Energy-Resolved Radiation and Resource-Limited Recovery Exact Memory Benchmarks, Loss-Closed Emission, and Explicit Streaming Protocols

We separate information availability from the resources needed to execute recovery in specified finite quantum-radiation models. An energy-labelled three-packet Clifford channel has unrestricted optimal half-diamond recovery error $3(1-p)/4$, where $p$ is the full-transmission probability. Under atomic single-pass delivery of $q$-qubit packets, its even-family optimum at capacity $q$ is $3/4-p/4$, whereas capacity $q+1$ attains unrestricted recovery. A controlled-$Z$ modification has a matching $q$-slot decoder. The reference state is explicitly locally Clifford-equivalent to the Smolin state; its unlocking mechanism is inherited.

We prove a sector-feasibility criterion for energy-preserving isometries and its stronger counterpart for state-independent prescribed probabilities. A repeated emission model has a unique terminal core, coherent vacuum opportunities, internal unitaries, and a loss-closed multiparticle packet space. Its absorption tail is bounded by a binomial tail, and its mean absorption time is at most $K/r_{\ast }$. An explicit collect-and-invert protocol has a complete-channel recovery bound with an event-wise memory ledger. Five ordinary emissions give a prepared-state entropy sequence $0,1,2,2,1,0$, without a terminal drain.

A localized-message transport family gives exact unrestricted recovery and analytically evaluated streaming channels. For inter-emission entangling dynamics, we specify coherent simulation, four complete decoder programmes, numerical semidefinite-program validation, and failure-preserving archival procedures. This computational supplement is not reported as executed: no numerical dynamics result, passing archive, solver certificate, or compilation record is asserted. The proved results are finite-model statements, not a derivation of Hawking radiation, generic scrambling, or gravitational islands.

A spectrum does not specify a quantum channel. Occupation statistics leave phases, temporal correlations, remnant states, and logical encoding undetermined. An entropy curve likewise does not determine a decoder or its execution resources.

Hayden–Preskill retrieval assumes rapidly mixing unitary dynamics and extensive access to radiation HP. Random-unitary evaporation models already distinguish internal and environment-coupling timescales and investigate information retrieval Piroli. Memory minimization is a global multistep optimization problem, even when classical memory is free Bisio.

Our mathematical contributions are:

1. exact transmission-dependent recovery and atomic-delivery memory optima for a finite benchmark;

2. realizability constraints for prescribed emission probabilities;

3. loss-compatible repeated emission, quantitative absorption, and explicit complete recovery programmes.

The underlying benchmark state is locally Clifford-equivalent to Smolin's unlockable state Smolin. We therefore claim neither a new unlockable state nor a new joint-measurement unlocking mechanism. Our additional statements concern complete-channel error, propagation loss, and a declared delivery interface.

Operator-spreading diagnostics have a separate role Nahum; they are not recovery certificates. Clifford components admit stabilizer simulation AaronsonGottesman, but arbitrary emission amplitudes and non-Clifford internal gates do not make the complete process a stabilizer circuit. Pure-loss purification provides relevant propagation and error-detection context Winnel, without establishing the finite deterministic recovery results proved here.

This revision repairs the computational specification by:

• distinguishing S0's final direct return from earlier measure-and-prepare paths;

• testing the entire loss-closed scattering domain with nonuniform constituent energies;

• adding a coherent multiparticle collect-and-invert regression;

• defining inter-emission entangling circuits and complete streaming programmes with priced occupancy;

• including recovery and independent channel-distance SDPs;

• retaining running and failed archives instead of writing an unearned passing manifest.

limitation. The supplement has not been executed in preparing this revision. There is no supplied passing audit, executed dynamics dashboard, SDP result, or compilation log. Consequently the manuscript's analytical statements and its unexecuted computational specification are explicitly separated. The requested minimum standard for an executed dynamics study remains pending.

We use $\hslash =c=k_B=1$, with base-two entropy. Energy conservation refers to the displayed finite Hamiltonians. Equal-energy rails are orthogonal scalar modes, not particle polarizations. Rotational covariance, an autonomous clock, apparatus work, detector absorption, and a field-to-qubit transducer are not derived.

Reflections remain inaccessible and do not return to the core. No companion manuscript is a proof premise. The finite models do not establish semiclassical evaporation, a typical Page curve, or a gravitational island formula.

All spaces are finite dimensional. A channel is completely positive and trace preserving (CPTP). We use

including external references WatrousBook.

For the accessible radiation channel ${\mathcal{N}}_t$, define

For a declared horizon, packets arrive once, in order. Each packet atomically occupies its entire declared binary width.

A passive supplier may buffer undelivered packets but may not measure, demultiplex, select, or decode them. The observer may pause delivery but may not revisit consumed packets, return intermediate quantum data to the supplier, use uncounted storage, share auxiliary entanglement, or export quantum output before consuming every horizon packet.

Classical memory is unrestricted. Every laboratory qubit is destructively measured or returned as the final output.

The class ${\mathfrak{P}}_N$ consists of finite deterministic protocols satisfying this assumption and $Q_{\mathrm{peak}}\le N$. Lower bounds permit arbitrary instruments. $Q_{\mathrm{peak}}$ counts delivered inputs, work, quantum records, retained data, and output. $Q_{\mathrm{stored},\mathrm{total}}$ additionally counts supplier buffers. The inaccessible core, reflected environment, and correctness reference are not observer storage.

A retained output is live memory. The no-early-export rule can make operational error nonmonotone in the mandatory delivery horizon.

The exact benchmark library contains $H,S,S^{†},X,Y,Z,\mathrm{CX},\mathrm{CZ}$, zero allocation, destructive computational readout, classical control, and return. A classically selected Pauli costs at most one gate. Parallel register readout takes one instruction round. Allocation, transfer, and return are instantaneous in this convention.

Preparing $I_2/2$ uses one recyclable slot and at most two gates: generate a random bit by $H$ and destructive readout, then reallocate and conditionally apply $X$.

The message occupies $Q_2$ among five qubits. Let

and define, rightmost factor first,

There are thirteen preparation gates. Put $F={\mathrm{CZ}}_{1,2}$ and $V_L=F^{L}V$. Only $L$'s parity matters.

For ${\mathcal{A}}_L(A)={\mathrm{Tr}}_{45}(V_{L}A{V}_L^{†})$,

For $\rho =(I+xX+yY+zZ)/2$,

where

The linear extension replaces $1,x,y,z$ by $\mathrm{tr}A,\mathrm{tr}(AX),\mathrm{tr}(AY),\mathrm{tr}(AZ)$.

Proof. Gate conjugation gives

The identity $Y=iXZ$ gives the signed $Y$ word. Conjugating by $F$, using $F{X}_{1}F=X_1{Z}_2$, $F{X}_{2}F=Z_1{X}_2$, and $F{Z}_{i}F=Z_i$, gives the odd row. The displayed channel normal form proves the expansion.

Every proper even-family subset is constant. The only nonconstant proper odd-family subsets are $\{2,3\}$ and $\{1,3\}$, carrying $X$ and $Z$, respectively. For example, the former is $\rho \mapsto (I-x{X}_2{Z}_3)/4$, an entanglement-breaking measure-and-prepare channel. Both chronological prefixes $\{1\}$, $\{1,2\}$ are constant. Applying $U{F}^L$ and discarding positions $1,3$ recovers the message from the full triple.

For even parity, the normalized Bell-input state is

Let $R_0=H{S}^{†}$ and $T=Y_R\otimes R_{0,1}\otimes I_2\otimes R_{0,3}$. Then

where $\beta$ runs over the four Bell states.

Proof. The right side is $(I+X^{\otimes 4}+Y^{\otimes 4}+Z^{\otimes 4})/16$. Conjugation by $R_0$ cycles $X\mapsto Y\mapsto Z\mapsto X$. The two rail rotations give signs $-,+,-$. Conjugation by $Y_R$ reverses the first and third signs.

Choose weights $w_0,\dots ,w_s$ summing to one and frequencies ${\omega }_x>0$. A raw packet has vacuum and occupied states $|x,a⟩$, $a\in \{0,1\}$, with energies $0,{\omega }_x$. Its declared representation has a label and one rail:

Fix $E_{\mathrm{in}}>3{\max }_x{\omega }_x$. Inject

Branch zero emits vacuum and retains all five qubits. Branch $x>0$ extracts $Q_1,Q_2,Q_3$ in successive packets. After $j$ extractions its core energy is $E_{\mathrm{in}}-j{\omega }_x$; branch zero retains energy $E_{\mathrm{in}}$. These maps are isometries on their declared domains because branches are orthogonal and extraction transfers a tensor factor. Their output energies equal their input energies.

Propagation is the rail-independent isometry

with vacuum mapped to two vacua. Thus $H_C+H_D+H_F=E_{\mathrm{in}}I$ on the evolved input space.

For $S\subseteq \{1,2,3\}$, set

Tracing the branch-labelled core and reflected positions gives orthogonal accessible blocks $(x,S)$ for nonempty $S$. Their rail channels are the corresponding reductions of ${\mathcal{A}}_L$. All empty surviving sets form one constant vacuum block. In particular, all flag weights are message-independent.

Define

and

Decoded constant and entanglement-breaking qubit channels have half-diamond error at least $3/4$ and $1/2$, respectively. For a Pauli channel $\Lambda ={\sum }_P{\lambda }_P{\mathrm{Ad}}_P$,

Proof. A constant channel's Bell-input output has Bell overlap $1/4$. A separable two-qubit state has Bell overlap at most $1/2$. Testing the Bell projector gives the lower bounds. For a Pauli channel the Bell errors are orthogonal, giving $1-{\lambda }_I$; the triangle inequality gives the matching upper bound.

For either parity, ${\epsilon }_{\mathrm{IT}}(t)=3/4$ for $t=0,1,2$. At horizon three,

Under Assumption ass:interface,

and

Proof. Constant early prefixes give the first claim. Flagwise Bell witnesses give the unrestricted lower bounds. The complete channels

attain them.

For the even $q$-slot lower bound, the first packet must leave only classical information before the second atomic arrival. A positive effect on its occupied rail has the form $E=(\mathrm{tr}E)(I+v_{x}X+v_{y}Y+v_{z}Z)/2$, $\left|v\right|\le 1$. The conditional reference–future state is

where $A=X_2{Z}_3$, $B=Y_2{X}_3$, $C=Z_2{Y}_3$, and $AB=-C$. For joint projectors $P_{\alpha \beta }=(I+\alpha A)(I+\beta B)/4$, it equals

This is separable. Subsequent operations preserve separability. Label-mixing effects do not avoid the argument because spectral blocks are already incoherent. Decoded Bell overlap is therefore at most $p/2+(1-p)/4$.

Measuring $Z_1,X_2,Z_3$ and preparing the logical $X$ eigenstate of their negative parity on full transmission attains $p{\Delta }_X+(1-p)\mathcal{T}$. The remaining upper bounds follow from the compiled programmes below.

Set

The even programme consumes packet labels, holds the first rail until the second arrival, applies $B_{12}$, and reads the early pair as $t,s$. It consumes both early rails before the third arrival. On full transmission it applies $C_{ts}{R}_3$ and returns rail three; otherwise it consumes the remaining inputs and prepares $I/2$.

Indeed,

The corrected path maps intertwine logical $X,Z$ with output $X,Z$, hence have the form $|\psi ⟩\mapsto |\psi {⟩}_3\otimes |{\chi }_{ts}{⟩}_{45}$. The early pair is $I_4/4$, giving path probability $1/4$.

For odd parity,

Separate early $Y_1$ and $X_2$ measurements suffice. The $\{2,3\}$ and $\{1,3\}$ failure flags use product parity readouts and logical eigenstate preparation.

The ledgers are

They include failure preparation. For a three-packet snapshot, $Q_{\mathrm{stored},\mathrm{total}}=3q$.

Every destructive readout is part of a complete instrument. If $K_h$ are the full instruction-history maps, then ${\sum }_h{K}_h^{†}{K}_h=I$. Thus successful branches are never renormalized into a deterministic claim.

Entropy calculations use input $|0⟩$; recovery uses a Bell reference with unchanged dynamics.

For $t\le 3$, define

Let ${\mathsf{f}}_t$ contain $v_t$ and the nonempty flag weights. For even parity the rail entropy is one for a singleton and two for a pair or triple. For odd parity the exception is $s_L(\{1,3\})=1$. Then

This follows from the orthogonal-block entropy identity. For $a=1-w_0$,

Propagation preserves entropy of the complete emitted region.

For the Bell experiment let $K_t=C_t{F}_{\le t}$ and

A constant qubit channel has a complementary Bell state isometric to ${\Phi }_{RE}\otimes \tau$, so its distance is $3/4$. This follows from a dilation $|i⟩\mapsto {\sum }_j\sqrt{{\lambda }_j}|j⟩|i,j⟩$ of a fixed output ${\sum }_j{\lambda }_j|j⟩⟨j|$.

Consequently the even benchmark has

Complementary full-transmission sectors are constant; failure sectors complement constant accessible channels. The vacuum sector is retained coherently as one sector.

This equality is special to this channel, not a general identification of Bell-input decoupling with worst-case recovery.

Let $C={⨁}_E{C}_E$. Outcome $\nu$ has radiation energy ${ϵ}_{\nu }$ and degeneracy $g_{\nu }$. Write

An energy-preserving isometry exists if and only if

for every $E$. A specified map is isometric precisely when

Prescribed probabilities $p_{\nu |E}$, independent of every state in $C_E$, are realizable precisely when they are nonnegative, sum to one, and

Their branch effects must equal $p_{\nu |E}{I}_{C_E}$.

Proof. The total-energy output subspace must contain $d_E$ orthogonal vectors, and this dimension condition is sufficient. Different total energies are orthogonal. Orthogonal radiation outcomes give the isometry equation.

State-independent expectations imply the scalar effect identity by polarization. A positive scalar identity has rank $d_E$. Conversely, choose an isometry into each sufficiently large branch and multiply it by $\sqrt{p_{\nu |E}}$.

Total dimension can suffice while individual branches are too small for a fixed probability law. A coarse count law and a law of individual rail configurations are different requirements.

Fix $K\ge 2$, $1\le c\le K$, and

The core is

Sector zero is the unique endpoint. Mode pair $r$ has energy ${\Omega }_r$. There is at most one quantum per retained mode pair.

Emission of $k\le \min (n,c)$ constituents occupies $I_{n,k}=\{n-k+1,\dots ,n\}$. Its energy is $E_n-E_{n-k}$.

Define the deduplicated occupation family

The loss-closed packet space is

with rail bits ordered by decreasing mode index and energy ${\sum }_{r\in S}{\Omega }_r$. A configuration is identified once, regardless of its hidden origin.

The label count and physical dimension are

For a span of length $d+1$, both endpoints are present; interior indices have two occupation choices or three physical choices. This proves the formulas.

The declared representation uses

The label is the surviving set; its actual rail bits are followed by zero padding. For stage A, $c=1$, $q_A=\lceil {\log }_2(K+1)\rceil +1$.

For rail-independent coefficients,

is an isometry on all of ${\mathcal{H}}_{\mathrm{loss}}$ and preserves energy.

Proof. Different assignments are orthogonal and their squared amplitudes sum to one. Different inputs are distinguished by the union of occupied output modes and rail bits. Both subsets remain in ${\mathcal{S}}_{K,c}$, and their energies add to the input energy.

For example, loss from $\{3,2,1\}$ can produce $\{3,1\}$. No emitted-interval tag survives for free. Accessible vacuum can mean no emission, complete reflection, or an already terminal core.

At opportunity $j$, specify $P_{j,n}(k)$ summing to one and a unitary $U_{j,n}$. Let $T_{n,k}$ transfer the first $k$ current factors to the ordered mode interval $n,\dots ,n-k+1$. Set

Each $W_j$ is isometric and energy preserving. Its coarse branch effect is $P_{j,n}(k)I_n$, for every rail state and reference. With propagation,

on the evolved initial sector.

Proof. Each extraction is isometric, distinct counts are orthogonal, and their effects sum to the identity. The branch dimension is $2^{n-k}{2}^k=2^n$. Its energy is $E_{n-k}+E_n-E_{n-k}=E_n$. Different sectors have different total energies. Compose with Proposition prop:scatter.

Arbitrary rail gates preserve this Hamiltonian because each sector is degenerate. This is not an excitation-number conservation claim for another Hamiltonian.

For initial sector $K$, let ${\pi }_t(n)=\mathrm{tr}({\Pi }_n{\rho }_{C_t})$, $s_t=1-{\pi }_t(0)$. Then

Branch effects are scalar and propagation does not change the core reduction. Distinct core energies have distinct emitted complementary energies, eliminating inter-sector coherences. Thus this recursion is independent of internal gates and loss.

If $1-P_{j,n}(0)\ge r_{\ast }>0$ for every nonterminal sector and opportunity, then

Terms $i>t$ are zero; for $r_{\ast }=1$, the tail is $1_{\{t<K\}}$. Absorption occurs almost surely, not necessarily by a deterministic finite horizon.

Proof. Couple the population recursion to independent uniform variables. A value at most $r_{\ast }$ forces a sector decrease of at least one. The waiting time for $K$ Bernoulli-$r_{\ast }$ successes dominates absorption. Its tail is the displayed binomial tail and its mean is the sum of $K$ geometric means $1/r_{\ast }$. This auxiliary probability construction does not measure the coherent quantum history.

For time-homogeneous stage A, $r_n=P_n(1)>0$,

For the synthetic law $r_n=(2^n+1{)}^{-1}$, $K=5$, the uniform bound is $165$ opportunities and the exact mean is $67$.

For a pure input and ${\mathcal{R}}_t=D_{\le t}{F}_{\le t}$,

The endpoint block is one-dimensional and the complement has dimension at most $2^{K+1}-2$. Also

Permanent inaccessible loss need not give zero accessible entropy.

For $K=5,c=1$, deterministic emission, initial state $V_L|0⟩$, and extraction of $Q_1,\dots ,Q_5$ in order, apply ${\mathrm{CZ}}_{4,5}$ between emissions three and four. At perfect transmission,

The third prefix retains Theorem thm:benchmark with width $q_A$ and $p={\gamma }_{1,5}{\gamma }_{2,4}{\gamma }_{3,3}$.

Proof. The first three channels are the benchmark channels with known descending mode labels. The remaining Bell-partner reduction is $I_4/4$, unchanged by the controlled $Z$. After the fourth extraction it is $I_2/2$; after the fifth the core is one-dimensional. No exceptional drain operation is used.

Initialize by a known isometry $J:M\to ({\mathbb{C}}^2{)}^{\otimes K}$ with known unitary preparation.

Full simulator histories include hidden reflections. Accessible histories contain only observed surviving sets and observer measurement outcomes.

Let $P_{\mathrm{all}}(t)$ be the probability that every one of the $K$ actual constituents has been emitted and transmitted. For complete no-loss histories $h$,

Distinct complete no-loss histories have orthogonal accessible occupation configurations.

A deterministic protocol has

and complete decoded channel

Thus its achieved error is $3(1-P_{\mathrm{all}}(t))/4$. At perfect transmission $P_{\mathrm{all}}(t)=1-s_t$.

Proof. Measure each surviving-set label, consume padding, and retain only actual transmitted rails. Each mode is emitted at most once. Collecting $K$ rails therefore certifies completion and no loss. Only then do observed vacua certify no-emission opportunities.

The complete history is reconstructible. Its collected map is $\sqrt{{\alpha }_h}{B}_{h}J$, with a known unitary $B_h$ on $K$ factors. Apply ${B_h}^{†}$, undo initialization, and consume the $K-1$ ancillary factors. Every corrected success is the identity channel. On incomplete collections, consume everything and prepare $I/2$.

Before an arrival there are at most $K-1$ undecoded rails or one decoded output. The atomic arrival adds $q_{\mathrm{loss}}$ slots. Label and padding consumption leave at most $K$ real rails. The inverse uses these existing slots. Early completion leaves one live output through subsequent arrivals; $q_{\mathrm{loss}}+1\le q_{\mathrm{loss}}+K-1$.

The total success effect is a scalar identity, so fixed-output failure has input-independent total probability. This proves the complete channel, including references.

Specified continuous gates and their inverses may be admitted as primitives. Discrete-library compilation requires an additional error and workspace budget. This theorem is an achieved bound, not a memory optimum.

Use stage A, identity inter-emission gates, transmission $\gamma$, and initialization

An initial sequence of $\ell$ adjacent swaps moves the message to extraction position $d=K-\ell$. Its mode is $n_{\ast }=K-d+1$. This is transport, not scrambling.

Let $T_d$ be its emission time, $a_t=\mathrm{Pr}(T_d\le t)$, $b_t=\mathrm{Pr}(T_d=t)$. The population recursion gives

Unrestricted recovery and a one-retained-qubit programme satisfy

A $q_A$-slot programme S0 implements

and has achieved error

No stochastic $q_A$-slot optimum is asserted.

Proof. If the message mode is transmitted by the horizon, it carries the message exactly. Otherwise the message remains in a different core sector or a reflected message mode, and the accessible channel is constant. These success and failure subspaces are distinguished in the complement, so their cross terms vanish. Bell witnesses give the lower bound.

S1 retains the observed message rail through later arrivals. S0 measures an earlier-arriving message rail in $Z$ and consumes it, retaining a classical outcome. If the message arrives at the final opportunity, S0 returns it directly without a logical measurement. An earlier record causes eigenstate preparation after the last packet; no record causes preparation of $I/2$. The stated complete channel follows.

For deterministic emissions,

The last lower bound follows because mandatory later delivery forces consumption of the earlier message qubit.

For constant $r$,

For pure input $|0⟩$, all rails remain zero, so $S(C_t)=S(D_{\le t}{F}_{\le t})=H({\pi }_t)$.

At $K=t=3,r=1/2,\gamma =3/4$, the following values are analytical, not solver outputs:

The common sector vector is $(1,3,3,1)/8$, ordered $3,2,1,0$. Its entropy is $3-\frac{3}{4}{\log }_{2}3$.

warning. At two opportunities, “emit then vacuum” and “vacuum then emit” can share the same final core and reflected configuration. For constant $r$, perfect transmission, and zero rails, their temporal off-diagonal element is $r(1-r)$. A mixture of simulator trajectories would incorrectly remove it.

This section defines the unexecuted computational study precisely. It does not supply numerical conclusions.

Use stage A with constant $r$, localized initialization, and initially perfect transmission. At every opportunity and sector, apply $L$ alternating nearest-neighbour layers before the emission decision. Layer parity alternates with opportunity and layer index.

Each two-qubit gate is

with independent angles uniform on $[-\pi ,\pi )$. These are specified entangling gates, not Haar-$U(4)$ samples. The supplement records every angle and gate matrix. Depths are nested using the same preselected layer arrays.

Controls are identity dynamics and initial SWAP transport. The depth parameter in the new family changes actual inter-emission gates, not just initialization.

The proposed first pilot uses $K=3$, $L=0,1,2$, $r=1,1/2$, two declared gate realizations, and horizons $1,\dots ,K$. The $K=4$ extension is a separate command, not an executed result.

Measure every observed occupied rail in $Z$, consume it, and retain the latest readout bit. After all horizon packets, prepare that bit as a computational basis state. If no occupied rail was observed, prepare $I/2$. The laboratory peak is at most $q_A$. This is a fixed achieved baseline, not the optimum of ${\mathfrak{P}}_{q_A}$.

Retain the first observed occupied rail as memory. For every later occupied rail $a$, apply ${\mathrm{CX}}_{b\to a}$, then $H_a$, measure $a$ as $z$, consume it, and apply ${Z_b}^z$. Vacuum dummy rails are consumed. Return $b$ after the horizon; if no memory exists, prepare $I/2$. The peak is at most $q_A+1$.

Retain the first two occupied rails. For each later occupied rail $a$, apply ${\mathrm{CX}}_{b_1\to a}$, ${\mathrm{CZ}}_{b_2,a}$, and $H_a$; measure and consume $a$, applying ${X_{b_1}}^z$. At the horizon, if two rails remain, apply ${\mathrm{CX}}_{b_1\to b_2}$, $H_{b_2}$, measure and consume $b_2$, and apply ${Z_{b_1}}^z$. Return the remaining rail. One or zero retained rails are handled by direct return or $I/2$ preparation. The peak is at most $q_A+2$.

Use Theorem thm:collect. At perfect transmission its achieved error is $3s_t/4$, with peak at most $q_A+K-1$. It is a correctness reference, not generally an optimal batch decoder.

The specified programmes are deterministic CPTP maps with the stated laboratory peaks.

Proof. Occupation-label readout determines actual occupied rails. No supplier-side operation or hidden emission record is used. At most $m=0,1,2$ memory qubits coexist with the next atomic packet. Every gate acts on existing memory and an existing incoming rail. Every measurement has both outcomes and its declared feedback. Padding and consumed rails are destructively removed. Finalization leaves exactly one output, or allocates one recyclable slot for fixed-output preparation. Instrument completeness proves trace preservation and includes all references.

No ordering of achieved errors across these heuristic programmes is assumed. More permitted memory cannot worsen an optimum, but different fixed heuristic programmes need not be monotone.

Use output-first unnormalized Choi matrices

For a decoder $R=\mathsf{J}(\mathcal{D})$,

For logical dimension $d$, let

Entanglement fidelity is $\mathrm{tr}(C_{\mathcal{N}}R)$. Its recovery SDP and dual are

Strict feasibility holds on both sides. This is fixed-noise fidelity recovery Fletcher. A verified dual value $U$ implies

by a normalized Bell-projector test. It is not diamond optimization.

For a difference $\Delta$ of CPTP channels, the SDP

has dual

Their common value is ${\Vert \Delta \Vert }_{\diamond }/2$ WatrousSDP.

Unrestricted diamond recovery is therefore

This affine finite-dimensional SDP is not a bounded-memory strategy optimization.

The implementation first checks identity, constant, dephasing, and the flagged even benchmark against analytical optima. For every new channel it records:

• raw recovery variables and affine/positivity residuals;

• a positive trace-preserving decoder obtained by explicit Choi normalization;

• an independent fixed-channel diamond primal and dual;

• a fidelity-dual lower-bound candidate;

• solver settings, statuses, and computed gaps.

Numerical positivity checks and scalar repairs are not interval certificates. Values remain numerical estimates unless stronger verification is supplied. An optimizer's status string alone is insufficient.

For each realized circuit and horizon, the supplement is designed to record

the exact finite energy expectations, unrestricted recovery estimates, complete A–D channels, and resource histories.

The predeclared closed-core spreading diagnostic is

Set $C_0=C_1=0$. The dashboard uses ${\sum }_n{\pi }_{t-1}(n)C_n(U_{t,n})$. This is an opportunity-local closed-core diagnostic, not a cumulative open-system OTOC or recovery certificate.

Opportunity, actual emitted count, and accumulated internal layers are distinct clocks. Recording the distribution of emitted counts does not grant the decoder access to hidden counts. Replotting against their mean is not conditioning on a physical emission-count trajectory.

Ensemble summaries must average per-realization quantities. Entropy or recovery of an averaged channel is a different object.

status. No dashboard values for the new entangling study are supplied. No conclusion about depth, memory improvement, or recovery coincidence is prewritten. The analytical transport table is the only results table above.

For compatible complete-channel errors,

For a common packet embedding and common feasible protocol class, the optimized errors differ by at most $e_{\mathrm{model}}$. Both statements follow from triangle inequalities and CPTP contraction.

For common isometry embeddings,

For changed branch probabilities,