Hawk01-B Complete Quantum-Code Preservation in a Sequential Evaporation Model: Radiation Correlations, Residual Leakage, and the Page-Curve–Recovery Gap

We study preservation of an arbitrary finite-dimensional quantum code, including all external reference correlations, in a declared sequential emission process. We translate the Kretschmann–Schlingemann–Werner information–disturbance theorem into half-diamond normalization and separate its worst-case guarantee from single-input entanglement and entropy diagnostics. We then construct an energy-conserving sequence that generates its own key resource, transfers an encrypted logical system through occupied radiation sectors, retains vacuum and unsuccessful-transfer branches, and releases every residual degree of freedom through an explicitly separate endpoint map. At each pre-endpoint prefix, the optimal recovery error and optimal constant-channel leakage distance are both $s_t(1-d^{-2})$, where $s_t$ is the probability that transfer remains pending. Individual emitted blocks are message-independent, whereas the joint radiation carries the logical coherence. Reference mutual information and its conditional increments are calculated exactly. Thermal occupation comparisons include explicit truncation errors and compatibility obstructions; no agreement with Hawking pair-production dynamics is inferred from local statistics. A second conserving evolution has the same radiation density operators for one specified preparation, and the same radiation entropy histories for every pure code preparation, but retains the code in an inaccessible memory. The counterexample discloses its different endpoint and does not challenge island or entanglement-wedge reconstruction results. Coherent access, destructive number measurement, complete-output erasure, and regulated implementation errors have separate certificates. The construction is a finite field-sector completion, not a microscopic theory of realistic black-hole evaporation.

Hawk01-A distinguished delayed-query retrieval from source spectra, observer access, and an assumed information-transfer mechanism CheungHawkA. Here the logical task is stronger:

Under what declared dynamical, conservation, endpoint, and access assumptions does radiation preserve every state of a specified quantum code, together with every external reference correlation?

Distinguishing selected preparations is not substituted for this arbitrary-input requirement.

We establish three connected results. First, residual leakage and arbitrary-input recoverability are related by the established channel information–disturbance framework, with all normalization factors displayed KSW2006,BenyOreshkov2010. Second, a consistent conserving sequence realizes complete code preservation using identifiable joint radiation correlations. Third, an explicit entropy-history separation shows why scalar radiation entropies, detached from their dynamical and reconstruction setting, do not determine that channel property.

The exact recovery interfaces in Hawk01-A and Em05-B, the key-transfer separation in Em03-E, and the readout/recovery/resource distinctions in En02-B–E provide useful source precedents CheungHawkA,CheungEm05B,CheungEm03E,CheungEn02B, CheungEn02C,CheungEn02D,CheungEn02E. Their models are not identified with the model below, and their conclusions are not transferred as black-hole theorems.

versus assumed. The isometries, conservation identities, channels, decoders, entropy formulas, and recovery optima below are mathematical results within the specified finite model. The degenerate radiation sectors, their couplings, transfer schedule, and endpoint map are model assumptions. They are not derived from the Schwarzschild geometry or the Hawking spectrum.

unpriced assistance. The construction starts from one logical input and fixed pure ancillary preparations. It does not require pre-collected radiation or a supplied correlated early key. It generates its early resource in the first emission and includes its purifier in subsequent residual and endpoint accounting.

implicit terminal extrapolation. The pre-endpoint construction is stopped above a declared $M_{\mathrm{cut}}$. A separate endpoint rule is then imposed. Setting that rule to complete evaporation is an additional assumption, not an extrapolation of fixed-background quantum field theory.

All Hilbert spaces in the channel theorems are finite-dimensional. Logarithms and entropies use base two. For states and channels, respectively, define

The diamond norm is the stabilized trace norm in the Schrödinger picture. Its dual is the completely bounded operator norm used in the Heisenberg-picture formulation of Kretschmann–Schlingemann–Werner KSW2006.

Let $A$ be a code of dimension $d\ge 2$, contained in one declared energy and charge sector. Let $S\cong A$ be an external reference, and write

A prefix isometry is

Its accessible and inaccessible channels are

The system $Z_t$ contains every accessible quantum system and classical record. The system $F_t$ contains the residual source, excluded radiation, detector environments, stored-resource purifiers, and every other inaccessible output.

Fixed mixed preparations are purified explicitly before defining $W_t$; their purifiers are assigned to $Z_t$ or $F_t$, not omitted.

The unrestricted deterministic recovery error and inaccessible leakage distance are

The decoder optimization is over CPTP maps. A resource-restricted decoder class defines a different optimum.

If a decoder satisfies $D_{\diamond }({\mathcal{D}}_t{\mathcal{N}}_t,{\mathrm{id}}_A)\le ϵ$, then, for every reference $S^{\prime }$ and every joint state ${\rho }^{S^{\prime }A}$,

Conversely, it is sufficient to verify this uniform state bound for a reference of dimension $d$.

Proof. Apply the definition of the stabilized trace norm to the channel difference. For a Hermiticity-preserving difference of channels, the diamond norm is attained on a state with reference dimension equal to the input dimension. Larger references do not increase the supremum Watrous2018.

distinct initial-state scopes. A formation-code statement preserves all variable states admitted by a declared initial code subspace of the regulated source. An injected-message statement preserves a message introduced into a separately specified source and early-radiation preparation. The construction below is a formation-code construction with fixed ancillaries. It does not claim preservation of arbitrary states of its entire enlarged ambient source Hilbert space.

For a finite-dimensional isometry $W:A\to Z\otimes F$, the following are equivalent:

1. there is a CPTP map $\mathcal{D}:Z\to A$ with $\mathcal{DN}={\mathrm{id}}_A$;

2. there is a density operator ${\tau }_F$ such that

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_F\text{for every }X\in \mathcal{B}({\mathcal{H}}_A);$$

3. for a Kraus family $\{K_{\alpha }\}$ of $\mathcal{N}$,

   $${K_{\alpha }}^{†}{K}_{\beta }=c_{\alpha \beta }{I}_A$$

   with scalar coefficients $c_{\alpha \beta }$.

If these conditions hold and $\mathrm{rank}{\tau }_F=r$, then

Proof. Dilate an exact decoder by $T:Z\to A\otimes G$. The composite $(T\otimes I_F)W$ dilates the identity channel. The Choi matrix of the identity has rank one, so every Kraus operator of that composite is proportional to $I_A$. Consequently

for one fixed normalized vector $\chi$. Tracing $AG$ proves (eq:constantcomplement).

Conversely, diagonalize ${\tau }_F={{\sum }_{j=1}}^r{\lambda }_j|j⟩⟨j|$, with ${\lambda }_j>0$, and write

The constant-complement identities on the matrix units $|i⟩⟨i^{\prime }|$ give

Thus there are $dr$ orthonormal vectors in $Z$, proving (eq:rankcapacity). On their span $L$, define $U|{\varphi }_{ij}⟩=|i{⟩}_A|j{⟩}_J$. For a fixed state ${\sigma }_A$, the map

is CPTP and satisfies $\mathcal{DN}(X)=X$.

Finally, in the environment basis associated with the Kraus family,

Constancy on every operator is equivalent to (eq:KL).

This is the arbitrary-code constant-complement form of exact Knill–Laflamme correctability KnillLaflamme1997,CheungEm05B,CheungHawkA. Testing only a small preparation ensemble does not establish its matrix-unit requirement.

We use the following published continuity theorem, rather than introducing a new general information–disturbance theorem.

After padding their dilation spaces when necessary, channels $\Phi ,\Psi$ admit Stinespring isometries with optimized operator-norm distance $\beta$ satisfying

This is Theorem 1 of Kretschmann–Schlingemann–Werner, expressed through the Schrödinger dual of its completely bounded norm.

For the complete complementary channels of Definition def:interface,

Equivalently,

The corresponding symmetric bounds follow by exchanging ${ϵ}_{\mathrm{rec}}$ and ${\delta }_{\mathrm{leak}}$. In particular, ${ϵ}_{\mathrm{rec}}=0$ if and only if ${\delta }_{\mathrm{leak}}=0$.

Proof. Fix a state ${\tau }_F$. A dilation of its constant channel is

where ${\chi }_{\tau }$ purifies ${\tau }_F$. Applying Lemma lem:KSW to $\mathcal{C}$ and ${\mathcal{T}}_{\tau }$ supplies, after padding, an isometry $T:Z\to A\otimes G$ such that

Tracing $GF$, and using the right inequality of (eq:KSWraw), gives

Optimize over $\tau$.

Conversely, fix a decoder with error $ϵ$ and dilate it. The resulting channel is within raw diamond distance $2ϵ$ of the identity. Lemma lem:KSW aligns its dilation with $|\psi ⟩\mapsto |\psi ⟩\otimes |\chi ⟩$, with operator distance at most $\sqrt{2ϵ}$. Tracing the recovered logical output and the decoder environment therefore bounds the original complement by a constant channel with half-diamond error at most $\sqrt{2ϵ}$. Optimize over the decoder.

This reproduces the information–disturbance argument of KSW Theorem 3 with the two infima explicit. The factors of two arise solely from (eq:normalization); no square-root constant is guessed.

The state ${\tau }_F$ depends on the channel and may depend on the decoder in the converse proof. The theorem does not state that every chosen constant output state gives the converse bound. An executable decoder and its resource cost remain separate from the unrestricted existence statement. Approximate-QEC complementary-channel duality offers additional fidelity-based and near-optimal recovery formulations BenyOreshkov2010,BenyOreshkov2011.

For a channel $\Lambda :A\to A$, define its normalized Choi state and Bell-input entanglement fidelity by

For every $\Lambda :A\to A$,

For any two channels with input dimension $d$,

Proof. The Bell input and Bell-projector effect give the lower bound in (eq:Bellconversion). The Choi trace-norm comparison gives (eq:Choiconversion); equivalently, the raw diamond norm is at most the trace norm of the unnormalized Choi matrix, which is $dJ$ Watrous2018. Finally, the trace distance to a pure state is at most the square root of one minus its overlap. Combine these statements.

Thus $F_e=1$ proves exact channel identity, but approximate Bell fidelity alone does not give a dimension-independent worst-case guarantee.

For the complementary Bell state

put

Then (eq:Choiconversion) gives

If only its mutual information is known, quantum Pinsker's inequality in bit units gives

These are diagnostic-to-channel conversions, not replacements for the diamond metric.

For any orthonormal pair $|0⟩,|1⟩$ in the code, define

Every decoder satisfies

Proof. Use a maximally entangled input on that two-dimensional subcode. If the decoder error is $ϵ$, its overlap with the corresponding Bell vector is at least $1-ϵ$. Dilate the decoder and project the global pure state onto that Bell vector. The normalized projected state is a Bell vector times an environmental vector, and its distance from the original pure state is at most $\sqrt{ϵ}$. Partial trace gives

Pinching the two-dimensional reference in the pair basis yields

The triangle inequality bounds the left side below by $\ell /2$.

This reproduces the Bell-projection witness reported in Em05-B, with its normalization and proof explicit CheungEm05B.

A consistent sequence begins with one fixed preparation and acts on the actual residual system:

No internal register is re-prepared between steps.

For each declared conserved quantity $Q$, conservation means

including the analogous Hamiltonian equation. Energy sectors and their dimensions are part of the model data.

We use orthogonal outgoing energy-eigenmode sectors:

1. a key block $R_1$ with $d^2$ orthogonal one-particle states $|r⟩$, all of energy $e_{\mathrm{K}}>0$;

2. each transfer block $R_t$, $2\le t\le N$, with vacuum of energy zero and $d$ orthogonal one-particle states $J_t|i⟩$, all of energy $e>0$;

3. an endpoint radiation space with the energy-labelled orthogonal states specified in Section sec:endpoint.

All these states have the same declared charge and angular-momentum labels within their respective occupation sectors; in the explicit neutral implementation these labels vanish.

Sufficient degeneracy is assumed in the residual source sectors. No Bekenstein–Hawking state-count formula is assigned to them.

One finite field realization uses distinct neutral scalar species of equal dispersion in the same regulated s-wave energy mode. No additional conserved flavor number is imposed, and the specified couplings may mix the degenerate species. This is an artificial but explicit field-sector choice, not an identification of known astrophysical particles.

Exact energy conservation below concerns this energy-eigenmode regulator. A finite-time wavepacket is not an exact energy eigenmode. Replacing these sectors by finite-time collected packets requires a separate spectral-width and channel-error certificate.

and energy budget. The initial source energy is $M$, and

The code is degenerate within that initial sector. Its logical Hamiltonian is zero after subtraction of the common sector energy.

. At prefix $t\ge 1$, the construction uses

The first summands record completed transfer time and key. The last summands contain a pending encrypted logical system. The growth of this particular ledger is disclosed; no claim that it models a decreasing black-hole microstate dimension is made.

The required finite isometries can be embedded in a larger fixed energy-sector Hilbert space and extended to block-energy-conserving unitaries on prepared source-plus-vacuum ports. Unused orthogonal states are included in that embedding. This finite-dimensional extension does not supply a local gravitational Hamiltonian.

Let $\zeta =e^{2\pi \mathrm{i}/d}$, and define

Write $r=(a,b)\in \{0,\dots ,d-1{\}}^2$.

For every $Y\in \mathcal{B}({\mathbb{C}}^d)$,

Proof. Averaging over $b$ removes every off-diagonal matrix unit. Averaging over $a$ distributes each surviving diagonal entry uniformly over the computational basis.

The first emission is

Here $|\mathrm{f},r;\psi ⟩$ is a linear isometric embedding of the logical vector into the pending residual summand labelled by $r$. All pending residual states have energy $M-e_{\mathrm{K}}$.

Equation (eq:firstemission) creates the key correlation inside the evolution. Its coefficient and the $d^2$ terms give ${V_1}^{†}{V}_1=I_A$. There is no externally supplied early register or classical random key. The first outgoing marginal is

The complementary output retains the encrypted code and the key.

Choose fixed probabilities ${\lambda }_t\in [0,1]$, $2\le t\le N$. For a pending residual input define

For an already completed branch $j<t$,

Completed residual states have energy $M-e_{\mathrm{K}}-e$.

Define

The label $s_t$ denotes pending-transfer probability. A zero-probability branch may be retained in the ambient ledger.

Equations (eq:firstemission)–(eq:completedmap) define a consistent sequence of isometries on the declared residual spaces. They conserve energy and the specified charges, retain every vacuum and unsuccessful branch, and perform no reset.

For $t\ge 1$, the accumulated isometry is

Every branch has total energy $M$. The mean pre-endpoint residual energy is

Proof. For pending inputs, the two terms in (eq:attemptmap) are orthogonal, and their squared norms sum to the input norm. For distinct key labels the residual summands are orthogonal. A newly completed branch has a different residual label from every previously completed branch. Thus the maps preserve all input inner products.

A pending successful term changes residual energy by $e$ and emits energy $e$. An unsuccessful or previously completed term emits vacuum and retains its residual energy. The first emission changes residual energy by $e_{\mathrm{K}}$ and emits that amount. The same reasoning applies to the declared charges.

Induction on $t$ proves (eq:accumulated): a pending amplitude is multiplied by $\sqrt{1-{\lambda }_t}$, and its newly completed amplitude is multiplied by $\sqrt{{\lambda }_t}$. The earlier amplitudes are unchanged. Their probabilities are (eq:weights) and sum to one. The energy average follows.

The model emits at most one logical transfer quantum across the attempt blocks. It is not a forecast that a physical black hole ceases ordinary emission once a message has been transferred.

For concision, identify each occupied block with its logical $d$-dimensional subspace and denote the associated all-other-vacuum embeddings by $J_j$.

For $t\ge 1$, with all emitted blocks accessible,

The complete residual complement is

where $I_{\mathrm{f},r}$ embeds the pending logical system into its residual summand. Every individual emitted block is message-independent:

Proof. Trace the orthogonal residual summands in (eq:accumulated). Different key labels also become diagonal because their residual supports are orthogonal. This gives (eq:accessibleprefix).

Conversely, tracing $R_1$ removes cross terms in the key. Tracing occupied/vacuum radiation removes cross terms between different transfer times and between transfer and pending branches. The resulting residual state is (eq:complementprefix).

The local formulas follow by tracing all other radiation blocks and applying Lemma lem:twirl.

The logical information is therefore not inferred from one-block distinguishability. It is present in the correlations between the generated key and an occupied transfer block.

Set

For ${\mathcal{T}}_d(Y)=\mathrm{Tr}(Y)I_d/d$,

Proof. The Weyl twirl represents ${\mathcal{T}}_d=d^{-2}{\sum }_r{\mathrm{Ad}}_{U_r}$. The identity term has weight $d^{-2}$. The triangle inequality and $D_{\diamond }({\mathrm{Ad}}_U,\mathrm{id})\le 1$ give the upper bound $c_d$. On the Bell input, the $d^2$ Weyl outputs are orthogonal Bell states, so the matching trace distance is $c_d$.

For any constant channel ${\mathcal{T}}_{\sigma }$, the Bell output is $I_S/d\otimes \sigma$. Its Bell-projector expectation is $d^{-2}$. The effect bound gives distance at least $c_d$.

For the complete pre-endpoint prefix,

The maximum Bell-input entanglement fidelity is

An explicit optimal decoder reads $r$, locates the occupied transfer block without reading its logical basis, applies ${U_r}^{†}$ to that block, and returns it. On the all-vacuum pending branch it prepares $I_d/d$.

Proof. The stated decoder has complete decoded channel

Lemma lem:depolarization gives error $s_t{c}_d$.

For any decoder, a completed branch contributes Bell overlap at most one. The accessible pending branch is input-independent. Its reference state remains $I_S/d$, so after any local decoding its Bell overlap is $d^{-2}$. Therefore

The Bell-projector lower bound proves that no decoder has error smaller than $s_t{c}_d$, and the stated decoder attains both optima.

For leakage, choose the constant state $\tau ={\mathcal{C}}_t(I_d/d)$. The completed summands cancel in ${\mathcal{C}}_t-{\mathcal{T}}_{\tau }$. On the pending summands the difference is $s_t$ times a reversible flagged Weyl encoding of ${\mathrm{id}}_d-{\mathcal{T}}_d$. Contractivity, followed by its left inverse, gives the upper bound $s_t{c}_d$.

To prove optimality against an arbitrary constant state, apply a CPTP map to the residual that retains a pending/completed flag, corrects $U_r$ and returns the logical qudit on pending branches, and returns a fixed state on completed branches. On the Bell input, the pending block of the actual state is $s_t|{\Phi }_d⟩⟨{\Phi }_d|$. Let $a$ be the pending weight of the mapped candidate constant state. Its pending Bell expectation is $a/d^2$.

Use first the effect equal to the Bell projector on pending and zero on completed. Its expectation difference is $s_t-a/d^2$. Use next the effect equal to that Bell projector on pending and identity on completed. Its expectation difference is $a(1-d^{-2})$. If $a\le s_t$, the first is at least $s_t{c}_d$; if $a\ge s_t$, the second is at least $s_t{c}_d$. The trace-distance effect bound, and hence the diamond norm, gives the required lower bound for every constant state.

The residual key is not ignored in the leakage calculation. Once transfer has succeeded, the remaining key and transfer-time labels are constant channels on the logical input, even though they remain correlated with accessible radiation. Constancy means absence of logical-input dependence, not absence of all physical correlations.

For any logical operator $O\in \mathcal{B}({\mathbb{C}}^d)$, define on the completed accessible sectors

Then

Thus Weyl $X$ and $Z$ coherence is accessed by key-conditioned joint observables, not by local number statistics. On a successful flag, the same expression normalized by that branch weight gives the exact logical expectation. The primary channel remains complete and unconditioned.

The decoder needs one surviving $d$-dimensional quantum logical carrier and classical key/time records. The key can be measured in its label basis and stored classically without destroying the recoverable logical coherence. This is different from measuring the occupied logical radiation in its occupation basis.

Define the branch distribution

with $0\log 0=0$, and put $L_d=\log d$.

For an arbitrary code state $\rho$ and $t\ge 1$,

For the input $|{\Phi }_d{⟩}^{SA}$,

The key emission has $I(S:R_1)=0$.

Proof. In (eq:accessibleprefix), completed blocks have conditional entropy $H(\rho )$, pending blocks have entropy zero, and the classical key has entropy $2L_d$. The orthogonal branch labels contribute $H({\mathbf{b}}_t)$. This proves (eq:radiationentropy). The complementary channel has the opposite placement of the logical state, proving (eq:residualentropy).

For the Bell input, $H(\rho )=L_d$ and the global state $SR{B}_t$ is pure. Therefore

and the complementary identity gives $I(S:B_t)=2s_t{L}_d$. The conditional mutual information is the difference of successive reference–radiation mutual informations. Since $p_t-p_{t-1}=w_t$, (eq:CMIincrement) follows.

The mutual-information identity

is an exact information ledger for this pure complete interface. It is not an independent argument for recoverability in another model. Here the diamond errors are separately calculated in Theorem thm:optima.

The construction is consistent with no-hiding statements: when a selected marginal is constant, its complete complement contains the corresponding logical information BraunsteinPati2007. It does not claim that arbitrary quantum information can be stored solely in correlations of two inaccessible marginals while disappearing from their complete complement.

Choose a fixed final residual energy $m_{\ast }$ satisfying

The information-free endpoint residual is one fixed state $|v⟩$ of energy $m_{\ast }$. For completed residual basis states, introduce endpoint radiation states $|j,r{⟩}_{\ast }$ of energy

For pending residual basis states, introduce $|\mathrm{f},r,i{⟩}_{\ast }$ of energy

All these radiation states are orthogonal, with the specified conserved labels.

Define

The endpoint map is an energy-conserving isometry and transfers every residual key, transfer-time record, and pending logical system into radiation. Its complement is

Hence

For input $\rho$,

and for the Bell input,

The endpoint conditional information increment is $2s_N\log d$.

Proof. The maps send an orthonormal residual basis to an orthonormal endpoint radiation basis with one fixed source factor. Every term preserves residual-plus-radiation energy. There is no residual record or purifier outside that fixed factor.

On completed branches, the earlier occupied block is decoded using $r$. On pending branches, the endpoint occupied subspace contains the encrypted logical system and its key label; apply ${U_r}^{†}$ there. The branch labels are orthogonal. This is an explicit exact decoder on the emitted state.

Alternatively, (eq:endpointcomplement) and Theorem thm:exact establish exact recovery. The full radiation map is an isometric encoding, so it preserves the nonzero spectrum of $\rho$, proving (eq:endpointentropy). The reference statements follow from purity, or from exact decoding. Their increment over (eq:MIradiation) is $2s_N\log d$.

For $m_{\ast }=0$ and a unique fixed source vacuum, this is a declared complete-evaporation endpoint. For $m_{\ast }>0$, it is an information-free remnant endpoint. Neither option is derived from semiclassical physics below $M_{\mathrm{cut}}$.

If a pure-preparation isometry has all non-source outputs included in accessible radiation and its final source space is one-dimensional, then the radiation preserves the entire input code exactly.

A final mass label, or vanishing final source energy, does not by itself imply this one-dimensional condition.

Proof. A one-dimensional source factor makes its complementary channel constant, so Theorem thm:exact applies. Conversely, an energy eigenspace can have dimension greater than one. A code can remain in that degeneracy without changing the energy label.

The endpoint rule is therefore a substantive part of the preservation claim. An inaccessible message-bearing auxiliary cannot be erased from the account by calling the visible source “evaporated.”

The source spectrum does not determine the transfer schedule ${\lambda }_t$. We make only a declared comparison with selected outgoing occupation states.

For a neutral scalar stationary energy mode, the fixed-background outgoing occupation is

This is the greybody-modified source interface developed in Hawk01-A and Em03-A CheungHawkA,CheungEm03A,Hawking1975. We do not repeat its field-theory derivation.

For $g$ identical degenerate scalar modes, define the product thermal state

Its normalized restriction to total occupation zero or one is

The discarded thermal probability is

For

one has

Thus the transfer block $R_j$ has the displayed bound with $g=d$ and $w=w_j$.

Proof. The vacuum and uniform one-particle sectors are orthogonal, so their probability difference gives (eq:Bernoullideviation). The conditional low-occupation state and the removed tail occupy orthogonal sectors; renormalization gives trace distance equal to the removed probability. Apply the triangle inequality.

The conditional thermal state in (eq:thermal01) is a calibration object, not a heralding rule for the model data. All model attempts remain in the primary channel.

Prescribed unconditional one-particle weights $a_j\ge 0$ can be realized by the transfer attempts exactly when

Before exhaustion, a realizing schedule is

After exhaustion all later $a_j$ must vanish, and later ${\lambda }_j$ are immaterial on the actual state.

Proof. The model emits at most one transfer excitation, and ${\sum }_j{w}_j=p_N\le 1$, proving necessity. For sufficiency, use (eq:calibratedschedule) and induction: the pending weight before attempt $j$ is $1-{\sum }_{k<j}{a}_k$, so $w_j=a_j$.

Consequently arbitrary exact local thermal restrictions cannot be promised independently at every step. A target $a_j=a_d(q_j)$ must satisfy (eq:sumcompatibility), and the parameters $q_j$ must first be obtained from an actual declared source calculation if an astrophysical calibration is intended.

The key block is deliberately not Hawking-like in a small-tail regime. For $g=d^2$, its state is ${I_g}^{(1)}/g$, and

The minimum follows by maximizing the one-particle probability at $q=1/(g+1)$. Thus this construction does not claim uniformly small thermal deviations for all emitted blocks.

and symmetry scope. The logical operations act inside a degenerate fixed-charge code. No claim is made about preserving coherent superpositions of arbitrary different charge sectors with the same rules. Conservation can impose additional recovery constraints in scrambling models and symmetry-covariant codes NakataWakakuwaKoashi2023,FaistEtAl2020. Those constraints are not removed by assuming sufficient degeneracy without disclosing it.

Outgoing number statistics, a one-block marginal, a near-horizon pair state, and local semiclassical dynamics are four different tests. The construction proves only its explicitly stated marginal and channel results.

A useful entropy form of the pair-production obstruction is elementary.

Let $R$ be previously emitted radiation, $b$ the next outgoing system, and $c$ its declared inaccessible partner. Suppose the actual post-step state satisfies

Then

In particular, if $h>2\eta$, radiation entropy cannot decrease at that step.

Proof. Weak monotonicity, equivalent to strong subadditivity after purification, gives

Insert (eq:pairhypotheses).

This proposition states explicit entropy hypotheses. Its parameter $\eta$ is not identified with Mathur's pair-amplitude correction parameter. Mathur's small-corrections theorem establishes positive entropy growth from a more specific near-Hawking pair assumption Mathur2009,MathurPlumberg2011. Thermal-looking outgoing marginals alone do not provide those hypotheses.

four distinct comparisons for the constructed process.

The model therefore does not evade the small-corrections theorem while retaining its assumptions. It changes the emission rule and does not assert small corrections to the full Hawking pair structure. At an entropy-decreasing endpoint, the positive-growth hypotheses of Proposition prop:pairentropy cannot all hold.

We now construct the counterexample rather than inferring it from similarity of spectra.

Let the preserving evolution above be denoted $G$. Fix a pure code preparation $|0⟩$. Construct a second evolution $L$ as follows:

1. retain the variable input $|\psi {⟩}_A$ in a separate inaccessible degenerate memory $F$;

2. run the entire $G$ sequence on an internal dummy code fixed initially to $|0⟩$;

3. emit the dummy sequence's residual through the same endpoint radiation embedding;

4. retain the variable memory $F$ at the endpoint.

The dummy code and all other ancillaries are fixed pure preparations, not additional variable inputs. Their common energies are included in the initial mass ledger.

Choose the endpoint memory's energy sector to have energy $m_{\ast }$ and degeneracy at least $d$. The preserving process occupies one fixed vector of this sector; the retaining process occupies the logical input in it. Thus the two processes can have the same macroscopic energy ledger while having different complementary channels.

There are conserving sequential evolutions $G,L$ with the following properties. For every prefix including the endpoint,

Consequently their full accessible radiation density operators agree for input $|0⟩⟨0|$.

Their scalar radiation entropy histories agree for every pure code input, although their radiation density operators need not agree for those other inputs. At the endpoint,

The nonpreserving accessible channel retains the input in an inaccessible $d$-dimensional memory; it does not destroy it globally.

Proof. The retaining evolution acts as identity on its variable memory and as the preserving sequence on its fixed dummy preparation. Tracing the memory gives (eq:retainingchannel). All maps remain isometries, and the variable code labels have equal energy, so the declared energy accounting is unchanged.

For the preserving sequence, Theorem thm:information shows that the radiation entropy depends on the input only through $H(\rho )$. Every pure code input has $H(\rho )=0$, so its pre-endpoint entropy is

The endpoint radiation is an isometric encoding and is pure for every pure input. Its entropy is therefore zero. The retaining sequence emits the fixed dummy history, giving exactly the same scalar values.

At the endpoint, the retaining radiation is constant and its complement contains the original input isometrically. A decoder on constant radiation can only prepare a fixed state. Lemma lem:depolarization gives optimal error $c_d$. The same lemma and a left inverse for the retained input give leakage distance $c_d$. The preserving endpoint result is Theorem thm:endpoint.

For $m_{\ast }>0$, $L$ is an information-bearing remnant model. For $m_{\ast }=0$, it postulates a degenerate zero-energy inaccessible memory, rather than a unique vacuum. If such memory is forbidden and all outputs are accessible, the retaining process is excluded by Proposition prop:endpointtest. The counterexample does not assert different recoverability under identical complete-evaporation and complete-access assumptions.

On the Bell input, the models are immediately distinguishable at the channel level: the preserving process has $I(S:R_{\mathrm{all}})=2\log d$, whereas the retaining process has $I(S:R_{\mathrm{all}})=0$ and $I(S:F)=2\log d$. Mixtures and reference-entangled inputs contain information absent from a list of pure-input scalar entropies.

Equal entropy histories are also weaker than equal complete record laws, and equal selected records are weaker than equal residual instruments. These distinctions parallel the complete-process and record interfaces in Psy02-B and En03-A CheungPsy02B,CheungEn03A.

The preceding finite counterexample concerns scalar entropy data without additional structure. It is not a counterexample to the island prescription.

Replica-wormhole calculations supply additional gravitational saddles and derive island entropy formulas in controlled settings. Quantum-extremal-surface and entanglement-wedge analyses additionally address reconstruction and its code-space dependence AlmheiriEtAl2020,Penington2020,EngelhardtWall2015. The replica-wormhole work of Penington, Shenker, Stanford, and Yang also gives a direct gravitational reconstruction argument using a Petz-map formula PeningtonEtAl2022. These advances are more than a fit to one scalar entropy curve.

For a concrete comparison, select the evaporating JT-plus-bath setting of Almheiri, Engelhardt, Marolf, and Maxfield AlmheiriEngelhardtMarolfMaxfield2019. Table tab:islandaudit records what must be extracted or added before making the particular finite-code diamond-norm statement of this paper. “Not derived here” is a boundary of this comparison, not a claim that the cited literature lacks reconstruction results.

audit for the selected controlled-gravity comparison.

For example, once a channel $\mathcal{N}:A\to Z$ and a full-rank code state $\sigma$ are supplied, a Petz candidate can be written as

where inverse powers are on $P=\mathrm{supp}\mathcal{N}(\sigma )$. The support-completion term makes this a CPTP map on the whole output space: the trace of the first term is $\mathrm{Tr}(PY)$.

Writing (eq:Petz) does not prove that it corrects a given evaporation code. One must evaluate or bound $D_{\diamond }({\mathcal{D}}_{\sigma ,\mathcal{N}}\mathcal{N},{\mathrm{id}}_A)$. A scalar entropy history supplies neither the channel on every matrix unit nor its adjoint, and therefore does not by itself specify this calculation.

The operational gap addressed here is consequently narrow: an entropy history alone is not a code-wide channel certificate. It does not deny reconstruction conclusions derived from entropy together with additional gravitational, algebraic, and code-space structure. Page-curve expectations likewise remain valuable unitary-evaporation diagnostics without being substituted for an arbitrary-input decoder error Page2013.

For a complete collection and storage channel ${\mathcal{A}}_t$,

Its complete complement includes the collection environment and every excluded mode.

Message-independent CPTP restriction of the accessible output cannot improve unrestricted recovery:

For consistent coherent radiation prefixes, ${ϵ}_{\mathrm{rec}}(t+1)\le {ϵ}_{\mathrm{rec}}(t)$.

Proof. Every decoder after ${\mathcal{A}}_t$ is a decoder for the original output by composition. For prefixes, a later decoder may discard the new block and use an earlier optimal decoder.

Storage degradation can violate the prefix premise. Information-theoretic access and executable coherent recovery are separate resource questions, as in En02-B–D CheungEn02B,CheungEn02C,CheungEn02D.

Let all emitted blocks be measured destructively in their fixed occupation/rail bases. The complete record retains key, transfer time, vacuum, and saturation or loss flags. No selected branch is omitted.

Define logical dephasing

For the pre-endpoint construction with only those complete classical number records accessible,

After the endpoint, with all endpoint rails also measured,

Thus complete evaporation in the model does not make an ordinary destructive detector a quantum-code decoder.

Proof. On a completed branch, the key identifies the cyclic shift in $U_r=X^a{Z}^b$. Correcting that shift in the classical number outcome leaves a measurement of the original logical computational basis. Its channel is ${\Delta }_Z$. Pending branches remain constant. Preparing the measured logical basis state and preparing $I_d/d$ on pending gives

Every completed branch is entanglement breaking. A separable state of a $d$-dimensional reference and logical output has Bell overlap at most $1/d$: the bound holds for pure product states by Cauchy–Schwarz and then by convexity. A constant pending branch has Bell overlap $1/d^2$. This proves the upper bound in (eq:numberfidelity).

Now