Hawk01-A Delayed-Query Information Retrieval from Astrophysical Hawking Radiation: Fixed-Background Baselines, Conserving Completions, and Observer-Access Bounds

We formulate delayed-query retrieval from Hawking radiation as an operational task on outgoing quantum-field modes in a four-dimensional Schwarzschild exterior. Starting from the affine-vacuum horizon boundary condition and the scalar scattering problem, we derive the normalized greybody-modified emission law and finite-packet covariance. For a trusted four-state qubit preparation, we prove the exact optimal retrieval formula, a message-independent semiclassical null, a sharp measure-before-query bound, and complete-attempt robustness bounds. We construct a finite-energy emission isometry on degenerate scalar occupation sectors whose late-radiation marginal agrees with a truncated Hawking-mode thermal state for every message. With a declared early key resource, its joint radiation permits partial quantum recovery; without that resource, the late radiation is exactly message-independent. A second conserving completion has the same late state and energy distribution but no retrieval advantage even with the early resource. Thus emission spectra, including the complete late marginal in this example, do not determine the assisted recovery channel. We calculate the exact coherent-access, destructive-number, and referenced-recovery optima of the construction. Finite-packet cutoffs, collection channels, certified model errors, and attempted-run sampling are treated explicitly. The results are direct field-mode statements and conditional completion theorems, not a derivation of microscopic black-hole dynamics, an astrophysical decoding experiment, or a resolution of the information paradox.

Hawking's particle-creation calculation determines a semiclassical emission law once the spacetime, state, field, and asymptotic time normalization have been specified Hawking1975,Unruh1976. It does not by itself determine the channel transferring a specified infalling quantum message into accessible outgoing correlations. Hayden–Preskill recovery introduces additional assumptions concerning unitary dynamics, mixing, prior radiation, and decoding access HaydenPreskill2007. These assumptions are not consequences of substituting an astrophysical mass into a finite radiation model.

This paper separates three mathematical objects:

1. the semiclassical outgoing field state;

2. a declared message-to-radiation channel;

3. an observer's collection, storage, and measurement channel.

All recovery statements refer to the complete composition of the second and third objects. An emission spectrum is not substituted for that composition.

The source normalization follows the fixed-background interface reported in Cheung's Em03-A CheungEm03A. The operational separations are informed by the delayed-query, recovery, extraction, and record interfaces in Psy04-A/B, En02-B–E, Em05-A/B, and En03-A CheungPsy04A,CheungPsy04B,CheungEn02B,CheungEn02C, CheungEn02D,CheungEn02E,CheungEm05A,CheungEm05B,CheungEn03A. The relevant results are proved below rather than transferred between those manuscripts' different models.

Except where SI units are displayed, we use

The time $u$ is retarded time normalized by the asymptotic Schwarzschild Killing field. Frequencies $\omega$ are positive angular frequencies conjugate to that time. For normal states and finite-dimensional channels, respectively, write

Diamond-norm statements are made on the explicitly regulated interfaces; no finite-dimensional tensor-product factorization of the exact gravitational theory is asserted.

The principal established conclusions are:

1. the fixed-background scalar source law, Theorem thm:source;

2. exact delayed-query scoring and observer monotonicity, Theorem thm:score;

3. a conditional semiclassical null and its complete-attempt correction, Theorem thm:null;

4. an explicit finite-energy, spectrum-constrained completion and a nonrecovering completion with the same late marginal, Theorem thm:completion;

5. exact access-dependent scores and quantum recovery error, Theorem thm:completionoptima;

6. regulator, model-error, and sampling certificates.

For $M>0$, consider the exterior $r>2M$ with metric

Set

The field in the proved source calculation is a real, free, massless, minimally coupled scalar satisfying ${\square }_g\varphi =0$.

Incoming modes from past null infinity are in their vacuum. The outgoing horizon sector is the affine-vacuum sector whose current two-point distribution in affine coordinate $U$ is

On that sector the affine/Killing relation is

The state is quasifree and has zero one-point function.

This is a stationary boundary-state prescription of the kind used to isolate the late-time source term Unruh1976,CheungEm03A. Equation (eq:affine) is exact in this prescribed stationary model. It is not asserted to be an exact affine map throughout an arbitrary collapse spacetime. A collapse application requires a separate peeling estimate and transient error control.

After spherical-harmonic decomposition, the radial equation is

Let $t_{\ell }(\omega )$ be the horizon-to-infinity transmission amplitude with unit incident flux, and define

Under Assumption ass:unruh, the Killing-coordinate current covariance is

For the Fourier convention $\hat{C}(\nu )={\int }_{\mathbb{R}}{e}^{\mathrm{i}\nu s}C(s)\mathrm{d}s$,

The horizon occupation at $\omega >0$ is

Proof. Pull back the affine current distribution with the two derivative factors $U^{\prime }(u)U^{\prime }(u^{\prime })$. Direct substitution of (eq:affine) gives (eq:current), with the boundary value preserved because $U^{\prime }(u)>0$.

For $\nu >0$, integrate the meromorphic boundary-value expression against $e^{\mathrm{i}\nu s}$ in upper-half-plane rectangles whose horizontal edges lie between successive poles. The vertical contributions vanish as their real parts tend to infinity. The upper contribution vanishes as its height tends to infinity. The poles are

and the principal part at each is $-[4\pi (s-s_k{)}^2{]}^{-1}$. Consequently each residue contributes $(\nu /2)e^{-k{\beta }_{\mathrm{H}}\nu }$. Summing the geometric series yields (eq:currentfourier) for $\nu >0$. Closing in the lower half-plane for $\nu <0$ gives the same expression. Its continuous value at zero is $1/(2{\beta }_{\mathrm{H}})$.

The positive-frequency vacuum contribution in this normalization is $\omega /2$. The negative-frequency contribution is therefore

which proves (eq:horizonoccupation).

For the geometry, field, and state above, $0\le {\Gamma }_{\ell }(\omega ;M)\le 1$, and the outgoing covariance is

For a normalized packet $h$ in one angular sector,

its mean occupation is

With one temporal mode per phase-space cell $2\pi$,

Proof. The real radial potential has conserved Wronskian flux. The scattering matrix relating the two asymptotic channels is unitary. In particular, transmission plus reflection probability equals one, proving the bound on ${\Gamma }_{\ell }$. The usual positive-frequency scattering solutions exist for $\omega >0$: the potential is integrable in $r_{\ast }$, and the corresponding Jost integral equations define the asymptotic solutions.

The outgoing annihilation operator is a linear combination of the incoming infinity mode and the incoming horizon mode. The infinity input contributes zero number occupation, the two inputs have no cross covariance, and Lemma lem:current gives the horizon occupation. Multiplying that occupation by ${\left|t_{\ell }\right|}^2$ proves (eq:outcovariance). Integrating the covariance against the packet profile proves (eq:packetoccupation).

For completeness, partition positive frequency into bins of width $\Delta \omega$. In a bin, the functions

form an orthonormal frequency-window packet family. Consecutive temporal centers are separated by $2\pi /\Delta \omega$. The expected occupation per packet is the bin average of the occupation density. Dividing by that temporal separation gives the bin rate

Summing $m$ and passing to the spectral density gives (eq:sourcerate). This counting convention is part of the rate definition, not an additional thermal assumption.

The theorem is the free-scalar fixed-background result. It does not prove evaporation backreaction, a collapse solution, an interacting Standard Model spectrum, or a message-recovery law. For other species, the appropriately normalized source term has the form

with the field-specific scattering problem and multiplicities specified separately Page1976,ArbeyAuffinger2019. Equation (eq:speciesrate) is not a claim that those additional scattering calculations have been performed here.

The outgoing source power in natural units is

For a selected species list, a quasistatic mass ledger is

In SI units the right-hand side is divided by $c^2$ when the powers are in watts and $M$ is in kilograms.

For the isolated massless scalar, the radial equation depends on $M\omega$. Define

Here ${\Gamma }_{\ell }(z;1)$ denotes the dimensionless scattering function at dimensionless frequency $z$.

If the sums in (eq:alpha)–(eq:nu) converge and the isolated quasistatic ledger is adopted, then

For $M(u)\ge M_{\mathrm{cut}}>0$, the cumulative expected scalar count is

The local surface-gravity adiabatic indicator is

Proof. Substitute $z=M\omega$ in the power and number integrals. This gives the first two identities. Integrating $\dot{M}=-\alpha /M^2$ gives the third. Using $\mathrm{d}u=-M^2\mathrm{d}M/\alpha$ in the number integral gives (eq:cumulativecount). Finally, $\kappa =1/(4M)$ gives $\dot{\kappa }=-\dot{M}/(4M^2)$, and the displayed ratio follows.

The ledger is not an exact evaporating metric. A small value of (eq:adiabaticindicator) is a diagnostic, not a proved channel-error bound. Every use of a sequence of fixed-background channels below requires an independently stated approximation allowance. The evolution stops at a declared $M_{\mathrm{cut}}$ well above the regime where the adopted semiclassical treatment is uncontrolled.

Restoring SI units,

Using the displayed SI constants and the conventional numerical mass $M_{\odot }=1.9885\times 10^{30}\mathrm{kg}$, the proposed study points give Table tab:masses. The conversion should not be confused with treating the solar mass itself as an exact SI defining constant NISTConstants,Mamajek2015.

-temperature scales, not identified target objects or completed detector forecasts.

For these numerical conversions we used $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, and $\hslash \simeq 1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$. The table does not import a lifetime estimate, assert the existence of a suitable primordial object, or replace the species and background calculations needed for a count forecast.

Let $A$ be a two-dimensional internal code carried by a normalized ingoing test-body or field preparation. The labels $x=(x_1,x_2)\in \{0,1{\}}^2$ are uniformly sampled, and

These are pure states. The preparation is a trusted input interface: the decoder does not receive a second copy, a preparation label, or a message-dependent classical side channel.

The internal states $|0{⟩}_A,|1{⟩}_A$ have equal energy, charge, and declared angular-momentum labels. The common center-of-mass wavepacket, injection trajectory, and arrival-time distribution do not depend on $x$. On the code, the specified smeared macroscopic source operators act as scalar multiples of the identity.

All other accessible preparation, prompt-scattering, capture, and exterior-probe records are included in the declared output channel, or excluded by an explicit access boundary whose leakage is separately bounded.

The assumption is stronger than equality of mean energy. Scalar macroscopic source operators do not prevent a separate internal-state probe from learning $x$. Consequently the assumption includes an instrument-level leakage audit, not only a stress-energy expectation test. The exterior source models and packet-effect probes in Em03-D/F/G are useful examples of why an outgoing, message-dependent field signal need not be information escaping from a black-hole interior CheungEm03D,CheungEm03F,CheungEm03G. Stimulated-emission and ordinary-scattering contributions also require separate treatment Wald1976.

The equal-energy code is an explicitly idealized physical interface. This paper does not claim that a suitable astrophysical injection apparatus has been constructed.

Let $B$ denote the initially inaccessible source degrees of freedom, and let $E$ be a declared stored early-radiation resource. Initially $A$ is independent of $BE$, whose state ${\omega }^{BE}$ may be correlated. A supplied linear physical evolution and access prescription defines a CPTP map

The complete output $Z_u$ contains every accessible quantum system and classical flag surviving to the decoding boundary:

A factor is absent if and only if its exclusion is part of the declared boundary. Initially correlated early radiation is not silently traced out and later restored to the decoder.

Only after that boundary is fixed, a uniform $J\in \{1,2\}$ is generated independently of $X$, all pre-query systems, and all accessible random seeds. The decoder then reports $B_{\mathrm{out}}\in \{0,1\}$. The score is

An attempt includes capture failures, loss, vacuum outcomes, saturation, storage failure, and decoding failure. Every attempt produces a declared binary answer, possibly by independent random guessing on a failure branch. A heralded score is a different task and must report both the heralding instrument and its probability.

This complete-output requirement is the channel analogue of the record and residual-instrument distinctions in En03-A and Psy02-B CheungEn03A,CheungPsy02B. Equality of a selected record does not establish equality of the surviving quantum instrument.

For any complete output ensemble, define the normalized states

For unrestricted binary measurements after receiving $J$,

For any CPTP channel applied to the trusted ensemble (eq:squareensemble),

If $Z^{\prime }$ is obtained from $Z$ by a message-independent CPTP map, then

Proof. For a fixed query $j$, the two requested-bit hypotheses have equal prior probability and states ${\tau }_{0\mid j}$, ${\tau }_{1\mid j}$. Let $H_j={\tau }_{0\mid j}-{\tau }_{1\mid j}$. For an effect $0\le M\le I$, the binary success probability is

Since $\mathrm{Tr}{H}_j=0$, the supremum is $\frac{1}{2}+\frac{1}{4}{\Vert H_j\Vert }_1$, attained by the positive spectral projector of $H_j$, with an arbitrary decision on its kernel. Averaging over $j$ proves (eq:helstromscore). This is the binary Holevo–Helstrom argument Watrous2018.

At the input,

Each trace norm is $\sqrt{2}$. Contractivity under a CPTP channel proves the upper bound and (eq:scoreprocessing). Random guessing proves the lower bound. An untouched qubit with query-dependent $X$ or $Z$ measurement attains the upper bound.

For two output ensembles on a common space,

In particular, for regulated channels,

Proof. For a fixed decoder, put

Then $0\le F_x\le I$, and its score is $\frac{1}{4}{\sum }_x\mathrm{Tr}(F_x{\rho }_x)$. The variational characterization of trace distance bounds the score difference by the right-hand side of (eq:scorecontinuity). Taking suprema in both directions proves the assertion. A half-diamond bound controls the trace distance for every input state, proving (eq:scorediamond).

Suppose the complete channel from the trusted qubit is measure-and-prepare:

with the measurement performed before the fresh query. Then

The bound is sharp, even with a two-outcome measurement and a one-bit classical record. It also holds when the complete pre-query output is an arbitrary classical record generated by a POVM on the trusted qubit.

Proof. Giving the decoder the entire label $y$ can only improve its performance. Write

For the classical label channel, the requested-bit probability difference is

where ${\sigma }_1={\sigma }_x$ and ${\sigma }_2={\sigma }_z$. Equation (eq:helstromscore) therefore gives

The final inequality uses $\left|r_x\right|+\left|r_z\right|\le \sqrt{2}\left|\mathbf{r}\right|\le \sqrt{2}$. Measuring along $({\sigma }_x+{\sigma }_z)/\sqrt{2}$ attains equality. For a continuous outcome space the same proof uses the trace measure of the POVM and its Bloch-vector density.

Measure-and-prepare channels are the entanglement-breaking channel class HorodeckiShorRuskai2003. The bound above is a trusted-input channel statement. It is distinct from an arbitrary classical encoder with a two-state storage bottleneck, although that bottleneck also has the sharp value $3/4$ in the task considered in Psy04-A CheungPsy04A. Neither statement is a universal “classical black-hole” bound.

In particular, if an early quantum resource survives outside the asserted classical cut, then the complete channel must include that resource before Theorem thm:classical can be applied. Entanglement breaking of one selected radiation factor does not establish entanglement breaking of the complete accessible output.

If a regulated complete channel is within half-diamond distance $ϵ$ of a measure-and-prepare channel, then

Proof. Apply Proposition prop:scorecontinuity and Theorems thm:score and thm:classical.

For the selected fixed-background calculation:

1. the initial state of the message-independent field, source auxiliaries, and stored early systems does not depend on $x$;

2. the evolution of all decoder-accessible field and apparatus degrees of freedom acts as the identity on the internal message code, or factors through a subsystem whose complete accessible output is independent of that code;

3. no prompt reflected carrier, preparation record, or other message-dependent branch survives outside this factorization;

4. the discarded message and inaccessible auxiliaries are traced out without postselection.

This assumption defines an exact, calculable null for a noninteracting internal test-code baseline. It is not a claim that every semiclassical infalling-field preparation obeys it.

Under Assumption ass:null, the complete accessible states are identical:

More generally, if a common state ${\rho }_{\ast }$ satisfies

then

Proof. In the identity-action case, tracing the message multiplies the accessible state by $\mathrm{Tr}{{\rho }_x}^A=1$. The stated factorization has the same consequence. Completeness excludes normalization by a message-dependent selected outcome. Thus the output is independent of $x$. Both differences in (eq:helstromscore) vanish. The approximate statement follows from Proposition prop:scorecontinuity, compared with the constant ensemble.

Suppose the complete output has orthogonal flags $f$, with message-independent probabilities $p_f$, and normalized conditional ensembles ${\rho }_{x,f}$. Then

In particular, if a message is captured with probability ${\eta }_{\mathrm{cap}}$, its captured accessible branch is constant, and an intact reflected qubit is available on the other branch, then

Proof. The requested-bit differences are block diagonal in $f$. Their trace norms are sums of the block trace norms, proving (eq:flagaverage). Apply the constant-branch score $1/2$ and the intact-qubit score $q_{\mathrm{Q}}$ to obtain (eq:reflectionfalsepositive).

Equation (eq:reflectionfalsepositive) is an explicit ordinary-access false positive for an unqualified “information recovery” claim. If flag probabilities depend on $x$, the flags themselves carry information and (eq:flagaverage) with fixed $p_f$ is not applicable; the complete ensemble must be used directly.

In a complete finite CPTP network, an output whose scheduled ancestors exclude the message input is independent of that input, including when a fixed initial environment is correlated with an untouched reference.

Proof. Pull an output observable backward in topological order. The adjoint of a trace-preserving map is unital, so an identity on all outputs of a gate pulls back to an identity on its inputs. The observable's support can expand only to declared predecessor ports of gates on its scheduled ancestry. If the message is not among those ports, every pulled-back observable is identity on the message input. Its expectation is therefore independent of the message state.

This is the finite-interface support statement used in Psy04-B and VII-J CheungPsy04B,CheungVIIJ. Scheduled ancestry can overestimate actual influence. It is not a physical light-cone theorem, and the proposition does not extend a linear channel calculation to unspecified nonlinear, message-dependent metric evolution.

Choose $k$ orthonormal outgoing one-particle functions $h_1,\dots ,h_k$, with all angular and species labels included. Assume their occupation integrals are finite. Let $b_a=b_{h_a}$ and define

with the appropriate angular-sector Kronecker factors.

The fixed-background state restricted to these modes is a normal, zero-mean, gauge-invariant Gaussian state determined by the full positive matrix $K$. Its displacement characteristic function is

After a passive mode transformation diagonalizing $K$, the state is a product of thermal oscillator states with mean occupations equal to the eigenvalues of $K$.

Proof. The boundary state is quasifree, and linear scattering and restriction to a finite CCR subspace preserve quasifreeness. Stationarity and positive-frequency gauge invariance give vanishing anomalous covariance. Wick's rule, or expansion of the Weyl exponential, gives (eq:weylcharacteristic). Diagonalize $K$ by a unitary transformation of the annihilation operators. Each diagonal mode has characteristic function $\exp [-(1/2+\mu ){\left|z\right|}^2]$, which is that of

These factors are trace class, and their finite tensor product is normal. Their characteristic function equals (eq:weylcharacteristic), proving the assertion.

Keeping only diagonal packet means generally discards information: orthogonal frequency-window packets need not have diagonal $K$ when the occupation density varies within a window. The finite covariance and occupation-cutoff interfaces in En02-E provide the corresponding source-record precedent CheungEn02E; no recovery theorem is inferred from Gaussianity.

Let $N_{\mathrm{tot}}={{\sum }_{a=1}}^k{b_a}^{†}{b}_a$, and let $P_L$ project onto total occupation at most $L$. The retained Fock subspace has dimension $\left(\genfrac{}{}{0ex}{}{L+k}{k}\right)$. Introduce an orthogonal overflow state $|\perp ⟩$.

On a packet state, define

With an external spectator or reference, the overflow branch retains its corresponding unnormalized reference state. The channel is trace preserving; it does not condition on low occupation.

Embed the original state in the Fock space enlarged by the overflow flag, and put ${\delta }_L=\mathrm{Tr}((I-P_L)\rho )$. Then

This statement holds with an untouched reference. If $[\rho ,P_L]=0$, the distance equals ${\delta }_L$. Moreover,

Proof. For a purification $|\psi ⟩$, write its projected and discarded components with squared norms $1-{\delta }_L$ and ${\delta }_L$. On their two-dimensional span, direct diagonalization gives

Partial trace is contractive, so the same upper bound holds for mixed states and with a reference. The added overflow block has trace norm ${\delta }_L$. The triangle inequality proves (eq:cutofferror).

If the state commutes with $P_L$, there are no off-diagonal low/high blocks. The removed high block and the added overflow block have orthogonal supports and each has trace ${\delta }_L$, so the trace distance is ${\delta }_L$. Finally, $N_{\mathrm{tot}}\ge (L+1)(I-P_L)$, proving (eq:cutoffmoment).

For the Gaussian state of Proposition prop:gaussian, the total-number cutoff commutes with the state. For message-dependent joint states it need not commute, and the square-root allowance must not be replaced by the Gaussian tail identity without proof.

An occupation-tail certificate does not certify discarded frequency or angular sectors. Those require separate output-state or channel bounds.

A single-mode pure-loss channel of transmissivity $\eta \in [0,1]$ has Kraus operators

On a number input $n$, the detected-number law is

For a thermal independent environment of occupation $n_{\mathrm{env}}$, the beam-splitter relation gives

The complete destructive number instrument on $k$ modes, with spectator $S$, is

The positive block sum converges in trace norm. Clipping or saturation must retain an explicit overflow outcome.

Equation (eq:numberinstrument) is a complete trace-preserving instrument. Different packet states can produce the same number record while having different delayed-query or quantum-recovery performance.

Proof. The traces of its positive diagonal blocks sum to one, which proves convergence and completeness. For the second assertion, the states

have the same joint number distribution and are orthogonal. A coherent measurement distinguishes them perfectly; the destructive number record does not distinguish them.

These are the complete-record distinctions made concretely in Em03-B/C and the extraction interfaces in Em05-A/B CheungEm03B,CheungEm03C,CheungEm05A,CheungEm05B. No independence of two packet records is assumed. A shared passive collection port must obey a contraction budget: if ${\mathbf{b}}_{\det }=T{\mathbf{b}}_{\mathrm{in}}+R\mathbf{e}$, then

It cannot be represented as two independent perfect copies of the same incident mode.

For an energy-eigenmode regulator, let ${\mathcal{H}}_B(M,\mathbf{q})$ be an internal source sector with energy $M$ and declared conserved labels $\mathbf{q}$. A one-step emission interface has the form

where the radiation occupation configurations $\alpha$ have energies $E_{\alpha }\ge 0$. Vacuum and no-emission configurations are included.

Writing

physicality requires

A specified occupation calibration on a code subspace is an operator constraint of the form

or a weaker expectation constraint on explicitly selected source states. Which of these is imposed must be declared.

Neither (eq:operatorcalibration) nor a diagonal probability distribution fixes the off-diagonal terms $K_{\alpha }\rho {K_{\beta }}^{†}$. Sector dimensions must also permit the isometry: a conservation label is not a proof that sufficiently many internal states exist.

A finite emission interface is used here as an effective completion on a prepared subspace. It is not identified with a microscopic black-hole Hilbert space or assigned dimension $e^{S_{\mathrm{BH}}}$ by assumption. Successive steps must act on the actual residual output; resetting that residual between steps would define a different dynamics.

Let $V:{\mathcal{H}}_A\to {\mathcal{H}}_Z\otimes {\mathcal{H}}_F$ be a finite-dimensional isometry with accessible channel $\mathcal{N}$ and complementary channel $\mathcal{C}$. The following are equivalent:

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

2. there is a state ${\tau }_F$ such that

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_F\text{for all }X\in \mathcal{B}({\mathcal{H}}_A);$$

3. for a Kraus family of $\mathcal{N}$,

   $${A_a}^{†}{A}_b=c_{ab}{I}_A$$

   with scalar coefficients $c_{ab}$.

Proof. Suppose a decoder exists and let $T$ be its Stinespring isometry. The composite $(T\otimes I_F)V$ dilates the identity channel. Its Kraus operators are all proportional to the identity: the Choi matrix of the identity is rank one, and a sum of positive rank-one Choi contributions can have that range only when every contribution lies in it. Thus

for a fixed ancillary vector $\chi$. Tracing all but $F$ proves (eq:constantcomplement).

Conversely, diagonalize ${\tau }_F={\sum }_j{\lambda }_j|j⟩⟨j|$, omitting zero eigenvalues, and write

Applying (eq:constantcomplement) to every matrix unit gives

On their span $S$, define the isometry $U|{\varphi }_{ij}⟩=|i{⟩}_A|j{⟩}_J$. For a fixed state ${\sigma }_A$,

is CPTP and returns $X$ when applied to $\mathcal{N}(X)$. Finally,

so constancy on every matrix unit is equivalent to (eq:KL).

This is the exact code-recovery condition in the Knill–Laflamme framework and the constant-complement interface reported in Em05-B KnillLaflamme1997,CheungEm05B. Constancy of the complement on only the four preparations in (eq:squareensemble) is insufficient: those states do not span the ${\sigma }_y$ direction.

The following construction supplies a concrete, conserving completion rather than an unspecified perturbation ${\rho }_{\mathrm{thermal}}+\delta {\rho }_x$.

Consider two identical neutral scalar species in the same s-wave energy-eigenmode regulator, both with frequency $\omega >0$. They are additional declared field degrees of freedom in this phenomenological example, not a claim that such particles have been observed. There is no imposed conserved flavor charge. Every radiation configuration below has zero angular momentum and zero electric charge.

The stationary outgoing mean occupation of either mode is

For $0<q<1$, the two-mode thermal state is

Its total radiation energy in this regulator is $n\omega$ on the $n$-particle sector.

A normalized time-localized packet is not an exact energy eigenmode. Exact conservation in this section refers to the energy-eigenmode regulator. Transferring the construction to finite-time packets requires an independently bounded spectral-width and dynamics error; no sharp-frequency, sharp-time identification is made.

Fix a finite integer $L\ge 0$ with

and define

This is the source state of a finite-energy completion, not a data-selection rule. All attempts in that model are retained.

Because the discarded thermal sectors are orthogonal,

Unbounded observables require their own moment audit; trace distance alone does not control energy or number expectation. Here that audit is explicit:

For each $n$, pair successive occupation vectors:

Both codewords have the same exact energy $n\omega$. For even $n$, the single unpaired vector is $|n,n⟩$. Set

and define the transferable weight

Thus

Let $r=(r_1,r_2)\in \{0,1{\}}^2$, and write

A stored early system $E$ and an internal key $K$ are initially in the fixed state

The key sectors are degenerate. A common early-system energy may be added to every term and does not affect the construction. The resource (eq:earlykey) is independent of $X$ and is explicitly priced as four orthogonal early states. Its preparation is an additional assumption; it is not derived from the Hawking spectrum. A shared classical key would suffice for the reduced channels below, so this construction is not offered as a theorem that specifically requires Hayden–Preskill entanglement.

The residual source contains orthogonal states $|b_{n,k}⟩$ of energy $M-n\omega$, and orthogonal two-dimensional residual subspaces with isometries $I_n$ for the unpaired even sectors, also at energy $M-n\omega$. Their existence is part of the declared source-sector dimension ledger.

On the prepared input subspace define

The residual subspaces used by distinct terms are orthogonal. $E$ is untouched. The common internal message energy has been included in the initial source energy; its logical Hamiltonian is zero.

The map $V_1$ is an isometry on the prepared subspace. Every term has total energy $M$, and the residual mass obeys the cutoff (eq:energycut). With input (eq:earlykey) and with $BK$ inaccessible, its accessible channel is

For every operator $X$,

There is a second energy-conserving isometry $V_0$, on a sufficiently large but otherwise identical declared residual sector space, whose accessible channel is

The two completions have the same entire late-radiation state, the same early marginal, and the same emitted-energy distribution for every input. Their assisted delayed-query scores differ whenever $p_L(q)>0$.

Proof. Each paired map $J_{n,k}{P}_r$ and each residual map $I_n$ is an isometry on the message. Orthogonality of the residual labels and (eq:weightscomplete) give ${V_1}^{†}{V}_1=I$. Distinct key inputs remain orthogonal. Every term carries radiation energy $n\omega$ and residual energy $M-n\omega$, so energy is conserved. The radiation is neutral and s-wave, and the stipulated residual labels conserve the declared charge and angular momentum.

Apply $V_1$ to the fixed $EK$ resource and trace the orthogonal residual sectors and $K$. This gives (eq:transferchannel). The Pauli twirl satisfies

Since $p_{n,k}/2=d_n$, the twirled paired sectors and unpaired terms assemble exactly into ${\sigma }_L(q)$. Tracing $R$ gives the uniform early marginal.

For the second completion, choose mutually orthogonal residual two-dimensional embeddings ${I_{n,j}}^{(0)}$ of energy $M-n\omega$, and set