapse scaling-limit analysis FredenhagenHaag, renormalized backreaction HollandsWald, and a dynamical mass-loss model are separate programmes.

Fix $B=[{\omega }_{-},{\omega }_{+}]\subset (0,\infty )$ and an orthonormal family $h_1,\dots ,h_J\in L^2(B,\mathrm{d}\omega )$. Define

Bounded covariance on $B$ extends the smooth-test formulas to these $L^2$ packets.

We distinguish

Thus $K=C^{\mathsf{T}}$. We use $K$ as the ordinary one-particle operator matrix. Put

The selected packet algebra has a unique normal, zero-mean, gauge-invariant Gaussian state ${\rho }_K$ on $\mathcal{F}({\mathbb{C}}^J)$. Its number covariance is $K^{\mathsf{T}}$ and its anomalous covariance vanishes.

If $K=W\mathrm{diag}({\lambda }_a)W^{†}$ and $z_a={\lambda }_a/(1+{\lambda }_a)$, then

where ${\mathcal{U}}_W|\alpha ⟩=|W\alpha ⟩$. Equivalently,

where ${\Gamma }_{\mathrm{s}}(T)={⨁}_{n\ge 0}{T}^{{\otimes }_{\mathrm{s}}n}$ is bosonic second quantization. The state commutes with $N_{\mathrm{tot}}={\sum }_i{b_i}^{†}{b}_i$, but need not factor in the declared packet basis.

Proof. The restricted quasifree characteristic function is

The transpose in (eq:K) follows directly by expanding the second moment.

For $c=W^{†}b$,

A geometric oscillator density with parameter $z_a$ has the required Gaussian characteristic function and mean ${\lambda }_a$. Its trace is one, including the vacuum case ${\lambda }_a=0$. The product and passive rotation therefore yield the restricted characteristic function. Uniqueness of finite-mode Weyl characteristic functions identifies the state.

In the eigenbasis, ${\Gamma }_{\mathrm{s}}(\mathrm{diag}{z}_a)$ has eigenvalues ${\prod }_a{z_a}^{n_a}$, and trace ${\prod }_a(1-z_a{)}^{-1}=\det (I+K)$. This proves (eq:density). Passive transformations preserve total number.

These formulas are finite-mode Gaussian normal-mode methods Weedbrook,Caruso, with conventions fixed by the proof.

The vacuum probability, total-number moments, and entropy are

where $g(x)=(x+1){\log }_2(x+1)-x{\log }_{2}x$, $g(0)=0$.

If ${\lambda }_{\max }>0$, then the absolutely convergent probability-generating function is

Here $s$ may be complex and $0^0=1$. If $K=0$, $G_K(s)=1$ for every $s$. Positivity of rational denominators alone is not a convergence criterion for negative $s$.

Proof. For each nonvacuum normal mode the geometric series converges absolutely precisely when $|s{z}_a|<1$. Multiplying finitely many such series gives (eq:pgf). Their elementary moments and entropy give the remaining formulas.

For positive semidefinite $K,L\in {\mathbb{C}}^{J\times J}$,

If $K,L⪰mI$, $m>0$, then also

Proof. Let $Z$ be a standard circular complex Gaussian vector. The coherent-state mixtures with amplitudes $\sqrt{K}Z$ and $\sqrt{L}Z$ have the characteristic functions of ${\rho }_K$ and ${\rho }_L$; degenerate Gaussian measures cover singular matrices. Since

convexity and Cauchy–Schwarz give the Frobenius bound.

Set $\delta ={\Vert K-L\Vert }_{\infty }$. Operator monotonicity of the square root gives

and the reverse inequality follows by interchanging $K,L$. Thus ${\Vert \sqrt{K}-\sqrt{L}\Vert }_{\infty }\le \sqrt{\delta }$.

For strictly positive matrices, $X=\sqrt{K}-\sqrt{L}$ solves $\sqrt{K}X+X\sqrt{L}=K-L$, and

The integral's Frobenius norm is bounded by ${\Vert K-L\Vert }_{\mathrm{F}}/(2\sqrt{m})$.

Replacing $K$ by $\mathrm{diag}K$ therefore requires an error budget:

Disjoint frequency supports make $K$ diagonal. Temporal translations of overlapping packets generally do not.

Let $E=\mathrm{span}\{h_i\}$, with projector $P_E$. Theorem thm:gaussian is an exact restriction, not a state approximation to all outgoing modes.

For a normalized target $g\in L^2(B)$, set $e=\Vert (I-P_E)g\Vert$. With $Q$ multiplication by $q_{\mathrm{out}}$, expansion into selected and unselected parts gives

This controls the named occupation observable, not a global continuum-state distance.

For integer $N\ge 0$, let

The finite physical representation is

The map

is deterministic and CPTP. Put

In the common space $\mathcal{F}({\mathbb{C}}^J)\oplus \mathbb{C}|\perp ⟩$,

The retained block is unnormalized; the failure flag carries its actual probability.

Proof. The low-block Kraus map is $P_N$ followed by inclusion. For a high-block orthonormal basis $e_{\alpha }$, add Kraus maps $|\perp ⟩⟨e_{\alpha }|$. Their effects sum strongly to $I$.

The number of occupation vectors with total at most $N$ is ${{\sum }_{n=0}}^N\left(\genfrac{}{}{0ex}{}{J+n-1}{n}\right)=\left(\genfrac{}{}{0ex}{}{J+N}{N}\right)$. Since $[{\rho }_K,N_{\mathrm{tot}}]=0$, no low–high coherence remains. The difference of the two states is a positive high block of trace $t_N(K)$ minus an orthogonal flag block of the same trace.

Write ${\pi }_n(K)=\mathrm{Pr}(N_{\mathrm{tot}}=n)$, reserving $t_N(K)$ for the cutoff tail. Then

For every $N\ge 0$,

For ${\lambda }_{\max }>0$,

The discarded mean occupation

satisfies

on the same positive $s$-domain. All these tails vanish for $K=0$.

Proof. Use Markov's inequality, then $1_{\{n>N\}}\le s^{n-(N+1)}$. For the first moment, use $n\le n^2/(N+1)$ on the tail and $\mathbb{E}{N}_{\mathrm{tot}}^2={\mu }^2+\mu +\mathrm{tr}{K}^2$. The derivative bound follows from the differentiated convergent generating series.

Let $T=K(I+K{)}^{-1}$. In the normalized occupation basis,

The repeated-row and repeated-column matrix $T[\mathbf{n},\mathbf{m}]$ contains row $i$ exactly $n_i$ times and column $i$ exactly $m_i$ times. The empty permanent is one.

For the one-particle block,

Extend compressed annihilators by zero on the flag: $b_{i,N}=P_N{b}_i{P}_N\oplus 0$. If

then the truncated number covariance is

It need not equal $K_{ji}$; the discarded moments are absent.

Proof. Equation (eq:permanent) is the matrix element of the symmetric tensor power in (eq:density); symmetrization sums over permutations and occupation normalization supplies the factorials. Its one-particle specialization is immediate.

The cutoff is invariant under passive mode rotation. In the normal-mode basis the retained, unnormalized state is diagonal, and its mean in mode $a$ is ${{\eta }_a}^{(N)}$. Rotating its one-particle matrix back and applying (eq:K) proves (eq:truncatedmoments).

These formulas permit regressions that distinguish $K$ from $K^{\mathsf{T}}$, unlike eigenvalues, vacuum probability, or total-number tails.

For any state ${\sigma }_{AR}$, with arbitrary reference $R$, put $p=\mathrm{tr}(Q_N{\sigma }_A)$. Then

If $\mathrm{tr}(N_{\mathrm{tot}}{\sigma }_A)\le E$, then $p\le E/(N+1)$. Without an input constraint, the cutoff's half-diamond distance from the identity embedding is one for every finite $N$.

Proof. Purify ${\sigma }_{AR}$ to $|\psi ⟩$ and set $|\varphi ⟩=(P_N\otimes I)|\psi ⟩$. The trace norm of $|\psi ⟩⟨\psi |-|\varphi ⟩⟨\varphi |$ is $\sqrt{4p-3p^2}\le 2\sqrt{p}$. Partial trace contracts it. The additional flag block has trace norm $p$. Markov's inequality gives the constraint estimate. A high-block input is sent to an orthogonal flag and attains distance one.

This explicitly realizes an occupation-constrained approximation; energy-constrained channel topologies are discussed in Shirokov.

For a positive approximation $L$, define ${\delta }_{K,L}=\min \{1,\sqrt{J{\Vert K-L\Vert }_{\infty }}\}$. In the common embedding,

Entrywise errors at most $\eta$ imply ${\delta }_{K,L}\le \min \{1,J\sqrt{\eta }\}$. A numerical convergence difference is not automatically a justified entrywise error bound.

Choose an ordered occupation basis, append the flag, and map these $d_N+1$ states to the first computational basis states of $q_{\mathrm{raw}}$ qubits. Unused binary states have zero prepared weight. A reverse representation map pinches valid versus invalid binary subspaces and sends invalid weight to the flag; it is CPTP by the same Kraus construction as the cutoff. Its action on valid encoded states is exact.

The representation does not realize bosonic CCR exactly in finite dimension. For bounded embedded observables the expectation error is at most $2t_N(K){\Vert A\Vert }_{\infty }$; number and energy require the separate moment estimates.

Either start with the horizon state and apply its scattering map, or start with outgoing covariance ${\Gamma }_{\ell }{n}_{\mathrm{H}}$. Route A uses the latter and does not apply the greybody barrier twice.

An additional detector amplitude $d(\omega )$, with an independent vacuum loss environment, would give $q_{\mathrm{detected}}=|d{|}^2{q}_{\mathrm{out}}$. This is separately labelled detector loss. Beamsplitter vacuum dilations are appropriate under those input assumptions Weedbrook,Caruso.

Let $E\subset L^2(B)$ have dimension $J$. Prepare arbitrary inputs in $E\oplus E$ in the two incoming channels, with orthogonal incoming modes in vacuum. The restriction of multiplication by $S_{\ell }(\omega )$ is an isometry into $E_D^{\prime }\oplus E_F^{\prime }$, where each output space is spanned by the relevant output components of the $2J$ incoming packet functions and has dimension at most $2J$.

On total input occupation at most $N$, its bosonic lift is an exact finite isometry into at most $4J$ output modes with total occupation at most $N$. Tracing one output gives a finite CPTP communication channel.

Proof. Restriction of a unitary one-particle operator is isometric. Its output components lie in the specified spans. On the $n$-particle sector, the symmetric tensor power is isometric and preserves total number. Take the direct sum for $0\le n\le N$, then partial trace.

This is a communication problem after a declared preparation. It does not replace unselected horizon modes by vacuum in the stationary Unruh-state calculation.

For unitary two-channel multiplication matrices $S,\tilde{S}$ on $B$, suppose

Their cutoff lifted isometries satisfy

and their complete channels, before or after the same partial trace, satisfy

Proof. Telescope the $n$-fold tensor-power difference. Every factor is an isometry, giving $n{ϵ}_S$. Symmetric restriction and direct sum preserve the bound. For referenced pure inputs, trace distance of the two isometric outputs is at most their vector-norm difference. Purification, convexity, and partial-trace contraction finish.

For mean occupation at most $E$, using a common cutoff and common flag extension gives the additional bound

It is not inferred from matching mean counts.

Frequency-dependent amplitudes generally leave the selected packet span:

A piecewise-constant unitary $S_a$ on subbands preserves the packet span when each packet belongs to one subband. Its discrepancy must include phases and be bounded as above.

For covariance only, contraction-valued amplitudes with $\sup |\tau -\tilde{\tau }|\le {ϵ}_{\tau }$ imply

This does not control coherent scattering phases.

Let $\Omega$ be multiplication by $\omega$, and let

The selected span need not be invariant under $\Omega$.

Compression of field energy to states with particles only in $E$ is $H_E$, with selected Gaussian expectation $\mathrm{tr}(HK)$.

Suppose packets lie in pairwise disjoint subbands whose halfwidths are at most $\Delta$. Several packets may belong to the same subband. Let $H_0$ assign its midpoint to each such packet. Then

Assigning the flag zero energy omits selected energy between zero and ${\omega }_{+}{m}_N(K)$.

Proof. On each $n$-particle sector, energy is the sum of $n$ one-particle operators; compression gives $H$. Using $C_{ij}=K_{ji}$ gives $\mathrm{tr}(HK)$. Multiplication by frequency minus the subband midpoint has norm at most $\Delta$; compression and the $n$-fold sum give the first two estimates. Finally $0⪯H_E⪯{\omega }_{+}{N}_{\mathrm{tot}}$, and both commute with the cutoff projector.

The compressed observable is not the energy of all radiation. It need not commute with the selected state. A surrogate's exact energy conservation is not exact conservation of the continuum field energy.

On the full band,

the cutoff Fock evolution error is at most $\min \{2,N|t|\Delta \}$.

Route A represents the entire flagged finite state as one raw block. Frequency orthogonality does not establish sharply localized arrivals. Any demultiplexing or streaming implementation needs its own delivery convention and resource ledger.

Let $\rho ,\sigma$ be ancillary states in a common space, independent of the message and its reference. For a fixed CPTP map $\Lambda :M\otimes A\to B$, define ${\mathcal{N}}_{\rho }(X)=\Lambda (X\otimes \rho )$ and similarly ${\mathcal{N}}_{\sigma }$. Then

The bound also holds after any common deterministic decoder.

Proof. For a referenced message state ${\xi }_{MR}$, the input difference is ${\xi }_{MR}\otimes (\rho -\sigma )$, with trace norm ${\Vert \rho -\sigma \Vert }_1$. Apply CPTP contraction and take the supremum over references.

Thus (eq:combined) supplies a complete-channel bound when this interface is supplied. It does not identify the encoding from radiation covariance or justify changing a purification, message correlation, or internal dynamics without a new comparison.

Define

This is the selected nonvacuum probability, not a rate or a mean. It is preserved exactly by the flagged cutoff when the flag is classified as nonvacuum. A binary vacuum measurement gives

Also

because $n-1_{\{n>0\}}\le n(n-1)/2$.

Only $a(K)$ is transferred to the benchmark. Different correlated Gaussian states may give the same parameter. The benchmark decoder is not given their covariance or full finite state.

The message is a qubit. On five qubits, define

Gate products act rightmost first.

A physical operational packet has basis $\{|\mathrm{vac}⟩,|1,0⟩,|1,1⟩\}$, represented by $|00⟩,|10⟩,|11⟩$. Its atomic raw width is two qubits. The unused binary state $|01⟩$ is assigned an invalid-input failure treatment.

At horizon three,

A coherent dilation retains a branch label in the core, with amplitudes $\sqrt{1-a},\sqrt{a}$, and extracts encoded factors $1,2,3$ on the occupied branch. Tracing the labelled core gives (eq:Nbench). This is a declared operational purification, not the physical purifier of the selected Unruh state.

A surrogate occupied-rail energy $ϵ>0$ and source energy $E_{\mathrm{in}}>3ϵ$ give core energies $E_{\mathrm{in}}-jϵ$ after $j$ occupied extractions. This finite conservation law is not identified with the Schwarzschild mass.

Three raw width-two packets arrive once, in order. A supplier may buffer, but may not measure, demultiplex, select, or decode them. Every arrival occupies its full width atomically.

All laboratory quantum data, work, records, and output count toward $Q_{\mathrm{peak}}$. Classical storage is unrestricted. There is no uncounted storage, auxiliary shared entanglement, return of intermediate quantum data, or quantum export before all horizon packets have been consumed.

Let ${\epsilon }_{\mathrm{IT}}$ optimize over all CPTP decoders. Let ${\epsilon }_{\mathrm{op}}(n,3)$ optimize over deterministic protocols obeying Assumption ass:atomic with $Q_{\mathrm{peak}}\le n$. Protocol memory is a global execution constraint Bisio.

A decoded constant qubit channel has Bell-input overlap $1/4$. A separable reference–output state has Bell overlap at most $1/2$. For a Pauli channel $\Lambda ={\sum }_P{p}_P{\mathrm{Ad}}_P$,

Proof. A constant channel gives $I_R/2\otimes \sigma$, whose overlap with a maximally entangled qubit state is $1/4$. A product pure state has overlap at most $1/2$; convexity gives the separable bound. For Pauli channels the Bell errors are orthogonal, giving the lower bound $1-p_I$, while the channel triangle inequality gives the same upper bound.

For $a=a(K)$, both parities satisfy

For even $L$,

For odd $L$, ${\epsilon }_{\mathrm{op}}(2,3)=3(1-a)/4$.

Proof. Conjugation gives

Hence ${\mathcal{A}}_L(\rho )=(I+x{P}_X+y{P}_Y+z{P}_Z)/8$. Every proper even subset is constant. Applying $U{F}^L$ to the full triple and discarding factors $1,3$ recovers the message.

The occupied Bell overlap cannot exceed one and the vacuum-branch overlap is $1/4$. Lemma lem:bell therefore gives the unrestricted lower bound $3(1-a)/4$. Recover on the occupied branch and prepare $I/2$ on vacuum to attain it.

At capacity two, no quantum slot remains from the first packet at the second atomic arrival. An arbitrary first-packet instrument therefore leaves only a classical outcome. On the occupied rail, its effect is $E=(\mathrm{tr}E)(I+v_{x}X+v_{y}Y+v_{z}Z)/2$, $|v|\le 1$. The normalized conditional reference–future state is

where $A=X_2{Z}_3$, $B=Y_2{X}_3$, $C=Z_2{Y}_3$, $AB=-C$. It equals

The reference factors are positive and the joint projectors are rank one, so the state is separable across $R\mid 23$. This also covers arbitrary first-packet label mixing: the input branch blocks are already incoherent and orthogonal. Later operations preserve separability. The Bell overlap is at most $a/2+(1-a)/4$, giving $3/4-a/4$.

Measuring $Z_1,X_2,Z_3$ and preparing the logical $X$ eigenstate of their negative parity realizes

which attains the bound. The explicit decoders below attain unrestricted recovery at the remaining capacities.

Set

three-slot protocol. Read and consume labels. Retain the first occupied rail through the second arrival. If both early labels are occupied, apply $B_{12}$, read both rails as $t,s$, and consume them. If all three labels are occupied, apply $C_{ts}{R}_3$ to rail three and return it after the final arrival. Otherwise consume all remaining data and prepare $I/2$.

two-slot protocol. The identity $B_{12}F=(H_1{S_1}^{†})H_2$ allows separate early readouts. Consume each early rail before the next arrival. On the all-occupied history apply the same final correction; otherwise prepare $I/2$ after consuming all packets.

two-slot protocol. Read $Z,X,Z$ on the occupied rails, retaining only classical outcomes. On the all-occupied history prepare the logical $X$ eigenstate of negative parity; otherwise prepare $I/2$.

The identities

verify the successful corrected channel. Equivalently, substituting the three logical Pauli representatives into the four early-readout Kraus maps gives the logical Pauli operators at the corrected output. The early pair is $I_4/4$, so all four outcomes have probability $1/4$. The complete decoded channels are $a\mathrm{id}+(1-a)\mathcal{T}$ and $a{\Delta }_X+(1-a)\mathcal{T}$, respectively.

To extend each protocol to every physical packet input, mixed vacuum/occupied histories are failures. Invalid binary packet states are also failures. All such histories are consumed before fixed-output preparation. Cross-history coherence is removed by the complete label-readout instrument.

accounting. Single-qubit gates, CNOT, CZ, allocation, destructive computational readout, classical control, and final return are admitted. Preparing $I/2$ uses one recyclable slot: allocate zero, apply $H$, destructively read a random bit, retain it classically, reallocate zero, conditionally apply $X$, and return. Preparation of an $X$ eigenstate uses zero allocation, conditional $X$, and $H$. These events are included in the ledger; a final direct return is not recorded as a destructive measurement.

The even reference state is locally Clifford-equivalent to Smolin's unlockable state Smolin. Indeed, for $R_0=H{S}^{†}$, conjugation by $Y_R\otimes R_{0,1}\otimes I_2\otimes R_{0,3}$ maps it to $(I+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4})/16$. The unlocking mechanism is inherited.

Since (eq:Nbench) is affine in $a$,

This is a channel bound for a fixed specified calibration interface. It is not a general conversion of count error into diamond error. The flagged cutoff adds no error to this particular binary statistic.

On $B=[c-w/2,c+w/2]\subset (0,\infty )$, choose

zero outside $B$. They are orthonormal and

The covariance $K$ is evaluated without deleting off-diagonal entries. Midpoint energy $cI$ has compressed error $w/\sqrt{12}$, bounded by $w/2$. These packets are not sharp arrival events.

For the synthetic, non-Schwarzschild covariance

the eigenvalues are $1/10,1/5$,

In the explicit order $(|1,0⟩,|0,1⟩)$, the one-particle density block is

For cutoff $N=1$, its number-moment matrix is the transpose:

These entries distinguish $K$ from $K^{\mathsf{T}}$.

The cutoff dimension including the flag is four, and

The calibrated recovery errors are

These are analytical values, not executed pilot results.

The first radial pilot uses $r_{\mathrm{h}}=1$, $\beta =4\pi$, $\ell =0$, $B=[0.08,0.24]$, the two packets above, and cutoff $N=2$. It constructs the full finite state and links it to the nonvacuum-calibrated benchmark by a parameter identifier.

The appendix separates synthetic tests from the radial pilot. It records complex amplitudes; flux residuals; separate boundary, quadrature, and ODE-tolerance comparisons; covariance and energy; finite Fock matrices and basis dictionaries; complete decoder matrices; resource histories; and optional SDP variables.

The analytical potential-tail bound is actually evaluated and archived for each radial boundary choice. Its floating-point evaluation is labelled as such. It excludes unverified integration error, and a vacuous bound remains recorded as vacuous.

BlackHawk may provide a version-pinned spectral comparison BlackHawk, while low-frequency scalar results Higuchi provide checks under their own hypotheses. Neither supplies the complex amplitude error needed here. No such comparison or numerical run is asserted.

The SDP formulations are given in Appendix app:sdp. Every requested solve first archives raw variables, constraint duals, settings, and solver statistics, before residual checks or another solve. Failure therefore preserves any obtained decoder.

For complex PSD constraints, the implementation uses realification:

It separately checks Hermiticity of every relevant original matrix. CVXPY constraint residuals and solver statistics are recorded as diagnostics CVXPYConstraints,CVXPYAdvanced,CVXPYSolvers. Independent original-matrix residuals determine acceptance.

The declared floating-point tests are:

1. finite values and acceptable solver status;

2. Hermiticity residual and Frobenius norm of the negative spectral part at most $3\times 10^{-4}$;

3. affine residual Frobenius norm at most $3\times 10^{-4}$;

4. independent primal–dual objective difference at most $5\times 10^{-4}$;

5. Choi positivity and trace preservation for every encoded or decoded channel;

6. agreement with analytical controls at most $5\times 10^{-4}$.

Raw and repaired decoder residuals are both retained. A numerical decoder repair is not an interval certificate. Agreement of objectives is insufficient without feasibility.

{}1.12

For the specified stationary model and finite orthonormal packet family, the selected state is ${\rho }_K$, its flagged cutoff has dimension $\left(\genfrac{}{}{0ex}{}{J+N}{N}\right)+1$, and its exact Gaussian state error is $t_N(K)$. A justified covariance approximation gives (eq:combined). Energy and coherent scattering have the separate estimates of Section sec:scattering. State error controls an encoded channel under the independent ancillary interface. The specified nonvacuum-calibrated benchmark has the exact recovery and memory optima of Theorem thm:recovery.

Proof. Apply Theorems thm:gaussian, thm:continuity, thm:cutoff, thm:finitescatter, thm:scattererror, thm:ancilla, and thm:recovery, together with Propositions prop:tails, prop:generalcut, and prop:energy.

The principal incorporation is a full finite-state construction, linked to a nonvacuum-calibrated recovery benchmark. No logical encoding, core, clock, or microscopic information-release dynamics is selected by the radiation covariance. The appendix supplies an unexecuted implementation with operational acceptance conditions; it supplies no empirical conclusion about the pilot.

For real $V\in L^1(\mathbb{R})$, the right Jost solution satisfies