Psy04-B Causal Process Structure for Delayed Random Access: Scheduled Ancestors, Storage Boundaries, and the Environmental-Memory Refusal

We place delayed random access in a complete finite causal network with explicit system ports, classical controls, and record outputs. We prove that scheduled ancestors upper-bound the support of backwards-propagated observables and hence unconditioned operational influence. This bound is not an equality, a statement about conditional correlations, or a certificate of quantum environmental memory. A decoder-relevant storage cut is defined using the complete joint accessible state, including correlated ancillary resources. Freshness of the late query is imposed separately from its absence from the encoding causal past. The cut yields the classical encoding–decoding factorization, from which the established sharp two-state success bound $3/4$ is independently proved. The standard one-qubit value $1/2+1/(2\sqrt{2})$ is realized by sequential local gates and destructive Enter readout. Explicit finite implementations distinguish retained storage from peak ancillary cost. Depolarizing storage gives an exact retention threshold, and parameter perturbations give a quantitative score allowance. Failures remain in the attempted-run law with neutral scores. Complete-law robustness and conditional concentration yield fixed-sample and anytime exclusion certificates. An already quantum carrier requires a different classical-environment null: a Markovian qubit already wins the present task, and a persistent classical register can produce a carrier-only causal-break witness. Finite-capacity bounds and causal-access checking specifications complete the construction. The contribution is the causal-cut and complete-process assembly, not discovery of the random-access benchmark, derivation of the Born rule, physical spacetime reconstruction, or environmental quantum-memory certification.

Keywords: random access codes; causal networks; scheduled ancestors; storage cuts; Enter calculus; complete instruments; bounded memory; classical environmental memory.

distinction The late query is outside the encoding causal past. The decoder receives that query later. A scheduled-ancestor bound is an upper bound on possible influence, not a quantum-memory certificate.

Psy04-A specifies a delayed random-access experiment with a complete storage boundary CheungPsy04A. The present manuscript asks how that boundary is represented as a cut in a complete causal process.

The established benchmark is unchanged: two uniform bits encoded in two classical states have optimal average late-query success $3/4$, whereas a qubit attains $1/2+1/(2\sqrt{2})$ AmbainisRAC. We prove these facts again for the exact class used here.

The new assembly consists of:

1. a complete finite network and scheduled-support theorem;

2. a decoder-relevant storage cut and its state assumptions;

3. explicit sequential Enter-compatible operations;

4. separation of retained memory from ancillary peak cost;

5. complete-record, robustness, and statistical interfaces;

6. a distinct null for an already quantum carrier with classical environmental feedback.

Finite quantum networks and memory-channel descriptions are established frameworks ChiribellaNetworks. The scheduled-support proof below is an elementary finite-channel specialization, not a new general theory of quantum causality.

Cheung's Enter calculus distinguishes a specified Born semantics from a separate characterization of an unknown measurement strategy CheungEnter,CheungI. Here we use the first distinction: allocation, coherent gates, destructive readout, and classical control are explicitly specified.

We do not derive Born's rule from a causal graph. Nor do we transfer the uniqueness theorem of I.tex to a class with apparatus-dependent hidden state. Our quantum calculations use printed CPTP maps and Born readout from the outset.

VII-J supplies a documentary scheduled-ancestor interface CheungVIIJ. Its finite support assertion is restated and proved below. The scalar input-stability and causal-accounting discussion in Em06-B is a methodological comparison CheungEm06B; no field equation, metric coefficient, or scalar response map is imported.

Psy02-B distinguishes complete process laws from mechanism identification CheungPsy02B. Psy03-B gives a classical-register example of reduced history dependence CheungPsy03B. Psy03-C distinguishes carrier contraction from persistent register information CheungPsy03C. Psy03-D separates supplied observations from hidden-rule inference CheungPsy03D. These are source interfaces, not unidentified premises.

First, placing the query event later does not by itself make its value unpredictable. A common random seed can correlate a late-generated query with an earlier encoder. Freshness is therefore an independent hypothesis.

Second, counting only systems whose marginals depend on the input is insufficient. An apparently input-independent ancillary marginal can participate in input-dependent joint correlations. The complete accessible joint state is the capacity object.

boundary. All positive conclusions are analytical and conditional on the printed hypotheses. No physical implementation, software compilation, LaTeX build, exhaustive execution, calibration, or machine-checked proof is asserted.

A complete finite network consists of:

1. a finite acyclic wiring diagram of system and classical ports;

2. a finite-dimensional quantum space, or a finite classical algebra, on each port;

3. a CPTP map at each operation vertex;

4. explicit ports for every classical control and reported outcome;

5. declared source preparations, terminal outputs, and inaccessible output systems.

A finite instrument $\{{\mathcal{I}}_o{\}}_o$ is represented by its recorded channel

where each selective map is CP and ${\sum }_o{\mathcal{I}}_o$ is trace preserving.

Adaptive control is a channel controlled by preceding classical record ports. A finite adaptive tree can consequently be represented as a finite network with classical registers and inactive branches explicitly completed.

Quantum channels, instruments, and their finite-dimensional norms use the usual conventions Watrous. Classical stochastic maps are included as channels on commutative algebras.

For an elementary operation, a conservative predecessor graph connects each input port to every output port. A finer predecessor declaration is valid only if the corresponding factorization or no-influence property of the operation has been justified.

A declared temporal order and a dependency graph are different objects. Two events may be ordered by the experimental clock without a system or control wire running from the earlier event to the later one.

Let $Y$ be a set of terminal output ports. Its scheduled ancestors ${\mathsf{Sch}}^{-}(Y)$ are the external input ports connected to $Y$ by directed paths in the declared predecessor graph.

At an intermediate frontier, ${\mathsf{Sch}}^{-}(Y)$ is interpreted as the set of frontier ports with directed paths to $Y$.

For a port set $S$, write $\mathcal{A}(S)$ for observables acting on $S$ and as the identity on all other ports of the same frontier.

Let $\mathcal{C}$ be the complete channel of a network in Definition def:network. For terminal outputs $Y$,

Consequently:

1. two initial states with equal reductions to ${\mathsf{Sch}}^{-}(Y)$ give equal output reductions on $Y$;

2. a trace-preserving intervention acting only on external inputs outside ${\mathsf{Sch}}^{-}(Y)$ has no unconditioned signalling effect on $Y$;

3. the same assertion holds jointly with an untouched reference, provided the intervention does not act on that reference.

Proof. Take an output observable supported on $Y$. Pull it backwards through a reverse topological order of the network.

If its support is disjoint from all outputs of an operation, the observable is the identity on those outputs. The adjoint of a trace-preserving map is unital, so that operation contributes an identity on its inputs.

Otherwise the observable may acquire support on the operation's declared input predecessors, but on no undeclared input. Under the conservative graph these are all of its input ports. Untouched tensor factors remain untouched. Fixed preparations introduce no unknown external input.

The recorded instrument is a single trace-preserving channel, so the same argument includes measurements, their records, and classically controlled continuations. Iterating leaves support only on ${\mathsf{Sch}}^{-}(Y)$, proving (eq:scheduledsupport).

Equal input reductions on that support give equal expectations for every output observable on $Y$. A trace-preserving local intervention outside the support preserves its reduction, even for initially correlated states.

For an untouched reference $R$, tensor the network with ${\mathrm{id}}_R$. The pulled-back support is contained in ${\mathsf{Sch}}^{-}(Y)\cup R$, and the same local-intervention argument applies.

This is the scheduled-support interface reported in VII-J CheungVIIJ, with the hypotheses and proof supplied for the present finite network.

An external port belongs to ${\mathsf{Inf}}_{\mathcal{C}}^{-}(Y)$ if some admitted initial state and two trace-preserving local interventions at that port give different unconditioned output laws on $Y$.

The inclusion need not be equality.

Proof. Theorem thm:schedule excludes influence outside the scheduled ancestors. For strict inclusion, a decoder can have a scheduled memory input but return the constant answer $0$. The memory input is then a scheduled ancestor with no actual influence on the answer.

A selectively normalized outcome branch is generally not trace preserving. Its adjoint is not unital, and the preceding proof does not apply to that branch in isolation.

Conditioning on a remote outcome can change a local conditional law without creating a signalling path. Likewise, observational correlations caused by common ancestors are not the interventions in Theorem thm:schedule.

Suppose an actual complete channel $\tilde{\mathcal{C}}$ and a scheduled ideal channel $\mathcal{C}$ satisfy

For two input states with equal reductions on ${\mathsf{Sch}}^{-}(Y)$, their actual output reductions on $Y$ differ in trace distance by at most $\min \{1,2e\}$. This estimate is reference stable.

Proof. Their ideal output reductions are equal. Insert that common reduction between the two actual outputs and use the triangle inequality. Each actual–ideal discrepancy is at most $e$. The half-diamond allowance includes arbitrary references Watrous.

A violation of a justified scheduled no-signalling bound would challenge the proposed complete network or its allowance. It would not, by itself, identify quantum environmental memory.

The process has encoding events $E_1,E_2$, storage events $S_1,\dots ,S_T$, a query-generation event $Q_{\mathrm{gen}}$, a decoder event $D$, and a scoring event $A$.

The declared temporal order is

A joint encoder may replace $E_1,E_2$. The order specifies a finite experiment, not a physical spacetime geometry.

For one attempted run:

1. $X=(X_1,X_2)$ consists of independent uniform bits;

2. all freely admitted pre-existing random resources $\Lambda$ are independent of $X$;

3. $J\in \{1,2\}$ is generated uniformly at $Q_{\mathrm{gen}}$, independently of $X,\Lambda$ and the complete pre-query device state;

4. no query value, correlated prediction of it, or query-dependent control reaches an encoding event;

5. the scoring archive has no feedback path to the current decoder;

6. timing, readiness, and stopping controls are either input independent or explicitly included in the complete access model.

In particular,

The decoder does receive $J$. The query is absent from the encoding past, not from the decoding past.

Under Assumption ass:fresh, the encoding state and every pre-query accessible record have a law independent of the realized $J$. Any pre-query prediction of $J$ succeeds with probability $1/2$.

Proof. The complete encoding subnetwork has no query-value input. Its operations depend only on the admitted preceding inputs and resources. Composition therefore supplies one encoding law for each $x,\lambda$, not a family selected by $j$.

Freshness makes $J$ independent of that law. For any pre-query predictor $\hat{J}$,

The same argument applies to predictions produced by a quantum instrument on a pre-query state.

This does not prohibit a carrier from encoding information about both input bits. It prohibits choosing an encoding using the actual future query value.

Let $\Lambda$ be a fair bit known to the encoder. The encoder stores $X_{1+\Lambda }$. Later the query generator outputs $J=1+\Lambda$.

There is no arrow from the query event back to encoding, yet one retained classical bit succeeds with probability one. The common cause $\Lambda$ makes the query predictable. This violates Assumption ass:fresh, not Theorem thm:schedule.

Choose a frontier after the final encoding operation and before storage.

The raw decoder-relevant frontier consists of all systems, classical registers, and persistent control states present at that frontier that have scheduled paths to the decoder's scored output $O$. Persistent apparatus state is represented as a frontier system, not hidden in an operation name.

The storage cut is complete when all routes from the encoding inputs to $O$ cross this frontier and no undeclared system, record, or parameter later re-enters the decoder.

The experimenter's archive is excluded only because it has no path to $O$. It remains part of the complete attempted-run record used by the scorer.

Membership in the raw frontier is conservative. An input-independent marginal is not automatically a free resource: its correlations with another frontier system can matter.

Conditional on an input-independent shared classical resource $\Lambda =\lambda$:

1. the wholly classical alternative has a complete input-bearing cut state $M$ with at most $K$ realized values;

2. any additional freely available post-cut resource has a specified input-independent law and does not carry additional input information jointly with $M$;

3. for the one-qubit comparison, the input-bearing cut state is a qubit, with no preshared entanglement or additional correlated decoder-accessible quantum system;

4. an uncounted auxiliary quantum system is free only when its joint cut state has the product form

   $${{\rho }_{x,\lambda }}^M\otimes {{\sigma }_{\lambda }}^F;$$

   merely requiring ${{\sigma }_{\lambda }}^F$ to be the same marginal for every $x$ is insufficient;

5. shared classical randomness is not mutated into an additional current-input-dependent storage variable;

6. the decoder receives realized classical memory values, not an exact observable description of their encoding probabilities.

For the main classical benchmark $K=2$.

The hypothesis applies to joint retained registers: controller state, pulse logs, accessible environmental variables, timing records, and preparation flags. An additional route cannot be declared absent merely because it is omitted from the carrier's name.

Let two qubits $M,F$ initially share

Encode $x_1,x_2$ by applying ${{\sigma }_x}^{x_1}{{\sigma }_z}^{x_2}$ to $M$. The four joint states are orthogonal Bell states, while both one-qubit marginals remain $I/2$.

A decoder with access to $M,F$ can identify both bits and answer either late query perfectly. The relevant cut contains two jointly accessible qubits, not one. Counting only input-dependent marginals would miss the resource.

Under the complete network, freshness, and cut-accounting hypotheses:

1. every wholly classical final decoder-output law has the form

   $$p(o\mid x,j)=\int \mu (d\lambda ){\sum _{m=1}}^K{a}_{\lambda }(m\mid x)d_{\lambda }(o\mid m,j);$$

2. every quantum output law has the form

   $$p(o\mid x,j)=\int \mu (d\lambda )\mathrm{Tr}\left[{{\mathcal{D}}^{\lambda }}_{o\mid j}({\mathcal{S}}_{\lambda }({\rho }_{x,\lambda }))\right],$$

   where ${\mathcal{S}}_{\lambda }$ is the admitted storage channel and $\{{{\mathcal{D}}^{\lambda }}_{o\mid j}{\}}_o$ is a complete decoder instrument.

All persistent input-bearing systems must be included in $M$ or in the quantum cut system.

Proof. Condition on $\lambda$. Compose the operations before the frontier. They produce a stochastic law $a_{\lambda }(m\mid x)$, or a density operator ${\rho }_{x,\lambda }$. Proposition prop:late excludes dependence on $j$.

By completeness of the cut, the remainder has no access to the encoding inputs except through that state. Compose storage and the decoder, integrating their fresh randomness or adjoining their product ancillary states. In the classical case this produces a stochastic kernel $d_{\lambda }(o\mid m,j)$. In the quantum case it produces the displayed complete instrument after the storage channel.

Systems exported to an inaccessible archive and never returned to the decoder can be summed out in the decoder marginal. No accessible correlation is removed by that step. Finally average over $\lambda$, whose law is independent of the fresh inputs.

The theorem is a conditional consequence of complete wiring and state accounting. A drawn separator alone does not establish its hypotheses.

Each of the following permits perfect retrieval but violates the declared comparison:

1. advance access to the actual $J$;

2. a predictable query as in Example ex:predictablequery;

3. a second accessible register retaining the other bit;

4. correlated ancillary assistance as in Example ex:dense;

5. an unrestricted exact-valued classical state storing the input word.

Proof. Store or recover the requested bit using the additional access indicated in each case. The added access either invalidates freshness, the cut, or its capacity accounting.

For a binary decoder answer, define

Every finite-alphabet law of the form (eq:classicalfactor) is a mixture of deterministic encoding and decoding tables with mixture randomness independent of $X,J$.

Proof. Condition on $\lambda$. Before the fresh input is used, sample an encoding-table entry for every possible $x$ according to $a_{\lambda }(\cdot \mid x)$, and a decoding-table entry for every possible $(m,j)$ according to $d_{\lambda }(\cdot \mid m,j)$. Use independent table samples conditional on $\lambda$.

Each table realization is deterministic. Averaging its entries reproduces the two kernels and their composition. Average also over $\lambda$.

This permits arbitrary stochastic strategies, not only one implementation of pseudorandom choices.

Every admitted wholly classical cut with $K\le 2$ satisfies

The bound allows stochastic encoding, storage, decoding, and input-independent shared randomness. It is attained by a sequential one-bit encoder.

Proof. It suffices to consider deterministic tables. For memory value $m$, let

An input $x$ encoded as $m$ has average success $1-d_{\mathrm{H}}(x,c_m)/2$. The best encoder for fixed answer words chooses a closest word, giving

There are four input words and at most two distinct answer words. At least two input words are not answer words and each has distance at least one. The distance sum is therefore at least two.

For attainment, store $X_1$ when it arrives, ignore the later $X_2$, answer query 1 correctly, and use a fixed answer for query 2. The score is

This is an established random-access-code benchmark AmbainisRAC. Sequential arrival does not tighten it: the upper proof already permits the more general batch encoder.

For a fixed encoder and an input-independent admitted classical storage kernel, optimal post-storage decoding cannot exceed optimal decoding from the pre-storage state.

Proof. A decoder given the pre-storage state can simulate the storage kernel internally and then use any post-storage decoder.

The classical null thus permits perfect storage. A noisy qubit is compared with this conservative null, not with an automatically equally noisy classical device.

We use the relevant types

A quantum name is an access right to one factor of a global state. Allocation $\mathsf{new}$ supplies $|0⟩$. A gate consumes and returns its named quantum arguments. Destructive readout consumes the measured name and returns a classical bit.

For a computational-basis readout of $S$,

and the explicitly specified transition has probability

and normalized residual ${M_{\kappa }}^S\psi /\text{norm}{M_{\kappa }}^S\psi$ on positive branches. This is the concrete semantics considered in CheungEnter,CheungI.

A conditional has the same quantum context in its two alternatives but executes only one alternative. Thus it does not duplicate quantum ownership. Classical deletion is represented by

There is no corresponding silent deletion of a quantum name.

For a finite program built from allocation, unitaries, destructive readout, and finite classical control, each complete history $h$ determines a linear path map $K_h$. For a pure input $\psi$,

If the reported output is $o$, its selective map is

These identities also hold with an untouched reference.

Proof. A complete history fixes the later classical controls. Compose allocation isometries, gate matrices, and destructive readout bras along that history. Conditional normalization factors cancel against the corresponding branch probabilities.

Completeness follows by backwards induction on the finite outcome tree. An allocation is isometric, a gate is unitary, and a readout satisfies

Grouping histories gives the selective maps. Tensor every path map with the identity on the reference.

This proof covers the programs used here directly. It does not assume the entire metatheory or measurement-characterization theorem of I.tex.

Let

Define the local encoding macros

They have the judgments

Write $z=J-1$. The decoder is

Its type is

For $J=1$ it reads the $X$ basis; for $J=2$ it reads the $Z$ basis.

The prepared and stored carrier may be returned to the late-query continuation

The continuation captures one quantum name, not the earlier input bits. Actual delivery of $z$ at the declared late event is an external scheduling hypothesis. A lambda term alone is not a physical freshness certificate.

Initialize $q$ in $|0⟩$, apply ${\mathsf{Enc}}_1(X_1,q)$, then ${\mathsf{Enc}}_2(X_2,q)$. The resulting state is

The destructive decoder (eq:decoderterm) satisfies

and hence

The only route from the first input to the second encoding event in this implementation is the carrier. No separate retained earlier-bit wire is required.

Proof. The first rotation takes the $+Z$ Bloch vector to

Conjugation by ${\sigma }_x$ preserves its $X$ component and reverses its $Z$ component. This gives (eq:qubitencoding).

The destructive branch maps are

Their effects are

Evaluation on ${\rho }_x$ gives the stated probability.

In both encoding macros the current input is consumed as a guard and the carrier is used once on the executed branch. There is no output wire for an earlier-bit copy.

The state and retrieval value are standard AmbainisRAC. The addition is their explicit local causal routing and ownership interface.

The mathematical encoding requires no surviving classical bit copy. An apparatus may nevertheless retain a pulse log, preparation setting, or timing signal. If such a system can later influence decoding, it belongs to the declared cut.

An input-independent phase reference is a specified control resource. Its input independence, absence of relevant correlations, and access rules must still be checked.

Under the one-qubit cut accounting of Assumption ass:capacity, arbitrary qubit encodings and binary POVMs satisfy

Input-independent shared classical randomness does not increase this optimum.

Proof. Condition on shared randomness. Write

where $s=(-1{)}^{x_1}$, $t=(-1{)}^{x_2}$.

For binary measurement $j$, let

The condition $-I\le A_j\le I$ implies

The scalar terms cancel in the uniform input average:

Therefore

By Cauchy–Schwarz and the parallelogram identity, the numerator is at most

The construction above attains equality. Average over the shared randomness.

Fresh product ancillas at decoding merely implement a POVM on the retained qubit. Correlated assistance is excluded separately by the cut, as Example ex:dense illustrates.

The carrier-preserving selective maps

can be implemented by:

1. rotating the carrier into the desired readout basis;

2. allocating a pointer qubit in $|0⟩$;

3. applying CNOT from carrier to pointer;

4. destructively reading the pointer;

5. reversing the basis rotation on the returned carrier.

The implementation retains one carrier but has peak quantum occupancy two.

Proof. In the rotated basis the pointer branch map on the carrier is $|b⟩⟨b|$. Conjugating by the basis rotation gives $P_{b\mid j}$. The selective map is therefore the displayed one. The pointer name is consumed by Enter; the carrier name is returned.

A record construction is not a free copy of an unknown quantum state. Placing a pointer across the storage cut changes its resource ledger. Tracing or measuring an inaccessible pre-cut pointer can also change the carrier channel. The record equation discussed in I.tex does not license omitting that resource or equating arbitrary interference contexts CheungI.

For $0\le \eta \le 1$, define

This is a linear CPTP map on arbitrary matrices.

The specified storage process consists only of ${\mathcal{P}}_{{\eta }_1},\dots ,{\mathcal{P}}_{{\eta }_T}$, with no syndrome-assisted recovery or other undeclared intermediate operation. Put

The stored code states are

The specified decoder has score

This is also optimal over encodings and binary decoders for this fixed composite storage channel.

Its strict violation of the perfect classical-storage benchmark occurs exactly when

For constant $\eta$, $r_T={\eta }^T$.

Proof. Depolarization multiplies every Bloch component by $\eta$, and its factors compose multiplicatively. Born readout gives (eq:retentionscore).

In the proof of Theorem thm:qubitoptimal, storage multiplies the nonconstant score term by $r_T$. The same code attains the resulting upper bound. Solving $Q_{\mathrm{Q}}(T)>3/4$ gives (eq:retentionthreshold).

The threshold is a retrieval-performance threshold. It is stronger than retaining a nonzero Bloch component and is neither a metric nor a spacetime statement. Psy03-C uses an analogous distinction in a different phase-label task CheungPsy03C; its constants are not transferred.

For

the propagator is ${\mathcal{P}}_{e^{-\kappa t}}$. When $\kappa >0$, strict separation survives exactly for

When $\kappa =0$, it survives at every finite delay.

Proof. The identity component is fixed and the traceless component decays as $e^{-\kappa t}$. Apply Theorem thm:retention.

For each $\eta$, let

and set

A two-qubit ancillary preparation

followed by controlled $U_a$ on the carrier and destructive readout of the ancillary register, has resolved selective maps

If the label is not available to later decoding, the carrier channel is ${\mathcal{P}}_{\eta }$.

Repeated steps can be represented with peak occupancy three qubits and one carrier retained between steps.

Proof. A unitary on the two fresh ancillary qubits prepares $|{\chi }_{\eta }⟩$ from $|00⟩$, by completing the specified unit vector to an orthonormal basis. The controlled unitary sends an input vector to

Enter consumes both ancillary names. Its branch map on the carrier is $\sqrt{w_a}{U}_a$, which proves the selective-map formula.

The Pauli identity

gives

The two classical label bits are either consumed by $\mathsf{dropBit}$ without feedback or exported to an experimenter-side archive with no return path. No ancillary quantum name is silently dropped. A fresh pair can be allocated at the next step.

This is a mathematical program representation, not an executed apparatus compilation.

For the preceding implementation,

is input independent. If the decoder nevertheless receives $a$, it can apply ${U_a}^{†}$ and undo the corresponding Pauli error.

Thus at $\eta =0$ the unlabelled carrier channel is complete depolarization, while the labelled channel permits exact correction before RAC decoding. The delay threshold assumes no such label access.

An input-independent label marginal does not establish a product joint state or authorize an omitted recovery wire.

The retained-memory claim and the peak implementation claim are consequently different:

Let $O$ be the decoder's scored output, taking values in

The complete attempted-run record may also contain

where $H_{\mathrm{ext}}$ includes preparation, storage, and diagnostic records kept by the experimenter.

The external archive need not factor through the memory: it deliberately contains the scoring copy of $X$. The capacity bound concerns the decoder marginal $p(O\mid x,j)$, whose access must satisfy (eq:classicalfactor). Any archived record returned to the decoder ceases to be an external spectator.

Define the predeclared score

Every attempted run receives a score. Timeouts and unavailable outputs have a declared outcome, not a retrospective deletion rule.

If the decoder-output law $p(O\mid x,j)$ factors through a $K\le 2$ cut as in (eq:classicalfactor), then

The decoder's neutral-output probability may depend on $m,j,\lambda$.

Proof. Replace every neutral decoder output by an independent fair binary answer. This replacement preserves the expected score and uses the same classical memory. Apply Theorem thm:classical.

The theorem does not permit an external input-aware scoring controller to replace unfavorable answers by neutral outcomes. Such a controller is a new route affecting the scored law, not a failure mechanism inside the declared cut.

Store $X_1$, answer query 1, and erase query 2. The observed-answer success is one, but the complete score is

Postselection changes the task.

Preparation success has probability $p$, observation success has probability $v$, and an observed binary answer is flipped with probability $ϵ$, where

The specified coins are independent of the fresh input, query, ideal outcome, and one another. Failures receive the neutral score. They create no uncounted input-dependent decoder access.

For the code and storage above, under Assumption ass:flags,

It exceeds $3/4$ exactly when

Proof. Conditional on a valid observation, assignment noise gives

Valid observations have probability $pv$. All remaining attempts contribute $1/2$. Substitute (eq:retentionscore).

Input-dependent or outcome-dependent losses require their own complete-law analysis. Neutral scoring remains well defined, but the multiplicative formula need not remain valid.

Consider two specified protocols with the same number of storage steps and respective parameters

Under the independent-flag model,

In particular,

Proof. For factors $u_{\ell },u_{\ell }^{\prime }\in [0,1]$, telescoping gives

Take absolute values and use factors at most one. Apply this identity to $p,v,1-2ϵ,{\eta }_1,\dots ,{\eta }_T$.

This is a score allowance for changes of declared data. It is not continuity of spacetime or a transfer of Em06-B's scalar constants CheungEm06B.

Use

For complete laws $P,P^{\prime }$ and a common score $f\in [0,1]$,

Proof. Use

and the defining total-variation supremum.

Suppose an actual quantum complete law is within total variation $e_Q$ of the specified protocol law.

Suppose every actual classical null law is within total variation $e_C$ of some complete law whose decoder output satisfies the two-state cut factorization and neutral-score convention. Then

A lower separation allowance is

It is positive exactly when

Proof. Apply Lemma lem:tv to the common complete-record score, then subtract the null upper allowance from the quantum lower allowance.

The errors describe actual comparison discrepancies. An unrestricted hidden wire cannot be renamed $e_C$ without a proof bounding its effect. A uniform input-law error can be charged through a complete-law comparison; the nominal uniform-input benchmark is not automatically unchanged.

Suppose two implementations have common declared input and output spaces, joint initial-state discrepancy at most $e_{\mathrm{prep}}$, and successive channel or recorded-instrument half-diamond discrepancies $e_1,\dots ,e_k$. Then their final complete-record laws differ by at most

For adaptive steps, the allowances must be uniform over the admitted histories and settings.

Proof. Insert ideal intermediate outputs one operation at a time. Trace distance contracts under CPTP maps. Each half-diamond error bounds the discrepancy on every allowed joint input, including an untouched reference Watrous. The triangle inequality gives the sum. Final record readout contracts trace distance to total variation.

Reference stability is a correctness property. It does not grant the decoder an uncounted reference.

Parameter allowances from Proposition prop:parameterstability may be included in $e_Q$, or handled by a separate worst-case parameter range. The same uncertainty should not be charged twice.

A violation of $3/4$ in the present delayed single-receiver task does not imply quantum non-Markovianity or quantum environmental memory.

The winning code can use identity storage and no persistent environmental register.

Proof. Use Theorem thm:qubit and identity interval channels. Every interval acts on the current carrier only. The process is Markovian on that carrier, while the score is $1/2+1/(2\sqrt{2})$.

The stored qubit retains input information, but no cross-interval environmental memory is needed.

Fix:

1. carrier dimension $d$;

2. classical environmental alphabet $\mathcal{E}=\{1,\dots ,L\}$;

3. a finite intervention schedule;

4. allowed instruments, feedback, controller storage, and complete record access.

The joint state is classical–quantum:

An environmental update is specified by CP maps ${{\mathcal{T}}_{e^{\prime }\mid e}}^{(t)}$ satisfying

with

Past-record and setting dependence is represented by the declared controller.

The environment has no additional direct access to the fresh input unless such access is expressly admitted. Its initial joint preparation is independent of the fresh input. No coherence between the classical environmental labels is permitted.

Write ${\mathfrak{N}}_{d,L,\mathcal{U}}$ for this class, where $\mathcal{U}$ collects the instrument, schedule, and access restrictions.

This is a valid finite classical-feedback specification, not identification of a particular physical environment.

If ${\mathfrak{N}}_{2,1,\mathcal{U}}$ admits the present code and imposes no compulsory intermediate disturbance, then

Thus $3/4$ is not an environmental-memory exclusion bound for this class.

Proof. A one-value environmental alphabet is trivial. The Markovian winning qubit process belongs to the null.

Let a hidden classical bit $E$ be uniform. Allocate a carrier in $|0⟩$, apply ${{\sigma }_x}^E$, and destructively read it in the $Z$ basis, obtaining $R_1$.

After that readout allocate a fresh carrier in the same $|0⟩$, irrespective of $R_1$. Apply ${{\sigma }_x}^E$ again and read $R_2$.

Then

but

The post-break carrier preparation is identical in both histories. The surviving information is the classical register $E$. All carrier states can remain diagonal.

The enlarged carrier–register process has a specified Markov update. The reduced carrier process has history dependence under this carrier-only break.

A classical persistent register can produce a carrier-only causal-break witness. This is the same interpretive boundary illustrated by the separate Psy03-B pilot CheungPsy03B. The operational causal-break approach concerns complete multitime interventions, not merely a one-time distinguishability curve PollockMarkov.

Let ${\omega }_0=\rho \otimes |0⟩⟨0|$ and ${\omega }_1=\rho \otimes |1⟩⟨1|$. For every carrier channel $\Lambda$,

The label may influence a future carrier observation through a later controlled operation, even when $\Lambda$ is complete depolarization.

Proof. The environmental supports are orthogonal, so the two joint states remain perfectly distinguishable. For the future observation, replace or allocate the carrier in $|0⟩$, apply the label-controlled ${\sigma }_x$, and measure $Z$. The resulting answer reveals the label.

In the causal graph the environmental label bypasses the contracting carrier channel. A carrier-only contraction argument has not removed that scheduled route. Psy03-C records this distinction in a different model CheungPsy03C.

Roy and colleagues study a three-party prepare-transform-measure QRAC with a quantum carrier, intermediate instruments, and a downstream receiver RoySequential. Their sharp-measurement analysis uses rank-one projective measurements at the intermediate party, and they also analyze unsharp instruments. Their classical-memory processes use environmental classical feedback, rather than making the complete processor a two-state classical system.

The relation to the present null is therefore structural: both must specify carrier dimension, complete instruments, and environmental feedback. The experimental restrictions are not identical.

distinct exclusion questions.

A stronger result would require a complete-record score $W$, a proved bound

and an admitted quantum-environment process exceeding that bound after uncertainty allowances. For arbitrary classical register size, the bound must be uniform over the admitted sizes.

No such new environmental optimization or certificate is claimed here. Linear dependence of a fixed witness on a process representation does not automatically make a joint optimization over processes and instruments convex.

Fix a basis on every retained carrier. Suppose initial states are diagonal and every selective map of every admitted instrument sends diagonal states to diagonal states. Suppose controls use only declared classical records.

Then every complete record law has a classical stochastic realization. If the complete input-bearing state on the storage cut has at most $K$ basis values, the classical cut factorization applies with that $K$.

Proof. For a selective map ${\mathcal{I}}_o$, define

These numbers are nonnegative. The complete instrument condition gives

On diagonal populations the selective map is precisely multiplication by this substochastic matrix. Products of the matrices reproduce the unnormalized history populations and probabilities. Induct through the finite controlled experiment.

This restates the relevant finite classical boundary, rather than importing Psy03-A's particular continuous-time generator CheungPsy03A. The coherent RAC preparation does not lie in this diagonal-preserving class.

Suppose two models have a common complete state description, identical initial states, and identical conditional complete instruments and interval maps after every admitted setting and history. Then their complete record laws are equal, including with an untouched reference.

For any two complete record laws $P_0,P_1$, a randomized test satisfies

For equal laws, $\alpha +\beta =1$.

Proof. Induct through the equal unnormalized conditional states. Their traces give equal probabilities for every record history.

For a test $\phi \in [0,1]$ declaring model 1,

Apply Lemma lem:tv.

Equality of a few reduced one-time channels is not the hypothesis of this proposition. An unrepresented persistent environment can correlate successive intervals.

This independently states the identification interface reported in Psy02-B CheungPsy02B. A positive robust gap separates declared complete-law families; it does not identify a unique hardware mechanism.

Let ${\mathcal{F}}_i$ contain the completed attempted-run history through run $i$. Scores are finalized by the predeclared complete-record rule. The unit of assessment is the attempted run, not each gate, storage step, or within-run observation.

Conditional on every pre-run history:

1. the next input bits are fresh uniform variables;

2. the next query is fresh, generated after encoding, and independent of pre-query device state;

3. the current-run complete classical cut still has at most two realized input-bearing states;

4. the current-run actual complete-law discrepancy has a predictable allowance $e_{C,i}$.

Pre-existing device memory may depend arbitrarily on earlier runs. Any mutation carrying current-run input information across the cut is counted in the current capacity.

The record history can include the experimenter's scoring archive. Freshness must hold conditional on that pre-run history, not merely in an unconditional average.

For $S_i\in [0,1]$, set

For real $u$,

For fixed $N$ and $r>0$,

and the same bound holds for the lower tail.

Proof. Convexity on $[0,1]$ gives

For fixed $p\in [0,1]$, let

Its value and first derivative vanish at zero, and

Integrating twice gives $K_p(u)\le u^2/8$.

Iterated conditional expectation bounds the centered sum's moment by $e^{N{u}^2/8}$. Markov's inequality yields

Optimize at $u=4r$; use negative $u$ for the lower tail. This is the bounded-variable argument underlying Hoeffding's inequality Hoeffding.

Under Assumption ass:history,

for every classical null. For predeclared $N$ and $0<\delta <1$, define

The rejection rule

has Type-I error at most $\delta$, uniformly over the declared classical null.

For constant $e_C$, it becomes

Proof. Condition on the pre-run state and history. The fresh classical cut gives the bound $3/4$; the conditional complete-law discrepancy adds at most $e_{C,i}$. Average any hidden pre-run state.

Since

Lemma lem:concentration proves the rejection bound.

For independent identically distributed attempts, the radius is a one-sided confidence allowance for a common mean. For adaptive attempts it bounds the average predictable mean $N^{-1}{\sum }_i{\mu }_i$, not an automatically stationary population parameter.

The test uses observed scores and the classical allowance. The design error $e_Q$ is not subtracted again from the measured mean. It supplies a prospective quantum lower score and power calculation.

If an alternative has conditional means

with constant null allowance $c$, then the fixed-sample rule has failure-to-reject probability at most

when $r_N(\delta )<g$.

A sufficient sample size for Type-I error at most $\delta$ and failure-to-reject probability at most $\beta$ is

Proof. Failure to reject implies

Use the lower tail in Lemma lem:concentration. The sample-size condition ensures $g\ge r_N(\delta )+r_N(\beta )$.

The robust gap in (eq:robustgap) is usable as this $g$ only when the actual conditional alternative allowances are justified uniformly.

Suppose the conditional null bounds hold indefinitely. Set

Then

Stopping at a first crossing is therefore valid.

Proof. At $N$, allocate ${\delta }_N=\delta /[N(N+1)]$. Apply Theorem thm:test, then use

and a union bound.

Repeated inspection with the unadjusted fixed-$N$ radius is not justified by Theorem thm:test. Multiple delays or protocols likewise need a confidence allocation or an independent assessment block.

If a pre-assessment calibration event has failure probability at most ${\delta }_{\mathrm{cal}}$, and the conditional assumptions hold on that event, total false-certificate probability is at most $\delta +{\delta }_{\mathrm{cal}}$.

A score array does not verify freshness, causal access, cut completeness, or capacity. It rejects their conjunction with the other null hypotheses at the stated level.

Let $X$ now contain $n$ fresh uniform bits, let the late query be uniform on $\{1,\dots ,n\}$, and let the classical cut admit $K\ge 1$ states.

For unrestricted batch encoding through the complete $K$-state cut,