Hawk03-A Encoded Partner Algebras, Complete Emission Channels, and Quantitative Firewall-Interface Tests

We formulate and prove a finite-dimensional interface package for testing proposals that identify an interior partner with an observable encoded in radiation. We first give a quantitative independent-factor AMPS benchmark, an entropy-growth bound under explicit near-pair hypotheses, and a sharp two-purifier projection inequality. We then construct one linear unitary finite model with a state-independent partner algebra, exact reference-preserving message recovery, a complete constant complement, and exact statistics for a declared pair-vacuum proxy instrument. The partner algebra is a subalgebra of the radiation algebra, not an additional commuting copy. A quantitative commutator obstruction proves why two commuting representations cannot both intertwine the same noncommutative logical algebra. An explicit purifier-distillation intervention changes the original pair test, while preserving the message channel.

All model-specific mathematical statements are proved locally. Standard entropy and Stinespring results are identified as external mathematical inputs. A separate stability theorem explains what additional physical bridges would turn the finite package into a certificate for named infalling measurements and recovery. No such gravitational bridge is constructed here. In particular, pair-proxy regularity is not Hadamard regularity, a message register is not a spacetime-localized infalling diary, and the construction is not a Page curve, an evaporation endpoint, or a resolution of the physical firewall problem.

The firewall argument concerns the joint compatibility of microscopic unitarity, a near-horizon effective description, and an uneventful horizon for an infalling observer AMPS,Apologia. A resolution must address their interaction in one physical model. Establishing an outgoing spectrum in one model and information recovery in another does not establish that compatibility.

The source records of Cheung separate precisely these obligations. Hawk01-A separates a fixed-background Hawking source from a microscopic message channel CHawk01A. Hawk01-B separates entropy histories from complete code preservation CHawk01B. Hawk02-A requires geometry, complete emission, comparison errors, and complementary leakage to be supplied independently CHawk02A. Em03-C explicitly does not construct an interior algebra CEm03C. Em03-H separates microscopic extraction from a physical infalling-diary interpretation CEm03H.

This manuscript completes a finite-interface mathematical milestone. It does not complete the missing gravitational programme. Its positive construction establishes that an encoded partner, reference-preserving message recovery, and a specified pair test can coexist in one finite model when the partner is not an independent factor of the radiation. Its obstruction results establish that this observation cannot be upgraded to an independent interior copy, or to intervention-independent vacuum statistics.

The distinction is essential. Holographic error correction and entanglement-wedge reconstruction motivate encoded-observable descriptions ADH,DHW,Cotler,Penington. They do not identify the finite construction below with a gravitational interior. Replica-wormhole and island calculations supply further external candidate settings, with their own dynamical and interpretive assumptions AMMZ,PSSY. None is a premise of our finite model. Conversely, the objections in Apologia are not answered merely by renaming an early-radiation observable as an interior observable.

The Cheung references are cited by identifier and supplied version. Where the supplied material consists of excerpts, the citation does not represent an independent verification of the full manuscript. Bibliographic entries do not assign an unprovided publication year, journal, DOI, or peer-review status.

The mathematical results below use the finite spaces, channels, operators, and hypotheses explicitly defined here. Results from different Cheung manuscripts are contextual comparisons, not silently imported premises. We cite standard quantum-information results where used and prove the model-specific consequences.

Throughout, we distinguish:

1. Constructed: a stated mathematical object and its claimed properties are established on the declared domain.

2. Obstructed: a stated collection of requirements is proved incompatible under explicit hypotheses.

3. Unresolved: a required construction or physical identification is absent.

Failure to construct an object is not a universal nonexistence theorem.

All Hilbert spaces in Sections sec:interfaces–sec:stability are finite dimensional. A continuum or gravitational extension requires additional hypotheses and is not implicit.

For a Hilbert space ${\mathcal{H}}_X$, write $\mathcal{B}({\mathcal{H}}_X)$ for its matrix algebra and $\mathsf{S}({\mathcal{H}}_X)$ for its density operators. Channels are linear, completely positive, trace-preserving maps. For a channel difference, the diamond norm is stabilized against an arbitrary untouched reference. We use the conventions

These distances are at most one for normalized states, channels, and probability laws. Standard channel facts used here can be found in Watrous.

An external reference is denoted by $Q$. It is not an output environment and is not traced out in a reference-preservation claim.

A complete finite emission interface consists of an input code ${\mathcal{H}}_C$, an accessible output ${\mathcal{H}}_R$, an inaccessible output ${\mathcal{H}}_K$, and an isometry

Every initially fixed auxiliary mixed state is purified, and its purification is assigned to the accessible or inaccessible output. The interface defines

The superscript $c$ refers to the complement of this specified dilation.

This definition is an output-accounting convention. It is not a theorem that physical inside and outside regions factorize. Gauge constraints, dressing, and observable algebras can obstruct naive subsystem identifications DG.

For the interface in Definition def:complete, define

The infimum defining recovery ranges over deterministic CPTP decoders $\mathcal{D}:\mathcal{B}({\mathcal{H}}_R)\to \mathcal{B}({\mathcal{H}}_C)$.

The quantifier is over all code inputs, including those entangled with $Q$. Agreement on a selected preparation or on diagonal matrix units is insufficient.

A finite instrument is a family of completely positive maps $\{{\mathcal{I}}_x{\}}_{x\in \mathsf{X}}$ whose sum is trace preserving. Its complete classical–quantum output channel is

Failure outcomes are included in $\mathsf{X}$.

Comparing only the probabilities $\mathrm{Tr}{\mathcal{I}}_x(\rho )$ compares records, not the conditional residual states. This distinction is recorded in Cheung's En03-A and Psy02-B CEn03A,CPsy02B.

Let $A$ denote a proposed partner, $B$ a selected outgoing packet, and $E$ previously accessible radiation. This section assumes a genuine tensor product

and one joint density operator on it. This is an explicit hypothesis, not a consequence of the names “interior” and “radiation.”

All logarithms in entropy expressions are base two. Set

For $0\le x\le 1$, define

with the usual zero conventions, and

If $\dim {\mathcal{H}}_B=d$ and $d_{\mathrm{tr}}({\rho }_{AB},{\sigma }_{AB})\le \epsilon \le 1$, then

Proof. We give the finite-dimensional argument corresponding to Winter's continuity bound Winter. Let $x=d_{\mathrm{tr}}(\rho ,\sigma )$. If $x=0$, the conclusion is immediate. Otherwise the Jordan decomposition gives states ${\rho }_{+}$ and ${\rho }_{-}$ such that

Consequently,

For a binary mixture with weights $1-t,t$, conditional entropy satisfies

Indeed, attaching a classical mixture flag $Z$ identifies the difference with $I(B:Z|A)$. Its lower bound is strong subadditivity LiebRuskai. Its upper bound follows from $I(B:Z|A)=S(Z|A)-S(Z|AB)\le H(Z)$, since classical conditional entropy conditioned on a quantum system is nonnegative.

Apply this mixture inequality to the two presentations of $\omega$ with $t=x/(1+x)$. The difference between their mixture gaps has absolute value at most $h_2(t)$. Since $-{\log }_{2}d\le S(B|A)\le {\log }_{2}d$, we obtain

The right-hand side is increasing for $0\le x\le 1$, so replacing $x$ by $\epsilon$ proves the assertion.

For every state ${\rho }_{ABE}$ on the stated tensor product,

Proof. Purify ${\rho }_{ABE}$ by $F$. Strong subadditivity gives $S(B|A)\ge S(B|AF)$. Purity of $ABEF$ gives $S(B|AF)=-S(B|E)$. Combining the two identities proves the claim.

Let $|\beta {⟩}_{AB}$ be a pure state, let

and suppose

Under the independent-factor hypothesis,

If a proposed radiation-purification certificate also requires

then its simultaneous validity requires

Proof. For the pure pair, $S(B|A{)}_{\beta }=-s$. Lemma lem:continuity gives $S(B|A{)}_{\rho }\le -s+g_d(\epsilon )$. Lemma lem:weak then gives (eq:ampsbound). Combining this lower bound with the asserted upper bound $-\mu$ gives (eq:ampsnecessary).

Microscopic unitarity alone does not imply $S(B|E)<0$ for every selected late packet. That inequality is a separate state-dependent purification hypothesis. Likewise, near-horizon smoothness does not become the trace-distance hypothesis of Theorem thm:amps without a physical derivation identifying the selected pair and its observables.

Suppose radiation grows by independent tensor factors, $E_t=E_{t-1}{B}_t$. At each step there is a joint state on $A_t{B}_t{E}_{t-1}$ satisfying the hypotheses of Theorem thm:amps, with pair entropy $s_t$, outgoing dimension $d_t$, and distance allowance ${\epsilon }_t$. Then

Proof. At step $t$, $S(E_t)-S(E_{t-1})=S(B_t|E_{t-1})$. Apply (eq:ampsbound) and sum.

This is a finite trace-distance version of the entropy obstruction under explicit pair and independence assumptions. It does not claim that these assumptions follow from our later model. Mathur's small-corrections argument and Hawk02-A's pair comparison must be read with their own hypotheses Mathur,CHawk02A.

Let $A,B,F$ be independent $d$-dimensional factors, $d\ge 2$. Define

and, on $ABF$,

For every density operator ${\rho }_{ABF}$,

The bound is sharp. Consequently, certificates $\mathrm{Tr}(P\rho )\ge 1-{\eta }_1$ and $\mathrm{Tr}(Q\rho )\ge 1-{\eta }_2$ require

Proof. Let

Both are isometries, $P=T_1{T_1}^{†}$ and $Q=T_2{T_2}^{†}$, and direct expansion gives ${T_1}^{†}{T}_2=d^{-1}I$ after identifying $A$ with $F$. The Gram operator of $T=(T_1{T}_2)$ is

Its largest eigenvalue is $1+d^{-1}$. The nonzero eigenvalues of $T{T}^{†}=P+Q$ are the same, proving ${\Vert P+Q\Vert }_{\infty }=1+d^{-1}$ and hence (eq:projectionmonogamy). A normalized top eigenvector attains the bound. The last statement follows by substituting the two lower bounds.

For a state on independent factors $ABE$, let an exterior-only channel map $E$ to a $d$-dimensional purifier register $F$. It cannot produce two Bell-test success probabilities violating (eq:projectionmonogamy).

Proof. Applying the channel on $E$ preserves the $AB$ marginal and produces a state on the independent factors $ABF$. Apply Theorem thm:twopurifiers.

The word “exterior-only” matters: a decoder that acts on $B$ or on the partner itself need not preserve the original $AB$ certificate.

For a complete finite interface $V:C\to R\otimes K$, the following conditions are equivalent:

1. A deterministic CPTP decoder satisfies $\mathcal{DN}={\mathrm{id}}_C$.

2. There is a state ${\tau }_K$ such that ${\mathcal{N}}^c(X)=\mathrm{Tr}(X){\tau }_K$ for every $X\in \mathcal{B}({\mathcal{H}}_C)$.

3. In any orthonormal code basis $\{|i⟩\}$,

   $${\mathcal{N}}^c(|i⟩⟨j|)={\delta }_{ij}{\tau }_K.$$

The resulting recovery preserves every external reference.

Proof. Conditions 2 and 3 are equivalent because matrix units span $\mathcal{B}({\mathcal{H}}_C)$.

Assume condition 3, and diagonalize ${\tau }_K={{\sum }_{a=1}}^r{\lambda }_a|a⟩⟨a|$ with ${\lambda }_a>0$. Write

The complementary matrix-unit identities imply

Thus the $u_{ia}$ form an orthonormal family. On their span, define an isometry

The channel that applies this isometry, traces $S$, and assigns an arbitrary fixed code state on the orthogonal unused subspace is CPTP. It maps $\mathcal{N}(X)$ to $X$ for every matrix unit and hence every $X$.

Conversely, dilate an exact decoder by an isometry $W_D:R\to C\otimes S$. The composite

is a dilation of ${\mathrm{id}}_C$. Purity of the decoded output for $|i⟩$ implies $T|i⟩=|i⟩|{\chi }_i{⟩}_{SK}$. Applying exact recovery to superpositions of any two basis vectors implies $⟨{\chi }_j|{\chi }_i⟩=1$. Hence all ${\chi }_i$ are the same vector $\chi$, and

Tracing $C$ and $S$ proves condition 2.

Finally, equality of channels on all matrix units remains equality after tensoring with ${\mathrm{id}}_Q$. This establishes reference preservation.

This is the constant-complement form of exact quantum error correction KL,KSW. Comparable locally stated criteria occur in Cheung's Hawk01-B, Hawk02-A, and Em03-H CHawk01B,CHawk02A,CEm03H. Their model-specific interfaces are not identified with ours.

We use the following external mathematical input from Kretschmann, Schlingemann, and Werner KSW: two channels can be represented by common, suitably enlarged Stinespring dilations whose optimal operator-norm separation $\beta$ satisfies

The dilation spaces may be enlarged; this statement does not prescribe a physically local decoder.

For Definition def:errors,

Proof. Fix a replacement channel whose distance from ${\mathcal{N}}^c$ is $\delta$. Its dilation is the embedding $|\psi ⟩\mapsto |\psi {⟩}_C|\chi {⟩}_{SK}$, where $\chi$ purifies the replacement state. The stated Stinespring comparison provides, after enlarging $S$, a decoder dilation bringing $V$ within operator norm $\sqrt{2\delta }$ of this embedding. Tracing its syndrome gives a decoded channel within half-diamond distance $\sqrt{2\delta }$ of ${\mathrm{id}}_C$. Minimize over replacement states.

Conversely, for a decoder of error $e$, compare a dilation of $\mathcal{DN}$ with a dilation of ${\mathrm{id}}_C$. Their operator-norm separation can be made at most $\sqrt{2e}$. Their complementary channels then differ by half-diamond distance at most $\sqrt{2e}$. Tracing the decoder syndrome leaves the original complete complement, while the identity dilation leaves a replacement channel. Minimize over decoders. The bound by one is automatic.

A single Bell-input test is not automatically a diamond-norm certificate. For input dimension $m$, if

then

For the upper bound, write a purified input as $(A\otimes I)|{\Phi }_m⟩$, with ${\Vert A\Vert }_{\infty }\le \sqrt{m}$, and apply the trace-norm ideal property. Convexity and Schmidt compression cover arbitrary inputs and references. This dimension factor must not be silently discarded.

Let all registers below be qutrits, with arithmetic modulo three. The map

is an isometry. Each one-register output channel is replacement by $I_3/3$. Nevertheless, the joint complement $K_1{K}_2$ contains perfectly distinguishable outputs for distinct computational inputs. In particular, separate constant residual marginals do not certify recovery from $R$.

Proof. The triples in (eq:qutritcode) are distinct for distinct $(r,j)$, proving orthonormality of the encoded basis vectors.

Tracing $K_1{K}_2$ on $|i⟩⟨j|$ forces $r+i=s+j$ and $r+2i=s+2j$, hence $i=j$ and $r=s$. The $R$ output is therefore ${\delta }_{ij}{I}_3/3$. Tracing the other two pairs gives the same identity for $K_1$ and $K_2$.

For input $|j⟩$, the joint complement is supported on

These supports are orthogonal for distinct $j$, since the difference between the two recorded coordinates is $j$. Thus two joint outputs have trace distance one. By the triangle inequality, every replacement state is at distance at least $1/2$ from one of them.

The accessible channel $R$ is constant. On a maximally entangled qutrit input, every decoded output has Bell overlap $1/9$, so its recovery error is at least $8/9$.

The encoding is a standard quantum-secret-sharing example CGL. Here it is used only to audit complement completeness, as required by the corresponding separation in Hawk02-A CHawk02A.

A message code and a structural logical space are different objects. We now keep that distinction explicit.

Let

A structural encoding is an isometry $J:{\mathcal{H}}_L\to {\mathcal{H}}_{\mathrm{phys}}$. For a fixed pair vector $|\beta {⟩}_{AB}$, its prepared message slice is the isometry

A representation of a logical operator $O$ intertwines if

The range of $j_{\beta }$ generally is not invariant under the full partner algebra. Consequently, compressing partner operators to the message slice is not a construction of that full algebra.

Suppose $J$ is unitary from ${\mathcal{H}}_L$ onto the physical space and $\{L_{x,a}\}$ are Kraus operators of a complete logical instrument. Then

define a complete physical instrument and satisfy

The identity holds with every untouched reference and under any finite adaptive composition of such instruments.

Proof. Unitary conjugation preserves ${\sum }_{x,a}{L_{x,a}}^{†}{L}_{x,a}=I$ and complete positivity. The displayed channel identity follows by multiplication. Tensoring all operators with $I_Q$ proves the reference version. Induction on unnormalized conditional states proves the adaptive version, including all failure outcomes.

Let $J:{\mathcal{H}}_L\to {\mathcal{H}}_{\mathrm{phys}}$ be an isometry, and let $X,Z\in \mathcal{B}({\mathcal{H}}_L)$ be contractions. Let ${\hat{X}}_1$ and ${\hat{Z}}_2$ also be contractions, satisfying

Then

In particular, if the physical representatives commute and $X,Z$ are logical Pauli operators on a qubit factor, then

Two exactly intertwining commuting representations cannot both represent the same full noncommutative logical factor.

Proof. First,

Each line has norm at most ${\epsilon }_1+{\epsilon }_2$. Subtract, apply the triangle inequality, and use ${\Vert J[X,Z]\Vert }_{\infty }={\Vert [X,Z]\Vert }_{\infty }$. For logical Pauli operators, ${\Vert [X,Z]\Vert }_{\infty }=2$.

The theorem concerns a domain carrying the stated noncommutative logical action. It must not be applied by assuming that the fixed pair slice $j_{\beta }({\mathcal{H}}_M)$ is invariant under arbitrary partner operations. Definition def:structural was introduced precisely to prevent that mistake.

The theorem identifies an algebraic obstruction, not a theorem about gravitational locality. In a physical application, one must derive both the logical domain and the proposed physical commutators. Dressing considerations are relevant to that separate task DG.

We now construct all positive finite objects in one model.

Fix:

1. a message dimension $m\ge 2$ and pair dimension $d\ge 2$;

2. bases for ${\mathcal{H}}_M\simeq {\mathcal{H}}_D\simeq {\mathcal{H}}_K\simeq {\mathbb{C}}^m$ and ${\mathcal{H}}_A\simeq {\mathcal{H}}_F\simeq {\mathcal{H}}_B\simeq {\mathbb{C}}^d$;

3. positive weights $p_j$, ${\sum }_j{p}_j=1$, and

   $$|\beta {⟩}_{AB}={\sum _{j=0}}^{d-1}\sqrt{p_j}|j{⟩}_A|j{⟩}_B;$$

4. a fixed unitary $U$ on ${\mathcal{H}}_E={\mathcal{H}}_F\otimes {\mathcal{H}}_D$;

5. the accessible final output ${\mathcal{H}}_R={\mathcal{H}}_E\otimes {\mathcal{H}}_B$ and complete residual ${\mathcal{H}}_K$.

There are no unassigned auxiliary purifications.

The word “pair” denotes the prepared finite vector above. The labels $F,B,D$ are output names, not spacetime locations.

Let $J_0:{\mathcal{H}}_M\otimes {\mathcal{H}}_A\otimes {\mathcal{H}}_B\to {\mathcal{H}}_F\otimes {\mathcal{H}}_D\otimes {\mathcal{H}}_B$ be the fixed relabeling and permutation

and define the structural unitary

Set

Start with the message in $K$, the ready output $D$ in $|0⟩$, and the independently prepared pair $|\beta {⟩}_{FB}$. Apply ${\mathrm{SWAP}}_{KD}$, then $U_{FD}$. The final state is exactly (eq:modelV).

These are specified finite unitary dynamics. If durations $t_1,t_2>0$ are desired, the swap is generated by

Writing $U={\sum }_{\alpha }{e}^{-i{\theta }_{\alpha }}{P}_{\alpha }$ with ${\theta }_{\alpha }\in (-\pi ,\pi ]$, the second pulse is generated by

Both generators are bounded. No spatial support, gravitational Hamiltonian, passive evaporation law, or black-hole mass balance is assigned to them.

A possible output schedule releases $F$ before the message-transfer pulse, releases $D$ after that pulse, and releases $B$ subsequently. The final unitary $U$ can be regarded as a declared processing operation on the collected $FD$ registers. This schedule is a finite interface schedule, not a spacetime embedding.

The map $V$ is an isometry. Let ${\tau }_F={\sum }_j{p}_j|j⟩⟨j|$. The complete accessible channel, its complement, and the $E$ restriction are

The decoder

followed by the fixed relabeling $D\to M$, satisfies ${\mathcal{D}}_E{\mathcal{N}}_E={\mathrm{id}}_M$. Thus recovery from $E$ alone is exact, with every reference. The complete complement of the $E$ channel is also replacement:

Proof. Since $J$ is unitary and $\beta$ and $|0⟩$ are normalized, $V^{†}V=I_M$. Taking the specified partial traces proves (eq:modelN)–(eq:modelNE) on every matrix unit, including off-diagonal units. Applying (eq:modeldecoder) gives ${\mathrm{Tr}}_F({\tau }_F\otimes X_D)=X_D$. For the $E$ complement, trace $E$; the unitary $U$ acts only on the traced output and disappears. The remaining $BK$ state is the displayed fixed state times $\mathrm{Tr}(X)$. Every equality is a channel identity, hence is reference stable.

Before the swap pulse, the channel from the message in $K$ to the released register $F$ is

It is replacement on all matrix units.

Proof. The initial pair preparation is independent of the message. Tracing $K,D,B$ gives the stated channel.

This proposition excludes pre-transfer imprint only for the complete declared early interface. It is not a theorem excluding every possible exterior imprint in a gravitational setting. Cheung's exterior-source constructions illustrate why that physical accounting is a separate obligation CEm03F,CEm03G.

Define representations on $R$ by

Set

The maps ${\pi }_A,{\pi }_B,{\pi }_M$ are unital faithful $\ast$-homomorphisms and exactly intertwine the structural unitary $J$. Their logical-factor images commute pairwise. Moreover,

The partner does not commute with the whole radiation algebra: for $d\ge 2$ its image is noncommutative.

Proof. Unitary conjugation preserves multiplication, adjoints, units, and injectivity. Operators acting on different logical factors commute before conjugation and therefore after conjugation. Since ${\pi }_A(O)$ acts only on $FD$, its image lies in the radiation algebra. Choose two noncommuting matrices on $A$; faithfulness preserves their nonzero commutator.

The partner is therefore encoded in radiation in an exact algebraic sense. It is not an additional independent system. Theorem thm:commuting prevents replacing this overlap by two commuting copies of the same full logical algebra.

Define

The two Kraus operators ${\Pi }_{\beta }$ and $I-{\Pi }_{\beta }$ define a complete projection instrument. We call its outcomes “proxy pass” and “proxy departure.”

For the same $J,V,U,\beta$, and for every reference-message state ${\rho }_{QM}$:

1. the emission dynamics are linear and arise from the specified finite unitaries;

2. recovery from $E$ is exact, with complete residual decoupling;

3. the partner algebra is the state-independent algebra of Theorem thm:partner;

4. the pair-proxy instrument passes with probability one and leaves the emitted state unchanged;

5. every declared logical $AB$ instrument, lifted by $J$, has exactly the same complete output as its logical counterpart, including references and adaptive histories.

These statements modify the independent-factor benchmark by identifying the partner algebra with a subalgebra of radiation. They do not establish physical horizon smoothness.

Proof. Items 1–3 follow from the specified evolution and Theorems thm:modelchannels and thm:partner. The range of $V$ lies in the range of ${\Pi }_{\beta }\otimes I_K$, so ${\Pi }_{\beta }V=V$. This proves item 4, including the selective residual state. Item 5 is Lemma lem:instrument, with the logical instrument acting as the identity on $M$. Because $A$ is represented inside $E$, no joint state on independent factors $A,B,E$ is asserted. The final sentence records the absence of a spacetime and field-state construction.

For an unreferenced message state ${\rho }_M$, let $s=S({\tau }_B)=H(p)$. The final state satisfies

The logical $AB$ state is the pure pair and satisfies $S(B|A)=-s$. Their sum is $-2s$, but this is not a violation of Lemma lem:weak: $A$ and $E$ are not independent factors. There are two descriptions of overlapping observables, not two purifiers on independent systems.

Choosing $p_j$ proportional to $e^{-\beta e_j}$ makes the selected $B$ marginal a finite Gibbs state. It does not derive a Hawking spectrum, greybody factors, dimensional emission rates, geometry, or transfer dynamics. Those separations are explicit in Hawk01-A, Hawk02-A, and Em03-E CHawk01A,CHawk02A,CEm03E.

A compatibility proposal must specify what happens after an exterior operation on the encoded partner. Computational difficulty alone is not an algebraic answer. Harlow–Hayden studies resource restrictions for particular distillation tasks HH; we impose no such restriction in the finite example.

Append a $d$-dimensional register $X$ in $|0{⟩}_X$. Perform:

1. undo $U_{FD}$;

2. apply ${\mathrm{SWAP}}_{FX}$;

3. reapply $U_{FD}$.

This is a specified exterior operation on $E$ and its ancilla.

After this operation:

1. the pair $|\beta ⟩$ is transferred exactly to $XB$;

2. the original decoded partner register $F$ is in $|0⟩$;

3. the original proxy-pass probability is

   $$\mathrm{Pr}({\Pi }_{\beta })={p_0}^2;$$

   for a maximally entangled pair it is $d^{-2}$;

4. the message remains exactly recoverable from $E$;

5. any inaccessible outputs added by this operation remain independent of the message.

Proof. In the decoded coordinates, the operation maps

This proves items 1 and 2. The original $FB$ marginal becomes $|0⟩⟨0{|}_F\otimes {\tau }_B$. Therefore

Undoing the reapplied $U$ still exposes the untouched message in $D$, proving item 4. All other decoded factors depend only on the fixed pair preparation, proving item 5.

For $d=2$, take

The exterior unitary $U_{FD}(X_F\otimes I_D){U_{FD}}^{†}$ preserves the message channel but changes the original proxy-pass probability to zero.

Proof. In logical coordinates the unitary acts only as $X_A$. Its overlap with the original pair is

It does not act on the logical message, so the message decoder is unchanged.

Theorem thm:compatibility concerns the declared prepared states. It does not assert that all radiation operations preserve their pair statistics. Transporting the partner representation to $X$ after Proposition prop:distill describes a changed observable assignment. It does not prove that the original infalling instrument remains vacuum-like, nor does it create an independent second partner.

A physical proposal must calculate the actual detector coupling, trajectory, stress-energy, and backreaction after the distillation operation. None is determined by the finite permutation above.

Let ${\mathcal{H}}_C\simeq {\mathbb{C}}^m$, $m\ge 2$, and fix a density ${\tau }_T$ with purification $|\chi {⟩}_{T{T}^{\prime }}$. There are complete isometric interfaces whose accessible channels are

For every pure code input, their accessible entropies are equal to $S({\tau }_T)$. Their complete accessible density operators also agree on the selected input $|0⟩$. Every diagnostic on $T$ agrees for all inputs. Nevertheless,

Proof. Complete isometries are

The accessible outputs have the stated form. For pure inputs the $D$ state is pure in both models, so the entropies are $S({\tau }_T)$. The remaining diagnostic assertions follow directly.

Tracing $T$ recovers the good input. The bad accessible channel is constant. For the maximally entangled code-reference input, every decoded bad output has the form $I_Q/m\otimes {\sigma }_C$ and Bell overlap $m^{-2}$. The Bell projection gives the lower bound.

If $D$ and $K$ are assigned degenerate bare energies and both models use the same $T{T}^{\prime }$ preparation, the comparison also agrees on the declared energy diagnostics. This adds no spacetime or emission-rate theorem.

This elementary separation matches the logic of Cheung's entropy-history and spectrum non-identification records CHawk01B,CHawk02A. It does not challenge an independently established reconstruction theorem.

Two complete one-outcome instruments can have identical record laws for all inputs while producing different residual states.

Proof. Take the channels ${\mathrm{id}}_2$ and ${\Delta }_Z(\rho )=P_0\rho P_0+P_1\rho P_1$, each with the single reported outcome $\ast$. Both report $\ast$ with probability one. For $\rho =|+⟩⟨+|$, their residual states are $|+⟩⟨+|$ and $I_2/2$, at trace distance $1/2$.

This is the same type of record–residual separation emphasized in En03-A CEn03A. In particular, a probability comparison must not be promoted to equality of complete instruments.

Fix any state ${\rho }_S$ and any family of measurements acting only on $S$. Append an unobserved oscillator with Hamiltonian $H_O=\omega N$, $\omega >0$. The states

give identical records for every such measurement, while their unobserved energy is $n\omega$.

Proof. The measurements act as the identity on $O$, so their probabilities depend only on ${\rho }_S$. The energy follows from $N|n⟩=n|n⟩$.

The proposition does not identify this oscillator with a gravitational stress tensor. It demonstrates why a finite record certificate alone cannot supply an omitted energy or ultraviolet bound.

The finite construction is useful only if any proposed physical successor proves its comparison interfaces. This section makes the required stability statement explicit.

Let ${\Lambda }_t,{\tilde{\Lambda }}_t$ be channels on identified successive interfaces and suppose

Then

The same conclusion holds for complete history channels if histories, conditional residual states, auxiliary outputs, and failures are retained and each one-step comparison is proved on that augmented interface.

Proof. Telescope the channel difference by inserting and subtracting one implemented step at a time. CPTP maps have diamond norm one, so pre- and post-composition cannot increase the norm of a step difference. Sum the half-diamond allowances. The proof is unchanged on an augmented history register.

The normalization agrees with the sequential stability logic in Cheung's VII-G CVIIG: if full-diamond allowances are used instead, the resulting state-distance budget includes their factor of one half.

For isometries $V,\tilde{V}:C\to R\otimes K$,

Consequently both accessible channels and their specified complements are within the same half-diamond allowance.

Proof. For a purified input, also including a reference, the output vectors differ in norm by at most ${\Vert V-\tilde{V}\Vert }_{\infty }$. The trace distance of their pure-state densities is no greater than that vector difference. Convexity covers mixed inputs. Partial trace is a channel and contracts trace distance and diamond distance.

Let $\{F_x\}$ be a physical POVM, $\{G_x\}$ a logical POVM, and $J$ an isometry. If

for every outcome, then for every logical state $\rho$,

If the prepared physical state is within trace distance $e_{\mathrm{prep}}$ of $J\rho J^{†}$, add $e_{\mathrm{prep}}$ to the allowance.

Proof. Each outcome probability differs by at most $a_x$. Sum their absolute differences and divide by two. Measurement contracts preparation trace distance, so the second assertion follows by the triangle inequality.

Lemma lem:povm controls classical probabilities. A guarantee about residual states, untouched-reference outputs, or subsequent adaptive experiments requires a complete instrument comparison. Neither a one-particle seminorm bound nor a list of effect expectations automatically supplies it. This retains the boundaries of Em06-B and Em03-D CEm06B,CEm03D.

For each declared time $t$ up to a stopping criterion, a candidate physical model supplies the following data.

1. A stated microscopic action or evolution, field content, initial data, physical observable algebra, regulators, renormalization prescription, and validity domain.

2. A complete physical isometric emission interface, with a fixed identification of every accessible and inaccessible output used in the comparison. Any gravitational factorization or algebra-to-factor bridge is separately justified.

3. A reference-uniform comparison on those identified outputs,

   $$d_{\diamond }({{\mathcal{V}}_t}^{\mathrm{phys}},{{\mathcal{V}}_t}^0)\le e_{\mathrm{em}}(t),$$

   where ${{\mathcal{V}}_t}^0$ is a complete comparison interface with proved recovery error $r_0(t)$ and complete leakage ${\delta }_0(t)$. The comparison map is linear and its transfer law is labeled derived or additionally assumed.

4. For each admitted infalling experiment, complete physical and comparison instruments on the identified output interface satisfying

   $$d_{\diamond }({{\mathcal{I}}_t}^{\mathrm{phys}},{{\mathcal{I}}_t}^0)\le e_{\mathrm{inst}}(t).$$

   Their comparison record laws are within $e_{\mathrm{reg}}(t)$ of a stated regular reference experiment with matched excitations and detector preparation.

5. Independently proved state and geometry certificates: appropriate field-state regularity; renormalized stress-energy and conservation; curvature and constraint estimates; total energy balance; and controlled backreaction through the stopping criterion. Any use of those estimates to bound $e_{\mathrm{reg}}$ or $e_{\mathrm{inst}}$ is proved in a declared topology.

6. A physical partner algebra and an operational assignment for joint radiation and infalling interventions, on the same admissible states. Reconstruction domains, commutators, state-preparation quantifiers, and post-intervention validity domains are stated.

The hypotheses are obligations, not results of this manuscript. In particular, item (H5) is not derived from (H3) or from a mass ledger.

Under Assumption ass:physical,

The channel bounds preserve arbitrary external references. If complete history comparisons are used, the corresponding history-record bound holds with their sequential budgets. For the exact benchmark of Section sec:model, $r_0={\delta }_0=0$.

These conclusions certify only the declared recovery and infalling experiments. Physical horizon regularity additionally depends on the independently proved content of (H5)–(H6).

Proof. Trace the full-output comparison over $K$ and apply the same comparison decoder. Contractivity and the triangle inequality give (eq:physicalrec), taking an arbitrarily near-optimal comparison decoder if necessary.

Trace instead over $R$ and compare to an arbitrarily near-optimal replacement channel for the comparison complement. This gives (eq:physicalleak).

For the instrument outputs, insert ${{\mathcal{I}}_t}^{\mathrm{phys}}{{\mathcal{V}}_t}^0$ between ${{\mathcal{I}}_t}^{\mathrm{phys}}{{\mathcal{V}}_t}^{\mathrm{phys}}$ and ${{\mathcal{I}}_t}^0{{\mathcal{V}}_t}^0$. Channel contractivity and the instrument comparison bound the first two differences by $e_{\mathrm{em}}$ and $e_{\mathrm{inst}}$. Reading only the classical record cannot enlarge the distance. Add the stated regular-reference allowance $e_{\mathrm{reg}}$.

All comparisons were in diamond norm, so the proof holds with ${\mathrm{id}}_Q$ tensored throughout. The history assertion follows from Lemma lem:telescoping. The exact benchmark values follow from Theorem thm:modelchannels. Nothing in this argument constructs the independent physical hypotheses (H5)–(H6).

Under the same output identifications, if only the comparison leakage ${\delta }_0$ is available, then