Hawk04-D From Finite Enter Interfaces to Conditional Evaporation Certificates: Controlled Limits, Mass Laws, and Quantum Recovery

We develop a proof-based extrapolation framework for the finite Enter interfaces of Hawk04-C. The operational core consists of a complete isometry, declared accessible and inaccessible outputs, quantum-return decoders, destructive readout, and explicit classical discard. We prove exact calibration formulas for a partial-SWAP schedule and a flagged sequential-transfer model, including their different information targets. We establish fixed-code approximation and growing-code certification statements, reconstruct a finite-band reservoir regulator, and give an independent thermal-pair truncation certificate. The regulator is not transferred to a collapsing-shell Bogoliubov model without a separate comparison theorem.

A supplied luminosity law is treated as an additional physical hypothesis. Under fixed massless-species and greybody scaling assumptions, we derive the inverse-square power law and the resulting cubic cutoff-lifetime formula, together with perturbation and discretization bounds. These conclusions stop at a positive validity cutoff and do not determine Planck-scale behavior. Entropy turnover, approximately thermal observations, complementary decoupling, and full-code recovery are separated by explicit counterexamples and dimension constraints. An access-preserving endpoint package yields a fixed-decoder certificate; terminal residual readout-and-discard leaves the existing radiation channel unchanged.

The resulting predictions are conditional on independently supplied dynamics, access, approximation, luminosity, and endpoint hypotheses. No gravitational evaporation channel, universal Page-time recovery transition, final-burst spectrum, or absence of physical remnants is derived. The contribution is a coherent certification architecture and a set of calibrated tests, not a solution of black-hole evaporation.

{{0.93} Central principle. A finite channel theorem can certify a physical prediction only after the physical model supplies a comparison at that channel interface. Thermal matching, a mass equation, and residual decoupling are separate hypotheses. None is supplied by the word “Enter.” }

Hawking's particle-creation calculation concerns quantum fields on a classical collapse background, while his information-loss proposal asks about a complete final description of the initial quantum state Hawking1975,Hawking1976. For an uncharged, nonrotating Schwarzschild black hole, the external semiclassical temperature is

A bosonic mode has a thermal denominator modified by an absorption factor:

These statements do not specify a complete logical-input-to-radiation channel. In particular, one occupation number is not a test of every logical coherence.

Page's random-subsystem analysis and unitary-radiation comparison concern additional assumptions about states and information in radiation PageAverage,PageInformation. Random-subsystem recovery also depends on the input message, initial correlations, and subsystem sizes, rather than on a universal count of “half the emitted quanta” HaydenPreskill. Island and replica-wormhole calculations address entropy in their specified gravitational settings AlmheiriEtAl. They are not replaced here by a finite circuit, and they are not used as a substitute for a missing complete-channel comparison.

The low-energy scalar-reservoir evaporation model of Gaikwad, Kaushal, Mandal, and Wadia has its own dynamics and physical assumptions GaikwadEtAl. Its geometry is not assigned to the finite-band source model below merely because both contain a scalar bath.

Cheung's Philosophy of Enter treats measurement as a declared interface: destructive computational-basis readout consumes the measured quantum name and returns classical data. It does not return the unknown quantum input with its reference correlations. A decoder has a different target and must specify its permitted access. The essay also separates complete energy, physical erasure cost, and residual leakage, and distinguishes an operational analogy from a physical identification CheungPhilosophy.

We implement that discipline by declaring the input code, its retained references, the accessible output, every excluded output, and the target channel before stating a recovery or observation result. Register consumption is a statement about the returned interface, not a proof that a physical environment has disappeared.

Cheung's journal article develops a linear quantum lambda calculus and an operational characterization of destructive measurement within an explicit strategy class CheungEnter. Here the finite Born instrument is defined directly. We neither reproduce that uniqueness argument nor enlarge its hypotheses by citation.

In the finite analogy:

• a prepared quantum code plays the role of an information-bearing source;

• the split $R\mid B$ specifies an access restriction;

• a transfer isometry exposes a new outgoing quantum factor;

• Enter converts a chosen output into a classical record;

• a terminal instruction consumes a named interface.

These are operational descriptions. A metric, horizon, stress tensor, radiation flux, and physical endpoint require additional constructions.

Hawk04-C supplies the six-row operational separation, while Hawk04-A and Hawk04-B supply complete-complement recovery, finite-regulator certificates, and an interacting-family negative control CheungHawk04C,CheungHawk04A,CheungHawk04B. The supplied excerpts of Hawk02-A, Hawk02-B, Hawk01-B, and En07-A/B/C motivate the staged organization used here CheungHawk02A,CheungHawk02B,CheungHawk01B, CheungEn07A,CheungEn07B,CheungEn07C. All elementary statements needed below have local proofs.

to the proposed plan. The following repairs are substantive.

1. The general complementary-Choi recovery allowance is $\sqrt{2de}+b$, not $2de+b$. A linear bound requires a different, independently justified certificate.

2. A sequence of changing input dimensions does not have a well-defined diamond comparison until input embeddings and recovery targets are specified.

3. The finite-band reservoir regulator is model specific. It is not a regulator theorem for a collapsing-shell Bogoliubov transformation.

4. An arbitrary shrinking core does not imply decoupling of the complete residual. A whole $n$-qubit code and a small injected diary are different recovery tasks.

5. Temperature matching alone does not determine a luminosity, emission clock, or lifetime.

6. Entropy turnover does not establish a referenced decoder or fix its transition time.

7. A fixed information-free residual can have nonzero energy and many physical degrees of freedom. It is not synonymous with no remnant.

8. A null detector result is not, without a statistical and population model, proof that remnants do not exist.

The exact channel principles have established antecedents in Stinespring dilation, quantum error correction, no-hiding, and information–disturbance theory Stinespring,KnillLaflamme,BraunsteinPati,KSW. No priority claim is made for those principles.

Let $C\simeq {\mathbb{C}}^d$, $d\ge 2$, be the entire logical input whose identity channel is the recovery target. Output Hilbert spaces may be separable; finite output spaces are stated explicitly whenever finite-dimensional optimization is used. Channels act on trace-class operators and are normal CPTP maps. An external reference $Q$ is untouched.

A complete interface is an isometry

Its complete, accessible, and complementary channels are

The complete inaccessible output $B$ includes all unreturned preparation purifiers, source registers, uncollected modes, apparatus outputs, work registers, and discarded-record stores in the declared model. The independent reference $Q$ is not part of $B$.

For a state $\sigma$, write

Use half-normalized distances

and define

Decoders are deterministic and return a quantum code. A normalized successful branch alone is not a decoder.

A reference of dimension $d$ suffices for differences of channels with input $C$, by Schmidt decomposition and convexity. Consequently all diamond statements below retain arbitrary references.

For a complete interface, choose any ${\tau }_C\in \mathcal{D}(C)$. The map

is CPTP and satisfies ${\mathcal{L}}_V\mathcal{V}={\mathrm{id}}_C$. Isometric embedding preserves the trace norm of every trace-class operator, including after tensoring with $I_Q$.

Moreover, the following are equivalent:

1. a CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_C$;

2. $\mathcal{E}={\mathcal{T}}_{{\sigma }_B}$ for one residual state;

3. $\mathcal{E}(|i⟩⟨j|)={\delta }_{ij}{\sigma }_B$ on all matrix units;

4. $$J_{\mathrm{n}}(\mathcal{E})=I_Q/d\otimes {\sigma }_B,J_{\mathrm{n}}(\Phi ):=({\mathrm{id}}_Q\otimes \Phi )(|{\Phi }_d⟩⟨{\Phi }_d|),$$

   where $|{\Phi }_d⟩=d^{-1/2}{\sum }_i|i{⟩}_Q|i{⟩}_C$.

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

An exact decoder on all of $R$ is

If ${\sigma }_B$ has finite rank $r$, then $\dim R\ge dr$.

Proof. Compression and positive-effect measurement followed by preparation are CP. Their traces in (eq:restore) add to $\mathrm{Tr}Y$, and direct substitution gives the left inverse. An isometry preserves all nonzero singular values, proving trace-norm preservation.

Matrix units span $\mathcal{L}(C)$, and their Bell expansion proves the equivalence of (ii)–(iv). Assume (ii), and diagonalize the nonzero residual support:

The index set is finite or countable. Constancy implies that all code vectors are supported in $R\otimes \mathrm{supp}{\sigma }_B$, so write

The complementary matrix-unit identities give

Define $A(|i⟩|\alpha ⟩)=|u_{i\alpha }⟩$ and ${\sigma }_K={\sum }_{\alpha }{p}_{\alpha }|\alpha ⟩⟨\alpha |$. These vectors are orthonormal, proving the normal form and the dimension bound. The two terms in (eq:Schmidt-decoder) are CP and have traces summing to $\mathrm{Tr}Y$; substitution returns $X$.

Conversely, dilate an exact decoder by a Stinespring isometry $L:R\to C\otimes G$ Stinespring. The composite $Z=(L\otimes I_B)V$ has pure decoded output $|i⟩$ on input $|i⟩$, hence

Recovery of each off-diagonal matrix unit requires $⟨{\xi }_j,{\xi }_i⟩=1$. All ${\xi }_i$ are one vector $\xi$. Tracing the decoded output and $G$ gives

Every equality holds on all operators and therefore on all reference extensions.

This is the full-input constant-complement structure of exact correction KnillLaflamme,BraunsteinPati; compare the locally proved formulations in Hawk04-A/B/C CheungHawk04A,CheungHawk04B, CheungHawk04C.

The residual alone recovers the whole code exactly if and only if the accessible channel is a replacement channel. A merely nonconstant complement obstructs exact recovery from $R$; it does not establish recovery from $B$. For $d\ge 2$, both factors cannot independently recover the whole code.

Proof. Interchange the output factors in Theorem thm:exact. A replacement channel cannot recover two distinct inputs.

Let $S$ be a measured finite register and $T$ the surviving quantum register. In the computational basis, put $M_x=⟨x{|}_S\otimes I_T$. Define the destructive instrument

The output names are $K,T$, not $S$; $K$ carries the diagonal classical algebra. Define

The instrument is CPTP and

For any CPTP operation on $B$, with all its outputs ignored, the untouched $QR$ marginal is unchanged. Thus residual Enter followed by Drop leaves $\mathcal{N}$ and ${\epsilon }_{\mathrm{rec}}(\mathcal{N})$ exactly unchanged.

A complete readout dilation is

It is an isometry. Tracing $E$ gives (eq:Enter). Discarding $K$ may be represented by moving its carrier to another inaccessible store. Those outputs remain in a complete dilation.

Proof. The measurement bras satisfy ${\sum }_x{M}_x^{†}{M}_x=I$. Summing the classical blocks gives the basis formula for partial trace. For a local channel ${\Lambda }_B$, its adjoint is unital, so every untouched observable $A$ obeys

All untouched expectations agree, including references. Orthogonality of the two outcome labels and measurement completeness give ${W_{\mathrm{read}}}^{†}{W}_{\mathrm{read}}=I$. Tracing $E$ removes the cross-outcome terms.

Conditional states can change when an outcome is supplied. That is a different access task. An observer restricted to $R$ also cannot execute Enter on $B$ without an additional permission CheungHawk04B,CheungHawk04C,CheungEn07A.

A finite acyclic instruction tree built from specified pure allocation, specified unitary constants, destructive Enter, classical control, classical discard, and a fixed returned hybrid type denotes a CPTP channel. Every quantum name is returned or explicitly consumed. There are refined path maps $K_h$ with

and each reported output block is the sum of its unnormalized path maps $X\mapsto K_{h}X{K}_h^{†}$. The denotation retains arbitrary references, and every branch terminates.

Proof. Allocation is isometric, unitaries are trace preserving, and readout bras are complete. Classical control selects a complete continuation on each block; discard sums blocks. Backward induction on the finite tree proves path completeness. Grouping histories proves the block formula. Tensoring path maps with $I_Q$ proves the reference statement. A finite tree has finite maximum branch length.

The theorem neither compiles the constants over a fixed gate library nor proves a physical endpoint. It uses the finite ownership discipline of Hawk04-B/C and the separate sequential-accounting comparison in En02-C CheungHawk04B,CheungHawk04C,CheungEn02C.

A replacement accessible channel has optimal recovery error $a_d$. A measure-and-prepare channel, followed by any decoder to $C$, has error at least $1-d^{-1}$. Computational readout and basis-state preparation attain the latter bound.

Proof. A decoded replacement has Bell output $I_Q/d\otimes {\tau }_C$ and Bell overlap $d^{-2}$. The Bell-projector test gives error at least $a_d$. Preparing $I_C/d$ attains it: the completely depolarizing channel is the uniform average of the $d^2$ Weyl unitary channels, and separating the identity term gives the matching upper bound.

A measure-and-prepare channel has separable Bell output. A normalized product vector has squared overlap with ${\Phi }_d$ at most $1/d$, by Cauchy–Schwarz; convexity extends this to separable states. The Bell test gives $1-1/d$. Computational dephasing is the uniform average of the $d$ phase unitaries ${Z_d}^k$. Its Bell outputs are orthogonal, giving diamond distance $1-1/d$ from the identity.

Measure-and-prepare channels are entanglement breaking HorodeckiShorRuskai. The benchmark does not prohibit classical assistance when a surviving quantum output is also returned CheungHawk04B,CheungHawk04C.

Let $C=C_A\otimes C_D$, with dimensions $a,m$, so $d=am$. The first $k$ transferred qubits of an $n$-qubit SWAP schedule correspond to $a=2^k$, $m=2^{n-k}$. Before measurement, the accessible channel is

up to fixed output dictionaries and blank factors. The complement returns $C_D$.

For this whole-code target,

If only the computational record on $C_A$ is returned, the channel is

and its optimal recovery error is

Thus, for $n$ qubits,

These are analytic identities, not reported numerical executions.

Proof. The decoder returning $C_A$ and preparing $I_{C_D}/m$ has decoded channel ${\mathrm{id}}_{C_A}\otimes {\mathcal{T}}_{I/m}$, with error $1-m^{-2}$. For the lower bound, use independent Bell states for $C_A,C_D$. After ${\mathrm{Tr}}_{C_D}$, the reference $Q_D$ is $I/m$ and independent of everything available to the decoder. The decoded full Bell overlap is at most $m^{-2}$: tracing $Q_D$ against its Bell projector gives $I_{C_D}/m^2$, and the remaining Bell-projector expectation is at most one. This proves the recovery lower bound.

The complementary channel is ${\mathrm{Tr}}_{C_A}$ with output $C_D$. Choosing replacement $I/m$ gives leakage at most $1-m^{-2}$. Fixing $C_A$ and testing a Bell state on $C_D$ gives the reverse bound by Lemma lem:benchmarks.

For the record channel, write $F_x=|x⟩⟨x|\otimes I_m$ and let ${\sigma }_x$ be the decoder output on record $x$. Its Bell overlap is

For attainability, prepare $|x⟩⟨x|\otimes I_m/m$. The decoded channel is ${\Delta }_a\otimes {\mathcal{T}}_{I/m}$, the uniform mixture of $a{m}^2$ orthogonal-Bell-output unitary channels, one of which is the identity. Its distance from the identity is $1-1/(a{m}^2)$.

In particular, emitting half the qubits of a completely unknown $n$-qubit input is not enough to recover that whole input. Immediate readout makes the task stricter still. This calibrates the schedule of Hawk04-C without confusing a whole star code with one small diary CheungHawk04C.

Let a memory cell and each outgoing cell contain $\mathbb{C}|\mathrm{vac}⟩\oplus C$. At step $j$, a specified unitary rotates

for all logical labels $i$, using the complementary rotation on the opposite occupied-vacuum states and the identity elsewhere. Choose $0\le {\theta }_j\le \pi /2$. Define

The amplitude of first transfer at step $s$ is $\sin {\theta }_s{\prod }_{\ell <s}\cos {\theta }_{\ell }$. Orthogonal transfer locations imply that, when $q_j<1$, these amplitudes define an isometry $A_j:C\to R_j$ after normalization.

After $j$ pulses, the complete map has the form

where the occupied ranges of $A_j,L$ are orthogonal to their vacuum flags. Its optimal recovery and leakage are

An optimal decoder compresses the occupied radiation code and prepares $I_C/d$ on its orthogonal complement.

If

each pulse conserves $H_0$. For one initially occupied cell,

For $K$ independent cells with the same $q_j$, this expectation is $K\omega (1-q_j)$, even for correlated logical labels.

Proof. Induction gives the pulse amplitudes above, and

This proves (eq:flagged) and its isometry property.

The accessible channel is

The stated decoder produces $(1-q_j){\mathrm{id}}_C+q_j{\mathcal{T}}_{I/d}$, with error $q_j{a}_d$. Any decoder's Bell overlap is at most $(1-q_j)+q_j/d^2$, proving optimality.

The complement is

Replacement $q_{j}L{I}_C{L}^{†}/d+(1-q_j)|\mathrm{vac}⟩⟨\mathrm{vac}|$ gives leakage at most $q_j{a}_d$. For an arbitrary replacement state, apply a CPTP pinching and retraction onto the occupied code block and vacuum flag. Let $u$ be its occupied weight. On a Bell input, the occupied Bell projector has expectation difference $q_j-u/d^2$. The effect equal to that projector on the occupied block and identity on vacuum has difference $u{a}_d$. If $u\le q_j$, the first is at least $q_j{a}_d$; if $u\ge q_j$, the second is at least $q_j{a}_d$. This proves the leakage lower bound.

Each pulse mixes equal-energy one-excitation states and is the identity on all other sectors. It commutes with $H_0$. The transfer probability is $1-q_j$, proving the energy formula. Linearity and addition of the independent number observables prove the $K$-cell statement.

These formulas reconstruct the single-cell benchmark reported in the supplied Hawk02-B excerpt and the flagged comparison in Hawk04-A CheungHawk02B,CheungHawk04A. They do not describe the whole-code partial-trace channel of Theorem thm:swap.

Every nonincreasing sequence $1=q_0\ge q_1\ge \cdots \ge q_N\ge 0$ is realized by the pulses above. Thus a sharp leakage decrease near a chosen step is a realizable design choice, not a derived Page-time law.

Proof. When $q_{j-1}>0$, take ${\cos }^2{\theta }_j=q_j/q_{j-1}$. After $q_j=0$, later pulses can be trivial.

Bare-energy conservation here is not a complete switching-work or cyclic controller-reset certificate. Full computational Enter on the logical labels also destroys their quantum coherence; measuring only an occupation flag is a different instrument.

On common output spaces,

Separate isometric embeddings of $R$ and $B$ preserve both diagnostics.

For common-output isometries,

For channels with input dimension $d$,

Proof. For each decoder, the triangle inequality and CPTP contraction change its error by at most the channel comparison distance. Take infima in both directions. Distance to the replacement set is likewise one-Lipschitz.

An isometry $J:H\to H^{\prime }$ has the CPTP retraction

Embedding candidate decoders or replacement states and then retracting proves invariance.

For referenced input $\rho$, expand the complete-output difference as

with reference identities understood. Its trace norm is at most $2{\Vert V_1-V_2\Vert }_{\infty }$. Partial trace is contractive.

Every pure input with a $d$-dimensional reference is $(L\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(L^{†}L)=d$ and ${\Vert L\Vert }_{\infty }^2\le d$. Its output difference is the Choi difference multiplied on both sides by $L,L^{†}$. Convexity completes the bound.

These stability statements agree with the supplied Hawk02-A interface and Hawk04-A/B proofs CheungHawk02A,CheungHawk04A, CheungHawk04B.

external approximate input. For finite-dimensional complementary channels, the KSW information–disturbance theorem gives

Indeed the full-norm inequality

becomes (eq:KSW) under the half normalization KSW (Theorem 3). We use this approximate theorem only on finite channels. No infinite-output attainment assumption is needed for the proxy lift.

Let $\hat{V}:C\to \hat{R}\otimes \hat{B}$ be a finite isometry, embedded factorwise into the output space of $V$. Suppose

For a specified proxy residual state $\hat{\sigma }$, set

Then

A finite decoder, extended CPTP to all of $R$, has error

Consequently ${\epsilon }_{\mathrm{rec}}(\mathcal{N})$ is also at most $a_d$.

For a specified proxy decoder with

its extension satisfies

Proof. The complete comparison gives allowance $b$ for both reduced channels. Equation (eq:Choi-bound) bounds proxy leakage by $de$. The triangle inequality proves (eq:leak-certificate). Finite decoder Choi matrices form a compact set, so an optimal proxy decoder exists. Apply (eq:KSW) to the proxy.

Let $P_R$ project onto the embedded finite accessible subspace. Extend the decoder by

with the compression identified with $\hat{R}$. This map is CPTP and agrees on proxy outputs. Postprocessing contraction adds $b$, outside the square root. The constant decoder supplies $a_d$. The direct certificate follows by applying the Choi bound to the specified decoded proxy channel, followed by the same lift.

This is the certificate of Hawk04-A/B with its normalization and continuum-lift distinction retained CheungHawk04A,CheungHawk04B. The dimension factor multiplies the normalized Choi discrepancy, not an allowance already expressed in diamond distance.

Fix $C$, common separable output spaces $R_{\ast },B_{\ast }$, and factorwise embeddings of a sequence $V_{\ell }$. Suppose the embedded isometries ${\overline{V}}_{\ell }$ converge in operator norm to $V_{\ast }$, with

Then $V_{\ast }$ is an isometry, and

The recovery and leakage diagnostics differ from their limit values by at most $\min \{1,{\beta }_{\ell }\}$.

Proof. Inner products pass to the limit, so $V_{\ast }^{†}{V}_{\ast }=I_C$. Apply Lemma lem:stability, including invariance under the local embeddings.

The theorem requires a complete isometry comparison. Convergence of selected outgoing occupation numbers is not its premise.

A growing-code study may use either:

1. Fixed-subcode convergence. For each fixed finite code $C^{(k)}$, supply embeddings $J_{n,k}:C^{(k)}\to C_n$ and compare $V_n{J}_{n,k}$ on that fixed domain, with compatible output identifications.

2. Whole-code certificates. For each $n$, test the target ${\mathrm{id}}_{C_n}$ separately, reporting its own complete proxy allowance $b_n$ and Choi discrepancy $e_n$.

Neither convention silently equates different identity channels.

For whole-code finite-proxy certificates,

Thus $b_n\to 0$ and $d_n{e}_n\to 0$ are sufficient for these upper certificates to vanish. The assertion $e_n\to 0$ alone is insufficient to make this particular certificate vanish.

Proof. Apply Theorem thm:certificate for each $n$.

The last sentence concerns the certificate, not a necessary condition for every successful recovery method. For infinite-dimensional input spaces, energy-constrained norms provide a different comparison framework Shirokov; no unrestricted infinite-input diamond limit is inferred here.

This section supplies one concrete realization of the approximation allowance. It reconstructs the finite-band regulator used in Hawk04-A/B CheungHawk04A,CheungHawk04B. It is independent of the thermal-pair calibration in the next section.

Let $H_S$ be a bounded self-adjoint operator on a finite source space. An optional quartic realization is

where $\{{\chi }_i,{\chi }_j\}=2{\delta }_{ij}I$. Then $0\le H_S\le 2B_{N}I$.

Fix $0<\mu <\Lambda <\infty$, and set

On ${\mathcal{F}}_{\mathrm{s}}(\mathfrak{h})$, let $H_B=\mathrm{d}\Gamma (h)$. For fixed self-adjoint source operators $A_a$, real $g_a$, and $f_a\in \mathfrak{h}$, define

Annihilation is conjugate-linear in $f$. Put

The bath is initially vacuum, and all finite source/carrier preparation is given by one fixed code isometry. Choose a finite collected packet space $\mathfrak{r}=\mathrm{span}\{p_1,\dots ,p_q\}$.

The Hamiltonian is self-adjoint on the free bath domain and generates a complete unitary group. If $H_S\ge 0$, it has the form lower bound

Proof. The field estimates give

and

Hence the interaction obeys

The square-root term has arbitrarily small relative operator bound with respect to $H_B$. Kato–Rellich therefore gives self-adjointness Kato; these are standard field-operator estimates HaslerHerbst. For normalized form-domain vectors,

The spectral theorem supplies the unitary group.

No ground-state or scattering theorem is imported from the field references.

Let $P_m$ project onto bath occupation at most $m$, and put

For fixed cutoffs, packets, and $T<\infty$, there is a finite prepared proxy isometry ${\hat{V}}_t$, embedded factorwise into the complete output, such that uniformly for $0\le t\le T$,

The same bound applies to both reduced channels and their diagnostic differences.

For $T>0$ and $0<b<1$, sufficient choices are

They give $b_{\mathrm{reg}}\le b$.

Proof. In the interaction picture,

On sectors with occupation at most $k$,

An $\ell$-fold product from vacuum has norm at most $(2G{)}^{\ell }\sqrt{\ell !}$. After integration over the ordered simplex, its Dyson term is bounded by $(2Gt{)}^{\ell }/\sqrt{\ell !}$. The series converges also after multiplication by any fixed polynomial in occupation. On these prepared vectors it solves the Schrödinger equation in the required domains and equals the self-adjoint evolution.

The compressed Hamiltonian $P_{m}H{P}_m$ has identical Dyson terms through order $m$. Inserted projections are contractions, so the two prepared isometries differ by at most $2Z_m(2GT)$.

Partition the frequency band into bins of width at most $\Delta$. Replacing frequency multiplication by bin midpoints gives $\Vert h-\tilde{h}\Vert \le \Delta /2$. On the occupation-cutoff space,

Both compressed Hamiltonians are bounded, so Duhamel's formula adds at most $Tm\Delta /2$.

The active one-particle space

is finite dimensional, invariant under $\tilde{h}$, and contains all couplings and collected packets. Its orthogonal complement remains vacuum. The total-occupation-cutoff space is finite dimensional and embeds into

This is an embedding, not an equality of the total cutoff with a tensor product. Source and carrier stay in the inaccessible factor. Lemma lem:stability converts the isometry allowance to (eq:breg).

Finally let $x=2GT$. The factorial estimate yields

Later term ratios are at most $1/2$, hence $Z_m(x)\le 2^{-m}$. The occupation and mesh terms are each at most $b/2$ under (eq:regulator-selection).

The theorem refines occupation and frequency proxies on a bounded time interval. It does not remove $\mu ,\Lambda$, and its constants are not uniform under their removal. It proves no collapse geometry, no local field-to-memory receiver, and no asymptotic completeness theorem. A different Hamiltonian requires a different analytical comparison.

Use $\hslash =k_{\mathrm{B}}=1$ in this subsection. For one oscillator frequency $\omega >0$, temperature $T>0$, and $\lambda =e^{-\omega /T}$, define

Its $T$-marginal is

The normalized cutoff preparation is

These preparations satisfy

For $L$ independent pairs with individual cutoffs $m_{\ell }$, the complete pure-state distance is

Appending these fixed preparations to the same logical isometry gives the same complete-channel allowance, independently of the code dimension.

Proof. The squared overlap of the complete pair preparations is $1-{\lambda }^{m+1}$, giving the pure-state distance. The discarded geometric tail has weight ${\lambda }^{m+1}$; the normalized retained distribution gives the marginal distance. Differentiating the finite geometric sum gives

Product overlaps multiply, and $1-{\prod }_{\ell }(1-t_{\ell })\le {\sum }_{\ell }{t}_{\ell }$. For any referenced logical input, the preparation difference is a tensor product with a trace-one logical output, so its trace norm is precisely the preparation trace norm.

This is a state-preparation certificate. It does not establish that a finite matrix is an exact truncation of a Bogoliubov propagator.

A proposed free-field bridge must provide:

1. the spacetime and state preparation, field algebra, mode normalizations, and incoming and outgoing access;

2. a canonical transformation with its complete partner and purifier accounting;

3. an implementability statement in the chosen representation;

4. a fixed-code complete-channel comparison to its finite proxy;

5. moment or flux control for the unbounded observables used in energy predictions.

Unitary implementability of bosonic canonical transformations is a nontrivial issue in infinite dimensions BruneauDerezinski. It is not supplied by choosing a lattice.

Nor can finite matrices satisfy the exact oscillator relation $[a,a^{†}]=I$, because the trace of a finite commutator is zero. Finite truncation therefore requires a stated approximation and cannot be described as an exact finite representation of the full CCR.

Even trace-distance convergence does not alone control an unbounded number observable. For example,

converges in trace distance to vacuum, but $\mathrm{Tr}(N{\rho }_n)=n$. Equation (eq:thermal-moment) supplies moment control for its particular geometric preparation; a collapse model must supply its own.

Choose a finite state ${\sigma }_K$, a purification $|\chi {⟩}_{KP}$, and $L\simeq C$. Let $R=C_R\otimes K$, $B=P\otimes L$, and define

Then

For every pure code input, their accessible spectra and entropies coincide. For $H_R=I_{C_R}\otimes H_K$, all accessible energy-measurement distributions coincide for every input. Nevertheless,

Taking ${\sigma }_K$ Gibbs makes these selected energy statistics thermal.

Proof. Both maps preserve inner products, and partial traces give the channels. For pure inputs, both outputs have the nonzero spectrum of ${\sigma }_K$. Every spectral effect of $H_R$ acts identically on the logical factor. Tracing $K$ decodes the return map; the retain map is a replacement, so Lemma lem:benchmarks applies.

The same separation is reported in the supplied En07-C excerpts and Hawk04-C CheungEn07C,CheungHawk04C. A complete channel, once specified, fixes its complementary information up to the standard environmental equivalence. The missing information here is the complete channel specification, not a failure of dilation theory Stinespring.

The ledgers have different defining data:

1. Complete energy. Supply the Hamiltonians and complete dynamics. Conservation means $U^{†}{H}_{\mathrm{tot}}U=H_{\mathrm{tot}}$, or, for an energy-intertwining isometry, $H_{\mathrm{out}}V=V{H}_{\mathrm{in}}$. Unitarity alone does not imply either statement.

2. Physical erasure/reset cost. Supply the memory, input class, reservoirs, work storage, controls, initial correlations, heat/work conventions, and resource-restoration conditions.

3. Residual leakage. Compute the dimensionless complete-channel diagnostic ${\delta }_{\mathrm{full}}$. It is not an energy, stress tensor, area, or bit count.

This separation is central to En07-A/B/C CheungEn07A,CheungEn07B,CheungEn07C.

Equality of a selected visible channel does not determine a unique heat or work ledger, even after a conserving spectator Hamiltonian is fixed.

Proof. Append independent qubits $W,L$ with $H_W=H_L=ϵ|1⟩⟨1|$, $ϵ>0$. Prepare $W$ excited and $L$ in ${\gamma }_L=\mathrm{diag}(1-p,p)$, $0<p<1$. Doing nothing and applying SWAP both conserve $H_W+H_L$ and leave the selected visible channel unchanged. The first has designated reservoir energy change zero, while the second has $Q_L=ϵ(1-p)>0$, supplied by the decrease of $W$'s energy. The consumed resource differs. These are not cyclic erasure protocols unless the spectator resources are restored.

Let a memory and finite reservoir start in ${\rho }_M\otimes {\gamma }_L$, with ${\gamma }_L=e^{-\beta H_L}/\mathrm{Tr}{e}^{-\beta H_L}$, $\beta >0$. After a specified unitary, define

Using natural logarithms,

Proof. Unitary entropy preservation and the initial product state give

Using $\log {\gamma }_L=-\beta H_L-\log Z$ gives

Combine the identities and use nonnegativity.

Equation (eq:Landauer) is the thermal entropy-balance identity associated with the additional implementation assumptions, not a cost derived from $\mathsf{dropBit}\circ \mathsf{Enter}=\mathrm{Tr}$ Landauer,ReebWolf.

Use $G=\hslash =c=k_{\mathrm{B}}=1$ only for the physical matching formulas in this subsection and the mass estimates below. Thus $T_{\mathrm{H}}(M)=1/(8\pi M)$.

On $M\in [M_{\mathrm{c}},M_0]$, with $M_{\mathrm{c}}>0$, a proposed model supplies a time normalization, outgoing flux observable, and power law

Here ${\eta }_{\alpha }=1$ for bosons and $-1$ for fermions; degeneracies are included in $\alpha$. The integrals and sum are finite. For the cubic-scaling specialization, the species are massless and fixed, and