Hawk02-C An Autonomous Source–Field Bridge with Exact Recovery and Finite-Packet Certificates

We construct a time-independent, excitation-conserving source–boson Hamiltonian with a finite logical code and derive its complete emission isometry. The prepared one-excitation sector is a regular Friedrichs model with identical, orthogonal logical-label channels. Its survival amplitude and emitted wavefunction determine the accessible channel and its complete complement.

For collection in a prescribed finite packet subspace, let $q(T)$ be source survival and ${\ell }_N(T)$ the uncollected one-particle weight. Both optimal recovery error and complete residual leakage, in half-diamond distance, are exactly $(1-d^{-2})[q(T)+{\ell }_N(T)]$. An identified complete packet approximation has distance at most $\sqrt{2{\ell }_N(T)}$, proved from its isometries rather than from discarded probability alone.

A positive-frequency semicircle form factor gives a bound-state-free prepared spectral measure, an explicit survival law, a finite-time survival bound, and predetermined Chebyshev packets with a factorial collection-tail certificate. A finite autonomous chain supplies a separate predictive complete-output approximation. Checked invariant-sector perturbations give a Duhamel allowance. These estimates provide a nonempty finite-time regime with implemented recovery error at most any prescribed $\epsilon \in (0,1)$.

The supplied execution manifest reports completion of all 18 finite-proxy predicates at $d=2$, $c=1$, ${\omega }_0=3$, $T=1$, with eight collected packets and a 24-packet reference chain. Its reported recovery/leakage formula is $0.2494586279164458$. The record supports selected finite-matrix identities and norm comparisons; it is not continuum simulation, numerical diamond-norm optimization, or machine proof verification. An arithmetic-aware interface separates exact truncation bounds from floating-point execution evidence.

The bridge is an engineered source–field model, not a gravitational Hawking-emission mechanism, microscopic black-hole Hamiltonian, Page curve, or spacetime-emergence theorem.

The construction follows \[

autonomous Hamiltonian and preparation complete source–field isometry
prescribed packet collection recovery, residual, and implementation certificates.

The field channel is derived from the stated dynamics. It is not selected by occupation-spectrum agreement.

The one-excitation operator belongs to the Friedrichs family Friedrichs,FacchiLigaboLonigro,Lonigro. No new Friedrichs-model class or erasure-channel mechanism is claimed Erasure. The contribution is a complete Hamiltonian-to-channel application with an explicit positive-energy example, finite-packet estimates, identified output embeddings, and arbitrary-reference recovery constants.

The ideal source–field evolution is autonomous: its Hamiltonian is time independent. This does not make preparation, collection, or decoding autonomous laboratory operations.

We distinguish:

1. Structural equivalence: an endpoint isometry is represented by a finite emission interface after declared output identifications;

2. Finite access: a predetermined finite packet algebra captures the logical radiation mode with proved missed-weight and complete-output allowances;

3. Predictive approximation: a finite Hamiltonian chosen independently of the calculated endpoint approximates the full evolution with a proved finite-time remainder.

The first does not imply the second or third.

Hawk02-A records the physical-acceptance and spectrum separations CheungHawk02A ({ass:physical, cor:spectrum}). Hawk02-B provides an externally scheduled finite exchange model, not the autonomous Hamiltonian constructed here CheungHawk02B. Theorem thm:C7 supplies an endpoint comparison with that model and, separately, a predictive finite-chain approximation.

Repeated-interaction and weak-coupling theories provide other approximation frameworks AttalPautrat,DerezinskiDeRoeck. No limit theorem is substituted for the finite-time estimate proved below.

The Cheung references identify the versions and statement locators supplied with the research programme. The supplied corpus locator is d2c1e6b3, retrieval-3. It is not a verified public archive manifest or evidence of a complete search of the series.

Every model-specific hypothesis and proof required here is stated locally. A documentary comparison does not identify another manuscript's Hilbert spaces, Hamiltonian, normalization, or unresolved object with ours.

Em03-F constructs a switched exterior source–scalar channel CheungEm03F ({thm:unitary, thm:fieldchannel}). Its propagator is not imported as this autonomous evolution.

The new computational evidence consists of the supplied hawk02c_diagnostics_manifest.json and detailed summary summary (21).txt ExecutionManifest,ExecutionSummary. They report execution at 2026-10-10T17:12:51.677880+00:00.

This revision incorporates those records without claiming:

• an independent rerun or source-hash recomputation;

• direct inspection of the binary matrix artifact;

• numerical optimization of a diamond norm or decoder;

• machine verification of the proofs;

• compilation of this revision.

Hawk02-B's execution record remains a different run.

1. The supplied summary says “17 checks,” but its manifest contains 18: sixteen identity/normalization predicates and two norm-comparison predicates. All 18 are reported passed.

2. A rectangular isometry is not a unitary on its output space. The complete split isometry has orthonormal columns; it is not a square unitary matrix.

3. A floating-point residual printed as zero is not exact-arithmetic verification. The reported small residuals are retained as numerical evidence only.

4. The source-resolvent argument is made first for nonreal spectral parameters. The uncoupled positive-frequency continuum outside the interaction band is not incorrectly declared to belong to the resolvent set.

5. The autonomous chain represents the coupled cyclic subspace, not all uncoupled field modes.

6. The previously undefined preparation macro J is defined.

7. The geometric-tail quantity computed by the script is a majorant of the exponential-series tail, not the tail's exact sum.

8. A complete packet distance computed as $\sqrt{1-h^2}$ suffers cancellation when $h$ is near one. An algebraically equivalent stable formula is supplied. The archived execution values are not retroactively changed.

9. An exact truncation bound below floating-point roundoff is not a certified error bound for the computed matrices. A separate arithmetic allowance is required for that conclusion.

10. Smaller proxy survival than a continuum upper bound is consistency, not evidence that finite truncation always lowers survival. No such monotonicity is asserted.

11. The reported $T=1$ run is not the $T=40$ analytical small-error example. The former has substantial source survival and does not demonstrate recovery error below $1/100$.

12. Historical “unexecuted” wording in the archived version-1 script is provenance. Its reported execution status is described in Section sec:repro.

Set $\hslash =1$. The code ${\mathcal{H}}_C={\mathbb{C}}^d$, $d\ge 2$, is finite dimensional. Output Hilbert spaces may be infinite dimensional and separable. Channels are normal CPTP maps on trace-class operators. We use

An untouched reference of dimension $d$ suffices for discrimination of channels with this finite input. For the complete isometric channels below, the span of the two output ranges is finite dimensional, so the corresponding comparison reduces directly to finite-dimensional channel discrimination Watrous (Chapter 3).

Write

For an accessible channel $\mathcal{N}:C\to A$ and specified complete complement $\mathcal{C}:C\to B$, define

Explicit optimizers will be constructed. No general infinite-dimensional compactness claim is needed.

Every fixed mixed auxiliary is purified. Every inaccessible purifier, uncollected mode, controller register, detector environment, and discarded carrier belongs to the complete residual appropriate to the claim. A later physical operation enlarges that residual before a postoperation leakage claim is made.

This locally restates the complete-output interface of Hawk02-A CheungHawk02A ({def:completeoutput}). Separate residual marginals do not generally determine their joint channel.

Let $V,W:{\mathcal{H}}_C\to {\mathcal{H}}_A\otimes {\mathcal{H}}_B$ be isometries on the same input and identified complete output spaces. Then

The allowance also bounds either partial-trace channel.

If $V^{†}W=z{I}_C$, then

If $z\in [0,1]$ is real, then

Proof. For every amplified trace-class operator $X$,

Its trace norm is at most $2{\Vert V-W\Vert }_{\infty }{\Vert X\Vert }_1$. Divide by two and use partial-trace contractivity.

For any pure code–reference input $\psi$, the output vectors have overlap

Their trace distance is $\sqrt{1-|z{|}^2}$. Convexity gives the same upper bound for mixed inputs, and every pure input attains it. The finite-input channel-discrimination characterization therefore proves (eq:scalaroverlap).

Finally,

when $z$ is real.

This theorem compares the specified dilations. Stinespring continuity concerns the existence of suitably identified nearby dilations; it does not identify arbitrary preselected physical environments KSW.

On ${\mathbb{C}}^d$,

Proof. Let $\mathsf{X}$ be the cyclic shift and $\mathsf{Z}$ the phase operator in a basis indexed modulo $d$. Summation on matrix units gives

Separating the identity term proves the upper bound $1-d^{-2}$.

Let

For every state $\sigma$,

The Bell-projector test supplies the matching lower bound for every replacement state.

Let $J:C\to A$ and $K:C\to B$ be isometries whose ranges are orthogonal to fixed unit vectors $a_0$ and $b_0$. For $r\in [0,1]$, define

Its accessible channel and complement are

and

An optimal decoder and residual comparison state are

At a zero-weight endpoint, the unused embedding need not be defined.

Proof. Orthogonality gives the two partial traces. The decoder is normal, completely positive, and trace preserving, and

Lemma lem:replacement gives error $r{c}_d$.

For any decoder on a Bell input, the successful branch contributes Bell overlap at most $1-r$. The erased branch contributes exactly $r/d^2$. The Bell test proves recovery optimality.

For the stated residual state,

which gives the leakage upper bound.

For any alternative $\tau$, apply a CPTP map to the residual that keeps the $K$ range as a logical sector and maps its orthogonal complement to a single failure flag. Let $t$ be the candidate state's weight in the logical sector. On the Bell input, the effect equal to the logical Bell projector and zero on the failure sector has expectation difference

The effect equal to that Bell projector plus the entire failure sector has expectation difference

If $t\le r$, the first is at least $r{c}_d$. If $t\ge r$, the second is at least $r{c}_d$. Contractivity proves the lower bound for every $\tau$. Endpoints follow from the constant or isometric channels.

These constants are proved in the half-diamond convention. Their agreement with a Bell-input infidelity in this model does not identify those metrics in general. Approximate-correction results based on worst-case entanglement fidelity supply a broader setting, not replacement constants BenyOreshkov.

Let

and

Define

The field space is ${\mathcal{F}}_s(\mathfrak{h})$, with vacuum $\Omega$. Multiplication by $\nu$ is denoted $M_{\nu }$ on either the scalar or labeled one-particle space, according to context.

The labels are distinct orthogonal field channels with identical dispersion and coupling. This engineered symmetry is an explicit model assumption, not a consequence of a local gravitational scalar field.

Let $L_a=|0⟩⟨a{|}_M$ and ${a_a}^{†}(g)=a^{†}(g\otimes e_a)$. The inner product is conjugate linear in its first argument; annihilation is conjugate linear in its test. Assume

The Hamiltonian is formally

Its precise operator definition is sectorwise.

The prepared code isometry is

No switching function or driving pulse enters the ideal evolution.

Define

For $n\ge 1$, its $n$-excitation space is

The zero-excitation space is $\mathbb{C}(|0{⟩}_M\otimes \Omega )$.

Under (eq:regularity), the restriction $V_n$ is bounded and Hermitian, with

Each

is self-adjoint. The direct sum

is self-adjoint on

It defines $U(t)=e^{-\mathrm{i}tH}$, and every excitation sector reduces this group.

Moreover,

In particular, $H\ge 0$ if ${\omega }_0\ge {\kappa }_g$.

On the one-excitation sector,

with domain $\mathbb{C}\oplus D(M_{\nu })$.

Proof. On the $(n-1)$-particle sector,

For excited-source components ${\psi }_a$,

The annihilation block is its adjoint. The Hermitian off-diagonal block has the stated norm. The bounded self-adjoint perturbation theorem gives $H_n$ Kato. The direct sum is self-adjoint and has the stated domain. Finite sector sums of sector-domain vectors form an operator core.

For the lower bound, use finite-particle energy-form vectors and complete the square:

Here $a_a(\nu )\Psi$ denotes the usual annihilation-density map, interpreted in the energy-form integral. The condition $g/\sqrt{\nu }\in L^2$ makes the shifted integral and its cross terms well defined. The cross terms equal (eq:V), and ${\sum }_a{L}_a^{†}{L}_a=P_{\mathrm{ex}}$. The identity may first be checked on finite-particle vectors with one-particle tests supported away from zero and then extended in the form norm. It implies the claimed lower bound, which extends to the self-adjoint direct sum.

In ${\mathcal{H}}_1$, each source label couples only to the one-particle field of that label. The creation block sends the source vector to $g\otimes e_a$, giving (eq:friedrichs).

The autonomous total energy $H$ and total excitation number are conserved. In general $[H,H_0]\ne 0$: photons of frequency $\nu$ exchange with a source of bare energy ${\omega }_0$.

On the prepared code,

Thus the initial product state is not generally a total-energy eigenstate, although its initial bare energy is sharp.

No work-free preparation, autonomous collector, or energy-conserving decoder is constructed. The conserved total energy is not a black-hole mass ledger.

Define

The prepared vector belongs to $D(h_g)$, which is invariant under its unitary group. Thus the component equations hold strongly:

Variation of constants gives, as an $L^2$ identity,

Consequently,

The kernel is bounded and continuous. These are exact equations, not a Markov approximation.

For $f\in \mathfrak{k}$, write

Set

The prepared emission isometry is

It satisfies

With the entire field accessible and memory residual,

Their exact errors are

When $p(t)>0$, an optimal decoder compresses $J_t=p(t{)}^{-1/2}{L}_t$ and replaces the complement by $I_C/d$.

Proof. The label decomposition of Theorem thm:C2 gives (eq:WSF). Unitarity gives $|A{|}^2+{\Vert f\Vert }^2=1$, and label orthogonality gives the operator normalization.

The ground/excited memory sectors and vacuum/one-particle field sectors are orthogonal. The two partial traces therefore give the displayed channels on every operator and reference extension.

Absorb the phase of $A$ into the residual embedding when its weight is nonzero. Apply Lemma lem:erasure with $r=q(t)$. Its endpoint rules avoid division by zero.

The explicit-decoder and Bell-witness architecture agrees with the supplied comparison records in Hawk01-B and Hawk02-B CheungHawk01B ({thm:optima}) CheungHawk02B. No constant or leakage convention is imported.

For every code input, the exact expectation ledger is

The interaction-energy term is not discarded in the general model.

Fix a rank-$N$ orthogonal projection $P_N$ on $\mathfrak{k}$, $N\ge 1$, before the input is supplied. Use the same projection for every label:

The canonical factorization is

Vacuum factorizes, and a one-particle vector separates into its two projected components.

The ideal assignment is

This is declared subsystem access, not yet a local detector or compact spacetime apparatus.

At time $T$, define

Suppress $T$ temporarily.

Let

and define the joint unnormalized residual embedding

Set

With outputs ordered as $A_N\otimes B_N$,

Its accessible channel and complete complement are

Hence

For $p_N>0$, put $J_{N,T}={p_N}^{-1/2}{L_N}^{\mathrm{col}}$. An optimal decoder is

For $p_N=0$, use ${\mathcal{T}}_{I_C/d}$.

There is an identified comparison isometry ${\hat{W}}_N$ on the same complete spaces that retains $A(T)$, places all emitted weight $p(T)$ in a collected mode, and leaves missed modes vacant. Writing

its complete distance is

The collected-channel distance is exactly ${\ell }_N(T)$.

Proof. Factorization gives (eq:Wcol). The two branches are orthogonal on both output sides, and

The residual embedding is orthogonal to $b_0$. Partial traces give the channels. Lemma lem:erasure proves (eq:packeterrors) and the decoder formula. An optimal residual comparison state is

If $p_N>0$, choose

If $p_N=0<p$, choose any normalized collected mode and multiply it by $\sqrt{p}$. If $p=0$, set ${\hat{f}}_T=0$. Use (eq:WSF) with this wavefunction. The resulting ${\hat{W}}_N$ is an isometry, and

Theorem thm:C1 proves the exact complete distance. Also,

because $\sqrt{p(p-{\ell }_N)}\ge p-{\ell }_N$.

The comparison collected channel minus the actual collected channel is

with the comparison embedding used at a zero-success endpoint. The two output supports are orthogonal, so the half-diamond distance is ${\ell }_N$.

For $p>0$, define

Then $h_N=1-d_N$, and

These are exact algebraic identities. They avoid subtracting nearly equal numbers in $1-{h_N}^2$. For $p=0$, both distances are zero.

The two terms in (eq:residualembedding) are coherent parts of one message-bearing residual branch. Tracing missed modes gives memory leakage $c_{d}q(T)$; tracing memory gives missed-field leakage $c_d{\ell }_N(T)$. The joint result is proved directly, not inferred from those marginals.

In general, individual residual marginals can be constant while their joint channel carries information. For a classical bit $a$, output a uniform random bit $k$ and $k\oplus a$ to two registers. Both marginals are uniform, but the joint output reveals $a$. Prepending code dephasing turns this into a channel counterexample to marginal-based joint constancy. This illustrates the supplied Hawk02-A warning CheungHawk02A ({prop:jointresidual}).

Later collectors and decoders can add residual systems. Equation (eq:packeterrors) applies precisely to (eq:AB), before such additions.

Flagging the $|1⟩$ component of $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ produces

In the common three-dimensional space, its trace distance from $|+⟩⟨+|$ is

The removed coherence prevents equality with discarded probability.

En02-E's exact cutoff equality uses its commuting Gaussian-state hypothesis CheungEn02E ({thm:cutoff}). It is not imported here. Em03-F's filtered-test packet bridge is likewise not a complete-channel embedding CheungEm03F ({thm:packetbridge}). Equation (eq:packetcomplete) is proved from the actual identified isometries.

For recovery, packet loss is already included exactly in (eq:packeterrors); its square-root complete comparison allowance is not added again to that ideal term.

Fix $c>0$ and ${\omega }_0\ge 3c$. Define

and

Its support is the positive band $I_c=[{\omega }_0-2c,{\omega }_0+2c]$. This is a regular $L^2$ coupling, not a flat distributional coupling. No differentiability at the band edges is assumed.

We have

The full Hamiltonian is therefore nonnegative.

For (eq:semig), the prepared source spectral measure is exactly

It has no pure-point or singular-continuous component. In particular, the prepared code has no bound-state component.

The survival amplitude is

where

The value at $T=0$ is one. For every $T>0$,

The emitted wavefunction is (eq:fieldsolution).

Proof. For $z\in \mathbb{C}\setminus \mathbb{R}$, set

Substitution $x=2c\cos \theta$ and the elementary Cauchy integral evaluation give

with branch $s(z)\sim z-{\omega }_0$ at infinity. Equivalently,

The Schur complement is valid for nonreal $z$ and gives

Uniqueness of the measure from its Cauchy transform identifies the entire prepared spectral measure. This argument does not declare real frequencies in the uncoupled continuum to be resolvent points. It proves the absence of hidden atoms and singular components in the prepared measure.

The spectral theorem gives (eq:exactA). The even moments are

and the odd moments vanish. Taylor expansion on the compact interval proves the series in (eq:bessel), which is the standard $J_1$ series DLMF (Section 10.2).

The zero extension of $w_c$ is absolutely continuous, vanishes at the endpoints, and satisfies

Integration by parts gives $|a_c(T)|\le 2/(\pi cT)$. Combine this with unitarity.

This explicit calculation specializes the regular Friedrichs spectral method. Broader spectral, resonance, and bound-state issues are treated in FacchiLigaboLonigro,Lonigro.

The exact survival law is not exponential:

It need not be monotone. An autonomous trajectory is therefore not identified with all prefixes of a monotone collision schedule.

Here $|f_T({\omega }_0+x){|}^2$ is even in $x$. The symmetry gives

These are expectation identities on the prepared trajectory, not the operator identity $[H,H_0]=0$.

Let $U_n$ denote Chebyshev polynomials of the second kind:

Define

Using $U_n(\cos \theta )=\sin ((n+1)\theta )/\sin \theta$ gives

Polynomial density in the weighted space, followed by multiplication by $\sqrt{w_c}$, proves completeness in $L^2(I_c,\mathrm{d}\nu )$. The band-edge zeros have measure zero.

Choose

This family is fixed by the Hamiltonian parameters, not by the unknown input or observed outcomes.

For the prescribed packets and $z=2cT$,

Proof. Writing $x=\nu -{\omega }_0$ gives

Replace the exponential by its degree-$(N-1)$ Taylor polynomial in $x$. The result belongs to $\mathrm{span}\{g(\nu )x^k:0\le k<N\}=P_N\mathfrak{k}$.

For real $y$,

Using $|a_c(s)|\le 1$, $|x|\le 2c$, and $\Vert g\Vert =c$, the norm error is at most

Orthogonal projection minimizes norm error. The other bound is ${\ell }_N\le {\Vert f_T\Vert }^2=1-q(T)$.

The argument differentiates neither $g$ nor its band-edge behavior.

For another regular coupling, an energy $E$ outside the coupled support may solve

When the vector is square integrable, an eigenvector is proportional to

with prepared spectral weight

These formulas follow directly from the eigenvalue equations. In-band eigenvectors require their own integrability conditions.

An atom does not imply $q(T)\ge {Z_E}^2$ at every time, because other spectral components can interfere. A valid uniform statement is: if an atom has weight $Z>1/2$, then

so all-field recovery error is at least $c_d(2Z-1{)}^2$ for every $T$.

If the remaining measure has an $L^1$ absolutely continuous density, its Fourier transform tends to zero. The long-time average of $q(T)$ is the sum of squares of the distinct atomic weights. For infinitely many atoms, finite-sum approximation in total mass justifies passage to the limit.

Theorem thm:C5 avoids these obstructions for its prepared code only. It does not assert decay for every regular coupling. For example, $g=0$ gives $q(T)=1$ and error $c_d$ for all $T$.

The field interaction is unbounded on full Fock space. The following comparison is made on its specified one-excitation sector.

Let $E_1(t)$ be bounded, Hermitian, strongly measurable, and satisfy

The ideal autonomous part is the self-adjoint $H_1$ on its stated domain. A required full physical propagator must separately exist and restrict to the specified sector dynamics.

A directly checkable class is

with bounded Hermitian diagonal blocks and bounded $B:{\mathbb{C}}^d\to \mathfrak{h}$. It includes source detuning, bounded dispersion changes, label mixing, and regular coupling changes, and obeys

The bounded interaction-picture construction defines a unique unitary sector propagator. For

it satisfies