Em03-E Microscopic Input Dependence and Certified Recovery in a Dynamical Radiation Model Specified emission dynamics, semiclassical matching, complementary channels, and operational access

We specify and solve a finite sequential radiation model in which a qubit message is encoded in a degenerate source-energy sector. Explicit bounded Hamiltonians randomize the message with an independent two-bit key, emit the key into a first cell, and transfer the encrypted qubit into a second cell with a tunable probability. A separately tunable dephasing interaction supplies a complete coherence-loss control. Every inaccessible system, including the key purification and the dephasing environment, is retained in the complementary channel.

The full radiation channel, its action on all message matrix units, and an explicit decoder are calculated. Each individual emitted cell is exactly independent of the message, while the collective output can transmit the message perfectly. For transfer probability $p$ and coherence parameter $\eta$, the optimal worst-case recovery error is $(3-p-2p\eta )/4$, and the optimal maximally entangled test fidelity is $(1+p+2p\eta )/4$. The distance of a named complementary channel from a replacement channel is evaluated exactly.

Independent finite occupation ladders are emitted by additional energy-preserving Hamiltonians. Their amplitudes are calibrated to clipped geometric packet laws from the Em03-A–C interface. We prove the occupation-state, count-law, and mean comparisons, while distinguishing those comparisons from agreement of the full information-bearing radiation state. An explicit finite coherent access construction and bounded-generator implementation estimates give end-to-end recovery and destructive-record certificates. Entropy, reference mutual information, and recovery fidelity are reported separately.

The construction is a designed finite dynamical information-transfer model. It does not derive the interactions from quantum gravity, identify the cells with a local Schwarzschild evaporation process, or resolve horizon-smoothness and evaporation-endpoint questions.

The construction follows the chain

The decisive distinction from Em03-C is that alternative radiation states are not supplied independently. The radiation channel is calculated from the Hamiltonians and preparations printed below.

Three levels of conclusion remain separate:

1. Finite dynamics: a message-dependent channel generated by a declared sequence of bounded interactions;

2. Semiclassical comparison: agreement or controlled disagreement with a named family of Em03-A–C occupation observables;

3. Black-hole identification: derivation of those interactions and degrees of freedom from a specified gravitational theory.

The first level is proved. A restricted comparison at the second level is proved. The third level is not claimed.

The information-transfer mechanism is the familiar Pauli randomization mechanism of a private quantum channel PrivateChannel. It is not presented as a new encryption scheme. The contribution is its explicit finite emission Hamiltonians, complete channel and complement calculations, exact recovery benchmarks, occupation calibration, and operational error accounting in one specified model.

General information–disturbance and approximate-correction theorems provide a broader setting KSW,BenyOreshkov. We do not import their constants into a different metric. The recovery and leakage constants needed here are proved directly. Hayden–Preskill models likewise retain unitarity, mixing, and radiation-access assumptions HaydenPreskill; none of those assumptions is derived by the present decoder.

. The supplied Em03-A, Em03-B, and Em03-C sources Em03A,Em03B,Em03C supply the stated semiclassical and operational comparison interfaces. No manuscript registry or companion retrieval account is a premise of the finite dynamics. In particular, no Em03-D source is assumed.

boundary. All results are analytical. Compilation, formal verification, numerical pulse execution, calibration, and experimental realization are not asserted. “Certified” means that an explicitly stated comparison follows from the printed mathematical hypotheses.

. All model Hilbert spaces are finite dimensional. We use

The diamond norm retains arbitrary external references. We write

Entropy is in bits. The binary entropy is

with $0{\log }_{2}0=0$. Channel and instrument conventions are standard Watrous; the particular formulas used below are proved in this manuscript.

The message $M$ is a qubit. Its external reference $R$ is not controlled by the source or receiver.

The source contains:

Here $Q$ is the source message qubit, $F_1,F_2$ are fuel qubits, $K_a,K_b$ are key qubits, and $A_1,A_2$ are finite occupation-source registers.

Choose integers $J_i\ge 1$, positive energy spacings ${{\omega }_i}^{\mathrm{cell}}>0$, and source energies

The register $A_i$ has basis

Its Hamiltonian is

The initial and the $n=0$ daughter states are distinct degenerate states.

For a payload energy ${\epsilon }_{\mathrm{cell}}>0$, set

The source factors $Q,K_a,K_b$ are energetically degenerate.

The encoding is

Its image lies in the common energy eigenspace

Thus different messages do not begin with different source energy.

There are two inaccessible key-purification qubits $C_a,C_b$, initially in $|00⟩$, and a dephasing qubit $Z_{\mathrm{e}}$, initially in $|0⟩$. Their free Hamiltonians are zero.

All initial environments are independent of $RM$. The key purification is not discarded from the complementary channel.

Each emitted cell has an occupation register and a payload register:

The occupation spaces are

The first payload space has basis

and the second has basis

Their free Hamiltonians are

All cells begin in

The occupied payload labels are degenerate internal states. They are additional model degrees of freedom, not unmentioned scalar-field packet modes.

The occupation registers are finite ladders. No canonical commutation relation is assigned to their boundary. Payload raising maps such as $|x_{\mathrm{o}}⟩⟨\mathrm{v}|$ are finite matrices, not continuum annihilation or creation distributions.

At the first emission the observer receives $r_1$; at the second emission the observer receives $r_1{r}_2$. Emitted cells do not interact again with the source.

The receiver never receives $B,C_a,C_b,Z_{\mathrm{e}}$. Unused cells and all source registers belong to the complementary output at the relevant prefix.

All formulas are in the interaction picture of

The finite energy spacings are model parameters. A positive-frequency wavepacket is not assumed to have the exact Hamiltonian ${{\omega }_i}^{\mathrm{cell}}{N}_i$.

For each operation choose a real smooth compactly supported pulse $g(t)$, with prescribed area

An interaction $g(t)G$, with fixed self-adjoint matrix $G$, has exact evolution $e^{-\mathrm{i}\alpha G}$. The operation intervals are disjoint unless explicitly stated to commute.

A normalized smooth pulse can be rescaled to any finite duration. For each chosen duration its amplitude and all its derivatives are finite. No universal hardware bandwidth is assumed.

Prepare the key purification by applying Hadamard gates to $K_a,K_b$, followed by

The resulting state is

Tracing $C=C_a{C}_b$ gives $I_K/4$.

These are actual finite controls. For example, Hadamard is generated exactly by a pulse of area $\pi$ multiplying $(I-H)/2$, and a controlled-NOT is generated by a pulse of area $\pi$ multiplying

These matrices act within zero-energy sectors.

Before encryption, apply

Define

On the initial environment this gives the isometry

The reduced source channel is

This is verified by the overlap $⟨e_1|e_0⟩=\eta$.

Put $P_{ab}=X^a{Z}^b$. Apply controlled $Z$ from $K_b$ to $Q$, then controlled $X$ from $K_a$ to $Q$. The resulting unitary is

Its generators are the two explicit projections

each with pulse area $\pi$.

The ordering fixes all phases. In particular, $P_{11}=XZ$ is not replaced silently by a differently phased Pauli matrix. Its conjugation action is nevertheless that of $Y$.

Define the partial isometry

Unprinted factors carry the identity. Use

On the prepared subspace,

Define

Use a pulse area $0\le \theta \le \pi /2$, and set

Then

The same source state $|0,0{⟩}_{F_{2}Q}$ appears for both successful message basis states. This is the coherence-preserving part of the emission.

Both $T_i$ are partial isometries with orthogonal initial and final subspaces, so $G_i$ are self-adjoint with norm one. Their action on every unpaired basis state is zero. The model therefore specifies the full Hamiltonians, not merely an isometry with an unspecified unitary extension.

For each $i$, choose a probability vector

Its semiclassical calibration is specified in Section sec:matching. Define

These are orthonormal. Set

The exact emission is

The generator is a norm-one finite matrix. Its terms, including the zero-occupation daughter term, are part of the specification.

Let $U_{\mathrm{int}}$ be the ordered key preparation, dephasing, and controlled-Pauli sequence. The first emission step applies key emission and occupation emission $i=1$. The second applies ciphertext emission and occupation emission $i=2$. Within each step the two printed emission interactions commute.

Write

The specified dynamics has a unique finite-dimensional unitary evolution. Its radiation prefixes are CPTP channels on all message operators:

A complementary channel is obtained by tracing the accessible prefix instead, retaining the whole source, the key purification, the dephasing environment, and unused cells.

Every printed interaction commutes with $H_0$. Consequently the expectation and full probability distribution of $H_0$ are conserved.

Proof. All generators are bounded matrices with smooth coefficients. Existence and uniqueness follow from the finite linear evolution equation; unitarity follows from self-adjointness. The pure-environment evolution composed with the encoding is an isometry. Partial trace gives CPTP channels and their complements.

The internal key and message controls act in degenerate sectors. For key emission, the source fuel decreases by ${\epsilon }_{\mathrm{cell}}$ while $C_1$ gains the same amount. For ciphertext emission the analogous equality holds for $F_2,C_2$; $Q$ is degenerate. Every term of occupation emission takes source energy $E_i$ to $E_i-n{{\omega }_i}^{\mathrm{cell}}$ and adds $n{{\omega }_i}^{\mathrm{cell}}$ in the cell. Thus every interaction commutes with $H_0$. The full evolution commutes with $H_0$, proving the conservation statement.

This is conservation of the declared free energy. It is not a thermodynamic theorem about switching work, reset cost, or the energy of a continuum wavepacket. Nor does it select the vectors $w_i$.

Define the independently emitted occupation densities

Let $J_{\mathrm{o}}:{\mathbb{C}}^2\to {\mathcal{H}}_{C_2}$ be $J_{\mathrm{o}}|x⟩=|x_{\mathrm{o}}⟩$. For a qubit operator $Y$, write

At prefix zero there is no accessible radiation factor. At prefix one,

At prefix two,

In particular, for $x=0,1$,

The other off-diagonal matrix unit is the adjoint of (eq:offdiagonalunit).

Proof. Occupation emission leaves the displayed diagonal densities after the daughter states are traced. These operations act on factors independent of the message.

Key preparation gives (eq:keypurification). The inaccessible key purification makes the radiation key diagonal with weight $1/4$ for every $a,b$. Tracing the still-internal message at prefix one gives (eq:N1).

After ciphertext emission, the successful and unsuccessful source fuel states are orthogonal. Their cross terms therefore vanish in the radiation channel. On success the source message has the common terminal state, leaving its full encrypted density in $C_2$. On failure $C_2$ is vacuum, and tracing the remaining source message contributes $\mathrm{tr}X$. Tracing $Z_{\mathrm{e}}$ gives ${\Lambda }_{\eta }$. This proves (eq:N2) on arbitrary operators. Equations (eq:diagonalunits)–(eq:offdiagonalunit) then follow directly.

Pauli averaging obeys

This follows by expanding $X$ in the Pauli basis: every nonidentity coefficient cancels.

For every $p,\eta$, each individual emitted cell is exactly independent of the message, including references:

Nevertheless, when $p=\eta =1$, the collective channel is exactly quantum-correctable.

Proof. The first marginal is unchanged by a later source interaction. The second follows from (eq:twirl), which applies to ${\Lambda }_{\eta }(X)$ because its trace is $\mathrm{tr}X$. The collective recovery assertion is proved in Theorem thm:recovery.

The private marginals use a mixed key whose purification is kept in the model. They are not a claim that an arbitrary pure encoding can hide a qubit in two pure shares with private marginals.

For computational-basis inputs,

For the states $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$,

Equal-prior optimal single-shot success probabilities are, respectively, $(1+p)/2$ and $(1+p\eta )/2$.

Proof. The common failure block cancels. The remaining key blocks are orthogonal, and Pauli conjugation preserves trace norm. Computational-basis inputs are unchanged by ${\Lambda }_{\eta }$. The difference of the two $X$-eigenstate densities is attenuated by $\eta$. Tensoring ${\sigma }_O$ preserves trace norm. The optimal binary discrimination formula follows by maximizing the effect against the positive part of the difference.

1. $p=0$: the whole radiation channel is a replacement channel. The source and environments retain the message.

2. $p=1,\eta =0$: computational-basis information is fully available collectively, but input coherence is lost to the dephasing environment. The channel is entanglement breaking.

3. $p=1,\eta =1$: the collective radiation permits exact referenced quantum recovery.

The emitted occupation statistics and the individual-cell privacy do not distinguish these three information tasks.

On the physical payload support, define

Here

The first term has trace $\mathrm{tr}(P_{\mathrm{good}}Y)$. Its Kraus maps are the count- and key-conditioned extraction maps. The second term is a replacement CP map of trace $\mathrm{tr}(P_{\mathrm{bad}}Y)$. Hence $\mathcal{D}$ is CPTP on every physical input, not only on emitted states.

The Pauli correction is applied before destructive logical readout. Discarding the count and key metadata does not discard the returned message.

The exact decoded channel is

Proof. In each successful key block the correction removes $P_{ab}$, while the failure block is replaced by $I_2/2$. The key weights sum to one, and the count densities have trace one. This proves the identity on all message operators.

For qubit Pauli channels

In particular, $d_{\diamond }(\Phi ,\mathrm{id})=1-a_I$.

Proof. The triangle inequality and $\Vert {\mathrm{Ad}}_P{\Vert }_{\diamond }=1$ give the upper bound. On a maximally entangled input, the four Pauli outputs are orthogonal Bell projectors. Their trace-distance difference is exactly the stated classical distance, proving equality.

Let

For a decoder $\mathcal{D}$, define

The first is a worst-case referenced criterion; the second is the specified Bell-input test.

Over all deterministic CPTP decoders with access to both full cells,

The explicit decoder $\mathcal{D}$ attains both values.

At prefixes zero and one, the corresponding optimal recovery error is $3/4$, and the maximal Bell-test fidelity is $1/4$.

Proof. The decoded channel (eq:decoded) has Pauli probabilities

Lemma lem:Pauli gives error $1-{\lambda }_I$; its Bell-test fidelity is ${\lambda }_I$.

For optimality, the orthogonal successful and failed ciphertext sectors have probabilities $p$ and $1-p$, independent of the message. The key and count labels give independent classical conditioning, not an additional message channel.

On a successful sector, undoing the known Pauli leaves

For any subsequent decoder, Bell overlap from the identity term is at most one. The dephased Bell state is separable across reference and output; local decoding preserves separability. A separable two-qubit state has Bell overlap at most $1/2$. Therefore every successful-sector decoder has overlap at most $(1+\eta )/2$.

A failed sector is input-independent. Every decoded constant qubit channel has Bell overlap $1/4$. Consequently every decoder has

Testing the Bell projector gives worst-case recovery error at least $1-F_e\ge 1-{\lambda }_I$, proving optimality.

At the earlier prefixes, (eq:N1) and the empty output are replacement channels. Every decoder again has Bell overlap $1/4$, giving error at least $3/4$. Preparing $I_2/2$ attains $3/4$ by Lemma lem:Pauli.

This is an optimality result for the declared accessible information, not an unrestricted physical black-hole decoder theorem.

Define the complementary dephasing channel

Let

The occupation-source complement is the independent density

Suppress the fixed terminal $F_1=0$ and all fixed initial factors. The fuel $F_2$ labels two orthogonal complementary sectors: success $F_2=0,Q=0$, and failure $F_2=1$.

Up to insertion of fixed factors, the prefix-two complementary channel is

The direct-sum notation refers to the orthogonal fuel sectors. The key purification $C$ and dephasing environment are part of this output.

At prefix one the complementary channel, apart from independent occupation-source and unused-cell factors, is

It is reversible when all its factors are accessible.

Proof. Tracing the emitted key eliminates different $a,b$ cross terms but retains both the source key and its purifying copy. Tracing the ciphertext eliminates success–failure cross terms because occupied and vacuum radiation states are orthogonal. On success, tracing the emitted qubit leaves ${\mathcal{C}}_{\eta }(X)$; on failure, the source qubit and its dephasing environment retain the isometric state $W_{ab}X{W}_{ab}^{†}$. Occupation registers contribute the independent diagonal daughter densities. This proves the complete formula.

At prefix one the encrypted qubit has not been emitted. Reading the retained key, undoing $P_{ab}$, and reversing the known dephasing unitary on $Q{Z}_{\mathrm{e}}$ recovers the input.

We compare with the named replacement channel

This is a specified center, not an unproved optimization over all replacement outputs.

For $\bar{e}=(|e_0⟩⟨e_0|+|e_1⟩⟨e_1|)/2$,

The Bell input attains this distance.

Proof. The channel difference is

The stabilized norm of the functional $X\mapsto \mathrm{tr}(ZX)$ is $\Vert Z\Vert =1$. The trace norm of the difference of the two pure-state projectors is $2\sqrt{1-{\eta }^2}$. This gives the stated upper bound.

On the Bell input the difference is

Its half trace norm is the same value.

The complete complement obeys

At prefixes zero and one, comparison with the respective complementary output on $I_2/2$ gives distance $3/4$.

For $\eta =1$,

This equality is specific to this model and these metrics.

Proof. The key blocks and fuel sectors in (eq:fullcomplement) are orthogonal, so their trace norms add. On a failed sector $W_{ab}$ is an isometry. Its comparison with the replacement output on $I_2/2$ is therefore the comparison of $\mathrm{id}$ with $\mathcal{T}$, of distance $3/4$. On success use Lemma lem:dephleak. This proves the weighted upper bound.

The same Bell input attains the $3/4$ value in every failure block and the value in Lemma lem:dephleak in every success block. Hence it attains the weighted sum. Independent tensor factors do not change trace norm.

At prefix zero the whole purified evolution is an isometric input channel. At prefix one the complement is an orthogonal mixture of isometric channels with known independent labels. The same calculation gives $3/4$. The final displayed equality follows from Theorem thm:recovery at $\eta =1$.

A comparison that omitted $C$ or $Z_{\mathrm{e}}$ would be a reduced environmental observation, not this complementary channel. It could give a misleading impression of decoupling.

Choose two orthonormal positive-frequency packet tests $h_1,h_2$ in the Em03-A–C scalar model, with disjoint frequency supports contained in compact bands away from zero. The stationary outgoing prescription then has vanishing off-diagonal occupation covariance.

Its selected state is the product of geometric densities

where

These are predecessor inputs, with state selection retained. They are not calculated from the finite Hamiltonians of this paper.

For the occupation emissions choose

We use $0^0=1$. This is a normalized probability vector.

Equation (eq:calibration) is an explicit calibration of interaction amplitudes to a clipped packet law. It is not a derivation of the Hawking law from the finite dynamics.

Embed $O_i$ into the oscillator number space by $|n{⟩}_{O_i}\mapsto |n⟩$. Let ${\hat{\sigma }}_O$ be the resulting oscillator density.

For every message input, including external references:

1. the joint occupation channel is the replacement channel

   $$X⟼\mathrm{tr}(X){\sigma }_O;$$

2. its occupation-state and joint count-law comparison with (eq:Hbaseline) is exactly

   $$\boxed{d_{\mathrm{tr}}({\hat{\sigma }}_O,{\rho }_{\mathrm{H}})=d_{\mathrm{TV}}(w_1\otimes w_2,p_{\mathrm{H}})=t_{\mathbf{J}}:=1-{\prod _{i=1}}^2(1-{q_i}^{J_i+1});}$$

3. the corresponding replacement channels have the same half-diamond distance $t_{\mathbf{J}}$;

4. the finite occupation mean is

   $$\boxed{{\mathbb{E}}_{\mathrm{model}}{N}_{O_i}={\nu }_i(1-{q_i}^{J_i}),{\nu }_i-{\mathbb{E}}_{\mathrm{model}}{N}_{O_i}={\nu }_i{q}_i^{J_i}.}$$

Each bounded occupation effect changes in probability by at most $t_{\mathbf{J}}$. No all-observable or local-field correspondence is asserted.

Proof. The occupation emissions act on independent factors and have the exact output densities in (eq:numberemission). Tracing all payload systems therefore gives the stated replacement channel.

The finite law is the deterministic clipping of the product geometric law. In the common countable number basis, mass above the finite rectangle is removed and added to its boundary. The positive inside change and negative outside change occupy orthogonal number records. Their half trace norm is the outside probability, which is (eq:Htail). The geometric marginals are independent for the chosen baseline.

For replacement channels,

Complete trace-norm contraction of partial trace gives the upper bound $\Vert \sigma -\tau {\Vert }_1$; a normalized input attains it. Thus the half-diamond distance is the state distance.

Finally,

An effect expectation is Lipschitz with constant one in trace distance, proving the bounded-observable statement.

The mean estimate is an explicit geometric-tail calculation. It does not follow for an arbitrary unbounded observable from a trace-distance estimate alone.

Define the message-independent payload references

After occupation embedding, the single-cell channel comparison with

has half-diamond distance exactly ${q_i}^{J_i+1}$.

This comparison has two qualifications:

1. ${\xi }_i$ describes new degenerate payload degrees of freedom. It is not supplied by the scalar Hawking baseline;

2. count and mean matching concern $O_i$, not every observable of $r_i=O_i{C}_i$.

The payload energy and payload occupation are part of the finite energy ledger, but are not hidden inside the matched scalar number observable.

For the named replacement center,

For every replacement channel ${\mathcal{R}}_{\sigma }$,

Thus exact private single-cell channels can coexist with a nonconstant collective radiation channel, but the collective channel cannot simultaneously be arbitrarily close to a replacement channel at fixed nonzero $p$.

Proof. In the comparison with ${\tau }_{\mathrm{rad}}$, the failure blocks cancel. The successful key blocks are unitary copies of ${\Lambda }_{\eta }-\mathcal{T}$. Its Pauli coefficient vectors are

Lemma lem:Pauli gives $\max \{1/2,(1+2\eta )/4\}$. Orthogonal key blocks and the common occupation factor preserve this distance, giving (eq:fullreplacement).

For an arbitrary replacement output, the triangle inequality between the two computational-basis radiation outputs and that common output gives

This explicitly separates occupation matching, finite-state matching on a named subsystem, and full message-channel matching. A nonzero $p$ is not inferred from the Hawking spectrum. It is selected by the independently specified ciphertext pulse.

For finite self-adjoint paths $H,\hat{H}$,

The induced channels, with references, fixed ancillas, and partial traces, have half-diamond distance bounded by this quantity, capped at one.

Compatible complete channel comparisons telescope: their half-diamond errors add, capped at one.

Proof. Variation of constants gives

The unitary factors have norm one. Expanding the induced channel difference in two terms bounds its complete trace norm by twice the unitary operator-norm difference. Pre- and postcomposition by channels contract that norm. Inserting one changed factor at a time proves telescoping.

Let the implemented encoding isometry obey $\Vert {\hat{V}}_{\mathrm{enc}}-V_{\mathrm{enc}}\Vert \le e_{\mathrm{enc}}$. Let the named purified initial environment be within trace distance $e_{\mathrm{prep}}$ of the ideal preparation. Let the finite dynamics have generator deviations $\Delta H_s(t)$ over the prescribed operation intervals. Define

The implemented prefix channels and the complements defined with the same complete environment interface differ from their ideal counterparts by at most ${ϵ}_{\mathrm{dyn}}$.

With the exact decoder,

A fixed Bell-test fidelity changes by at most ${ϵ}_{\mathrm{dyn}}$.

Proof. Encoding, independent preparation, and dynamics telescope by Lemma lem:stability. Either accessible or complementary partial trace is contractive. Decoder postcomposition and the triangle inequality give the recovery bound. The Bell projector is an effect, so its expectation changes by at most trace distance.

A bound only on a mixed environmental marginal is enough for the corresponding radiation-channel comparison. It is not, by itself, a comparison of chosen full complementary channels: their purifications and environmental identifications must also be specified.

The exact energy conservation of Theorem thm:evolution persists under a perturbation only if that perturbation also commutes with $H_0$. A small channel error is not an exact conservation law.

The emitted cells are finite systems. We construct their whole coherent access map directly. We do not extrapolate Em03-C's two-mode field-capture theorem to an unspecified radiation block.

Encode occupation $n$ into

qubits. Encode $C_1$ using one occupied/vacuum bit and two key bits, and encode $C_2$ using one occupied/vacuum bit and one data bit:

The total receiver width is

Unused binary words are invalid, not postselected away. Let $J_{\mathrm{acc}}$ denote this fixed isometric encoding.

For each physical cell choose its all-vacuum vector $|g⟩$. For every other physical basis label $u$, swap the disjoint pairs

The all-vacuum input is fixed. Define

The rank-one projections are orthogonal. A pulse of area $\pi$ on ${\sum }_u{R}_u$ gives all swaps exactly, because $e^{-\mathrm{i}\pi R_u}=I-2R_u$.

With a zero receiver register, the two-cell transfer leaves the same all-vacuum cell state for every basis input and induces

It preserves every input coherence and every external reference.

Proof. The final cell factor is $|g⟩$ for every input basis vector. Linearity gives a common cell environment for every superposition. Tracing that common environment leaves the stated isometric channel, including referenced inputs.

Let receiver preparation have trace-distance error $e_A$, independent of the cells and their references. Allow a joint bounded transfer perturbation $\Delta H_A(t)$, including cross-cell and spectator coupling. Then