En07-B Access-preserving endpoint certificates under destructive Enter Exact recovery, approximate certification, and the limits of channel-only closure

We give a self-contained finite-interface formulation of the endpoint obligation left open by Hawk04-B and En07-A. Complete isometric evolution, recovery from a declared outgoing factor, destructive classical readout, operational discard, and thermodynamic implementation are different assertions. We prove the complete-output restoration theorem, the exact constant-complement criterion with an explicit deterministic decoder, reference-preserving readout-and-drop identities, record-only reconstruction bounds, and a featureless-output obstruction. We state explicitly the external Kretschmann–Schlingemann–Werner input used for approximate recovery, derive its half-diamond normalization, and provide dimension-explicit Choi certificates and access-preserving endpoint stability estimates.

Two defects in the proposed programme are corrected. All quoted errors use half-diamond distance; the corresponding unhalved diamond bounds have twice the stated values. Moreover, approximate constancy of the complete complement supplies a recovery upper bound, whereas approximate constancy of the accessible output supplies a recovery lower bound. These hypotheses cannot be interchanged.

The mathematical release is an exact and approximate endpoint certificate, not a constructed gravitational endpoint. No microscopic gravitational preparation, outgoing receiver, bulk–boundary dictionary, or evaporation endpoint is supplied. The analytic admission obligations recorded in the supplied Cheung manuscripts are unchanged. Companion manuscripts are documentary comparisons, not premises of the local proofs.

The physical information problem arises from the relation between black-hole emission and the quantum evolution of the complete initial state Hawking1975,Hawking1976. This paper addresses only the following finite-code question:

This is not the assertion that every physical black-hole question reduces to finite-dimensional channel theory. It is the recovery obligation for the finite interface specified below. A physical application must justify that interface in its own model.

The documentary predecessors are Cheung's Hawk04-A, Hawk04-B, and En07-A CheungHawk04A,CheungHawk04B,CheungEn07A. The present proofs do not assume their theorem statements. Where a standard mathematical theorem is imported, it is named explicitly and cited to an external source.

“complete” means. The inaccessible output must include every output omitted from the observer's accessible description: the remaining source, preparation purifiers, discarded field modes, unused cells, apparatus environments, and any other unreturned quantum registers. A mixed preparation is not made complete by ignoring its purification. A selected residual marginal is not necessarily the complementary channel.

“self-contained” means here. The exact statements and elementary certificates are proved below. We use the usual finite-dimensional Kraus/Stinespring structure and standard diamond-norm facts Stinespring,Watrous. Approximate information–disturbance conversion uses the external KSW theorem KSW; its normalization and consequences are proved here, but its continuity theorem is not reproved.

boundary. The supplied manuscript identifiers and versions are cited as provided records. Their citations do not assert journal publication, independent proof verification, or a scan of the entire Enter series. In particular, statements about an unconstructed analytic object are not converted into nonexistence theorems.

Every operational error in this paper is a half-diamond distance:

Consequently a lower bound $a$ for $d_{\diamond }$ is a lower bound $2a$ for the unhalved diamond norm.

Thus the record-only bound is $1-d^{-1}$ in half-diamond distance, and $2(1-d^{-1})$ in unhalved diamond norm. The featureless-output bound is normalized in the same way.

For an isometry $V:C\to R\otimes B$,

The first implication uses approximate information–disturbance conversion. The second is the featureless-output obstruction proved in Proposition prop:featureless.

In particular, the Stage D sentence in the proposed programme cannot apply the featureless-radiation lower bound to a nearly constant endpoint complement. The repaired endpoint estimate is Theorem thm:endpoint.

Let $C,R,B$ be finite-dimensional Hilbert spaces and let $d=\dim C\ge 2$. A complete code interface is an isometry

Its complete, accessible, and complementary channels are

The designation $R$ is the declared accessible output; $B$ is its complete inaccessible complement.

A quantum reference $Q$ is arbitrary and untouched. Every channel equality below is an equality on all operators, not only on a selected preparation ensemble. It therefore remains an equality after tensoring with ${\mathrm{id}}_Q$.

One sufficient preparation/evolution construction is

followed by a complete output regrouping, where $J$ is an isometry, $\Omega$ is a fixed normalized preparation including all necessary purifiers, and $U$ is unitary. Such a construction is not supplied merely by naming its factors.

A quantum-return decoder is a deterministic CPTP map

where $C_{\mathrm{out}}$ has a fixed identification with $C$. Its error is

A subsequent observation of $C_{\mathrm{out}}$ is a measurement channel $\mathcal{M}\circ \mathcal{D}\circ \mathcal{N}$, not the channel $\mathcal{D}\circ \mathcal{N}$ itself.

“Deterministic” includes every success and failure branch. Conditioning on success and renormalizing the successful output is not the decoder interface in Definition def:decoder.

For a density operator $\sigma$ on an output space, define

The complete residual leakage is

For comparison, define the accessible featurelessness diagnostic

These are different quantities.

The number ${\delta }_{\mathrm{full}}$ is a channel-distance diagnostic. It is not an operation, an energy, or a heat.

For a difference of channels with input dimension $d$, the diamond norm can be evaluated on states with a reference of dimension $d$; a pure state suffices. CPTP pre- and postprocessing contract diamond distance, and channels have diamond norm one Watrous (Chapter 3). We use these standard norm facts explicitly.

The sets of density operators and finite-dimensional CPTP maps are compact. For channels this follows from their positive Choi matrices and the fixed partial-trace constraint. Hence both infima in Definitions def:decoder and def:leakage are attained.

For Hermitian operators, the trace-norm contraction used below also follows directly. If $X=X_{+}-X_{-}$ is its Jordan decomposition and $\Phi$ is CPTP, then

The same argument applies after adjoining a reference.

The structure proved in this section is the standard exact error-correction/no-hiding structure KnillLaflamme,BraunsteinPati. Its explicit proof fixes the access factors and decoder used in the rest of the paper.

For the complete isometry $V$, the map

is CPTP and satisfies

Moreover, for every code–reference operator $X$,

Proof. The compression term is completely positive. The second term is a positive-effect measurement followed by fixed-state preparation and is also completely positive. Their traces sum to $\mathrm{Tr}Y$, because $V^{†}V=I_C$.

On the encoded range, the second term vanishes and $V^{†}(VX{V}^{†})V=X$. This proves the left-inverse identity on every operator, and hence with every reference.

An isometric embedding preserves the nonzero singular values of an operator. Applying this to $I_Q\otimes V$ proves the trace-norm equality.

The domain of $\mathcal{L}$ is $R\otimes B$. No decoder on $R$ has been obtained from this theorem. Its inverse compression is also not a physical compilation or laboratory resource certificate.

Fix an orthonormal code basis $\{|i⟩{\}}_{i=0}^{d-1}$, put $E_{ij}=|i⟩⟨j|$, and define

For a channel $\Phi$, its normalized Choi operator is

For $V:C\to R\otimes B$, the following conditions are equivalent:

1. A deterministic CPTP decoder satisfies $\mathcal{D}\circ \mathcal{N}={\mathrm{id}}_C$.

2. $\mathcal{E}={\mathcal{T}}_{\sigma }$ for one density operator $\sigma$ on $B$.

3. $\mathcal{E}(E_{ij})={\delta }_{ij}\sigma$ for all $i,j$.

4. $J_{\mathrm{n}}(\mathcal{E})=I_Q/d\otimes \sigma$.

5. For every $O\in \mathcal{B}(B)$, $V^{†}(I_R\otimes O)V$ is a scalar multiple of $I_C$.

When these conditions hold, there are an auxiliary space $K$, a state ${\sigma }_K$, and an isometry $A:C\otimes K\to R$ such that

An exact deterministic decoder on all of $R$ is

If $r=\mathrm{rank}\sigma$, then $\dim R\ge dr$.

Proof. Matrix units span $\mathcal{B}(C)$, proving the equivalence of (ii) and (iii). Expanding

proves the equivalence with (iv).

If (ii) holds, then

Trace duality gives (v). Conversely, under (v), evaluating the compressed observable at ${\tau }_C$ determines its scalar as $\mathrm{Tr}(O\mathcal{E}({\tau }_C))$. Since observables separate operators,

Thus (v) implies (ii).

Assume (ii). Diagonalize

The $B$ marginal of each $V|i⟩$ is supported on $\mathrm{supp}\sigma$, so write

The coefficient of $|b_a⟩⟨b_b|$ in $\mathcal{E}(E_{ij})={\delta }_{ij}\sigma$ gives

There are therefore $dr$ orthonormal vectors in $R$.

Let $K$ have basis $\{|a⟩{\}}_{a=1}^r$, define

Then $A$ is an isometry, and tracing $B$ gives the asserted normal form for $\mathcal{N}$.

Both terms of (eq:exact-decoder) are completely positive. Their traces add to $\mathrm{Tr}Y$. Substitution of the normal form shows $\mathcal{DN}(X)=X$ for every $X$.

Conversely, suppose (i). Choose a Kraus family $\{D_{\alpha }\}$ for $\mathcal{D}$ and its isometric dilation

The composite isometry $Z=(L\otimes I_B)V$ dilates the identity on $C$. The decoded marginal of $Z|i⟩$ is the pure state $|i⟩⟨i|$, so

Exact recovery of $E_{ij}$ then gives $⟨{\xi }_j,{\xi }_i⟩=1$. All normalized vectors ${\xi }_i$ are one vector $\xi$. Tracing $C\otimes G$ yields

which is (iii).

All identities are operator identities; tensoring with ${\mathrm{id}}_Q$ proves reference preservation.

If $\mathcal{N}$ is constant on the whole code, $B$ exactly recovers the code. If $\mathcal{E}$ is constant, $R$ exactly recovers it. For $d\ge 2$, the two outputs cannot both be constant, and cannot both independently recover the whole unknown code.

Proof. Interchange $R$ and $B$ in Theorem thm:exact. An exactly recoverable channel is not constant on two distinct code states. If one output is exactly recoverable, its complement is constant and is consequently not itself exactly recoverable.

The isometry

has the computational dephasing channel on each output. Both outputs are nonconstant. Neither returns the entire unknown quantum code. This is not a contradiction to Corollary cor:nohiding; quantum coherences are absent from each marginal.

Let $X|j⟩=|j+1\mathrm{mod}d⟩$ and $Z|j⟩=e^{2\pi ij/d}|j⟩$. Put $W_{ab}=X^a{Z}^b$, for $0\le a,b<d$, and define

This is an isometry. The declared accessible factor is $R$; the complete complement is $K\otimes G$, not $K$.

Proof. Orthogonality of the key basis and unitarity of each $W_{ab}$ give ${V_{\mathrm{key}}}^{†}{V}_{\mathrm{key}}=I_C$. Tracing both key registers gives

The last identity follows by applying the sum to each matrix unit: the phase sum kills off-diagonal units and the shift sum averages diagonal units.

Tracing $R\otimes G$ gives the selected key marginal

Thus that selected residual is constant even though $R$ does not recover the code. The actual complement has matrix elements

It retains input dependence in the joint key coherences. Indeed, because $R$ is constant, the complete $KG$ output exactly recovers the code by Corollary cor:nohiding.

The register $G$ can be the purifier of a mixed key preparation. Deleting it from the output ledger would falsify the exact-complement hypothesis. This example is a finite negative control, not a gravitational preparation.

Let $B$ have computational basis $\{|j⟩{\}}_{j=0}^{m-1}$, and let $K$ be a classical record register with the same labels. For any unmeasured register $A$, define

where

The output has no surviving quantum register named $B$. The map ${\mathsf{Drop}}_K$ is partial trace over the record.

This is the destructive interface fixed in the supplied Cheung framing CheungEn07A,CheungHawk04B. Here it is a definition, not an imported uniqueness theorem for all measurement devices.

For every operator $X$ on $A\otimes B$,

The equality holds with arbitrary external references. The coherent readout map

is an isometry and dilates ${\mathsf{Enter}}_B$.

Proof. Expand

Enter retains the terms $X_{jj}$ and attaches orthogonal classical labels. Dropping the labels gives ${\sum }_j{X}_{jj}={\mathrm{Tr}}_{B}X$.

Also ${W_B}^{†}{W}_B=I_B$, and

Tracing $E$ gives Enter; tracing $K\otimes E$ gives ${\mathrm{Tr}}_{B}X$. A reference can be included in $A$, so the proof already contains every referenced version.

With both $K$ and $E$ accessible, $W_B$ has a CPTP left inverse by Theorem thm:global. With only the classical $K$ returned, that conclusion no longer follows. Coherent copying of computational labels is not a record-only quantum decoder.

The observable equality $\mathsf{Drop}\circ \mathsf{Enter}={\mathrm{Tr}}_B$ does not imply that the complete physical input has been annihilated. Under a complete isometric realization, distinct code states remain distinct on the complete output.

Proof. Lemma lem:erasure-interface supplies an isometric realization whose omitted output retains the required coherent degrees of freedom. Theorem thm:global preserves their complete trace distances. Thus the same observable discard channel is compatible with complete preservation, and its equality with partial trace cannot entail complete annihilation.

This proves underdetermination of the physical conclusion, not a theorem that every physical gravitational evolution must be isometric.

Suppose a channel from the code to a returned quantum register factors through a classical record, with no surviving input-dependent quantum memory. Then its half-diamond distance from ${\mathrm{id}}_C$ is at least

More explicitly, if

where each ${\sigma }_j$ is a density operator on $C_{\mathrm{out}}$, then

The bound is attained by computational dephasing.

Proof. The normalized Choi output is

It is a separable density operator across the reference–output cut. For density operators $\alpha ,\beta$,

Convexity gives $\mathrm{Tr}(P_{{\Phi }_d}{J}_{\mathrm{n}}(\Lambda ))\le 1/d$. Testing the Bell projector therefore gives

For sharpness, write computational dephasing as

The triangle inequality and the fact that channels have diamond norm one give

The preceding Bell lower bound gives equality.

The measure-and-prepare form is the standard entanglement-breaking structure HorodeckiShorRuskai; the elementary Bell witness above also proves the needed obstruction directly.

The theorem does not apply to a decoder that returns an unmeasured $C_{\mathrm{out}}$ while destructively reading only a syndrome. In the exact normal form of Theorem thm:exact, the syndrome state is independent of the code. Reading and forgetting that syndrome can leave the returned code untouched.

Likewise, a coherent register retained alongside a classical record invalidates the record-only premise. Unrestricted classical storage does not by itself invalidate it.

If an available unitary $U$ has transformed ${\rho }_{QC}\otimes |0⟩⟨0{|}_S$, applying $U^{†}$ to the same required registers restores that joint state, including $Q$. This is reversible uncomputation. It need not remain available after a required register has been discarded. The distinction is consistent with reversible-computation methods Bennett1973.

It is not a rule for cloning an unknown output before uncomputing, and it is not a reusable-memory thermal reset certificate.

If, for one density operator $\omega$ on $R$,

then every deterministic decoder satisfies

In particular, a constant accessible channel has optimal recovery error $1-d^{-2}$.

Proof. For any density operator $\sigma$ on $C$,

whose Bell overlap is $d^{-2}$. The Bell-projector test gives

CPTP contraction gives

The reverse triangle inequality proves the asserted lower bound.

When $\mathcal{N}$ is constant, a decoder can ignore its input and prepare $I_C/d$. The resulting channel is

The same triangle argument as in Theorem thm:classical gives

The Bell witness gives the matching lower bound.

A replacement channel need not have a thermal output. Conversely, knowing that one radiation state or one packet marginal is thermal does not prove that the entire code-to-radiation channel is a replacement channel. Proposition prop:featureless supplies no temperature, Hamiltonian, horizon area, or Hawking state.

There are complete isometric interfaces having identical zero radiation entropy on every pure code input, while their optimal outgoing recovery errors are respectively $1-d^{-2}$ and $0$.

Proof. Take $R\cong B\cong C$ and define

For every pure code input, both accessible outputs are pure and have entropy zero. The first accessible channel is constant, so Proposition prop:featureless gives optimal error $1-d^{-2}$. The second is the identity and has error zero.

This counterexample is not an entropy-history or Page-curve calculation. It shows why such a calculation does not automatically identify a recovery channel. Page's random-subsystem entropy analysis is a separate mathematical question Page1993, as is the documentary entropy/recovery separation in Hawk01-B CheungHawk01B.

If ${\mathcal{N}}_s=P_{t\to s}\circ {\mathcal{N}}_t$ for a CPTP restriction $P_{t\to s}$, then

Proof. For any decoder ${\mathcal{D}}_s$, the channel ${\mathcal{D}}_t={\mathcal{D}}_s{P}_{t\to s}$ is an admissible decoder for ${\mathcal{N}}_t$ and has exactly the same decoded code channel. Take the infimum over ${\mathcal{D}}_s$.

This is an availability statement. It is not a streaming-memory, gate-count, work, or execution-time optimum. The supplied En02-B and En02-C records make those distinctions in their own interfaces CheungEn02B,CheungEn02C; their model-specific optima are not transferred here.

The three ledgers are:

The inequality signs mean that these are different kinds of data, not that their numerical values can never coincide.

For a specified Hamiltonian $H_{\mathrm{tot}}$, complete energy is evaluated from $\mathrm{Tr}(H_{\mathrm{tot}}{\rho }_{\mathrm{tot}})$. A heat or work ledger requires an implementation, designated reservoirs and work storage, and sign conventions. Residual leakage is the dimensionless norm diagnostic in Definition def:leakage.

Equality of observable channels, including equality with partial trace or a reset channel, does not determine a unique heat, work, or cyclic reusable-memory reset cost.

Proof. Let $M$ be a qubit memory and $A$ an independent blank qubit in $|0⟩$. A swap gives, including arbitrary references and after regrouping,

The observable memory channel is always the same reset channel. Nevertheless, for every $ϵ>0$ we may choose

The swap conserves $H_M+H_A$, but if the initial excited-state probability is $p$, then

Thus the observable channel fixes neither these energy transfers nor the cost of restoring the consumed blank. The input diary has moved to $A$.

Even after the memory Hamiltonians have been fixed, an independent unobserved process can change the total implementation ledger without changing the memory channel. For example, choose a fuel qubit $F$ initially excited and a designated thermal reservoir qubit $H$, both with gap $ϵ$. At inverse temperature $\beta =(k_{\mathrm{B}}T{)}^{-1}$, the latter has excitation probability

Swapping $F$ and $H$ increases the designated reservoir energy by $ϵ(1-r)$ while consuming fuel. It acts independently of $QM$. Repeating such independent spectator processes changes the designated reservoir energy increase while leaving the observed reset channel unchanged.

These are not cyclic erasure protocols unless the blanks, fuel, and controls are also restored. They prove underdetermination, not a violation of a thermal erasure inequality.

This reproduces the separation illustrated in the supplied En07-A record by a local counterexample, rather than treating that record as a premise CheungEn07A.

external thermal inequality. In the specified thermal setting of an initially uncorrelated memory and Gibbs reservoir undergoing complete unitary evolution, the reservoir heat satisfies

with entropy in natural logarithms. A decrease of one unbiased classical bit corresponds to $Q_{\mathrm{bath}}\ge k_{\mathrm{B}}T\ln 2$. These statements belong to the additional thermal setup, not to the equality $\mathsf{Drop}\circ \mathsf{Enter}={\mathrm{Tr}}_B$ Landauer1961,ReebWolf. Finite reservoir effects, reset accuracy, correlations, and consumed resources require their own accounting.

No thermal machine implementing the present Enter interface has been constructed. The implementation package required for a successor is listed in Appendix app:thermal.

The supplied Em03-E statement conserves a declared free Hamiltonian $H_0$ CheungEm03E. Conservation of $\mathrm{Tr}(H_0\rho )$, or of its full probability distribution, is not conservation of a thermodynamic Helmholtz free energy and is not a reset-cost theorem. Similarly, a completed apparatus record and a later apparatus reset are separate tasks CheungEm03B.

For a finite-dimensional channel $\mathcal{N}$ and a complementary channel $\mathcal{E}$ from the same Stinespring isometry, set

The KSW information–disturbance theorem gives

We import this theorem from KSW, using Schrödinger-picture diamond norms, equivalently the Heisenberg-picture completely bounded norms of the adjoint maps.

For the complete interface $V$,

Equivalently,

Proof. In (eq:ksw-full), $q=2{\epsilon }_{\mathrm{rec}}(\mathcal{N})$ and $r=2{\delta }_{\mathrm{full}}(V)$. The left inequality becomes ${\epsilon }_{\mathrm{rec}}(\mathcal{N}{)}^2\le 2{\delta }_{\mathrm{full}}(V)$. The right becomes ${\delta }_{\mathrm{full}}(V)\le \sqrt{2{\epsilon }_{\mathrm{rec}}(\mathcal{N})}$. Rearrange and use the universal half-diamond upper bound one for a pair of channels.

A proved upper bound ${\delta }_{\mathrm{full}}\le \zeta$ proves existence of a decoder with error at most $\sqrt{2\zeta }$, capped at one. It does not prove a lower bound ${\zeta }^2/2$: an upper allowance is not a measured lower bound on leakage.

Likewise, KSW's auxiliary dilation padding is not permission to give a physical inaccessible factor to the observer. The decoder still has domain $R$. Its laboratory implementation and resource budget remain separate.

For channels $\Phi ,\Psi$ with input dimension $d$, put

Then

Proof. The lower bound is the Bell input.

For the upper bound, use the standard stabilization fact stated in Section sec:interface. Any normalized pure state on $Q\otimes C$, with $\dim Q=d$, has the form

Thus ${\Vert S\Vert }^2\le d$, and

Convexity handles mixed states. Divide by two and take the supremum.

Define

Then

For a declared decoder $\mathcal{D}$, define

Then

In particular,

Proof. Apply Lemma lem:choi to $\mathcal{E}$ and each replacement channel, whose normalized Choi operator is $I_Q/d\otimes \sigma$, and take infima. Apply it also to $\mathcal{DN}$ and ${\mathrm{id}}_C$. The last estimate follows from Theorem thm:approx.

A single residual state $\sigma$ with a certified small Choi discrepancy gives a valid upper certificate. Testing only a small input ensemble does not give such an upper certificate unless its extension to all code matrix units or to the complete channel is proved.

If two code states satisfy

then ${\delta }_{\mathrm{full}}(V)\ge x/2$. If $O=O^{†}$, $\Vert O\Vert \le 1$, and

then ${\delta }_{\mathrm{full}}(V)\ge w/4$.

Proof. For every $\sigma$, the triangle inequality gives

Take the infimum. Also $w\le {\Vert \mathcal{E}({\rho }_0-{\rho }_1)\Vert }_1=2x$.

These witnesses detect some logical dependence. They do not locate a recoverable whole diary in the residual.

Let $V,W:C\to R\otimes B$ be isometries and suppose $\Vert V-W\Vert \le \kappa$. Then

and both corresponding marginal channel distances are at most $\kappa$. Moreover,

and

Proof. For every referenced operator $X$,

with reference identities understood. Its trace norm is at most $2\kappa {\Vert X\Vert }_1$. Taking the stabilized supremum and dividing by two proves the complete-channel estimate. Partial trace gives the marginal estimates.

For every $\mathcal{D}$, the triangle inequality and contraction give

Take infima and interchange $V,W$. The residual estimate is the same argument with each replacement state $\sigma$.

Consequently, if a complete finite approximation has error at most $a$ in half-diamond distance and an approximate decoder has certified error at most $b$, the same decoder has error at most $\min \{1,a+b\}$ on the compared accessible channel. The approximation must concern the declared complete channel and access identification; a selected occupation or entropy approximation is not enough.

The endpoint is a construction obligation. This section states a sufficient finite-interface contract and proves what follows if that contract is met. It does not assert that it has been met in a gravitational model.

For complete interfaces $V_t:C\to R_t\otimes B_t$, an access-preserving comparison into endpoint spaces consists of specified isometries

The compared complete interface is

The comparison of an accessible output is performed on $R_t$ alone. It does not use $B_t$.

In a physical application, the allowed access operations, output-factor meanings, and receiver realization must be justified in the same model. Product embeddings are a sufficient mathematical access-preserving class, not a claim that every physically meaningful identification has this form.

Actual dynamics may transfer information from an initially inaccessible system to outgoing radiation. What is excluded is obtaining recovery merely by changing the comparison map so that a still-inaccessible diary register is renamed as part of the accessible output.

With the access-preserving embeddings of Definition def:access,

where

Proof. Any isometry $j$ has a CPTP left inverse

for a fixed state $\tau$ on its domain. A decoder after ${\mathrm{Ad}}_j\mathcal{N}$ composes with ${\mathrm{Ad}}_j$ to give a decoder after $\mathcal{N}$; a decoder after $\mathcal{N}$ composes with $L_j$ to give a decoder after ${\mathrm{Ad}}_j\mathcal{N}$. The resulting decoded channels coincide, so their optimal errors coincide.

For the complement, embedding any replacement state gives

because the output isometry preserves trace norms, including references. This proves one direction. For any replacement state $\omega$ on the larger space, postprocessing by $L_k$ gives

Taking infima proves the opposite direction.

A finite-code endpoint package consists of:

1. Complete preparation. A justified finite code $C$, its embedding, and a fixed complete preparation are supplied in one microscopic model. All purifiers are retained in the complete description.

2. Complete evolution. The model supplies unitary evolution on its complete physical system and therefore the complete interfaces $V_t$, not merely radiation marginals.

3. Outgoing access and endpoint. The model supplies the meaning of $R_t,B_t$, an access-preserving comparison, and a map $V_{\infty }:C\to R_{\infty }\otimes B_{\infty }$ such that

   $$\Vert {\hat{V}}_t-V_{\infty }\Vert \le {\kappa }_t,{\kappa }_t⟶0.$$

   The spaces in this finite theorem are finite dimensional. A continuum or infinite-output application needs its corresponding normal-channel and convergence theorem, rather than an unproved truncation.

4. Complete endpoint complement. For one endpoint state ${\sigma }_{\infty }$, either

   $${\mathcal{E}}_{\infty }={\mathcal{T}}_{{\sigma }_{\infty }},$$

   or the declared approximate allowance

   $$d_{\diamond }({\mathcal{E}}_{\infty },{\mathcal{T}}_{{\sigma }_{\infty }})\le \zeta$$

   is proved, where ${\mathcal{E}}_{\infty }={\mathrm{Tr}}_{R_{\infty }}{\mathrm{Ad}}_{V_{\infty }}$.

H3 is a sufficient, relatively strong convergence contract. It is not claimed necessary for every possible endpoint construction. No hypothesis is established by the terminology “boundary”, “bulk”, “recycle bin”, or “holography”.

Assume the package in Definition def:four. Then $V_{\infty }$ is an isometry.

If the endpoint complement is exactly constant, there is a deterministic decoder ${\mathcal{D}}_{\infty }:R_{\infty }\to C_{\mathrm{out}}$ such that

The finite-time decoder

uses only $R_t$ and satisfies

If the endpoint complement has the approximate allowance $\zeta$, one may choose ${\mathcal{D}}_{\infty }$ so that

and then

All conclusions preserve external references in the stated diamond-error sense.

Proof. Each ${\hat{V}}_t$ is an isometry. Operator-norm convergence implies

Under exact endpoint constancy, Theorem thm:exact gives an exact endpoint decoder, including its deterministic extension to the unused output subspace.

Under approximate constancy, ${\delta }_{\mathrm{full}}(V_{\infty })\le \zeta$, and Theorem thm:approx, together with attainment of the decoder minimum, gives the asserted ${\mathcal{D}}_{\infty }$.

The map ${\mathcal{D}}_t$ is CPTP and has domain $R_t$. By Lemma lem:stability,

Therefore

For the exact case take $e_{\infty }=0$. Cap the bounds at one. Every step uses complete channel distance, so the reference statement is already included.

If the exact endpoint complement is one fixed pure state, then ${\mathcal{N}}_{\infty }$ is an isometric encoding.

Proof. In Theorem thm:exact, the residual rank is one. Thus $K$ is one dimensional and ${\mathcal{N}}_{\infty }(X)=AX{A}^{†}$ for an isometry $A:C\to R_{\infty }$.

If one physical gravitational model justifies the four obligations in Definition def:four, with an interface to which the preceding theorem applies, its declared outgoing endpoint recovers the finite code exactly, or within the stated approximate allowance. A subsequent terminal record is distinct from this quantum recovery.

Proof. Apply Theorem thm:endpoint to that justified complete interface. Apply Lemma lem:erasure-interface and Theorem thm:classical to any subsequent destructive record interface, with its actual surviving registers.

This is the local implication also documented in Hawk04-B's :gravity CheungHawk04B. Its gravitational hypotheses remain unsupplied in the present paper.

Suppose the compared family is operator-norm continuous near $t_0$, the same-model geometry and access construction are valid on an open interval $I$ containing $t_0$, and a verified endpoint decoder satisfies

where ${\kappa }_t=\Vert {\hat{V}}_t-V_{\infty }\Vert$. Then some open interval $J\subseteq I$ containing $t_0$ has the deterministic decoder errors

Proof. Continuity of ${\hat{V}}_t$ implies continuity of ${\kappa }_t$. The strict inequality therefore persists on a sufficiently small neighborhood of $t_0$ inside $I$. Apply Theorem thm:endpoint.

No such $t_0$, same-model interval $I$, or physical field-to-memory receiver has been constructed here. In particular, this conditional statement does not establish a positive noncommuting recovery window for one of the documented finite baths.

There are finite complete preparations and unitary programmes with the same code, same access factorization, and same conserved complete Hamiltonian, which terminate after finitely many instructions and have complete isometric endpoints, but have opposite exact outgoing-recovery verdicts.

Proof. Prepare the code in $B\cong C$ and a blank $|0⟩$ in $R\cong C$. Fix a common self-adjoint matrix $h$ on both factors and the complete Hamiltonian

One programme applies $I_{RB}$ and halts. The other applies ${\mathrm{SWAP}}_{RB}$ and halts. Both commute with $H_0$ and preserve its complete probability distribution. Their complete endpoints are $V_{\mathrm{keep}}$ and $V_{\mathrm{send}}$ from Proposition prop:entropy.

Both preserve the code and every reference on the complete output. The first has a constant accessible output and recovery error $1-d^{-2}$. The second has constant complete complement and recovery error zero. The access factor $R$ has not changed.

This is an insufficiency theorem for the listed finite properties, not a counterexample to a gravitational evaporation theorem.

Recovery error is not invariant under a comparison that moves an inaccessible diary-bearing factor into the accessible output.

Proof. For $V_{\mathrm{keep}}$, the declared accessible factor $R$ has error $1-d^{-2}$. If $B$ is instead made accessible, its channel is the identity and its error is zero. A comparison that swaps these access roles therefore changes the recovery question. It is not one of the local product comparisons in Definition def:access with the declared roles fixed.

comparison is still required. The SYK–scalar evaporation model of Gaikwad, Kaushal, Mandal, and Wadia has its own preparation, low-energy, and large-$N$ assumptions GaikwadEtAl. Its geometrical conclusions and its reconstruction of microstate labels using repeated processes are not automatically a deterministic one-copy decoder for an arbitrary reference-entangled finite code. Such a bridge requires the four same-model obligations above.

Likewise, a horizonless asymptotic geometry by itself does not prove complete residual constancy. A CPTP decoder by itself does not construct a local radiation receiver.

The following ledger distinguishes what this manuscript proves from what a physical successor must construct.

Hawk04-A is the documentary predecessor for complete preservation and exact correction; Hawk04-B and En07-A fix the quantum-return, readout, and discard framing CheungHawk04A,CheungHawk04B,CheungEn07A. En04-A separates residual response from non-Clifford and non-local-magic quantities CheungEn04A. No magic quantity is substituted for ${\delta }_{\mathrm{full}}$, and no companion theorem is a premise of Theorems thm:global–thm:classical.

The Em03-B record distinguishes a completed apparatus record from a later reset; Em03-E conserves its declared free Hamiltonian CheungEm03B,CheungEm03E. Neither is a thermal implementation of this paper.

The En02-B and En02-C records distinguish information-theoretic availability from executable streaming recovery CheungEn02B,CheungEn02C. Their memory bounds are tied to their delivery, storage, and output contracts, not to an unrestricted resource optimum for the present fragment.

The flagged cutoff statements in En02-E retain their commuting-state hypotheses CheungEn02E. A discarded probability in that setting is not a generic coherent cutoff allowance in diamond norm.

Hawk02-E is a regulated SYK–scalar certification framework; Hawk03-A is a finite firewall-interface package; Em03-H supplies complementary channels and certificates for its fixed quartic bath CheungHawk02E,CheungHawk03A,CheungEm03H. These records do not supply H3 or H4 for a gravitational endpoint. They can motivate tests such as those in Section sec:approx, but their Hilbert spaces, preparations, and accessible registers must not be identified merely because their channel notation looks similar.

No new finite-bath simulation, independent numerical rerun, four-Majorana analysis, or recurrence computation is reported here.

En06-B's increasing-stencil transport and exact instrument identity are documentary finite-network templates for moving registers while retaining references CheungEn06B. Its constant $c$ is an input; it is not a derived physical speed or a quantum field theory.

En05-C's tensor-free regional comparison does not, by state restriction alone, select a tensor environment or complementary-state channel CheungEn05C. A gauge-algebra application needs its own access and dilation construction.

XVII-A reconstructs a finite effect and record on an assumed background. Its finite time-band algebra result is not an entanglement-wedge theorem CheungXVIIA. Em06-A and Em06-B are scalar-only limits with metric data as inputs CheungEm06A,CheungEm06B. None of these supplies the endpoint package of Definition def:four.

A useful successor must declare:

1. the complete channels and complete purifications compared;

2. the reference-preserving error topology and allowance;

3. the accessible registers on both sides of the comparison;

4. a deterministic decoder and its permitted operations;

5. the common interval of geometric, access, and recovery validity;

6. any coherent receiver that turns field access into a laboratory quantum return.

Selected entropy, a prepared transfer flag, a classical two-state control, or fixed-proxy recurrence supplies none of these automatically.

This section is a documentary status ledger, not a proof about the existence or nonexistence of an analytic gravitational coefficient. None of the distributional objects in that programme is an input to the local channel theorems above.

The underlying citations are CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB, CheungXVIN,CheungXVIO,CheungXVIP. The version distinctions matter: a later partial construction does not retroactively change what an earlier release contained, and a historical stop does not negate a later explicitly stated partial advance.

The retained analytic target includes all branches in the grouped sum

with both $K_5$ placements and the residual variable retained. A reference for $K_6+K_8+K_9$ alone is not the full characteristic contraction. An internal retarded Fourier pole is not a new cross-line prescription.