En07-A Enter, Reversible Recycling, and Thermodynamic Access: Complete Residuals Without Holographic Identification

This paper is a framing and separation note, not a new quantum recovery theorem. It places reversible recycling, operational forgetting, Landauer erasure, and holographic terminology within the reference-preserving access formulation of Hawk04-B.

For a complete isometry $V:C\to R\otimes B$, we locally restate and prove complete-access restoration and the equivalence between exact recovery from $R$ and constancy of the complete complementary channel to $B$. Destructive computational-basis readout returns a classical record, not an arbitrary unknown quantum code. Readout followed by classical discard is exactly partial trace, including external references. Its coherent dilation remains isometric, but this mathematical dilation is not a thermodynamic implementation certificate.

Three ledgers remain distinct: conservation of a declared complete Hamiltonian, the heat and work of a specified erasure implementation, and information-theoretic residual leakage. Channel equality alone does not determine implementation cost. If radiation is within half-diamond distance $\eta$ of a replacement channel, its optimal deterministic recovery error is at least $\max \{0,1-d^{-2}-\eta \}$. An exact flagged-transfer example and a two-state control make the information destination explicit without assigning a temperature.

No new Landauer bound, thermodynamic advantage, area law, bulk–boundary dictionary, Page curve, physical receiver, or gravitational evaporation endpoint is constructed. Companion manuscripts and supplied catalogue descriptions are documentary comparisons, not premises of the local channel proofs. No inherited interacting gravitational quantum coefficient is admitted; XV-B's admission-dependent exit is not opened, XVI-N's separate obligations are not discharged, and O.1 remains NOT_CLOSED.

{{0.93} Governing distinctions.

The residual is an output ledger, not a deletion mechanism. A channel interface does not, by itself, specify a thermal machine. }

The terms “recycling,” “erasure,” and “holography” can refer to different mathematical and physical tasks. This note fixes their interfaces before comparing them. Its immediate source is the supplied Hawk04-B version 2 manuscript CheungHawk04B, with the predecessor Hawk04-A providing the complete-complement formulation and featureless-radiation comparison CheungHawk04A.

The present task is not to derive a new radiation decoder. It is to prevent three changes of meaning:

1. replacing a quantum-return decoder by a classical readout;

2. replacing a declared marginal by a claim of physical deletion or thermodynamic reset;

3. replacing a channel-correction criterion by an area law or a gravitational bulk–boundary identification.

Throughout, $\mathsf{Enter}$ means destructive computational-basis readout. Its measured quantum names are consumed and its observable output is classical. A decoder instead has a quantum-return interface

A prescribed observation may subsequently consume $C_{\mathrm{out}}$, but that subsequent operation has a measurement channel, not ${\mathrm{id}}_C$, as its target.

Unitary uncomputation, destructive readout-and-drop, and residual leakage are not three versions of the same operation. The first can coherently reverse a specified computation when the required registers are available. The second is generally many-to-one at its declared observable interface. The third is a numerical channel diagnostic, not an operation at all.

Sections sec:channels–sec:examples provide local proofs of the channel facts needed by the framing. These facts are standard exact-correction, no-hiding, and entanglement-breaking structures KnillLaflamme,BraunsteinPati,HorodeckiShorRuskai. Their inclusion makes the present arguments independent of unavailable companion proofs.

The thermodynamic inequality discussed in Section subsec:landauer is an explicitly external result, with additional physical assumptions Landauer,Bennett,ReebWolf. It is not deduced from the readout-and-drop identity.

The microscopic energy statements from Hawk04-B and Em03-E are recorded with their source-model restrictions. No new Hamiltonian, field-to-memory receiver, or reservoir regulator is constructed here. The companion comparisons in Section sec:companions and Appendix app:analytic are documentary, not theorem premises.

Published references are distinguished from supplied unpublished manuscripts and excerpts. The supplied records are not represented as independently verified journal articles, and the bibliography does not invent missing publication or archival metadata.

Let $C\simeq {\mathbb{C}}^d$, $d\ge 2$, be the logical code. Output Hilbert spaces are separable. States are normal density operators, and channels are normal completely positive trace-preserving maps on trace-class operators, abbreviated CPTP. The finite programs below use finite-dimensional encodings.

Define

The diamond distance retains arbitrary external references. For finite input dimension $d$, a reference of dimension $d$ suffices in channel discrimination Watrous.

For an isometry

write

The output $R$ is declared accessible. The residual $B$ is complete relative to that declaration: it contains every inaccessible output of the specified pure dilation, including preparation purifiers and apparatus factors when present.

For a state $\sigma$, define

The two diagnostics are

Decoders are deterministic. A normalized postselected branch is not, by itself, a candidate in (eq:rec).

For a fixed code basis, put

and define the normalized Choi state

Agreement on the populations $E_{ii}$ alone is not agreement on the quantum code.

CPTP preprocessing and postprocessing contract $d_{\diamond }$. On fixed accessible output spaces,

Replacement channels satisfy

Moreover, ${\delta }_{\mathrm{full}}(\mathcal{C})=0$ if and only if $\mathcal{C}$ is constant.

Proof. CPTP maps have completely bounded trace norm one. The contraction statements follow by composing the reference-stable norm comparison.

For any fixed decoder, the triangle inequality changes its error by at most $d_{\diamond }({\mathcal{N}}_1,{\mathcal{N}}_2)$. Taking infima in both directions proves recovery stability.

On a referenced input state, a replacement channel returns the reference marginal tensored with its replacement state. This proves the replacement-channel equality; a product input already attains it.

If leakage is zero, choose states ${\sigma }_n$ with $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\sigma }_n})\to 0$. Evaluation at ${\tau }_C=I_C/d$ gives ${\sigma }_n\to \mathcal{C}({\tau }_C)$ in trace norm. The replacement-channel equality and the triangle inequality then give

The converse is immediate.

Suppose the unitary channels ${\mathrm{Ad}}_{U_j}$ have mutually orthogonal Bell outputs. For real coefficients $a_j$,

In particular,

Every channel from $C$ has recovery error at most $a_d$.

Proof. A Bell input gives a diagonal signed output with eigenvalues $a_j$, proving the lower bound. Each unitary channel has diamond norm one, so the triangle inequality gives the upper bound.

For any state $\sigma$, the Bell output of ${\mathcal{T}}_{\sigma }$ is $I_Q/d\otimes \sigma$. Its overlap with the target Bell projector is $d^{-2}$. Testing that projector therefore gives

The $d^2$ Weyl unitaries have orthogonal Bell outputs and uniform twirl ${\mathcal{T}}_{{\tau }_C}$. The first part gives the matching upper bound.

A decoder that ignores its input and prepares ${\tau }_C$ attains the same error after any channel.

The following restatement is the restore structure in Hawk04-A and Hawk04-B CheungHawk04A (thm:global) CheungHawk04B (thm:global).

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

is CPTP and satisfies

For every code–reference trace-class operator $X$,

In particular, if

where $J$ is an isometry, $|\Omega ⟩$ is fixed and normalized, and $U$ is a complete unitary, the entire code is preserved.

Proof. The first term of (eq:restore) is completely positive. The second is a positive-effect measurement followed by fixed preparation and is also completely positive. Their traces sum to $\mathrm{Tr}Y$. On the range of $V$, the second term vanishes and the first returns $X$.

An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity. This proves the trace-norm equality. The specified preparation and unitary compose to an isometry.

The map $\mathcal{L}:RB\to C$ is a complete-access quantum decoder. It is not generally a radiation-only decoder $R\to C$. It is also not a unitary inverse on every state of $RB$: the second term is an off-code completion. On the specified encoded range, complete coherent access allows reversal of the preparation and evolution.

This corrects the possible shorthand “the restore map is not a decoder.” Its mathematical task is decoding; the issue is which input interface it requires.

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

1. there is a CPTP map $\mathcal{D}:R\to C$ with $\mathcal{DN}={\mathrm{id}}_C$;

2. $\mathcal{C}={\mathcal{T}}_{\sigma }$ for some state $\sigma$;

3. $\mathcal{C}(E_{ij})={\delta }_{ij}\sigma$ for every matrix unit;

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

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

An exact decoder is

Proof. Linearity gives equivalence of (ii) and (iii). Expanding the Bell projector gives equivalence of (iii) and (iv).

Assume (ii), and write the nonzero spectral decomposition

The index set is finite or countable. The constancy condition puts every $V|i⟩$ in $R\otimes \mathrm{supp}\sigma$. Write

The complementary matrix-unit identities imply

Consequently

extends to an isometry. With ${\sigma }_K={\sum }_{\alpha }{p}_{\alpha }|\alpha ⟩⟨\alpha |$, tracing $B$ gives (eq:normalform).

Both terms of (eq:decoder) are completely positive, and their traces sum to $\mathrm{Tr}Y$. Substitution into (eq:normalform) gives exact recovery of every matrix unit.

Conversely, dilate an exact decoder by a Stinespring isometry $L:R\to C\otimes G$. The normal channel dilation is the standard mathematical structure theorem Stinespring,Watrous. Put $Z=(L\otimes I_B)V$. Since the decoded marginal for basis input $|i⟩$ is pure,

Exact recovery of $E_{ij}$ forces $⟨{\eta }_j,{\eta }_i⟩=1$. All the normalized vectors ${\eta }_i$ therefore coincide. Tracing the decoded register and $G$ gives

which is (iii).

These are the standard exact-correction/no-hiding conditions KnillLaflamme,BraunsteinPati, restated in the complete output convention of Hawk04-A and Hawk04-B CheungHawk04A (thm:exact) CheungHawk04B (thm:exact). No priority claim is made.

If $\mathcal{N}$ is constant on the entire code, $B$ exactly recovers that code. If $\mathcal{C}$ is constant, $R$ exactly recovers it. For $d\ge 2$, both complementary outputs cannot independently recover the entire unknown code.

Proof. Interchange $R$ and $B$ in Theorem thm:exact. An exactly recoverable output has a constant complement, and a constant channel cannot recover two distinct inputs.

A nonconstant $\mathcal{C}$ distinguishes some logical inputs. It need not contain an independently recoverable copy of the entire code. Joint access to $RB$ can be necessary. The examples in Section sec:examples exhibit this without changing the no-hiding theorem's hypotheses.

A specified complete unitary may be uncomputed when its required registers remain available. This is reversible recycling of a coherent computation, not deletion of its unknown input.

A local operation on $B$, with all outputs ignored, cannot improve the pre-existing channel $\mathcal{N}$. Exact recovery from $R$ requires a constant complete complement. A fixed mixed complement gives the syndrome form (eq:normalform); it does not assert that every physical work register has already been reset to a pure blank state.

These channel facts are compatible with irreversible erasure costs. They establish neither those costs nor a holographic encoding.

For the finite fragment, use types

with observable algebras

A hybrid state is block diagonal on its classical factors. A bit is not an unrestricted qubit under another name.

The primitives are pure allocation, specified unitary constants, destructive computational readout, and classical discard:

Return commands designate the retained interface; they are not additional dynamics.

Every live quantum wire is returned or explicitly consumed. It is never duplicated. Classical control acts separately on classical blocks. Only finite acyclic instruction trees are considered.

The fragment is defined here directly. The supplied Landmark I characterization concerns Born probabilities and positive-branch destructive residual rays within its own declared strategy class CheungI (thm:main). That full characterization is not reconstructed or imported. Likewise, the En03-A interface and scheduled-locality material is documentary CheungEn03A.

Let $S$ be $s$ measured qubits and $T$ the surviving quantum register. Put

The observable instrument is

The names in $S$ are consumed; $T$ survives.

For a pure input $\psi$, the branch probability and positive-probability residual are

No normalized residual is assigned to a zero-probability branch. Complete-channel calculations use unnormalized branches and retain every outcome.

For a finite encoded residual $B$, destructive readout followed by discard equals partial trace:

Consequently,

For the instrument (eq:instrument), the map

is an isometry. Tracing $E$ gives the observable instrument. Its complete coherent output $KET$ preserves the input, whereas the classical interface alone generally does not.

Proof. The diagonal $B$-block sum is exactly partial trace:

This identity holds for every joint state and therefore with arbitrary references.

For the dilation,

Tracing the orthogonal $E$-labels removes cross-outcome terms and gives (eq:instrument). Complete restoration follows from Theorem thm:global.

This is the local restatement of Hawk04-B's partial-trace and apparatus results CheungHawk04B (thm:drop, prop:measurement-dilation).

The physical carrier $K$ in (eq:read-dilation) is quantum. It becomes a commutative observable interface after the omitted factor $E$ is traced out. The omitted information is preserved in the complete coherent output; it is not claimed to be independently recoverable from $E$ alone.

For example,

Neither marginal generally retains the unknown phase. Joint $KE$ access does. Neither marginal channel is constant on the entire qubit code, so this is not a counterexample to Corollary cor:nohiding.

Let $\Lambda :B\to B^{\prime }$ be any normal CPTP map. For every joint state,

In particular, unobserved residual readout-and-drop leaves ${\epsilon }_{\mathrm{rec}}(\mathcal{N})$ exactly unchanged.

Proof. The Heisenberg adjoint ${\Lambda }^{\ast }$ is unital. For every bounded observable $A$ on $QR$,

Thus all expectations on $QR$ are unchanged. The recovery task is determined by that unchanged channel.

This is an unconditional statement. Supplying a measurement outcome and conditioning on it changes the access task.

A physical-memory reset has a different output type. For a declared memory $M$, define

On a joint state,

By contrast, discard returns no memory register at all.

One may model “discard, then allocate a blank” as an observable implementation of (eq:reset). That syntax still does not say how a reusable physical memory, its blank-state preparation, its controllers, or its environment are restored between runs.

Resetting one apparatus factor cannot retroactively change a completed record already delivered to an untouched register. This follows from Proposition prop:local. Resetting the very register that holds the only retained record, however, removes that retained record from the subsequent interface.

The matching distinction is recorded in Em03-B CheungEm03B. It is a statement about specified interfaces, not a thermodynamic cost theorem.

Every finite-dimensional CPTP map admits a semantic implementation by pure allocation, a specified unitary, destructive readout of unused work qubits, discard of their outcomes, and quantum return of its designated output.

Every well-typed finite acyclic instruction tree in the declared fragment denotes a CPTP hybrid channel, including untouched references, and every evaluation branch terminates.

Proof. For Kraus operators $D_j$ of a finite channel, define

Completeness of the Kraus family makes $L$ an isometry. After qubit encoding and sufficient pure allocation, its orthonormal columns extend to a unitary. Readout and discard of $G$ give

while the designated logical output remains quantum.

For a finite instruction tree, allocation and unitary nodes are complete channels, and readout satisfies ${\sum }_x{M}_x^{†}{M}_x=I$. Classical discard sums unnormalized blocks. Induction gives path maps $K_h$ with

Grouping paths by reported outputs gives the hybrid denotation. Tensoring every path map with a reference identity proves reference preservation. A finite acyclic tree has finite maximum branch length.

This is a semantic existence statement, not finite-gate compilation, efficient synthesis, laboratory realization, or a proof about an unspecified larger calculus. An infinite-dimensional decoder supplied by Theorem thm:exact need not have a finite program.

For finite encodings, the interfaces are:

The local decoder can be written as:

w := new; (Cout,G) := DecoderUnitary(R,w); k := Enter(G); Drop(k); returnQ(Cout);

All unused work wires belong to $G$. The consumed qubits are work or syndrome qubits, not the recovered diary output.

Applying it locally to the coherent body gives the joint observable channel

The observer's first-factor marginal is $\mathcal{DN}$. A strict one-register observable implementation is:

(R,B) := CoherentBody(C); k := Enter(B); Drop(k); Cout := Decode(R); returnQ(Cout);

Its channel is again $\mathcal{DN}$, by Lemma lem:erasure-interface. This added readout is not inferred from an autonomous Hamiltonian.

The joint observable return $C_{\mathrm{out}}B$ does not silently become the complete physical dilation of the decoder. Decoder work, measurement carriers, and omitted apparatus outputs must be added to that complete ledger.

For a prescribed terminal observation $\mathcal{M}$, contraction gives

The converse for one chosen observation is false: computational dephasing preserves computational readout but removes off-diagonal matrix units.

Let $\mathcal{E}$ be a measure-and-prepare channel from $C$, and let $\mathcal{D}$ be any CPTP decoder to $C$. Then

The bound applies to every finite adaptive process whose only retained output is classical. Computational readout followed by basis-state preparation attains it.

Proof. A measure-and-prepare channel has the form

Its Bell-input output, and any locally decoded version, is separable across reference and output HorodeckiShorRuskai. For normalized product vectors,

Convexity gives Bell overlap at most $d^{-1}$ for every separable state. Testing the target Bell projector proves (eq:classical).

A channel to a finite classical register has POVM form. Any decoder from it prepares an output state for each record. Finite adaptive processing before the final record does not change this measure-and-prepare form.

For attainability, let

Computational dephasing is

These $d$ unitary channels have orthogonal Bell outputs. Lemma lem:unitary-mixture gives $d_{\diamond }({\Delta }_d,{\mathrm{id}}_C)=1-d^{-1}$.

This is Hawk04-B's quantitative record-only obstruction CheungHawk04B (thm:classical). A surviving quantum memory or quantum side information supplied to the receiver changes the hypothesis.

For

radiation is constant and has recovery error $a_d$. The complete residual exactly recovers the diary. Its computational record alone cannot do so: Theorem thm:classical applies. Readout-and-drop leaves radiation unchanged.

Proof. The channels are a replacement channel and the identity, up to local isometric dictionaries. Equation (eq:ad) gives the radiation error. The readout assertions follow from Theorem thm:classical and Lemma lem:erasure-interface.

For a qubit, set

Radiation has optimal recovery error $1/2$. Computational readout of $B$ gives a uniform bit $x$, independent of the logical input. Supplying that bit and applying $Z^x$ to $R$ gives exact quantum recovery. The bit alone is not a replacement for the diary.

Proof. The unnormalized branch is

Its probability is $1/2$, and correction by $Z^x$ followed by summation returns $\rho$. Without the record, the channel is ${\Delta }_Z=(\mathrm{id}+{\mathrm{Ad}}_Z)/2$. The two inputs $|+⟩$ and $|-⟩$ have identical dephased outputs, giving recovery error at least $1/2$ by the triangle inequality. The identity decoder attains $1/2$ by Lemma lem:unitary-mixture.

For a normalized input, the complete residual is

It is not constant. Only its computational record is diary independent.

These are local destination tests corresponding to Hawk04-B's residual and assistance examples CheungHawk04B (ex:bin, ex:assistance). They concern quantum access and classical assistance, not thermodynamic branch-reuse advantages.

The next example reproduces the information-theoretic structure of Hawk04-B's four-Majorana control without reconstructing its Hamiltonian CheungHawk04B (thm:control).

Let $0\le c\le 1$, $z=\sqrt{1-c^2}$, and choose $0\le \theta \le \pi /4$ with $\cos (2\theta )=c$. Define

and the isometry

Then

The complete output is exactly reversible. For $0<c<1$, neither local output exactly recovers the code.

Proof. Orthogonal $B$-labels give (eq:controlN). Put ${\sigma }_i=|r_i⟩⟨r_i|$. Their trace distance is $z$.

For any qubit decoder, its Bell overlap is

The two diagonal output tests pull back to complementary effects. Optimal binary discrimination therefore gives

The Bell-projector test yields error at least $(3-z)/4$.

The two states are interchanged by $Z$. Symmetrizing an optimal discrimination measurement makes its two error probabilities equal to $(1-z)/2$. Measuring and preparing the inferred logical basis state then realizes

Its Pauli probabilities are

Lemma lem:unitary-mixture gives error $1-p_I=(3-z)/4$, proving (eq:controlR).

Tracing $R$ gives

The identity decoder has error $(1-c)/2$. The outputs of $|+⟩$ and $|-⟩$ have trace distance $c$. If a decoder has error $\epsilon$, contraction and the triangle inequality give $1\le 2\epsilon +c$. This proves (eq:controlB).

The channel ${\Delta }_c$ is Pauli covariant. Averaging a replacement state over Pauli conjugations cannot increase its distance from this channel and gives $I_2/2$. That replacement is therefore optimal. Relative to the uniform Pauli representation of ${\mathcal{T}}_{I_2/2}$, the coefficient differences in the order $I,Z,X,Y$ are

Lemma lem:unitary-mixture gives

which is (eq:controlleak). Complete restoration is Theorem thm:global.

For the supplied dynamical control, $c$ is the collected coherent-state overlap $c_R(t)$, and the residual channel is locally isometric to ${\Delta }_{c_R(t)}$. The preceding proof checks its channel formulas, not the Hamiltonian-to-overlap calculation. No noncommuting recovery window, scrambling estimate, or out-of-time-order-correlator bound follows.

An energy ledger begins with a named Hamiltonian, not with the words “discard” or “recycle.”

In Hawk04-B's regulated source–scalar model, the declared Hamiltonian has the form

with

For its vacuum bath preparation, the supplied energy proposition records, on every finite interval,

These are source-model statements CheungHawk04B (prop:energy); their Hamiltonian construction is not repeated here.

The notation matters. The bath-energy quantity called $E_B$ in that proposition is an expectation of $H_{\mathrm{bath}}$. It is not an energy automatically assigned to every factor of the complete residual $B$ in the channel partition. Selected source, carrier, uncollected-mode, and apparatus ledgers must not be identified by a shared letter.

Em03-E records conservation of a declared free Hamiltonian in its different finite model CheungEm03E (thm:evolution, thm:release). A free Hamiltonian is not the thermodynamic Helmholtz free energy $F=E-TS$. Neither conservation statement determines switching work, memory-reset cost, or the cost of restoring a controller.

An external switching schedule, readout apparatus, or reset mechanism would need its own Hamiltonian and work accounting. Those components are not hidden inside (eq:energy-source).

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

The channel data do not specify the physical Hamiltonians, thermal reservoir, initial resource states, controls, or cyclic restoration requirements. Energy conservation for a declared Hamiltonian is a separate assertion.

Proof. A concrete reset dilation makes the underdetermination explicit. Let $M$ be a qubit memory and let $A$ be an independent blank qubit in $|0⟩$. A swap gives, including arbitrary references,

with the factors regrouped. The observable memory channel is the same reset channel for every choice of a common energy gap

The swap conserves $H_M+H_A$, but for input excited-state probability $p$, the energy changes are

Thus even the energy transfers are not fixed by the reset channel. The blank resource is consumed, and the diary has moved to $A$, not disappeared.

One can also append an independent unobserved process without changing any channel on $QM$. For example, take a fuel qubit $F$ initially in $|1⟩$ and a thermal qubit $H$ with the same gap $ϵ$, initial excited-state probability

Swapping $F$ and $H$ conserves their total energy and increases the designated reservoir's energy by $ϵ(1-r)$, while consuming fuel energy. Whether that energy increment is included in a heat ledger depends on the specified implementation and reservoir convention; it is invisible to the memory channel.

Likewise, an independently driven, thermally isolated spectator with equal endpoint Hamiltonians can absorb actuator work while leaving the observed channel unchanged. Hence observable channel equality cannot fix the total implementation ledger.

These augmented implementations are not cyclic erasure protocols unless the consumed blank, fuel, and controls are also restored. They therefore establish underdetermination, not evasion of a thermal erasure bound.

The scope sentence adopted from the supplied Em03-B record is:

Thermodynamic costs are not fixed by channel notation.

Here it is supported by the local implementation counterexample, not imported as a new physical theorem CheungEm03B. Em03-C's joint-apparatus discipline is compatible with this distinction, but no apparatus model from that manuscript is substituted for (eq:read-dilation) CheungEm03C.

Landauer's principle concerns physical erasure under specified thermodynamic conditions, not the bare map $B\to \mathsf{I}$ Landauer. Reversible computation and record erasure are distinct tasks Bennett.

For example, the externally established quantum-statistical formulation of Reeb and Wolf starts with an independent memory and a Gibbs reservoir, uses a joint unitary, and defines heat as the reservoir energy increase ReebWolf. With entropy measured in natural units, its Landauer consequence is

under that formulation's hypotheses.

For a uniformly distributed classical bit erased to a fixed pure state, the entropy decrease is $\ln 2$, giving the familiar $k_{\mathrm{B}}T\ln 2$ lower bound. This is not a universal price for every command that discards one syntactic bit:

• a biased or already known bit has a different entropy decrease;

• retained correlations and accessible side information change the resource problem;

• heat and work are not interchangeable without an appropriate Hamiltonian and endpoint specification;

• finite reservoirs, imperfect reset, and attainability have additional restrictions.

The exact reset of a full-rank memory using a finite full-rank thermal reservoir is not presumed available merely because the entropy inequality can be written.

Equation (eq:external-landauer) is a cited external result. No Landauer inequality is derived here from Lemma lem:erasure-interface, Proposition prop:finite, or the source energy identity (eq:energy-source).

Quantum side-information erasure is an established different resource framework delRioEtAl. Its possible work advantages must not be described as violations of a bound whose independent-memory hypotheses no longer hold. No such resource advantage is constructed for the present residual or for the Hawk04-B dynamics.

The quantity ${\delta }_{\mathrm{full}}(\mathcal{C})$ is dimensionless and compares complete referenced channels. It has no temperature, Hamiltonian, time scale, or work-storage convention in its definition.

The next proposition restates the featureless-radiation bound recorded in Hawk04-A CheungHawk04A (prop:thermal). The word “thermal” in that source label does not supply a temperature to this channel inequality.

If, for some radiation state $\sigma$,

then

Proof. For any deterministic decoder,

Contraction and the triangle inequality give

by Lemma lem:unitary-mixture. Take the decoder infimum and use nonnegativity.

The hypothesis is uniform referenced channel closeness on the entire code. A thermal-looking spectrum, matching number statistics, or agreement on one input state does not establish it. Featureless radiation in this precise sense is incompatible with sufficiently accurate quantum recovery; that incompatibility is not a heat bound.

The following local check restates the flagged erasure comparison attributed to Hawk04-A CheungHawk04A (prop:erasure). Its complete dilation shows that “erasure” here means loss of access, not physical deletion.

Let

where $C_R$ and $C_B$ are copies of $C$, and let $0\le s\le 1$. Define

Then $V_s$ is an isometry, and

Exactly,

The complete output recovers the code for every $s$.

Proof. The two flag sectors are orthogonal, so the squared norm is preserved and the displayed partial traces follow.

The decoder that returns the successful code block and prepares ${\tau }_C$ on the erasure block realizes

Its error is $s{a}_d$.

For any decoder, the failure block is independent of the input. On the Bell input, its decoded target overlap is $d^{-2}$. The successful block has overlap at most one. Therefore every decoder has Bell overlap at most $1-s+s/d^2$, giving error at least $s{a}_d$.

For leakage, dephase a candidate replacement state between code and flag and twirl its code block over the Weyl unitaries. The phase averaging leaves ${\mathcal{C}}_s$ unchanged; the Weyl averaging uses its covariance and input-unitary invariance of diamond distance. Convexity thus permits the replacement state

On the Bell input, the half trace-norm difference is

If $q\le s$, this equals $s-q/d^2\ge s{a}_d$. If $q\ge s$, replacing $\left|s-q/d^2\right|$ by its lower bound $s-q/d^2$ gives a value at least $q{a}_d\ge s{a}_d$. Thus every replacement has distance at least $s{a}_d$.

For $q=s$, the channel difference is the code-block embedding of $s({\mathrm{id}}_C-{\mathcal{T}}_{{\tau }_C})$. Lemma lem:unitary-mixture gives the matching upper bound. Complete recovery is Theorem thm:global.

Equation (eq:flagged) assigns no heat to a flag. Its parameter is a transfer probability, not a temperature, an evaporation time, or a thermodynamic reset efficiency.

A deterministic instrument has completely positive branch maps ${\mathcal{F}}_h$ with

Its complete retained-record channel is

Deleting failures and normalizing a success branch changes the problem. If success probability depends on the unknown input, the normalized conditional rule is not a linear CPTP decoder on that input.

Example ex:assistance shows that a supplied classical record can assist a quantum correction. It establishes neither reusable thermodynamic garbage nor a heat advantage. Any such claim would additionally need the success and failure probabilities, side-information access, initial resources, switching work, heat convention, and the cost of restoring every resource used over repeated cycles.

Theorem thm:exact concerns one specified isometry $V:C\to R\otimes B$ and recovery of its entire logical matrix algebra. It is not an area law, a Ryu–Takayanagi relation, an entanglement-first-law identity, an island formula, or a bulk–boundary code.

No spacetime dimension, metric, horizon area, Newton constant, boundary conformal field theory, or gravitational equation appears among its hypotheses. Naming $R$ “boundary” and $B$ “bulk” does not add those data.

There are genuine relations between quantum error correction and reconstruction in established holographic settings AlmheiriDongHarlow, and between entropy, geometry, and relative entropy under their respective gravitational hypotheses RyuTakayanagi,JLMS. The present refusal is a refusal of identification without those hypotheses, not a denial of those external results.

Operator-algebra reconstruction of a proper logical subalgebra is also a different target from recovery of the entire code. Constancy of a complement for the whole matrix algebra must not be imposed as the universal criterion for every weaker operator-reconstruction task.

If the outputs are finite dimensional and exact recovery holds with $\mathrm{rank}\sigma =r$, then

This is a Hilbert-space dimension bound, not a statement about geometric area.

Proof. The proof of Theorem thm:exact constructs the $dr$ orthonormal vectors $|r_{i\alpha }⟩$ in $R$. No geometric variable enters their counting.

There are complete isometries with identical zero radiation entropy on every pure code input but radiation recovery errors $0$ and $a_d$, respectively.

Proof. Take

Both radiation marginals are pure for every pure input. The first channel is the identity; the second is constant. Their optimal recovery errors are therefore $0$ and $a_d$.

This is the selected-entropy diagnostic separation in Hawk04-B CheungHawk04B (prop:diagnostics). It does not claim that all entropy theorems are uninformative. It identifies an insufficient diagnostic.

Page's radiation-information analysis and the island literature involve their own statistical or gravitational structures Page,AlmheiriEtAl. No Page curve is inferred from Proposition prop:entropy, Example ex:control, or the flagged-transfer parameter.

The supplied XVII-A finite sine-effect reconstruction is a separate finite boundary benchmark CheungXVIIA (thm:release). It does not identify the present code with a boundary CFT, an entanglement wedge, or an evaporating black hole. The scalar-only limits of Em06-A and Em06-B retain a metric input and likewise supply no such identification CheungEm06A,CheungEm06B. These comparisons are not premises.

Suppose one physical gravitational model supplies:

1. a justified finite quantum code and complete preparation or purification;

2. complete unitary evolution;

3. a justified outgoing access construction and an access-preserving isometric endpoint $V_{\infty }:C\to R_{\infty }\otimes B_{\infty }$;

4. proof that the complete endpoint complementary channel is constant.

Then $R_{\infty }$ exactly recovers the code and arbitrary reference correlations. If the complete endpoint residual is one fixed pure state, the radiation channel is an isometric encoding.

Proof. Apply Theorem thm:exact to $V_{\infty }$. A pure residual makes the syndrome space one dimensional.

This is the conditional implication in Hawk04-B CheungHawk04B (cor:gravity). The endpoint in (iii) is an additional construction; it does not follow from finite-time unitarity or finite program termination.

No output identification may move a diary-bearing inaccessible factor into accessible radiation merely to obtain recovery. Access-preserving identifications respect the named factors.

This section records the supplied source relationships. It does not claim a scan or verification of the complete Cheung series. Similar terminology does not identify models, cutoffs, observables, resource classes, or admission contracts.

and resource interfaces. Landmark I version 2 characterizes Born probabilities and destructive positive-branch residual rays inside its declared strategy class CheungI. En03-A records its own interface and scheduled-locality statements CheungEn03A. The present instrument and fragment are directly defined, not deduced from an unprovided larger calculus. These companion statements are not a premise.

The supplied En02-B perfect-code benchmark excludes exact recovery with one live laboratory qubit under its specific streaming contract CheungEn02B (thm:B5). En02-C distinguishes unrestricted information-theoretic availability from executable recovery and records a peak-one streaming obstruction for its own benchmark CheungEn02C (thm:C2, thm:C3). These motivate the distinction between consuming syndrome qubits and consuming the recovered quantum output. They are not resource optima for the decoder in this note and are not a premise.

En02-E's flagged-cutoff comparisons retain their commuting-state hypotheses CheungEn02E. A discarded probability is not a generic allowance for a coherent channel cutoff. No cutoff theorem is transferred into the present proofs. This comparison is not a premise.

and apparatus. Em03-B records the distinction between an effective apparatus readout, later reset, and thermodynamic costs CheungEm03B. Em03-C retains joint spectators, apparatus outputs, and references in its coherent comparisons CheungEm03C. The dilation (eq:read-dilation) is specified locally; neither apparatus is silently substituted for it. These records are not a premise.

Em03-E calculates complete channels and a complement for one designed finite dynamical model and distinguishes its declared energy conservation from physical black-hole identification CheungEm03E. Em03-H records complementary upper and lower certificates for a separately fixed quartic-bath interface CheungEm03H (thm:complementary). Neither is a Page curve or a gravitational dictionary for the present residual. These records are not a premise.

and scrambling. The supplied En04-A record separates its specified non-Clifford or non-local-magic quantities from complete residual response CheungEn04A (cor:noimplication). The present leakage definition therefore does not acquire a positive value merely because a program is called “magic” or “scrambling.” No such resource measure is used in the proofs. This comparison is not a premise.

and scalar-limit comparisons. XVII-A treats one finite boundary-effect reconstruction; Em06-A and Em06-B treat scalar-only limits with metric inputs CheungXVIIA,CheungEm06A,CheungEm06B. Their source records explicitly retain the missing gravitational identifications. They neither prove nor disprove the general holographic literature and are not a premise.

and transport sheets. En05-A, En05-B, En06-A, and the supplied description of En06-B concern their own schedules, transport interfaces, or limiting instruments CheungEn05A,CheungEn05B,CheungEn06A,CheungEn06B. A prescribed speed $c$ is not a temperature or a derived gravitational constant. VII-J's finite communication certificates on a fixed cone do not derive matter-driven evolution of that cone CheungVIIJ. None supplies a reset cost, area law, or recovery theorem for the Hawk04-B residual. These records are not a premise.

The supplied companion record attributes the following catalogue descriptions to OpenAI SuppliedCatalogue. Their attribution, proof content, and publication status are not independently verified here. The descriptions are not used as mathematical results.

No catalogue family is a premise. No family closes O.1, admits an inherited gravitational coefficient, or supplies the missing analytic maps described in Appendix app:analytic.

Under the channel and finite-interface definitions of this paper:

1. complete isometric evolution admits the restoration map (eq:restore);

2. exact whole-code recovery from $R$ is equivalent to a constant complete complementary channel;

3. destructive finite readout followed by discard equals partial trace, including references;

4. the specified complete readout dilation is isometric;

5. record-only reconstruction has error at least $1-d^{-1}$;

6. finite semantic realizations return a quantum code without measuring that designated output, and finite acyclic evaluation terminates;

7. replacement-channel proximity gives (eq:featureless);

8. the exact flagged-transfer identity (eq:flagged) and the two-state control formulas (eq:controlR)–(eq:controlleak) hold;

9. observable channel equality does not fix a unique implementation energy, heat, work, or reset ledger;

10. the selected-entropy counterexample and the conditional outgoing-recovery implication hold.

These are local restatements, examples, and separation arguments. They are not a new dynamical recovery theorem.

Proof. Theorems thm:global and thm:exact prove (i)–(ii). Lemma lem:erasure-interface proves (iii)–(iv). Theorem thm:classical proves (v). Proposition prop:finite proves (vi). Proposition prop:featureless proves (vii). Proposition prop:flagged and Example ex:control prove (viii). Proposition prop:no-landauer-from-channel proves (ix). Proposition prop:entropy and Corollary cor:gravity prove (x).

The physical and documentary boundaries are separate from that mathematical collection:

• No new Landauer bound is derived. No optimal heat, work, or cyclic reset cost is determined for $\mathsf{Enter}$, $\mathsf{Drop}$, or the Hawk04-B Hamiltonian.

• No thermodynamic advantage is claimed for residual, success-branch, or failure-branch reuse.

• No area law, Ryu–Takayanagi relation, entanglement first law, island, Page curve, or bulk–boundary dictionary is constructed.

• No positive noncommuting recovery window, physical field-to-memory receiver, quantum-gravity unitarity theorem, or gravitational evaporation endpoint is established.

• Corollary cor:gravity remains conditional on all four same-model hypotheses.

• No exact finite-gate compilation, numerical execution, or resource-optimal implementation is asserted.

{{0.93} Unchanged inherited statuses.

No interacting gravitational quantum coefficient ${R_1}^{(1)}$ is admitted by this note. No missing characteristic product is defined. XV-B's admission-dependent exit is not opened. XVI-N's separate universal or prepared-source obligations are not discharged. O.1 remains NOT_CLOSED.

These are scoped documentary non-transfer statements, not universal nonexistence theorems. }

Complete preservation concerns the complete output. Recovery concerns declared access. Readout concerns a specified record. Thermodynamic erasure concerns a physical implementation and its resource restoration. Holographic reconstruction concerns additional geometric and dynamical identifications.

The residual ledger keeps these questions from being answered by changing the meaning of “recycle bin.”

of the contribution. This is a self-contained framing and separation note. No novelty is claimed for Stinespring dilation, exact quantum correction, no-hiding, entanglement-breaking channels, reversible computation, or Landauer's principle.

status. Hawk04-B version 2 is supplied as a complete source. Other Cheung manuscripts are cited through the supplied records or excerpts, with version identifiers where provided. Missing journal or archival metadata are not invented. Documentary summaries do not certify the full companion proofs.

results. Standard channel structure is identified by ordinary published references. The thermal inequality (eq:external-landauer) is explicitly external. Holographic references provide context only; none is a premise of a local channel proof.

. The mathematical claims are the analytical arguments printed here. Successful pdfLaTeX execution, proof-assistant verification, numerical optimization, independent technical review, laboratory realization, and journal acceptance are not asserted.

The map $\mathcal{L}:RB\to C$ is a CPTP decoder with complete access. It is incorrect to call it “not a decoder” without qualification. What does not follow is a decoder on $R$ alone. Its off-code completion is not a globally reversible unitary on every input state.

The equality

consumes the residual's observable interface. A reset returns a specified physical memory state, as in (eq:reset). Reuse across repeated cycles further requires the restoration of resources and apparatus. Those tasks are not specified by the discard equality.

A coherent unitary inverse acts on quantum registers. It is misleading to say that arbitrary quantum uncomputation is “reversible at the classical interface.” A classical record generally omits the phase information needed for that inverse. A complete coherent dilation can preserve the input while its classical marginal does not.

Residual leakage is a distance to replacement channels. It does not consume a register, perform a measurement, reset a memory, or dissipate energy. Consequently it is not a third operational variant of uncomputation and discard.

The value $k_{\mathrm{B}}T\ln 2$ refers to an entropy decrease of one uniformly distributed classical bit under an appropriate thermal erasure specification. It is not a universal cost per syntactic bit, and it does not follow from a channel being non-injective. Side information, finite reservoirs, and non-cyclic resource consumption must be accounted for rather than hidden in the word “recycling.”

The bath-energy expectation in Hawk04-B's energy proposition must not be identified with a total energy of the channel residual merely because both were denoted using $B$. Similarly, conservation of a free Hamiltonian is not conservation of thermodynamic free energy.

The supplied Hawk04-B version 2 has a conclusions and release section, labelled sec:release; it does not contain a theorem labelled thm:release. Theorem thm:release is the release collection defined in the present note, not a fabricated cross-reference to the predecessor.

A coherent cutoff also need not have error equal to discarded probability. The latter identity requires the appropriate commuting-state or flagged interface. The present proofs do not use such an unqualified cutoff conversion CheungEn02E,CheungHawk04B.

The analytic programme in the supplied XIV–XV records has different objects, domains, signs, cutoffs, and observables from the channels of this paper. A finite quantum channel does not define an absent distributional product.

The supplied historical sequence records:

1. XIV-H's unresolved internal-amplitude requirement at H2 CheungXIVH (thm:H2);

2. XIV-I's construction of the named internal retarded-pole amplitude module at I2, followed by the unresolved full joint characteristic map at I3 CheungXIVI (thm:I2, thm:I3);

3. XIV-J's reconstruction-descent condition and supported-sector off-characteristic reference, with the full joint map still unresolved at J2 CheungXIVJ (prop:descent, thm:reference, thm:J2, cor:noadmission).

Thus H2 is a predecessor stop, not a claim that XIV-I's named internal construction never occurred. That internal construction does not supply the later characteristic product.

The supported reference covers $K_6+K_8+K_9$, not the complete six-branch target. A normal-jet formula away from a characteristic locus is not an extension across it, and a conditional extension interface is not a prescription.

The supplied admission-dependent continuation still requires:

is a documentary identifier for that separate problem; it is not asserted here. No Ward defect is formed from an absent coefficient.

The supplied XV-B release contains abstract interface theorems but does not construct the inherited full characteristic contraction CheungXVB (thm:release). Saying that its admission-dependent exit remains closed does not deny those abstract results.

XVI-N's specified-instance massless obstruction retains its own route and domain hypotheses CheungXVIN (thm:massless-obstruction). It is neither the missing joint-map construction nor a universal completion of the separate programme. The present note changes none of those statuses.

No corpus coverage claim, catalogue entry, measured record, finite-program termination theorem, or channel dilation substitutes for these analytic obligations.

A successor seeking more than the present framing should supply the relevant package explicitly.

thermodynamic erasure or recycling. Specify:

1. the physical memory and its Hamiltonian;

2. the thermal reservoir, temperature, and initial state;

3. all initial correlations and accessible side information;

4. controls, switching interactions, work storage, and endpoint Hamiltonians;

5. the heat convention and complete energy balance;

6. success and failure branches without hidden postselection;

7. the reset accuracy and state or channel input class;

8. restoration costs for blanks, fuel, records, and controllers;

9. the finite-time, finite-bath, or asymptotic regime;

10. the theorem connecting that implementation to the declared observable channel.

quantum recovery. Specify the complete code-to-output channel, all logical matrix units, the accessible subsystem, a feasible deterministic CPTP decoder, and a reference-preserving error allowance. A terminal record is not a substitute.

holographic or gravitational identification. Specify a same-model microscopic theory, justified code, geometric and boundary data, observable algebras, access construction, receiver where execution is claimed, dynamics, approximation topology, and endpoint theorem. A code-dimension bound or an entropy history is not a bulk–boundary dictionary.

These requirements are proposed interfaces for future claims. They are not asserted accomplishments of En07-A.

KnillLaflamme

E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Physical Review A 55 (1997), 900–911. doi: 10.1103/PhysRevA.55.900. Preprint: arXiv:quant-ph/9604034.

BraunsteinPati

S. L. Braunstein and A. K. Pati, “Quantum information cannot be completely hidden in correlations: Implications for the black-hole information paradox,” Physical Review Letters 98 (2007), 080502. doi: 10.1103/PhysRevLett.98.080502. arXiv:gr-qc/0603046.

Stinespring

W. F. Stinespring, “Positive functions on $C^{\ast }$-algebras,” Proceedings of the American Mathematical Society 6 (1955), 211–216. doi: 10.1090/S0002-9939-1955-0069403-4.

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

HorodeckiShorRuskai

M. Horodecki, P. W. Shor, and M. B. Ruskai, “Entanglement breaking channels,” Reviews in Mathematical Physics 15 (2003), 629–641. doi: 10.1142/S0129055X03001709. Preprint title: “General Entanglement Breaking Channels,” arXiv:quant-ph/0302031.

Landauer

R. Landauer, “Irreversibility and heat generation in the computing process,” IBM Journal of Research and Development 5 (1961), 183–191. doi: 10.1147/rd.53.0183.

Bennett

C. H. Bennett, “The thermodynamics of computation—a review,” International Journal of Theoretical Physics 21 (1982), 905–940. doi: 10.1007/BF02084158.

ReebWolf

D. Reeb and M. M. Wolf, “An improved Landauer principle with finite-size corrections,” New Journal of Physics 16 (2014), 103011. doi: 10.1088/1367-2630/16/10/103011. arXiv:1306.4352.

delRioEtAl

L. del Rio, J. Å{}berg, R. Renner, O. Dahlsten, and V. Vedral, “The thermodynamic meaning of negative entropy,” Nature 474 (2011), 61–63. doi: 10.1038/nature10123. arXiv:1009.1630. Cited as a separate quantum-side-information resource framework.

RyuTakayanagi

S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Physical Review Letters 96 (2006), 181602. doi: 10.1103/PhysRevLett.96.181602. arXiv:hep-th/0603001. Context only; not a premise of the channel proofs.

AlmheiriDongHarlow

A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT,” Journal of High Energy Physics 04 (2015), 163. doi: 10.1007/JHEP04(2015)163. arXiv:1411.7041. Context only; no holographic dictionary is imported.

JLMS

D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh, “Relative entropy equals bulk relative entropy,” Journal of High Energy Physics 06 (2016), 004. doi: 10.1007/JHEP06(2016)004. arXiv:1512.06431. Context only.

Page

D. N. Page, “Information in black hole radiation,” Physical Review Letters 71 (1993), 3743–3746. doi: 10.1103/PhysRevLett.71.3743. arXiv:hep-th/9306083. Context only; no Page-curve result is derived here.

AlmheiriEtAl

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, “The entropy of Hawking radiation,” Reviews of Modern Physics 93 (2021), 035002. doi: 10.1103/RevModPhys.93.035002. arXiv:2006.06872. Context only; no island formula is applied in this note.

CheungHawk04A

N. Cheung, Hawk04-A: Unitary Information Preservation and Radiation Recovery: A Complete-Complement Formulation of the Black-Hole Information Problem, version 1. Supplied unpublished manuscript record, Hawk04-A-v1.tex, n.d. Source labels include thm:global, thm:exact, prop:thermal, and prop:erasure. Relevant channel facts are proved locally.

CheungHawk04B

N. Cheung, Hawk04-B: Black-Hole Information as an Access Problem: Enter, Quantum Recovery, and Complete Residuals, version 2. Supplied complete unpublished source, Hawk04-B-v2.tex, n.d. Immediate framing source: complete channels, destructive readout, classical obstruction, regulated model, energy ledger, four-Majorana control, endpoint distinctions, and release boundary.

CheungI

N. Cheung, Landmark I, version 2. Supplied unpublished excerpts and source record, n.d. In particular, thm:main, “Characterization.” The declared strategy-class hypotheses are not reconstructed here.

CheungEn03A

N. Cheung, En03-A, version 1. Supplied unpublished excerpts, En03-A.tex, n.d. Interface and scheduled-channel locality comparisons. Not a premise of the independently defined finite fragment.

CheungEn02B

N. Cheung, En02-B, version 1. Supplied unpublished excerpts, En02-B-v1.tex, n.d. In particular, thm:B5: the specified perfect-code streaming memory benchmark. No resource optimum is transferred.

CheungEn02C

N. Cheung, En02-C, version 1. Supplied unpublished excerpts, En02-C-v1.tex, n.d. In particular, thm:C2 and the supplied thm:C3 description: information-theoretic availability and executable streaming recovery are distinct.

CheungEn02E

N. Cheung, En02-E, version 2. Supplied unpublished excerpts, En02-E-v2.tex, n.d. Finite Gaussian packet and flagged-cutoff comparisons. Commuting-state cutoff restrictions are retained.

CheungEm03B

N. Cheung, Em03-B, version 2. Supplied unpublished excerpts, Em03-B-v2.tex, n.d. Effective apparatus-model readout, visible failures, later reset, and the disclaimer that channel notation does not fix thermodynamic costs. No apparatus realization is imported.

CheungEm03C

N. Cheung, Em03-C, version 1. Supplied unpublished excerpts, Em03-C-v1.tex, n.d. Joint apparatus, spectator, coherent-comparison, and complete record interfaces. Documentary comparison only.

CheungEm03E

N. Cheung, Em03-E, version 1. Supplied unpublished excerpts, Em03-E-v1.tex, n.d. In particular, thm:evolution, thm:complement, and thm:release. Declared finite dynamics and energy conservation are not a physical evaporation identification or reset-cost theorem.

CheungEm03H

N. Cheung, Em03-H, version 8. Supplied unpublished excerpts, Em03-H-v8.tex, n.d. In particular, thm:complementary. Fixed quartic-bath complementary certificates; not a Page curve or a gravitational dictionary.

CheungEn04A

N. Cheung, En04-A, version 1. Supplied unpublished excerpts, En04-A.tex, n.d. In particular, cor:noimplication. Specified non-Clifford and non-local-magic quantities are not identified with complete residual leakage.

CheungXVIIA

N. Cheung, Landmark XVII-A, version 1. Supplied unpublished excerpts, XVII-A.tex, n.d. In particular, thm:release. Finite boundary sine-effect reconstruction and binary readout; no continuum CFT or entanglement-wedge identification is imported.

CheungEm06A

N. Cheung, Em06-A, version 1. Supplied unpublished excerpts, Em06-A-v1.tex, n.d. Scalar-only first-coefficient limit with metric input. Documentary separation only.

CheungEm06B

N. Cheung, Em06-B, version 1. Supplied unpublished excerpts, Em06-B-v1.tex, n.d. Scalar-only family concentration; optional metric gate not entered. Documentary separation only.

CheungEn05A

N. Cheung, En05-A, version 1. Supplied unpublished excerpts, En05-A.tex, n.d. Finite scheduled locality and metric-cone comparison. Not a thermodynamic or gravitational-dynamics premise.

CheungEn05B

N. Cheung, En05-B, version 1. Supplied unpublished excerpts, En05-B.tex, n.d. Matched finite transport families with prescribed operational speed. Documentary separation only.

CheungEn06A

N. Cheung, En06-A, version 1. Supplied unpublished excerpts, En06-A.tex, n.d. Increasing-stencil causal comparisons and retained device-closure limitations. Documentary separation only.

CheungEn06B

N. Cheung, En06-B. Supplied unpublished descriptive source record, n.d. A separately described transport or instrument-limit comparison; a version-complete proof is not supplied here. No theorem is used as a premise.

CheungVIIJ

N. Cheung, Landmark VII-J, version 1. Supplied unpublished excerpts, VII-J.tex, n.d. Finite communication certificates on a fixed cone. Certified matter communication is not matter-driven cone dynamics.

CheungXIVH

N. Cheung, Landmark XIV-H: Internal kernel relations on the operator route, version 1. Supplied unpublished excerpts, XIV-H.tex, n.d. In particular, thm:H2. Historical internal-amplitude stop; documentary separation only.

CheungXIVI

N. Cheung, Landmark XIV-I, version 1. Supplied unpublished excerpts, XIV-I.tex, n.d. In particular, thm:I2, thm:I3, and thm:release. Named internal amplitude construction does not supply the full joint characteristic map.

CheungXIVJ

N. Cheung, Landmark XIV-J, version 1. Supplied unpublished excerpts, XIV-J.tex, n.d. In particular, prop:descent, thm:reference, thm:J2, thm:conditionalextension, and cor:noadmission. Supported-sector off-characteristic reference is not full admission.

CheungXVB

N. Cheung, Landmark XV-B, version 1. Supplied unpublished excerpts, XV-B.tex, n.d. In particular, thm:release. Abstract descent and extension interfaces do not construct the inherited full characteristic contraction.

CheungXVIN

N. Cheung, Landmark XVI-N, version r12. Supplied unpublished excerpts, XVI-N.tex, n.d. In particular, thm:massless-obstruction. Conditional specified-instance obstruction; separate universal and route obligations are not discharged here.

SuppliedCatalogue

OpenAI, as attributed in the supplied companion record, OpenAI catalogue excerpts. Supplied documentary descriptions of Families 276, 279, 227, 119, and the non-transfer concerning Family 356. Attribution, full proof content, and publication status are not independently verified here. No catalogue entry is a theorem premise.