Hawk05-C Conditional Page-Curve Extrapolations under Enter Compatible capacities, conserved-sector proxies, two return types, and cutoff certificates

We develop conditional Page-curve extrapolations from a finite Haar proxy while retaining the access and output distinctions of the Philosophy of Enter. Nonuniform coherent cell release gives the Page–Sen mean, whereas destructive readout gives an append-only classical record with a different exact entropy mean. Their entropy difference is basis-dependent relative-entropy coherence, not a whole-code recovery error or a thermodynamic cost.

A compatibility theorem identifies a missing condition in a proposed clock replacement: a fixed-dimensional tensor-release proxy requires the sum of its two logarithmic capacities to remain constant. Consequently, identifying radiation capacity with a coarse entropy production law larger than black-hole entropy loss is not merely a reparametrization of Hawk05-B. We give an explicit rounded, fixed-dimensional realization for the compatible case and an exact direct-sum calculation for a conserved additive-charge sector. A positive-mass inverse-square clock yields a conditional capacity crossover, its cutoff criterion, and a peak-location stability bound.

Physical entropy transfer, ensemble typicality, and quantum recovery receive separate quantitative allowances. A proposed Petz decoder is made into a deterministic channel on its entire input space and can be certified by its decoded Choi discrepancy. No occupation law, entropy slope, restricted observable algebra, or finite termination supplies the missing physical comparisons. The result is a proved finite extrapolation framework and a collection of explicitly conditional evaporation sketches, not a microscopic gravitational Page curve, an island construction, or spacetime emergence.

scope. All channel and random-state results below are finite dimensional. The semiclassical clock is an additional supplied model. The physical channel, receiver, and typicality allowances remain unfilled until justified in one compatible microscopic theory. The manuscript does not replace an unconstructed gravitational interface by a finite proxy and then declare the identification proved.

Cheung's Philosophy of Enter distinguishes destructive readout, quantum return, physical implementation, and analogy. Enter consumes a measured quantum name at its interface and returns a classical record. A decoder instead acts on declared accessible systems and returns a quantum target; whole-code correctness includes untouched reference correlations. Consumption is not physical annihilation, and an input–output channel does not determine a thermal reset machine CheungPhilosophy.

The present paper adopts ordinary finite Born instruments explicitly. Cheung's operational characterization applies inside its stated strategy class; it is not used to derive a gravitational field instrument or enlarge that class CheungEnter,CheungI. The four questions guiding every extrapolation are:

Hawk05-A separates classical accumulation from quantum recovery. Hawk05-B then constructs a finite coherent Haar history with the Page–Sen mean and a different exact classical-record mean Hawk05A,Hawk05B. We retain those objects as proxies, not gravitational identifications.

We distinguish:

1. a finite channel theorem;

2. a conditional identification and its quantitative allowance;

3. a physical conclusion after the identification has been proved.

This paper proves statements at level (i), proves implications at level (ii), and does not assert that the gravitational hypotheses required for level (iii) have been discharged.

The Page–Sen average is a random-state theorem PageAverage,Sen. A black-hole time profile requires additional entropy and luminosity assumptions, as in Page's time-dependent analysis PageTime. Island and entanglement-wedge calculations have further geometric and dynamical hypotheses Penington,IslandReview. None is instantiated by a finite capacity plot.

{}1.12

The proposal's unspecified “2026 algebraic reading” is not treated as an established universal statement or attributed to an unidentified recent paper. Section sec:algebras instead states a finite access theorem and separately identifies an external asymptotic proposal with its own hypotheses.

Let $C$ be a variable code with $\dim C=d\ge 2$, and let $Q$ be an untouched reference.

A complete interface is an isometry

with accessible channel and complete complement

Every unreturned purifier, measurement environment, lost mode, and admitted implementation factor belongs to $B$.

Use the normalizations

and define

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

All entropies are in bits unless a natural logarithm is printed explicitly.

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

Proof. For a fixed state ${\omega }_C$, the map

is CP, trace preserving, and left-inverts ${\mathrm{Ad}}_V$. An isometric embedding preserves all nonzero singular values, also after reference extension. Trace-distance contraction on referenced states gives diamond contraction. Insert an arbitrary decoder in the triangle inequality, take infima, and interchange the compared channels to prove stability.

An accessible output exactly recovers the whole code if and only if its complete complement is replacement:

In that case there are a state ${\sigma }_K$ and an isometry $T:C\otimes K\to R$ such that

One may take $\dim K=\mathrm{rank}{\tau }_B$. For a pure logical input,

Thus exact recovery does not force purity of the raw accessible output.

Proof. If the complement is constant, diagonalize ${\tau }_B={\sum }_{\alpha }{\lambda }_{\alpha }|\alpha ⟩⟨\alpha |$ on its nonzero support and write

The complementary equations on matrix units imply $⟨u_{j\beta }\mid u_{i\alpha }⟩={\delta }_{ij}{\delta }_{\alpha \beta }$. They define $T(|i⟩|\alpha ⟩)=|u_{i\alpha }⟩$. Compression by $T^{†}$, tracing $K$, and preparing a fixed state on the unused accessible subspace gives a deterministic decoder.

Conversely, dilate an exact decoder by $W:R\to CG$. The composite $(W\otimes I_B)V$ dilates the identity channel. Its Choi operator has rank one, so each Kraus operator is a scalar multiple of $I_C$. Hence the composite is $|\psi ⟩\mapsto |\psi {⟩}_C|\chi {⟩}_{GB}$ for fixed $\chi$. Tracing $CG$ proves complement constancy. The entropy statement follows from (eq:normalform).

This is the standard exact correction criterion KnillLaflamme,CheungEm05B, not a new theorem supplied merely by the Enter terminology.

A replacement accessible channel has optimal recovery error $a_d$; every channel has recovery error at most $a_d$. If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\omega })\le \eta$, then

For a complete interface, the information–disturbance theorem gives

Proof. On a Bell input, a decoded replacement produces $I_Q/d\otimes {\sigma }_C$, with Bell overlap $d^{-2}$. Preparing $I_C/d$ attains the lower bound: depolarization is the uniform average of the $d^2$ Weyl conjugations, one of which is identity. This also gives the universal constant-decoder cap. The accessible-replacement lower bound follows by contraction and the reverse triangle inequality.

For the last assertion use the finite information–disturbance inequality

and substitute the half-diamond normalization KSW.

For a basis $\{|y⟩\}$ of a measured factor $A$, define

The measured name $A$ is not returned. Its complete dilation can be

with $L$ unreturned. A continuation with joint access to $YL$ would have a different quantum interface.

For every purely classical output channel on a $d$-dimensional code,

Computational-basis measurement attains the bound. For an unedited append-only record,

Moreover,

on all referenced operators.

Proof. Any decoder following a classical-output channel is measure-and-prepare, so its Bell output is separable HorodeckiShorRuskai. Every separable state has Bell overlap at most $1/d$, since a product state has overlap $\mathrm{Tr}({{\rho }_Q}^{\mathsf{T}}{\sigma }_C)/d\le 1/d$. The Bell-projector test gives error at least $b_d$. Basis-state preparation gives dephasing, the average of $d$ phase conjugations, and has matching error $b_d$.

The entropy assertion is the classical chain rule. For the last assertion, expand $X={\sum }_{y,z}{X}_{yz}\otimes |y⟩⟨z{|}_A$. Readout followed by discard returns ${\sum }_y{X}_{yy}$, the partial trace. The blocks may include any reference.

In particular, a record-only endpoint cannot have a constant complete complement. Terminal residual readout and discard do not improve the already-returned accessible channel CheungEn07A,CheungEn07B.

Fix cell dimensions $s_1,\dots ,s_N\ge 2$ and a final core dimension $b_{\ast }\ge 1$. Let

Prepare a fixed isometry $J:C\to {\mathcal{H}}_{\mathrm{int}}$, choose one logical-input-independent unitary $U\in U(D)$, and then release the cells in order. Earlier outputs are untouched.

At cut $k$,

Haar statements refer to one Haar sample of $U$, not independently sampled states at different cuts. The pure input is fixed independently of $U$; $U$ is known when a decoder is invoked.

A finite conserving implementation may give each occupied memory and outgoing cell a common gap $\omega$, with a vacant zero-energy sector. The preparation occupies all memory cells. Extend $U$ on the all-occupied, degenerate internal sector and transfer each cell by SWAP. These operations commute with the declared sum of occupied-sector energies. This establishes a register conservation law, not a gravitational mass law Hawk05B,CheungHawk02B.

For integers $r,b\ge 1$, put

and define

A Haar pure state on ${\mathbb{C}}^r\otimes {\mathbb{C}}^b$ has mean reduction entropy $\mathsf{P}(r,b)$.

Proof. The Gaussian–Wishart evaluation in Appendix app:page gives (eq:PageMean). This is the established Page–Sen formula PageAverage,Sen.

For every fixed pure input independent of $U$, coherent radiation and immediate destructive cell readout have the exact means

The classical mean increases strictly at each release. The coherent mean increases on cuts with $r_k\le b_k$ that remain on that side, and decreases on cuts with $r_k\ge b_k$ that remain on that side. If the capacity crossing is traversed, a maximal cut is among the two cuts bracketing it.

If $b_{\ast }=1$, the coherent endpoint mean is zero while the record endpoint mean is

If the schedule stops before a crossing, no turnover is guaranteed.

Proof. Haar invariance makes $UJ|\psi ⟩$ a Haar unit vector, and Lemma lem:pagesen applies at every cut.

Write its product-basis components as normalized independent complex Gaussians. A radiation prefix has $b_k$ squared Gaussian components. The $r_k$ prefix probabilities therefore have a Dirichlet distribution with all parameters $b_k$. Its moment identity is

Differentiate at $t=1$. Using ${\psi }_0(\ell +1)={\mathsf{H}}_{\ell }-{\gamma }_{\mathrm{E}}$ gives the record mean. Later SWAPs act on unmeasured factors, so immediate readout has precisely these prefix probabilities.

For a same-side coherent step, write the smaller and larger dimensions as $m,n$, and transfer an integer factor $s\ge 2$ so that the next early-side dimensions are $sm,n/s$. Remaining on that side implies $n\ge s^{2}m$. The Page-mean increment in natural units is

Exchange the two factors for the late side. The record increment is $({\mathsf{H}}_{b_{k-1}}-{\mathsf{H}}_{b_k})/\ln 2>0$. The endpoint assertions follow by substitution.

A large cell can jump over the equal-capacity point. The precise finite maximum need not be located at a time satisfying $r_k=b_k$, because that equality may occur at no cut.

Let $T(r,b)={\log }_{2}m$. Then

For a single sampled Haar history and every $z>0$,

No independence of cuts is assumed.

Proof. Haar purity is $\mathbb{E}\mathrm{Tr}{{\rho }_m}^2=(m+n)/(mn+1)$, obtained by the two-copy swap test on $\mathbb{E}|v⟩⟨v{|}^{\otimes 2}=(I+\mathsf{F})/(D(D+1))$. Since $S(\rho )\ge -{\log }_2\mathrm{Tr}{\rho }^2$, Jensen's inequality gives

The upper entropy bound is the rank bound.

The harmonic sum obeys

Its difference from $\ln r$ is at most $\ln (1+1/b)\le 1/b$, proving (eq:recordgap). Finally, every capacity deficit is nonnegative, and its expectation is bounded by (eq:pagegap). Apply Markov's inequality to their sum, which dominates their maximum.

Let ${\Delta }_R$ dephase the declared radiation basis. Its diagonal probabilities equal the destructive prefix record probabilities.

For every state ${\rho }_R$,

For the Haar proxy,

This is not a whole-code recovery diagnostic.

Proof. The diagonal logarithm satisfies $\mathrm{Tr}({\rho }_R{\log }_2{\Delta }_R{\rho }_R)=\mathrm{Tr}({\Delta }_R{\rho }_R{\log }_2{\Delta }_R{\rho }_R)$. Positivity ensures the required support containment. The relative-entropy identity and nonnegativity follow. Subtract (eq:PageMean) from (eq:RecordMean) for the two cases.

For a Bell input to a $d$-dimensional identity channel, the accessible marginal is $I/d$, so the marginal gap is zero. Nevertheless, destructive basis readout has recovery error $b_d$. Thus a vanishing gap does not mean no quantum-recovery loss.

The quantity is the relative entropy of coherence in the selected basis Baumgratz. It is not a basis-independent count of information destroyed, a residual leakage, or a Landauer cost.

Let $\mu$ be a distribution of fixed unitaries on the same internal space. Define

If $d_{\mathrm{TV}}(\mu ,{\mu }_{\mathrm{Haar}})\le \tau$, then at a fixed cut

These statements apply to fixed inputs independent of the sampled unitary.

Proof. Entropy ranges are $0\le S(R)\le {\log }_{2}m$ for a pure complete proxy and $0\le H(Y)\le {\log }_{2}r$. The expectation difference of a function in $[0,L]$ is at most $L{d}_{\mathrm{TV}}$, by integration of its level sets.

Total variation is a strong sufficient assumption, not a claim that physical scrambling achieves it.

A weaker hypothesis suffices for a leading capacity law. For a fixed pure input, let

The purity swap witness has operator norm one, so

The proof of Proposition prop:deficits then gives

Two-copy information can control this leading deficit; it does not by itself establish either exact entropy mean.

If the sampled unitary is forgotten and the accessible channel itself is Haar-averaged, then

That channel is replacement and has recovery error $a_d$. The Page average is ${\mathbb{E}}_{U}S({{\rho }_R}^U)$, not $S({\mathbb{E}}_U{{\rho }_R}^U)$. Likewise, the expression “the complement is diamond-close to Haar” has no meaning until a comparison channel, ensemble, input quantifiers, and treatment of the random control are specified.

For an integer $L\ge 1$, define

If states on an $L$-dimensional space have trace distance at most $\eta$, then their entropy difference is at most $c_L(\eta )$ Audenaert.

At a declared cut, supply:

1. a physical accessible channel and a proxy accessible channel on the same code, after justified access-preserving identifications;

2. a common accessible comparison dimension $L$, with $d_{\diamond }({{\mathcal{N}}_{\mathrm{phys}}}^U,{{\mathcal{N}}_{\mathrm{proxy}}}^U)\le \eta$;

3. a physical detector channel compared with the proxy basis readout, with channel allowance $\nu$, and common record dimension $K$;

4. a distribution $\mu$ of proxy unitaries with TV allowance $\tau$, if an exact Haar-mean comparison is desired;

5. real target capacities $A,B$ and integer proxy dimensions $r,b$ with

   $$|{\log }_{2}r-A|\le e_R,|{\log }_{2}b-B|\le e_B.$$

The physical and proxy families must be coupled on these interfaces. If a temporal claim is made, the cuts must belong to compatible complete histories.

Under Definition def:comparison, for the declared fixed pure input,

The record-only recovery obstruction remains independent of these entropy allowances.

Proof. The channel comparison bounds the compared output trace distances, including references, by $\eta$. Detector composition contraction and the triangle inequality give record distance at most $\min \{1,\eta +\nu \}$. Apply the finite entropy continuity bounds and take expectations. Proposition prop:TV replaces the proxy ensemble mean by its Haar mean. Proposition prop:deficits compares those means with ${\log }_{2}m$ and ${\log }_{2}r$. Finally, $\min$ is one-Lipschitz in the maximum norm:

This proves both budgets. The classical recovery lower bound is Proposition prop:classical.

The dimensions $L,K$ are those of the actual common comparison spaces. If errors populate additional sectors, their dimensions cannot be deleted from the continuity modulus without a support estimate.

Compatible preparation, evolution, collection, and implementation channel errors telescope by replacing one factor at a time. This yields a sum of channel allowances, capped at one. A metric discrepancy, power error, or selected occupation error cannot be inserted into that sum without a conversion theorem CheungHawk04D,CheungEn02C.

For the proxy of Section sec:nonuniform,

at every cut. If target capacities $A(t_k),B(t_k)$ satisfy the rounding bounds of Definition def:comparison, then

In particular, for any two cuts $j,k$,

This is a necessary condition for that fixed-dimensional model, not for every possible evaporation model.

Proof. Equation (eq:dimensions) gives $r_k{b}_k=D$. Subtract the two target capacities from the corresponding logarithms and use the triangle inequality.

Changing support sectors, adding fresh resources, or admitting additional inaccessible factors can produce other models. Such changes require a new complete history and state calculation. They do not preserve the fixed tensor-release theorem merely by keeping the word “Haar.”

Only in this subsection and its clock examples set $G=\hslash =c=k_{\mathrm{B}}=1$. For a Schwarzschild mass label, the Bekenstein–Hawking entropy in bits is

This is a supplied semiclassical entropy law Bekenstein,Hawking. Identifying $B(M)$ with ${\log }_2\dim B_{\mathrm{proxy}}$ is an additional modelling declaration.

Let $M_0>M_{\mathrm{cut}}>0$. Supply a positive continuous power law $P(M)$ on $[M_{\mathrm{cut}},M_0]$ and define

Its cutoff clock is the inverse of

Continuity and positivity make this inverse unique.

Supply a positive continuous capacity-production ratio $\xi (M)$ and define the target, not the fine-grained entropy,

The candidate envelope is

For the supplied laws above:

1. $A(M(t))$ increases strictly and $B(M(t))$ decreases strictly;

2. there is exactly one crossover in the retained clock interval if and only if

   $$A(M_{\mathrm{cut}})\ge B(M_{\mathrm{cut}});$$

3. a strict interior turnover occurs if and only if that inequality is strict;

4. if $A(M_{\mathrm{cut}})<B(M_{\mathrm{cut}})$, the envelope is increasing throughout the interval.

The crossover is a capacity-envelope time, not automatically a physical recovery transition or the exact maximum of a finite Page mean.

Proof. Differentiating gives

The difference $A-B$ starts at $-B(M_0)$, increases strictly in time, and is continuous. The intermediate value theorem proves existence and uniqueness exactly under the endpoint condition. The minimum selects the increasing branch before the crossing and the decreasing branch afterward.

Assume

The flux coefficient $\lambda$ is not the surface gravity ${\kappa }_{\mathrm{H}}$.

Within the positive-cutoff interval,

The formal crossover mass and time are

They belong to the retained model only if $M_{\mathrm{cut}}\le M_{\mathrm{P}}$. No value after $t_{\mathrm{cut}}$ is predicted.

Proof. Integrate $M^{2}dM=-\lambda dt$, then solve $A(M)=B(M)$. The cutoff condition is Theorem thm:crossover.

The dependence on the entropy-production ratio is the same algebra that appears in Page's semiclassical time profile PageTime. A half-capacity crossing is not generically half the mass or half the formal lifetime.

The inverse-square clock itself can reparametrize a fixed tensor-release proxy. The assignments

cannot both be exact assignments in that proxy over a nontrivial mass interval when $\xi \ne 1$. More quantitatively, they must satisfy

Proof. Here

It is constant on a nontrivial interval exactly when $\xi =1$. Subtract its initial value and apply (eq:sumdifference).

Thus taking a coarse entropy-production ratio different from one does not simply “replace the clock” in Hawk05-B. It changes the capacity model, and potentially its state ensemble.

Choose $\xi =1$, assume $B_0:=B(M_0)\ge 2$, and put

Use $n$ qubit cells and one sampled Haar unitary on their $2^n$-dimensional internal space. Release a cell whenever $k(t)$ increments. This gives one nested finite history.

For this declared construction,

and

Its exact coherent and record means are

They obey

If $B(M_{\mathrm{cut}})\ge 1$, at least one residual qubit remains at the cutoff. No terminal drain is included.

Proof. Before the cap in $k(t)$, the radiation rounding error is less than one and

lies between $-1$ and $1$. After the cap, both target errors are at most one because $A+B=B_0$ and $A\ge n$. Apply Proposition prop:deficits. If $B(M_{\mathrm{cut}})\ge 1$, then $A(M_{\mathrm{cut}})\le B_0-1$, so $k(t_{\mathrm{cut}})\le n-1$.

This is a completed proxy calculation combining a clock with two return types. Its physical extrapolation is exactly Theorem thm:transfer, with $\eta (t),\nu (t),\tau$ still requiring justification.

Suppose a supplied actual power on the same mass interval satisfies

Then its time to any mass $M$ in that interval satisfies

Separately, suppose the inverse-square envelope has an interior crossover and a function $f(t)$ satisfies ${\sup }_t|f(t)-F(t)|\le E$. Every attained global maximum $\tilde{t}$ of $f$ obeys

Proof. Integrate the bounds on $1/P_{\mathrm{act}}(M)$. For the second assertion, the envelope has slope $2c_{\mathrm{BH}}\lambda \xi /M$ on its early branch and negative slope of magnitude $2c_{\mathrm{BH}}\lambda /M$ on its late branch. Both magnitudes are at least $v$, so

At a maximizing $\tilde{t}$, $f(\tilde{t})\ge f(t_{\mathrm{P}})$, which implies $F(t_{\mathrm{P}})-F(\tilde{t})\le 2E$.

The power allowance controls a clock. The entropy allowance controls a peak of a specified function. Neither supplies the other.

In the stationary massless scalar setup, with the additional time–frequency counting convention, the supplied occupation is

Its flux is

Here $\alpha$ denotes the declared modes or species and $g_{\alpha }$ their counted multiplicities. The stationary field derivation and its validity hypotheses are external physical inputs Hawking,PageEmission,CheungEm03A. Using them at an evolving $M(t)$ additionally assumes a quasistatic approximation.

If ${\Gamma }_{\alpha }(\omega ;M)={\gamma }_{\alpha }(M\omega )$ for a fixed massless mode set, and the following integrals converge, then

This change of variables explains the inverse-square clock. It says nothing by itself about $r(t)$ or $b(t)$.

For $0<u<1$, a bosonic number mean $u$ is compatible with both

and the geometric state

They have different spectra and support dimensions. Therefore an occupation mean does not determine a Hilbert-space dimension or a Haar state ensemble.

Proof. Both states have mean $u$, by a direct sum of their number probabilities. The first has rank two and the second has infinite rank. Equality of one number moment does not identify their states.

Quasifreeness and covariance data can identify geometric or multimode Gaussian packet states under the additional assumptions in Cheung's En02-E and Em03-B records CheungEn02E,CheungEm03B. Those assumptions are stronger than a selected mean.

For a known geometric state, truncation through level $K$ has tail $z^{K+1}$. The normalized truncated marginal has trace distance $z^{K+1}$ from the original marginal. The normalized truncation of its standard pure purification instead has trace distance $\sqrt{z^{K+1}}$. These follow from the orthogonal tail and pure-state overlap. They are state-specific estimates, not a uniform whole-code diamond certificate.

If one separately assumes independent geometric outgoing modes and the same counting convention, define

and a coarse entropy flux

The geometric-state entropy is $g_2(u)$, obtained by summing its number probabilities. Mode independence makes these entropies additive. This is a coarse state model, not the fine-grained entropy of a variable-code evaporation channel.

Under the same massless scaling,

Since $-\dot{B}=8\pi \lambda /(M\ln 2)$, its declared ratio is

Equations (eq:power) and (eq:xi-greybody) require different state and counting assumptions. If this ratio is not one, using $A_{\mathrm{cg}}$ as ${\log }_{2}r$ changes the fixed-dimensional proxy, by Proposition prop:xi.

A greybody factor suppresses a selected transmitted occupation. It does not universally specify an “early-time suppression” of a Page curve, nor prove typicality of the source code.

Let $C={{⨂}_{j=1}}^K{C}_j$, with $\dim C_j=d_j$, and supply probabilities $q_j\in [0,1]$. On each factor use the flagged transfer isometry

The occupied logical sector and vacancy are orthogonal. Let $S\subseteq \{1,\dots ,K\}$ denote the surviving accessible factors and put

For arbitrary input ${\rho }_C$,

Its exact whole-product-code recovery error is

These are product-erasure formulas, not Haar Page formulas.

Proof. The accessible output is an orthogonal direct sum over flags $S$, with weights $w_S$ and states ${\rho }_{C_S}$. The flag entropy is the sum of the independent binary entropies, which proves (eq:productentropy).

Compress every occupied block and append $I_{C_j}/d_j$ on vacancy. The decoded product channel is

Write each replacement as a Weyl average. The identity-conjugation coefficient is ${\prod }_j[1-q_j(1-{d_j}^{-2})]$; separating that term gives the upper bound (eq:productrecovery).

For the lower bound use the product Bell input. On flag $S$, the missing references are independent maximally mixed states and no decoder can attain full Bell overlap larger than ${\prod }_{j\notin S}{d_j}^{-2}$. Average over flags. The resulting overlap bound is the same product coefficient, proving optimality.

This extends the supplied multi-cell calibration in Hawk02-B CheungHawk02B. A proposed greybody-to-$q_j$ map must still be stated and justified. At $q_j=0$ for every $j$, the code is released, not retained. Entropy turnover alone does not prove that a particular process retains the code; a retaining comparison must be constructed separately.

A conserved total additive charge need not leave a tensor-product sector. At a cut, suppose the justified sector is

embedded in the physical tensor product. Distinct $\lambda$ have orthogonal radiation charge sectors and orthogonal residual charge sectors. Retain only nonzero blocks, and set

For a Haar pure vector in ${\mathcal{H}}_{\mathsf{q}}$, the mean radiation entropy is

Destructive basis readout resolving each radiation charge sector has mean entropy

The sector geometry and weights cannot be replaced by two aggregate dimensions in the unrestricted Page formula.

Proof. Use independent Gaussian coordinates in all sector blocks. Their squared block norms $p_{\lambda }$ have the Dirichlet law with parameters $D_{\lambda }$. Conditional normalized directions are independent Haar vectors in the individual blocks and independent of the weights.

The residual trace kills cross-charge coherences, so

Dirichlet differentiation gives

Add ${\sum }_{\lambda }{w}_{\lambda }\mathsf{P}(r_{\lambda },b_{\lambda })$; the ${\mathsf{H}}_{D_{\lambda }}$ terms cancel, yielding (eq:sectorpage).

Every individual radiation basis outcome in sector $\lambda$ collects $b_{\lambda }$ Gaussian coordinates. Its probability has a Dirichlet parameter $b_{\lambda }$, repeated $r_{\lambda }$ times. The same moment differentiation gives (eq:sectorrecord).

This direct-sum structure is the standard symmetry-resolved random-state setting SymmetryPage. Its derivation is included to make the missing tensor-product hypothesis visible.

If the sector probabilities are instead fixed at specified $p_{\lambda }$, with independent conditional Haar directions, the means become

and