Hawk02-E Coherent Diaries and Complete-Residual Certification in a Regulated Interacting SYK–Scalar Model

We specify a fixed-realization interacting SYK source coupled to a continuum of free scalar reservoir modes with declared infrared and ultraviolet cutoffs. A normalized Euclidean-filtered pure source state is prepared before a logical diary arrives. A single message-independent, parity-preserving swap loads an even-parity qubit diary into two source modes. The displaced carrier is retained in the complete residual, including its correlations with the remaining source.

We construct the self-adjoint continuum Hamiltonian, determine the complete code isometry, and derive a reference-preserving finite approximation for every prescribed finite packet algebra and finite observation interval. If $G$ is an explicit interaction bound, $m$ is a total reservoir occupation cutoff, and $\Delta$ is a frequency-mesh bound, the approximation allowance is

The finite approximation supplies all logical matrix units and both channel outputs. A complementary Choi-state certificate ${\hat{e}}_J$ gives

with the code dimension factor retained. Independently, a specified finite CPTP decoder can be certified directly on its normalized Choi operator. We prove the exact microscopic energy ledger, an obstruction for commuting source couplings, and continuity requirements for an information endpoint.

The manuscript establishes a coherent microscopic interface and a quantitative certification procedure. It does not exhibit a small dynamical residual certificate for the interacting model, derive a gravitational geometry from that model, or establish a gravitational evaporation endpoint. Those missing results are identified separately rather than assumed inside a purported evaporation theorem.

{{0.93} Claim boundary. The proved results concern one regulated interacting microscopic model and its complete channels. Recovery during controlled gravitational evaporation requires an additional small dynamical certificate and a gravitational correspondence on the same time interval. Complete evaporation requires an additional information endpoint and a gravitational endpoint. None of these conclusions is supplied as an assumption disguised as a proof. }

The relevant question is not whether an engineered source can transfer a quantum register to a reservoir. It is whether one independently specified microscopic evolution supports all of the following:

These are separate obligations, and their validity intervals must overlap.

The supplied Cheung manuscripts maintain the distinctions necessary for this question. Hawk01-B separates entropy histories from complete code preservation. Hawk02-A separates a recovery certificate from a microscopic gravitational transfer law. Hawk02-C and Hawk02-D provide autonomous benchmarks without identifying their sources with black holes CheungHawk01B,CheungHawk02A,CheungHawk02C,CheungHawk02D. Hawk03-A distinguishes an algebra encoded in radiation from an additional independent interior factor CheungHawk03A. The operational manuscripts distinguish observable readout, subsystem recovery, coherent collection, decoder execution, and complete-record error accounting CheungEn02B,CheungEn02C,CheungEn02E,CheungEm03C.

The present contribution advances the microscopic interface beyond the quadratic benchmark: the source Hamiltonian is an interacting quartic SYK Hamiltonian. The reservoir approximation is proved for its actual unitary dilation, including all logical coherences and the complete residual. The construction does not turn a quadratic survival formula into an interacting recovery estimate.

Chen, Qi, and Zhang analyze a SYK system coupled to Majorana chains. Their preparation is thermofield double, and their retrieval test uses a classical control bit ChenQiZhang. That is a relevant architecture, but not the initially pure, reference-entangled diary task adopted here.

Gaikwad, Kaushal, Mandal, and Wadia study a SYK source coupled to scalar fields, with a low-energy influence-functional description, partial-evaporation solutions, and horizonless solutions GaikwadEtAl. Their bath preparation, disorder averaging, couplings, and switching prescriptions are substantive parts of that model. In particular, their Model (b) uses a prescribed late-time reduction of the coupling to preserve the low-energy description. Their reconstruction of initial spin labels uses repeated experiments with different couplings. It is not identified here with single-run recovery of an arbitrary reference-entangled diary.

We therefore choose a regulated scalar-reservoir architecture, but do not import the geometry of GaikwadEtAl. The vacuum preparation and finite-band regulator below are explicit differences. A comparison with that paper would have to control these differences rather than suppress them in notation.

Pure Euclidean-filtered SYK states motivate the preparation KourkoulouMaldacena. The SYK low-energy reparametrization sector motivates a prospective nearly-${\mathrm{AdS}}_2$ description MaldacenaStanford. Neither result supplies a uniform gravitational dictionary for the inserted code defined below.

Recovery maps and gravitational reconstruction are already available in important settings NakayamaMiyataUgajin,PSSY. The JT-plus-bath analysis of Almheiri, Engelhardt, Marolf, and Maxfield also supplies an explicit evaporating gravitational setting AEMM. The missing claim in this manuscript is narrower: those results are not a certified channel correspondence for the microscopic preparation, interaction, packet access, and complete residual specified here.

Cheung references identify the supplied manuscript codes and versions. Their use here is documentary and comparative. No proof depends on an unprovided theorem from those manuscripts.

External mathematical inputs are stated explicitly. Model-specific constructions and estimates are proved below. No exhaustive novelty claim is made.

The supplied OpenAI catalogue contains proposed mathematical ingredients, not verified premises of this paper SuppliedCatalogue. Its spectral-gap, area-law, geometric, and entropy claims are not used to establish SYK recovery or gravitational evaporation.

We use $\hslash =1$. The code $C$ has finite dimension $d$. The main microscopic example has $d=2$.

For states and channels, write

Channel comparisons retain arbitrary external references. Finite-dimensional input permits a reference of dimension $d$ for channel discrimination Watrous.

For an isometry

define the accessible channel and complete complementary channel by

The replacement channel for a state ${\sigma }_B$ is

The two operational quantities are

The source is prepared in a declared pure state before the diary arrives. The reservoir is initially independent of the diary. The diary may be entangled with an untouched reference $Q$. No early radiation or purification of the source is supplied as a decoding resource.

The displaced diary carrier is not discarded. After insertion, it can be correlated with the source. It remains in $B$ at every later time.

access. The first mathematical task permits coherent batch access to a declared finite packet subsystem. It does not assume an efficient decoder or a compact local receiver. Any physical receiver requires its own reference-preserving channel comparison.

Let $N=2n\ge 4$, and let ${\chi }_1,\dots ,{\chi }_N$ be Majorana operators satisfying

They act on

This normalization is fixed throughout; conversion to another SYK normalization must rescale the couplings explicitly.

Freeze a real array $\mathbf{J}=(J_{ijkl}{)}_{i<j<k<l}$. Set

Products of four distinct Majoranas are self-adjoint and have norm one. Consequently,

The added constant fixes an energy reference. It is not a gravitational mass normalization.

No disorder average is taken. Theorems below hold for each frozen array. They do not imply that every array has a useful recovery regime or a gravitational interpretation.

Define commuting source spins

For a fixed vector $s^{\prime }\in \{-1,1{\}}^n$, set

Then $O_{s^{\prime }}$ is even, self-adjoint, and has norm at most one.

Fix real coefficients $\alpha ,\zeta$, and define

Thus

This polynomial coupling is independently specified before channel certification. It is not selected after seeing a favorable decoder.

For notational flexibility, the analytical estimates also permit a fixed finite family of even self-adjoint operators $A_a$. The model used for the scalar comparison may take all $A_a=A$.

Fix

The reservoir one-particle space and Hamiltonian are

On bosonic Fock space,

The vacuum is denoted ${\Omega }_B$.

For species $a$, define

where ${\mathbf{e}}_a$ is the species basis vector. Let

with $a(f)$ conjugate-linear in $f$. Then

The $1/\sqrt{2\omega }$ factor specifies a regulated scalar-mode coupling. Its normalization is a model input, not an inferred bulk-to-boundary correspondence.

Freeze real $g_1,\dots ,g_r$, and define

Introduce the explicit constants

For the common operator (eq:coupling-A), these satisfy

and the corresponding bound for $G_{\omega }$ follows from (eq:f-weighted).

The estimates below take finite observation time and refine a frequency discretization and a particle cutoff at fixed $\mu ,\Lambda$. They do not prove the limits $\mu \downarrow 0$ or $\Lambda \uparrow \infty$. Indeed the explicit constants in (eq:f-norm)–(eq:f-weighted) diverge in those limits. Failure of these estimates to be uniform is not a theorem that every renormalized or alternative infrared construction is impossible.

We use the standard Kato–Rellich theorem: a symmetric perturbation with arbitrarily small relative operator bound preserves self-adjointness on the original domain. The same field-operator estimates underlie standard spin–boson constructions HaslerHerbst.

The operator (eq:H) is self-adjoint on ${\mathcal{H}}_S\otimes D(H_B)$, bounded below, and generates the unitary group $U(t)=e^{-\mathrm{i}tH}$. As quadratic forms,

Proof. The standard annihilation estimate, proved by Cauchy–Schwarz on each particle sector, is

The creation identity

therefore gives

For every $b>0$, the spectral theorem bounds $\Vert {H_B}^{1/2}\psi \Vert$ by $b\Vert H_B\psi \Vert +c_b\Vert \psi \Vert$. Thus $V$ has relative operator bound zero. The bounded source term does not change the domain, so Kato–Rellich establishes self-adjointness.

For a normalized vector in the form domain,

Since $H_S\ge 0$, this proves (eq:lower-H). The unitary group follows from self-adjointness.

The carrier introduced next is tensored with the identity during this evolution. A formal infinite Hamiltonian sum is not needed.

Choose a normalized simultaneous spin eigenvector

with a declared total fermion parity. For $\beta >0$, define

This is the Euclidean-filtered preparation motivated by KourkoulouMaldacena. Its exact finite-realization properties used here are elementary.

The state (eq:KM-prep) is well defined and has the same parity as $|B_s⟩$. Its normalization obeys

If $H_S={\sum }_{E}E{P}_E$, its energy probabilities are

Moreover,

Proof. The spectral bounds $0\le H_S\le M_{N}I$ give the normalization inequalities and positivity of $Z_s$. The quartic Hamiltonian is even, so its exponential preserves parity. The spectral decomposition gives $p_E$. Differentiating the finite-dimensional spectral sum proves the mean and variance formulas.

A preparation instrument may use the contraction $K_{\mathrm{succ}}=e^{-\beta H_S/2}$ and the complementary operator

Its success probability on $|B_s⟩$ is $Z_s(\beta )$. The preparation occurs before the diary is supplied. The task may start after a declared successful preparation; an attempted deterministic protocol must instead retain the failure flag and include its preparation error. No input-dependent postselection is used.

No finite-$N$, fixed-realization thermal or gravitational correspondence is inferred from Proposition prop:preparation.

Introduce a carrier $D$ containing two fermionic modes. Encode the logical qubit by

Both logical basis vectors have even parity. The logical reference $Q$ is an abstract quantum register.

Choose two source modes $T$ before studying recovery. Let $U_{\mathrm{sw}}$ be the second quantization of the one-particle unitary exchanging those source modes with the carrier modes. It is number conserving and parity preserving.

Define the insertion isometry

Fix the fermionic occupation ordering once and for all.

The map $W$ is an isometry implemented by one message-independent unitary. It preserves every logical matrix unit and arbitrary reference entanglement.

After insertion, the carrier marginal is independent of the diary, but the carrier can be correlated with the remaining source. It must therefore remain in the complete residual.

If the carrier Hamiltonian is degenerate and preparation interactions are absent before and after the swap, the change in mean source energy satisfies

for every logical density operator $\rho$.

Proof. Composition of the isometry $J_D$, a fixed pure ancillary preparation, and a unitary proves $W^{†}W=I_C$. Linearity gives the action on all matrix units; tensoring the unitary with $I_Q$ proves the reference statement.

Exchanging the entire two-mode subsystems transfers the old source two-mode marginal to the carrier. Its correlations with the remaining source are also transferred, rather than erased. Fermionic permutation signs are fixed by the mode ordering and the even logical code; they do not create a message-dependent operation.

Both source energy expectations lie in $[0,M_N]$, proving the final inequality.

In Hawk02-D the destination source was vacuum, and the carrier became a fixed vacuum. That conclusion does not apply to the Euclidean-filtered source here. The carrier is generally not vacuum and is not removable from the residual ledger. Initial carrier independence alone does not establish its later joint independence.

The energy bound is deliberately crude. It does not prove that the insertion remains within a gravitational low-energy window. An infalling interpretation and sharper backreaction control remain additional obligations.

Fix an orthonormal packet family $p_1,\dots ,p_q$ in ${\mathfrak{h}}_{\mu ,\Lambda }$, independently of any recovery optimization. Put

The bosonic Fock factorization is

Define

The complete code isometry is

where $\mathcal{R}$ is the fixed regrouping into $R\otimes B$.

The vacuum reservoir has no inaccessible preparation purifier. If a mixed reservoir preparation is substituted, its purification must be added to $B$; that is a different input specification.

For ${\Xi }_i(t)=V_t|i⟩$,

These formulas determine every reference-entangled input.

Let $E_0(\rho )=\mathrm{Tr}(H_{S}W\rho W^{†})$. The initial interaction mean is zero because the reservoir is vacuum.

For every logical input, define

Then

Uniformly in the logical input and time,

The source current, with positive sign for loss from the source, is

Consequently,

No sign is asserted for either current at every time.

Proof. The initially prepared vectors belong to $D(H)$. Autonomous unitary evolution preserves the total Hamiltonian expectation. The initial bath and interaction means vanish, giving (eq:energy-conservation).

The form inequality (eq:lower-H) gives

which proves (eq:bath-energy-bound).

The particle-sector bounds established in Section sec:regulator also show that these prepared evolved vectors have finite moments of every fixed particle-number order on bounded time intervals. Since $\omega$ is bounded on the one-particle band, the commutator expectations can be obtained by finite-particle approximation. Differentiation gives ${\dot{E}}_S=\mathrm{i}⟨[H,H_S]⟩=\mathrm{i}⟨[V,H_S]⟩$. Differentiating (eq:energy-conservation) proves (eq:interaction-energy).

Source energy loss and reservoir energy gain differ by the interaction-energy derivative. A source-only mass-loss formula requires an additional approximation or a declared dressing convention. Equation (eq:energy-conservation) supplies neither a positive outward stress tensor nor a gravitational metric.

For a prescribed differentiable switching schedule $g_a(t)$, a separate time-dependent model has the work term

A switch used in a gravitational comparison must retain this work. It is not an autonomous endpoint operation.

This section proves the finite-channel approximation for the interacting model itself. It does not assume source relaxation, Markovian reduced dynamics, or a small complementary channel.

Let $N_B=\mathrm{d}\Gamma (I)$, and let $P_m$ project onto reservoir sectors with at most $m$ particles.

In the interaction picture with respect to $H_S+H_B$, write

where

These transformations preserve the norms in (eq:G).

For $x\ge 0$, define

This series is finite for every finite $x$.

For an initial vector consisting of an arbitrary normalized source–carrier state and the reservoir vacuum, the interaction picture Dyson term of order $\ell$ has norm at most

The expansion converges in norm, uniformly on bounded time intervals and uniformly over normalized code-reference inputs.

For the complete prepared isometry,

where $P_m$ acts on the reservoir before regrouping.

Proof. On sectors containing at most $k$ particles,

An $\ell$-fold product acting on vacuum is therefore bounded by $(2G{)}^{\ell }\sqrt{\ell !}$. The ordered integration simplex has volume $t^{\ell }/\ell !$, giving the claimed term bound.

The ratio of consecutive scalar bounds is $2Gt/\sqrt{\ell +1}$, so the series converges. The same proof holds with an untouched reference and carrier. Each term is supported in sectors with at most $\ell$ particles.

For completeness, multiplication of the scalar term bound by any fixed power of $\ell$ still gives a convergent series. Thus the Dyson series converges in every fixed particle-number graph norm. With the finite frequency band and bounded source operators, the integral equation and its derivative hold on the prepared vectors. The resulting Schrödinger-picture solution is the unique solution generated by the self-adjoint Hamiltonian of Theorem thm:selfadjoint. This identifies the expansion with the actual evolution.

Terms of order $\ell \le m$ vanish under $I-P_m$. Summing the remaining bounds proves (eq:occupation-tail).

On $P_m({\mathcal{H}}_S\otimes {\mathcal{H}}_B)$, set

It is bounded and self-adjoint, since the reservoir energy is at most $m\Lambda$ there and $\Vert P_{m}V{P}_m\Vert \le 2G\sqrt{m+1}$. Let ${V_t}^{(m)}$ be its complete prepared isometry, embedded into the original output space.

For $0\le t\le T$,

Proof. The free evolution preserves particle number. In interaction picture, the exact and truncated Dyson terms acting on vacuum coincide through order $m$: no intermediate state at those orders can exceed $m$ particles.

At every higher order, both terms satisfy the bound of Lemma lem:dyson; inserted projections are contractions. The difference is therefore bounded by twice the tail series. Returning to Schrödinger picture does not change the norm.

Partition $[\mu ,\Lambda ]$ into finitely many intervals $I_b$ of width at most $\Delta$. Let ${\omega }_b$ be their midpoints, and define

Then

Let $\mathfrak{f}$ be the finite-dimensional space spanned by

for every bin, species, and declared packet. Zero vectors are omitted and the remaining vectors are orthonormalized.

This space contains the coupling form factors and the entire packet space $\mathfrak{r}$. It is invariant under ${\tilde{h}}_B$. All modes in ${\mathfrak{f}}^{\perp }$ remain vacuum in the discretized evolution. Thus restriction to ${\mathcal{F}}_{\mathrm{s}}(\mathfrak{f})$ is exact for that discretized model.

Let

on the active finite-mode space. Its Hilbert space has finite dimension. If $p=\dim \mathfrak{f}$, the reservoir dimension is

The source and carrier dimensions are included separately.

For the occupation-truncated prepared isometries,

Proof. On sectors with at most $m$ particles,

The source and interaction terms are unchanged. Duhamel's formula for bounded self-adjoint operators gives the stated unitary difference. The invariant active-mode space realizes the discretized prepared evolution exactly. Restriction to the code preserves the bound.

If $V_1,V_2:C\to R\otimes B$ are isometries into a common output space, then

The same bound holds for the complete output channels.

Proof. For a pure code-reference input, the two output vectors differ in norm by at most $\Vert V_1-V_2\Vert$. Their pure-state trace distance is no greater than that norm. Convexity handles mixed inputs, and partial trace is contractive. Taking the reference-input supremum proves the result.

At fixed $\mu ,\Lambda$, for the preparation, interaction, and packet access specified above,

bounds the half-diamond distance between exact and finite complete output channels uniformly for $0\le t\le T$. It also bounds the distances between their accessible channels and between their complete complementary channels, after the specified common embeddings.

For every $T<\infty$ and $b>0$, finite $m$ and a finite mesh can be selected so that $b_{\mathrm{reg}}\le b$.

Proof. Combine Lemmas lem:occupation-dynamics, lem:frequency, and lem:isometry. The finite output has the same packet space as the continuum output. The displaced carrier and all uncollected active modes are retained; unused orthogonal modes are fixed vacuum. Thus both partial traces refer to the same declared partition.

Convergence follows because $Z_m(2GT)\to 0$ as $m\to \infty$, followed by $\Delta \to 0$.

Suppose $T>0$, $0<b<1$, and $x=2GT$. Choose an integer $m\ge 1$ such that

and choose the mesh so that

Then $b_{\mathrm{reg}}(T;m,\Delta )\le b$.

Proof. For $\ell \ge m+1$, consecutive terms of (eq:Z-tail) have ratio at most $1/2$. Also,

Thus $Z_m(x)\le 2^{-m}$, and $2Z_m(x)\le b/2$. The mesh term is also at most $b/2$.

The frequency bins are a spectral discretization, not finite spatial boxes. The theorem makes no locality or detector-support claim. It proves a reference-preserving channel approximation for the declared spectral packets.

The occupation cutoff is an analytical approximation to dynamics, not a physical process that destroys excess particles.

Let

For a channel $\Phi$, define the normalized Choi operator

For channels $\Phi ,\Psi$ with input dimension $d$,

The output may be infinite dimensional provided the channel outputs are normal states.

Proof. A pure input with a $d$-dimensional reference can be written as

Its channel-output difference is

Its trace norm is at most $d$ times the Choi trace norm because ${\Vert L\Vert }^2\le d$. Convexity and the finite-reference characterization of channel diamond distance complete the proof.

The information–disturbance theorem supplies

in half-diamond normalization KSW. This is an external channel theorem. It does not estimate ${\delta }_{\mathrm{full}}$ from the interacting Hamiltonian.

Fix $t\le T$, and compute the finite complementary channel ${\hat{\mathcal{C}}}_t$ from ${\widehat{V_t}}^{(m,\Delta )}$. Let ${\hat{\sigma }}_B$ be any rigorously specified finite residual density operator and put

Embed ${\hat{\sigma }}_B$ into the complete residual space, with the unused modes in vacuum. Then

If a validated calculation provides ${\hat{e}}_J(t)\le e_{\mathrm{cert}}(t)$, that certified upper bound may replace ${\hat{e}}_J(t)$.

Proof. Theorem thm:regulator gives

The triangle inequality and Lemma lem:choi give

Taking the replacement-state infimum proves (eq:residual-certificate). Equation (eq:KSW) proves (eq:recovery-certificate).

The regulator theorem already bounds the complementary channel in diamond distance. Its allowance is therefore added after the Choi conversion. Replacing it by a Choi-state estimate and multiplying it by $d$ would be an unnecessary weakening.

Theorem thm:certificate is a proved implication with a specified finite computation. It does not exhibit a time at which $e_{\mathrm{cert}}$ is small. No such numerical or analytical certificate has been supplied for this interacting model. The manuscript consequently does not claim an actual interacting recovery window.

For $d=2$, use the logical Pauli matrices $X,Y,Z$.

Choose ${\sigma }_B={\mathcal{C}}_t(I_C/2)$. Then

Hence

The same statement holds for the finite approximation.

Proof. The Bell projector has the expansion

Subtracting $(I_Q/2)\otimes {\mathcal{C}}_t(I_C/2)$ removes the identity term. Each Pauli matrix has trace norm two. The triangle inequality gives the first estimate, and Lemma lem:choi gives the second.

Both $X$ and $Y$ involve off-diagonal logical matrix units. A calculation limited to two diagonal preparations does not verify this test.

The finite complementary Choi operator can be used in the optimization

This is a finite semidefinite program. Its value is the optimized normalized Choi trace distance, not automatically the optimized diamond distance.

Any rigorously feasible tuple supplies an upper certificate. An exploratory floating-point optimum, an unverified positive semidefiniteness test, or a solver residual is not a proof of feasibility.

A direct decoder certificate is also available.

Let $\hat{\mathcal{D}}$ be a rigorously CPTP decoder on the finite packet output space, and define

Complete $\hat{\mathcal{D}}$ to the full packet Fock space by

where $P_R$ projects onto the finite packet occupation space. Then

Proof. Compression by $P_R$ is completely positive. The second term is a positive-effect measurement followed by state preparation. Their traces add to $\mathrm{Tr}Y$, so the extension is CPTP.

The approximate channel is supported on $P_R$. Lemma lem:choi gives its decoded error $d\hat{r}(t)$. Postprocessing contraction and Theorem thm:regulator give the additional $b_{\mathrm{reg}}$.

For decoder search, its unnormalized Choi operator $J_D$ must obey

Composition with the known finite channel is linear in $J_D$; the trace-norm objective can be expressed using positive and negative parts as in (eq:complement-SDP). A complete feasible decoder, not only a successful branch, is the required output.

candidate. For a finite channel $\hat{\mathcal{N}}$, a full-rank code state $\tau$, and $P=\mathrm{supp}\hat{\mathcal{N}}(\tau )$, a candidate is

Inverse powers are taken on $P$. The first term is completely positive and has trace $\mathrm{Tr}(PY)$; the completion therefore makes the map CPTP. Writing it down does not prove a small recovery error. Proposition prop:decoder supplies a direct test. This agrees with the distinction emphasized in the supplied Hawk01-B comparison CheungHawk01B.

Suppose an implemented channel and decoder satisfy the declared comparisons

in half-diamond distance at their respective interfaces. Then a finite direct decoder certificate gives

capped at one. A decoder supplied by the residual theorem instead replaces $d\hat{r}(t)+b_{\mathrm{reg}}$ by the bound in (eq:recovery-certificate).

Proof. Insert the intermediate channels one at a time. Use the triangle inequality and contraction of diamond distance under CPTP pre- and postprocessing.

Loss already included in a complete complementary channel is not added again as an independent collection allowance. A separate term is appropriate only for a separate proved comparison. This avoids the double-counting problem noted in Hawk02-C CheungHawk02C.

The reservoir covariance is exactly

This follows directly from the canonical commutation relations. It supplies a microscopic Gaussian influence kernel, in the sense of the influence-functional framework FeynmanVernon. It is not yet a gravitational effective action.

For real integrable histories $j_a(s)$ on a finite interval, consider the interaction-picture bath Hamiltonian

Its propagator has the form

where

is the Weyl unitary and ${\theta }_j(t)$ is real. For two histories,

Proof. The commutator of $H_j(s)$ and $H_j(u)$ is a scalar multiple of the identity:

Consequently chronological Weyl multiplication yields one Weyl displacement and a real common phase. The displacement follows by integrating the linear generator. The vacuum characteristic function is

The Weyl product phase has unit modulus, proving the overlap formula.

Replacing the quantum source operators $A_a(s)$ by prescribed classical histories is an additional approximation. When they do not commute, their time ordering and both logical branches must be retained. A scalar-history saddle or a diagonal reduced density operator is not the complete channel (eq:matrix-units).

Suppose

for every pair of couplings. For any insertion isometry $W$, the accessible reservoir channel is entanglement breaking. Every deterministic decoder therefore satisfies

For a qubit diary, the lower bound is $1/2$.

Proof. Choose a common source eigenbasis, with spectral projectors $P_{\lambda }$. The complete propagator is block diagonal:

Tracing the source removes terms between different projectors. Tracing the carrier does not restore them. The accessible channel therefore has measure-and-prepare form

where

are positive effects summing to $I_C$.

Such a channel breaks entanglement with every reference. A decoder preserves that property. A separable state on $Q\otimes C$ has overlap at most $1/d$ with $|{\Phi }_d⟩$: this is immediate for pure product states by Cauchy–Schwarz and extends by convexity. Testing the Bell projector gives trace distance at least $1-1/d$, hence the stated diamond lower bound.

Noncommutativity removes this particular obstruction; it does not prove recovery. The theorem generalizes the negative-control role of the effective single-observable scalar source discussed in the supplied Em03-F CheungEm03F. The noncommuting exterior construction in Em03-G is a different model and is not used as a decoder for the present Hamiltonian CheungEm03G.

An averaged channel can be constant while every realization retains distinguishable logical information.

Proof. For a qubit, let

Their equal mixture is the replacement channel $\mathrm{Tr}(\rho )I/2$. For each realization, the outputs of $|0⟩$ and $|1⟩$ are orthogonal. By the triangle inequality, its distance from any replacement channel is at least $1/2$. Thus averaging has erased information in the description, not proved its absence in each realization.

Suppose a declared disorder distribution and a measurable family of residual states satisfy

For $0<p<1$,

On the complementary event,

Any decoder dependence on $\mathbf{J}$, and any realization information needed to implement it, must be declared.

Proof. Apply Markov's inequality to the nonnegative random trace distance, and then apply Theorem thm:certificate realization by realization.

This proposition uses an expectation of distances. It does not use the distance of an averaged channel. A result uniform on a time interval requires a separate uniform estimate or a proved discretization and continuity argument.

A fixed source marginal does not imply complete residual decoupling.

Proof. The isometry