Hawk05-B A Finite Page-Curve Derivation under the Enter Discipline Haar-scrambled coherent radiation, destructive records, and the limits of gravitational identification

We derive the finite random-state Page curve within an explicitly declared Enter-compatible model. A single logical-input-independent Haar unitary scrambles a prepared finite code; a conserving sequence then transfers quantum cells into a growing accessible radiation register. For every fixed pure input chosen independently of the unitary, the exact mean radiation entropy is

We also prove a simultaneous, all-prefix leading Page law for one sampled unitary. Destructive Enter defines a different process: its append-only records have nondecreasing Shannon entropy and cannot recover an arbitrary unknown quantum code. Their exact mean entropy is

which does not turn over.

Recovery is addressed separately. We give exact full-code recovery errors, a conservative Haar subcode-decoupling certificate, corrected flagged-transfer calibrations, and same-model endpoint and entropy comparison bounds. The Haar hypothesis, coherent access, and emission schedule are explicit additional assumptions, not consequences of Enter. The construction derives a finite Page curve and specifies a conditional route to a gravitational comparison; it does not construct a microscopic gravitational evaporation channel, an island formula, or the unresolved characteristic quantum coefficient of the supplied analytic programme.

of completion. The mathematical Page-curve target is completed for the finite random-state model stated below. A gravitational Page-curve derivation cannot be obtained from the supplied channel and manuscript summaries alone: the required gravitational dynamics and access-preserving comparison are not supplied. That missing identification is retained as a hypothesis, not replaced by a finite circuit or claimed as a nonexistence theorem.

Cheung's Philosophy of Enter separates destructive readout from quantum recovery. Destructive computational-basis readout consumes the measured quantum name and returns classical data; a decoder instead returns a quantum target through declared access. Consumption is an interface statement, not physical annihilation. The philosophy also distinguishes complete energy, erasure cost, and residual leakage, and distinguishes an analogy from a physical identification CheungPhilosophy.

We implement these distinctions by specifying the code, untouched reference, complete preparation, accessible outputs, excluded outputs, and returned type before stating entropy or recovery claims. Ordinary finite Born instruments are defined explicitly. Cheung's calculus and Manuscript I provide background; their measurement-characterization hypotheses are not enlarged by citation CheungEnter,CheungI.

We distinguish:

1. a designed entropy turnover, such as $h_2(q)$;

2. the random-state Page law, with entropy approximately the logarithm of the smaller subsystem dimension;

3. a gravitational radiation-entropy history obtained from a specified physical theory and radiation interface.

The first does not establish the second, and the second does not identify the third.

Page proposed the exact random-subsystem average; Sen proved it PageAverage,Sen. Page's radiation discussion supplies a separate physical interpretation PageRadiation. Existing gravitational entropy analyses have additional dynamical and geometric content Penington,AlmheiriEtAl. They are not instantiated by naming finite registers “radiation” and “black hole.”

The supplied Hawk05-A is a useful foundation, but its flagged calibration theorem contains an obsolete reversed-weight equation. That false equation must be removed from the theorem statement, rather than retained alongside its correction.

{}1.12

The corrections to destructive records, complete complements, and endpoint identification agree with the distinctions in Cheung's Hawk and En07 manuscripts CheungHawk02B,CheungHawk04D,CheungEn07B,CheungEn07C. All elementary mathematical statements used below are proved locally.

All spaces are finite dimensional. Let $C$ be the variable logical code, with dimension $d\ge 2$. An external reference $Q$ is untouched.

A complete interface is an isometry

Its two channels are

The complete complement $B$ includes every unreturned preparation purifier, lost output, measurement environment, and implementation factor admitted in the model.

For states and channels use

The diamond norm retains arbitrary reference extensions Watrous (Chapter 3).

Define replacement, recovery error, and complete leakage by

We abbreviate

Compactness ensures that these finite-dimensional infima are attained. Entropies are in bits:

Every isometric channel ${\mathrm{Ad}}_V$ has a CPTP left inverse. It preserves the trace norm of every referenced operator. CPTP pre- and postprocessing contract $d_{\diamond }$, and

Proof. For a fixed code state $\omega$, the channel

is CP, trace preserving, and satisfies ${\mathcal{L}}_V{\mathrm{Ad}}_V={\mathrm{id}}_C$. Isometric embeddings preserve nonzero singular values, also after tensoring with $I_Q$. Trace-distance contraction on referenced states gives diamond-distance contraction. Insert any decoder into the triangle inequality, take infima, and interchange the compared channels to obtain the stability bound.

The accessible output exactly recovers the entire code if and only if

for one state ${\tau }_B$. If this holds and $r=\mathrm{rank}{\tau }_B$, there are a state ${\sigma }_K$, $\dim K=r$, and an isometry $T:C\otimes K\to R$ such that

An exact decoder on all of $R$ is

In particular, $\dim R\ge d\mathrm{rank}{\tau }_B$. Both complementary factors cannot independently recover the whole code when $d\ge 2$.

Proof. Suppose $\mathcal{C}={\mathcal{T}}_{{\tau }_B}$, and diagonalize its nonzero support:

Write

The complementary identities on all matrix units give

Consequently $T(|i⟩|\alpha ⟩)=|u_{i\alpha }⟩$ is isometric and yields (eq:normalform). The displayed decoder is CPTP and recovers every matrix unit.

Conversely, dilate an exact decoder by $W:R\to CG$. The composite $(W\otimes I_B)V$ dilates the identity channel. The Choi operator of the identity has rank one. Every Kraus operator of that composite is therefore proportional to $I_C$, so the composite has the form

with fixed $\chi$. Its $B$ channel is replacement. All identities extend to references by linearity. If both factors recovered, the complement of either recoverable factor would be both replacement and recoverable, which is impossible on two distinct inputs.

This is the standard exact-error-correction criterion, with the complete complement specified KnillLaflamme,CheungEm05B. It is not a new error-correction principle.

A replacement accessible channel has recovery error $a_d$; every accessible channel has recovery error at most $a_d$. For

put $J_{\mathrm{n}}(\Phi )=({\mathrm{id}}_Q\otimes \Phi )({\Phi }_d)$. Then

If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\tau })\le \eta$, then

Proof. A decoded replacement has Bell output $I_Q/d\otimes {\omega }_C$ and Bell overlap $d^{-2}$, giving the lower bound $a_d$. Preparing $I_C/d$ attains it. Indeed, depolarization is the uniform average of the $d^2$ Weyl conjugations; separating its identity term gives diamond distance $1-d^{-2}$.

The first Choi inequality uses the Bell input. For a pure referenced input, write $|\psi ⟩=(L\otimes I)|{\Phi }_d⟩$, where $\mathrm{Tr}(L^{†}L)=d$. Then ${\Vert L\Vert }_{\infty }^2\le d$. Its output difference is obtained from the normalized Choi difference by multiplication with $L\otimes I$ on both sides, giving the second inequality. Convexity handles mixed inputs. The final assertion follows by postprocessing contraction and the triangle inequality.

The finite information–disturbance theorem gives

Proof. Use the Stinespring continuity theorem and its information–disturbance consequence in raw diamond normalization KSW (Theorems 1 and 3):

Here the replacement state is optimized, and the converse requires existence of an appropriate replacement state, not every state. Substitute ${ϵ}_{\mathrm{raw}}=2{\epsilon }_{\mathrm{rec}}$ and ${\delta }_{\mathrm{raw}}=2{\delta }_{\mathrm{full}}$. The independent constant-decoder cap is Lemma lem:benchmarks.

Adding fixed pure spectators or access-preserving isometric output embeddings changes none of the optimized distances above: compression with a CPTP off-support completion gives the reverse comparison. This observation permits us to suppress fixed vacant factors in the finite model.

Fix integers $s\ge 2$ and $N\ge 2$. Let

be a fixed isometry, with $2\le d\le D$. Choose a single unitary $U\in U(D)$. For ensemble statements only, $U$ is distributed according to normalized Haar measure.

Haar randomness is an additional ensemble assumption about a family of devices. In each device, $U$ is fixed, independent of the variable logical input, and known to the decoder. It is not resampled between emission steps.

If $U$ is instead an unknown discarded random control and the accessible channel is averaged over it, that is a different, replacement-channel task.

This extra typicality hypothesis is not derived from the Enter philosophy, energy conservation, or Hawking occupations.

For each memory cell $M_j$ and radiation cell $R_j$, use

The preparation occupies all memory cells with the internal state $J|\psi ⟩$ and leaves every radiation cell vacant.

Let $P_X$ project onto the occupied sector of cell $X$. For an energy gap $\omega >0$, declare

Extend $U$ to act on the all-occupied memory sector and to be identity elsewhere. Emit cell $j$ by the full SWAP between $M_j$ and $R_j$.

The extended $U$ and all emission SWAPs commute with $H_0$. At cut $k$, the effective quantum output is

with

Up to fixed vacant factors and tensor permutations, its complete isometry is

where $F_k$ is the canonical tensor regrouping. Earlier emitted cells are never acted on again. At $k=N$, the residual is a fixed pure vacant state.

Proof. The all-occupied memory sector has one $H_0$ eigenvalue, $N\omega$; thus arbitrary $U$ on that sector commutes with $H_0$. SWAP preserves $P_{M_j}+P_{R_j}$. After $k$ swaps, the first $k$ internal tensor factors are in radiation, while the remaining factors are in memory. All other factors are fixed vacancies. This is (eq:Vk). At the last cut no internal memory factor remains.

The energy distribution of $H_0$ is conserved. The ideal mean radiation and memory energies are $k\omega$ and $(N-k)\omega$. This register Hamiltonian is not a black-hole mass law, and implementation work and control reset are not priced by it.

Fix a pure logical input $|\psi ⟩$ independently of $U$. Then $UJ|\psi ⟩$ is a Haar-distributed unit vector in ${\mathbb{C}}^D$. Define

All cuts in (eq:Sk) belong to one state and one sampled $U$. We do not replace this history by independently sampled states at different cuts.

The fixed-pure-input entropy problem is distinct from recovery of every state of $C$, including reference-entangled states.

For a positive integer $\ell$, let

Put

Under Definition def:haar, for every fixed pure input independent of $U$,

The value is independent of that fixed input and of the code embedding $J$. It satisfies

It increases strictly up to the middle and decreases strictly afterward. For odd $N$, the two central cuts have equal maximal mean entropy. For even $N$, the unique maximal mean occurs at $k=N/2$.

Proof. Haar invariance makes $UJ|\psi ⟩$ a uniform unit vector. The two reductions of a bipartite pure state have the same nonzero spectrum. The calculation in Appendix app:page therefore applies with $m=m_k$, $n=n_k$, and gives (eq:page). This is the Page–Sen average, rederived there with normalization explicit PageAverage,Sen.

The endpoint and symmetry statements follow immediately. To verify monotonicity, consider consecutive cuts lying on the early half, so $m^{\prime }=sm$, $n^{\prime }=n/s$, with $m^{\prime }\le n^{\prime }$. Their difference, in natural entropy units, is

The first term is at least $1-1/s$. Since $n\ge s^{2}m$, the difference is at least

Symmetry proves the decreasing half and the central statements.

The midpoint arises from the equal-dimension equation $r_k=b_k$, the equal-size emission rule, and the Haar ensemble. It is not a universal “half the emitted quanta” law. Different cell capacities, conservation sectors, or dynamical ensembles require a new calculation.

Define the capacity tent

For a Haar unit vector in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$,

Proof. For $D=mn$, unitary invariance and symmetry give

where $\mathsf{F}$ swaps the two complete copies. The normalization follows by taking the trace. The purity swap test uses ${\mathsf{F}}_m\otimes I_{n^2}$. Its traces against $I$ and ${\mathsf{F}}_m\otimes {\mathsf{F}}_n$ are $m{n}^2$ and $m^{2}n$, respectively.

For one sampled $U$, set

Then

Consequently, with

for every $z>0$,

In particular, for fixed $s$ and inputs ${\psi }_N$ independent of $U_N$,

Proof. A reduction has rank at most $m_k$, so its entropy is at most ${\log }_2{m}_k$. The elementary Rényi bound $S(\rho )\ge -{\log }_2\mathrm{Tr}{\rho }^2$ follows by Jensen's inequality with weights equal to the eigenvalues. Thus

This proves (eq:gap).

For even $N$, the infinite extension of the sum of $s^{-|N-2k|}$ is $1+2/(s^2-1)$. For odd $N$, it is $2s/(s^2-1)$, which is no larger. Therefore

Markov's inequality and ${\max }_k{\Delta }_k\le {\sum }_k{\Delta }_k$ prove (eq:allprefix). Take $z=ϵN{\log }_{2}s$ to obtain (eq:leadingpage).

No independence between cuts was used. This is a temporal statement about one nested sequence, although the exact finite statement is an ensemble mean and an individual realization can fluctuate.

For the fixed pure input,

If the unitary is not known and the accessible channel itself is averaged over Haar measure, then

Its recovery error is $a_d$ at every cut, including $k=N$.

Proof. The Haar twirl of $JX{J}^{†}$ is $\mathrm{Tr}(X)I_D/D$. Take the residual partial trace. The recovery conclusion is Lemma lem:benchmarks.

An unknown discarded scrambling control cannot be omitted from complete output accounting and then treated as known by a decoder.

For a register $A$ with basis $\{|y⟩\}$, define destructive readout by

The returned quantum names are those of $K$, not $A$. A complete dilation is

where $L$ is unreturned. The complete evolution remains isometric; the returned classical record alone is dephased.

Finite adaptive compositions are CPTP: composing the unnormalized path maps $K_h$ gives ${\sum }_h{K}_h^{†}{K}_h=I$. This is the ordinary finite instrument setting also used in Cheung's hybrid and sequential interfaces CheungVIIG,CheungEn02C.

Apply Enter in the occupied computational basis immediately after each emission of Section sec:model. Let $Y_{\le k}=(Y_1,\dots ,Y_k)$ be the complete unedited record.

For a fixed pure input and fixed $U$,

The averaged cumulative record entropy is

It increases strictly with $k$. At the endpoint,

For every fixed $U,\psi$, not merely on average,

Proof. Later operations are SWAPs on other cells. The projectors defining the full outcome history therefore correspond to the commuting tensor-basis projectors on the original scrambled state. Their Born law is (eq:recordlaw); no postselection is used.

Represent a Haar vector by independent standard complex Gaussians divided by their Euclidean norm. The sums of squared moduli over the $b_k$ entries associated with each prefix are independent Gamma variables of shape $b_k$. After normalization, the $r_k$ prefix probabilities have the Dirichlet distribution with all parameters $b_k$. For real $t$ near one,

Differentiation at $t=1$ yields

where ${\psi }_0=(\ln \Gamma {)}^{\prime }$. This proves (eq:recordmean). Its increment is $({\mathsf{H}}_{b_k}-{\mathsf{H}}_{b_{k+1}})/\ln 2>0$.

For a fixed process, Shannon's chain rule gives

and the conditional entropy is nonnegative.

Every purely classical channel with code input dimension $d\ge 2$ satisfies

The bound is sharp for computational-basis measurement. Its complete complement is never constant.

Proof. Following a classical channel by any quantum decoder gives a measure-and-prepare channel. Its Bell output is separable HorodeckiShorRuskai. For product states,

Convexity extends the bound to separable states. The Bell-projector test gives error at least $1-1/d$.

Basis measurement followed by basis-state preparation is dephasing. Dephasing is the average of the $d$ phase conjugations, so separating its identity term gives matching error $1-1/d$. A constant complete complement would imply exact recovery by Theorem thm:exact, a contradiction.

The bound concerns the entire adaptive record, not just one cell. A quantum-output decoder cannot repair entanglement breaking merely by preparing a quantum state from classical data.

For any returned block-diagonal output ${\rho }_{YK}={\sum }_y{p}_y|y⟩⟨y|\otimes {\rho }_{K|y}$,

Thus an unchanged positive-entropy record prevents purity of the raw hybrid output.

Furthermore, for all referenced operators,

Terminal residual Enter followed by drop leaves the already-returned accessible channel unchanged.

Proof. The spectrum of a diagonal block is $p_y$ times the spectrum of ${\rho }_{K|y}$, proving the entropy formula. Writing an operator as ${\sum }_{y,z}{X}_{yz}\otimes |y⟩⟨z{|}_A$, readout followed by discard returns ${\sum }_y{X}_{yy}$, precisely the partial trace. The blocks may contain an arbitrary reference.

This identity is an interface equality, not physical annihilation or a reset-cost theorem CheungEn07A,CheungEn07B,CheungEn07C.

Take $d=D$ and identify $J$ with a unitary input dictionary.

For every fixed $U$, at a cut with $r=r_k$, $b=b_k$, the coherent channels satisfy

If the returned radiation is destructively read out, its classical channel satisfies

In particular, a midpoint mean-entropy peak does not imply accurate recovery of the whole $D$-dimensional input.

Proof. Conjugating the logical target by $U$ leaves the optimization unchanged. The coherent channel becomes ${\mathrm{Tr}}_B$. The decoder appending $I_B/b$ has error $1-b^{-2}$, by the Weyl twirl on $B$. For the reverse bound, use the product Bell input ${\Phi }_r\otimes {\Phi }_b$. After $B$ is lost, its reference is independent of every decoder input. The decoded full Bell overlap is at most $b^{-2}$. This proves the recovery formula.

The complementary channel is ${\mathrm{Tr}}_R$. Replacement by $I_B/b$ has distance at most $1-b^{-2}$. Fixing the input on $R$ and testing a Bell input on $B$ gives the matching lower bound for every replacement.

For record-only access, the measurement effects on the transformed code are

If ${\sigma }_y$ is any decoder output, its full Bell overlap is

The decoder preparing $|y⟩⟨y{|}_R\otimes I_B/b$ attains the bound: its channel is ${\Delta }_R\otimes {\mathcal{T}}_{I_B/b}$, the average of $r{b}^2$ phase/Weyl conjugations, one of which is the identity. Separating that term proves the matching upper bound.

Theorem thm:fullcode is the whole-code SWAP calibration documented in Cheung's Hawk04-D, here applied to the same Haar-prepared entropy model CheungHawk04D. It is not a Hayden–Preskill small-diary claim HaydenPreskill.

For the full-code Bell input, the marginal logical state is $I_D/D$. For every $U$,

This referenced-input marginal entropy is monotone and ends at ${\log }_{2}D$, not zero. Its final reference entanglement is correctly preserved by coherent recovery. The pure-unreferenced Page theorem does not apply to that marginal.

For every $U$ on the full code, the input $U^{†}|0,\dots ,0⟩$ produces zero entropy at every cut. It depends on $U$, so it violates the fixed-input hypothesis. This explicitly prevents promotion of Theorem thm:uniform to a uniform entropy assertion over all inputs.

For a Haar embedding $V:C\to {\mathbb{C}}^r\otimes {\mathbb{C}}^b$, $D=rb$, define

Put

For the fixed code dimension $d$,

Consequently,

For $0<\alpha <1$, with probability at least $1-\alpha$,

These are whole-code, reference-preserving bounds.

For fixed $d$ and $D\to \infty$, ${\Lambda }_B\to 0$ whenever $b/r\to 0$. More generally, the sufficient regime is $d^{3}b/r\to 0$. This last dimension dependence is a conservative certificate, not an optimal random-code threshold.

Proof. Appendix app:fourth proves the fourth-moment identity (eq:HS). The operator acts on a space of dimension $db$, so

Cauchy–Schwarz in expectation, followed by Lemma lem:benchmarks, gives

Markov's inequality gives the leakage statement, and Proposition prop:ksw gives recovery.

Since

the stated sufficient regimes follow. At $b=1$, the expression vanishes exactly.

This theorem supplies a channel estimate not inferred from the Page entropy slope. It deliberately retains the input-dimension factor in the Choi-to-diamond conversion.

An early-access comparison follows by interchanging $r$ and $b$:

With probability at least $1-\alpha$,

and therefore

Early replacement and late recovery are thus separate proved comparisons in the Haar subcode model.

The normal form (eq:normalform) represents the recovered logical algebra on the accessible support as $T(\mathcal{B}(C)\otimes I_K)T^{†}$. Its commutant on that support is $T(I_C\otimes \mathcal{B}(K))T^{†}$, as follows by commuting with all logical matrix units. The syndrome algebra is not a second recoverable copy of the same unknown logical state. An abstract algebra of the same dimension may exist there without carrying that input.

This is the distinction relevant to the supplied partner comparisons in Hawk03-A CheungHawk03A. No independent-factor entropy bound is used to estimate complete leakage or identify its maximum.

There are two conserving sequential complete interfaces with the same accessible and residual spaces such that:

1. for one fixed pure code input, their accessible states agree at every prefix;

2. the two ensembles have the same exact Page mean (eq:page) for every fixed pure code input;

3. the endpoint recovery errors are respectively $0$ and $a_d$, and endpoint complete leakages are respectively $0$ and $a_d$.

Thus even the extensive Page mean does not determine endpoint whole-code recovery.

Proof. Append a degenerate inaccessible register $L\simeq C$. In the returning model, run Section sec:model on the variable input and fix $L$ in $|0⟩$. In the retaining model, store the variable input in $L$ and run the same radiation schedule on the fixed dummy input $|0{⟩}_C$.

All transfer operations conserve (eq:H0); $L$'s logical labels are degenerate. Both are complete isometries into the same spaces. For the input $|0{⟩}_C$, the radiation histories agree exactly for the same $U$. Haar invariance proves that the returning model's mean entropy for any fixed pure input equals the dummy model's mean, namely (eq:page).

At the endpoint the returning radiation is $UJ|\psi ⟩$, with fixed complete residual. The retaining radiation is fixed at $UJ|0⟩$, while the residual contains $|\psi {⟩}_L$. The endpoint recovery errors follow from exact correction and the replacement benchmark. The retained logical copy has leakage $a_d$: a CPTP left inverse gives the identity-channel lower bound, and an embedded maximally mixed replacement gives the upper bound.

This is an ensemble Page-history extension of the disclosed-memory separation in Cheung's Hawk01-B CheungHawk01B. The comparison does not impose identical assumptions of complete outgoing access and a unique empty physical endpoint.

Let ${\sigma }_K$ have purification $|\chi {⟩}_{KP}$, and put $R=C_{R}K$, $B=PL$, with $L\simeq C$. The isometries

have identical accessible spectra for every pure input, but recovery errors $0$ and $a_d$. For $H_R=I_{C_R}\otimes H_K$, their selected energy laws agree. One may take ${\sigma }_K$ to be Gibbs.

Proof. The accessible channels are $X\mapsto X\otimes {\sigma }_K$ and $X\mapsto \mathrm{Tr}(X)|0⟩⟨0|\otimes {\sigma }_K$. The spectrum, recovery, and energy conclusions follow directly.

The supplied En07-C and Oai265-A records contain related separations CheungEn07C,CheungOai265A. Oai265-A is used only as a documentary comparison; no catalogue family is a premise.

For comparison with Hawk05-A and Hawk02-B, let each memory and emission cell have space $\mathbb{C}|\varnothing ⟩\oplus C$. At opportunity $j$, rotate

with the complementary rotation on $|\varnothing ,i⟩$. The occupied labels are degenerate, so the rotation conserves the sum of occupied-sector energies.

Set

The complete isometry has the form

where the normalized emission amplitudes define an isometry $J_n$.

For $q=q_n$,

Immediate cell readout returns histories consisting of either no emission or an emission time $j$ and basis label $i$. For $u_i=⟨i|\rho |i⟩$, their probabilities are

Its recovery error and complete-record entropy are

If $q_{n-1}>0$ and $t_n=q_n/q_{n-1}$, then

If $q_{n-1}=0$, the increment is zero.

Proof. The channels are orthogonal-block channels

The decoder compressing the occupied block and preparing $I_C/d$ on vacancy gives the radiation upper bound $q{a}_d$. On a Bell input, no decoder has overlap greater than $1-q+q/d^2$, proving optimality. Interchanging occupied probabilities gives the residual recovery error. The exact residual leakage is proved in Appendix app:flag; it is $q{a}_d$, not residual recovery. The entropy formulas follow from the orthogonal blocks.

Successive local readouts may be commuted past later rotations on other cells for the purpose of computing the complete outcome law. Their probabilities are therefore $p_j{u}_i$ and $q_n$.

Preparing $|i⟩$ on a reported label and $I_C/d$ on no emission gives decoded channel $(1-q_n)\Delta +q_n{\mathcal{T}}_{I_C/d}$. Its error is at most $(1-q_n)b_d+q_n{a}_d$. The Bell test gives the same lower bound, irrespective of emission time. The history entropy follows by splitting the no-emission branch and then its label. At the next opportunity only the former no-emission outcome is refined, giving the displayed increment.

The recovery formula agrees with the supplied Hawk02-B calibration CheungHawk02B. Erasing the emission-time label gives a coarse snapshot record with entropy $h_2(q_n)+(1-q_n)H(u)$, but does not change the recovery optimum: time can be resampled with input-independent conditional probabilities $p_j/(1-q_n)$. The refined and coarse records have different entropy ledgers.

For a pure unreferenced input the coherent entropy is $h_2(q)$, bounded by one bit regardless of the number of cells. Taking ${\theta }_1=\pi /4$, ${\theta }_2=\pi /2$ gives coherent entropies $(0,1,0)$ for a basis input and cumulative classical entropies $(0,1,1)$. Other prepared turnover sequences, including the supplied En02-D example, are comparisons rather than premises of the Haar Page derivation CheungEn02D.

A physical endpoint package must provide:

1. a justified finite code and complete prepared isometry, including preparation purifiers;

2. complete logical-input-independent unitary evolution on a justified domain;

3. access-preserving identifications into common comparison spaces, including a justified quantum receiver if physical radiation is to be collected;

4. a proved complete-complement replacement allowance.

These are the same-model obligations in Cheung's En07 endpoint formulations CheungEn07B,CheungEn07C. They are not consequences of selected thermal occupations.

After the declared identifications, suppose actual and proxy complete channels satisfy

If the proxy complement obeys ${\delta }_{\mathrm{full}}(\hat{\mathcal{C}})\le \zeta$, then

A fixed proxy-optimal decoder has actual error at most

If a feasible state ${\tau }_B$ instead supplies the normalized Choi discrepancy

one may take $\zeta =\min \{1,de\}$.

Proof. Partial traces contract the complete-channel comparison. The triangle inequality proves leakage. Apply a fixed proxy-optimal decoder to the actual accessible channel, use contraction, and then use (eq:ksw). The constant decoder gives the independent $a_d$ cap. Equation (eq:choi) gives the Choi conversion.

The finite model of Section sec:model meets these obligations mathematically: $J$ fixes preparation, its unitary extension and SWAPs fix evolution, tensor regrouping fixes access, and the terminal residual is a fixed pure vacancy. These are not gravitationally justified objects.

For an output dimension $L$, define the monotone entropy modulus

Suppose one physical model and one receiver supply complete access-preserving comparisons to a fixed sampled finite proxy $U$, at every declared cut $k$. Let the common accessible comparison dimension be $L_k$, including any necessary padding, and suppose