Oai265-A A Conditional Audit of Two-Dimensional Area-Law Records Boundary accounting, tensor-network existence, reference-preserving recovery, and temporal non-identification

We complete a source-qualified audit programme for the external OpenAI Family 265 area-law and polynomial-PEPS claims without using either claim as a theorem premise. The frozen class consists of finite induced square-lattice domains, bounded finite-range interactions, a unique ground state, and a positive full-system spectral gap. We prove the boundary conversions, an independently verified two-family entropy-cancellation lemma, and a quantitative terminal certificate that identifies exactly which geometric and information-theoretic constants would price a uniform area-law constant. The missing estimates are not supplied by a decoder resource ledger or a channel error budget.

We distinguish PEPS representability from tensor production, contraction, and physical preparation. Explicit models show that dropping uniqueness permits volume-law ground vectors even with a unit gap above the ground space, that a uniform full-system gap does not imply a uniform restricted-Hamiltonian gap, and that an entropy upper bound does not imply a strict area-law equality. We also correct two overstrong exclusions: complementary mutual information and Rényi entropies of order greater than one do have elementary bounds conditional on a von Neumann area law.

Locally proved channel counterexamples reproduce the separation between selected entropy and reference-preserving recovery: complete isometries can agree on every pure-input accessible spectrum while their optimal recovery errors are $0$ and $1-d^{-2}$. Uniformly gapped bond-dimension-two instantaneous ground states permit both turnover and no turnover; their preparation dynamics are separate data. A finite qubit-memory certificate is included only as a statistical comparison, with an explicit procedure-dependent forty-shot nonacceptance result. All stages receive a proved result, an obstruction, or an exact unresolved disposition. The single active successor is quantitative constant extraction within the frozen class. Family 265 remains unverified here, and no Page curve, gravitational dictionary, or inherited quantum-coefficient admission is constructed.

The plan has seven stages, numbered $0$ through $6$. Completing this plan means:

1. freezing the external statements without promoting them to premises;

2. auditing the five-stage proof architecture against its stated interfaces and available evidence;

3. proving the finite accounting and separation statements that can actually be established from the supplied material;

4. recording every missing universal estimate;

5. selecting one successor target, rather than silently asserting its solution.

In particular, this manuscript does not claim to finish the unverified proof of a universal two-dimensional area law. Nor does it claim an explicit universal constant, an efficient PEPS algorithm, or a formalization of the external proof.

The central local results are Theorems thm:terminal, thm:recovery, thm:separation, and thm:release. They are self-contained mathematical statements on explicitly declared finite interfaces.

The records named 265-01 and 265-02 are attributed to OpenAI, with manuscript edition date 24 September 2026 OpenAI265A,OpenAI265B,OpenAIOverview. The companion page for 265-01 expressly distinguishes that external record from Cheung's programme Companion265.

Cheung's supplied Oai265-A version 1 separates static entropy records, quantum recovery, and radiation histories. The present development preserves that distinction and reproves the finite statements it uses CheungOai265A. It is a development under the requested identifier, not a claim that a new revision has been registered or approved by N. Cheung.

The other Cheung sources are used as comparisons with their supplied scopes retained:

• En02-A/B/C distinguish a task, its complete channel, and its resource ledger CheungEn02A,CheungEn02B,CheungEn02C;

• En07-B/C distinguish complete preservation, accessible recovery, selected entropy, and thermal data CheungEn07B,CheungEn07C;

• Oai265-B separates finite-confidence coverage from physical calibration and reports a procedure-dependent small-sample limitation CheungOai265B;

• XXII-A/B/D/F distinguish an operational relation, a supplied geometry, a conditional implication, and a nonempty physical instantiation CheungXXII;

• Landmark I's destructive readout has its own finite measurement interface, not a many-body ground-state construction CheungI.

No Cheung theorem is inserted as a premise into an external Family 265 proof stage.

The catalogue, the companion identification page, and published literature used below were consulted. Attempts to retrieve the full external PDFs through their available repository routes did not produce a readable full proof in this audit. The detailed five-stage description and the quoted entropy-cancellation passage therefore remain source-qualified supplied material.

Accordingly, Section sec:audit is an interface audit, not a claim that every line of the two external papers was checked. No omitted lemma, constant, or proof dependency is reconstructed from a summary.

The statement “no Lean note exists” is too broad to authenticate from the available evidence. The supported statement is narrower:

No complete, source-matched formalization certificate for 265-01 or 265-02 is supplied or authenticated here. The supplied description of Canonical265.lean concerns moment canonicalization in OAI.MUB6.Certificate.PairSemantics, not the lattice Hamiltonian theorem.

This is the provenance warning already recorded in the supplied Oai265-B material CheungOai265B. A numerical filename, an unrelated namespace, a scope note, or a formalization work item is not execution or coverage of the present theorem.

Let $\Lambda \subset {\mathbb{Z}}^2$ be finite. Its graph is the induced graph with edge set

The graph distance $d_{\Lambda }$ is $+\infty$ between distinct connected components. For $A\subseteq \Lambda$, write

and

Thus $Z_A$ includes endpoints on both sides of the cut. Its definition is necessary for the numerical factor below.

Fix

The Hilbert space is

We use the interaction convention

There is one aggregated term per support $X$. An arbitrary number of separately bounded copies on the same support is not permitted to bypass the strength bound. Scalar identity terms can be removed without changing the state or gap.

The ground vector has norm one, is unique up to phase, and satisfies

All entropy logarithms are natural:

The entropy is dimensionless; changing to thermodynamic units would introduce an independently specified factor $k_{\mathrm{B}}$.

The external statement under audit asserts that a finite constant $C=C(q,R,J,\Delta )$ exists such that every instance (eq:H)–(eq:gap), on every finite induced domain, satisfies

The same $C$ is independent of domain size, shape, and region. This is a documentary record, not a theorem proved or assumed here OpenAI265A.

There is no subsystem-gap, frustration-free, translation-invariant, product-path, continuum, or efficient-algorithm hypothesis in this frozen package.

For every $A\subseteq \Lambda$,

and

Consequently, if an individual state has an independently established bound (eq:AL), then

Proof. Every inner-boundary vertex is incident to a crossing edge, and every crossing edge has exactly one endpoint in $A$. The maximum degree is four. This proves the first two inequalities. Counting both endpoints of the crossing edges gives

Substitution proves the conditional entropy statements.

The factor $2$ would generally be false if $Z_A$ meant only the inner endpoint set. Similar notation is not a license to change a boundary convention.

For (eq:H), finite-range terms lie in individual connected components. If the full ground state is unique, it is the tensor product of the unique component ground states. In particular, if ${\partial }_{\Lambda }A=\varnothing$, then $S_{\Omega }(A)=0$.

Proof. A finite-diameter support cannot meet two distinct components. Hence $H$ is a tensor sum of component Hamiltonians. Ground energies add and ground-space dimensions multiply. Uniqueness forces each component ground space to be one-dimensional. A zero-boundary set is a union of components, whose joint state is pure and uncorrelated with the remaining components.

The lemma handles zero-boundary cases without division by zero and without an appeal to the external area-law assertion.

A finite $\mathsf{AL}$-record specifies the graph, tensor factors, Hamiltonian, normalized ground vector, declared gap certificate, boundary convention, and any independently supplied entropy allowance $C_0$ such that

on its stated region class.

It is a record of that state and those inequalities. It contains no logical encoding, radiation factor, emission sequence, or gravitational identification unless those objects are added explicitly.

No universal assertion about all such records is needed to construct or use an individual record. Product-state and dimer examples below have direct certificates.

The supplied proof description has the following architecture. The audit column states what must actually be verified before a stage can be used.

In particular, one must not replace the first stage by an area law for arbitrary regions, introduce a restricted-Hamiltonian gap in stage 2, or replace an unconditional information inequality by a normalized postselected branch.

Let $K=H-E_{0}I$ obey (eq:gap). Then

is well defined and satisfies

These spectral statements provide no graph-local decomposition or tail estimate for $Q$.

Proof. On the ground vector the spectral value is zero. On an orthogonal eigenvector with eigenvalue $\lambda \ge \Delta$, the spectral value is $\lambda /(\lambda +\Delta )\in [1/2,1]$. The assertions follow from the spectral theorem. No support conclusion occurs in this argument.

Likewise, the local operators $I-P_{\mathrm{supp}{\rho }_X}$ annihilate a pure state, but their sum need not have a unique ground state or a positive comparison gap. For a full-Schmidt-rank entangled pair, each one-site support projector is the identity, so all such one-site constraints vanish. Constructing useful positive quasi-local constraints is substantive additional work.

For disjoint systems, write

We use ordinary entropy subadditivity, strong subadditivity, and finite-dimensional channel conventions LiebRuskai,Watrous.

For a density matrix $\rho$, $p>0$, and $n\ge 1$,

and

For a bipartite pure vector with marginal ${\rho }_X$,

where inverse powers act only on the positive spectral support. Also,

Proof. Diagonalize $\rho$. Tensor-product eigenvalues multiply and entropies add. The derivative follows by differentiating the finite sum over positive eigenvalues. The vector-norm identity follows from the Schmidt decomposition. Substitution into the four entropy terms proves conditional-information additivity.

These identities neither justify an unspecified interchange of replica limits nor supply the stage-3 norm or energy estimates. A replica tensor product is not a gravitational replica saddle.

Let $\psi$ be pure on a finite site set $\Lambda$, and suppose

is a partition. Fix an order on each finite ${\mathcal{I}}_f$, and put $B=\Lambda \setminus A$. If

then

Proof. Let $A_f={\bigcup }_{i\in {\mathcal{I}}_f}{X}_i$ and $E_f={\sum }_{i\in {\mathcal{I}}_f}{\epsilon }_i$. The chain rule gives

Purity gives $S(B)=S(A)$ and $S(B{A}_f)=S(D{A}_{1-f})$. Subadditivity gives

Add the inequalities for $f=0,1$, cancel all tile entropies, and divide by two.

This is the supplied cancellation passage, with its finite argument verified locally OpenAI265A. The lemma alone does not construct its partition or establish the displayed mutual-information bounds.

Let $r=\lfloor R\rfloor$, and put

For every $x\in \Lambda$, the number of allowed supports in (eq:H) containing $x$ is at most ${\nu }_r$, and

Proof. Every such support lies in the graph ball of radius $r$ about $x$. That ball is contained in an ambient ${\ell }^1$ ball with $v_r$ lattice sites. There are at most $2^{v_r-1}$ subsets containing the centre. Some are not allowed supports, which only improves the bound.

The count is deliberately crude, but explicit. It illustrates why one must specify interaction multiplicity before pricing any universal constant.

Define the extended-real optimal allowance

where the supremum runs over the frozen instances and cuts of positive boundary. We set a supremum over an empty class to zero. This definition does not assert that the supremum is finite.

For $a>0$,

The quantity is nondecreasing in $J$ and $R$, and nonincreasing in $\Delta$. For $R=0$, it is zero. For $R\ge 1$ and $0<\Delta \le J$,

Proof. Scaling every Hamiltonian by $a$ is a bijection between the two classes and preserves its ground state. The monotonicity statements follow from class inclusion.

When $R=0$, all terms are on-site. A unique ground state is a product, so all cut entropies vanish. For the lower bound, take a two-site induced edge and

Its gap is $J$, and its one-edge cut entropy is $\log q$.

A proposed formula depending on $J$ and $\Delta$ through their absolute energy units, rather than dimensionless ratios, must explain why it violates or compensates for this rescaling.

For $t\in \mathbb{N}$, write

Let $\psi$ be pure on a finite induced square-lattice domain. For every positive-boundary cut $A$, suppose one supplies a partition satisfying Lemma lem:cancellation, together with constants

independent of the domain and the cut, such that:

1. $D\subseteq {Z_A}^{[t]}$;

2. the tiles have integer levels $j\ge 0$;

3. the number of level-$j$ tiles is at most $g_1|{\partial }_{\Lambda }A|2^{aj}$;

4. each level-$j$ information charge obeys ${\epsilon }_i\le K{2}^{-(a+\delta )j}$.

Then

Proof. Each endpoint of a crossing edge contributes at most $v_t$ sites to ${Z_A}^{[t]}$. Since $|Z_A|\le 2|{\partial }_{\Lambda }A|$,

Thus $S(D)\le 2v_t|{\partial }_{\Lambda }A|\log q$. Summing the information charges gives

Apply Lemma lem:cancellation.

The theorem prices a terminal certificate. It does not establish its four hypotheses from a spectral gap. Those hypotheses identify a concrete route by which the external proof's constants could be exposed.

More generally, independently proved bounds

give the exact terminal allowance

En02-A's one-shot environment and register accounting is a task ledger, not an entropy coefficient. Only its supplied labels and statements are attributed here; no unsupplied mesh formula is invented CheungEn02A.

En02-B's tree result prices a specified decoder module:

with its stated gate, depth, and record bounds CheungEn02B ({thm:tree}). These are decoder resources, not a PEPS virtual dimension, a boundary collar width, or an entropy charge.

The earlier Oai283-A development normalizes a full-diamond allowance $1/2$ as a half-diamond allowance $1/4$. That normalization is correct but gives none of $t,g_1,K,\delta$ in Theorem thm:terminal Oai283Draft.

The plan's phrase “the only missing number” requires correction. Within the documentary conclusion, $C$ is existential. Within this audit, the derivation and quantitative uniformity of the external proof stages also remain unverified. An unspecified constant is not the sole outstanding proof obligation.

For dimensional consistency, a possible successor conjecture is: there exist absolute $K_0,p>0$ such that, for $R\ge 1$,

This is an unproved proposal, not a consequence of the catalogue, not a bound extracted from its preprint, and not a hypothesis used elsewhere in this paper. Even this deliberately generous dependence must be proved rather than inferred from the word “finite.”

On an open $L\times L$ square, for the stated unique uniformly gapped nearest-neighbour class, the external companion asserts existence of a nonzero PEPS ${\Psi }_L$ such that

with uniform constants for fixed local parameters. The tensors may depend on $H$ and $\Omega$. No efficient tensor-production or contraction algorithm is asserted in the frozen record OpenAI265B. This statement is not assumed in the proofs below.

For a nonzero pure PEPS with edge dimensions $D_e$, its normalized state satisfies

In particular, a maximum bond dimension $D$ gives $S(A)\le |{\partial }_{\Lambda }A|\log D$.

Proof. Cut all virtual edges crossing the partition. Expanding their indices writes the physical vector as a sum of at most ${\prod }_{e\in {\partial }_{\Lambda }A}{D}_e$ product vectors across the cut. Its Schmidt rank is no larger, and the entropy is at most the logarithm of that rank.

This is a representational fact about tensor networks VerstraeteCirac. It neither supplies tensors nor computes their contraction.

If $D_L\le C_{\mathrm{P}}{L}^{c_{\mathrm{P}}}$, the direct cut argument gives

The extra $\log L$ is not the uniform constant asserted in 265-01. No equivalence between the two external statements is proved by this cut argument.

For $D\ge 2$, define the monotone entropy modulus

where

Set ${\omega }_1(u)=0$.

For normalized pure vectors $\psi ,\varphi$ with $\Vert \psi -\varphi \Vert \le \epsilon \le 1$,

Proof. The trace distance of the pure states is at most their vector distance. Partial trace contracts trace distance. Apply the sharp finite-dimensional entropy continuity inequality to the smaller side of the pure bipartition Audenaert. The capped modulus is its maximum over distances at most $\epsilon$.

For macroscopic cuts on an $L\times L$ square, vector error $L^{-1}$ allows an entropy discrepancy of order $L\log q$, not a size-independent calibration of every subsystem entropy.

Let $p_1\ge p_2\ge \cdots$ be a Schmidt probability distribution. For $D\ge 1$,

Keeping and renormalizing its first $D$ Schmidt components gives a vector approximation with squared error at most $2{\sum }_{j>D}{p}_j$.

Proof. For $j>D$, $p_j\le 1/j\le 1/(D+1)$. Hence

If the discarded weight is $\tau$, the optimal normalized truncation has overlap $\sqrt{1-\tau }$, so the squared vector distance is $2(1-\sqrt{1-\tau })\le 2\tau$.

The estimate can require exponentially large $D$ as a function of $S/{\epsilon }^2$, and separate cut truncations need not glue into one PEPS. Thus an entropy upper bound alone is not the polynomial-PEPS construction theorem.

An algorithmic theorem must fix at least:

1. finite encodings and bit lengths for the local matrices;

2. the gap and uniqueness promises and how they are supplied;

3. the accuracy, normalization, and global phase convention;

4. output tensor dimensions and entry precision;

5. running time and success probability;

6. which contraction, normalization, or verification tasks are included.

Existence of tensors with arbitrary real entries does not itself bound their bit complexity or give a method to find them.

Generic PEPS contraction has strong complexity obstructions SchuchEtAl. Those generic results do not, without a promise-preserving reduction, prove hardness of the particular unique, uniformly gapped ground-state class frozen here.

For a normalized $\varphi$ and a Hamiltonian obeying (eq:gap), put $\eta =⟨\varphi ,H\varphi ⟩-E_0$. Then

and

In particular, the sufficient total-energy tolerance for vector error $L^{-1}$ is $\eta \le \Delta /(2L^2)$.

Proof. Take the expectation of (eq:gap). If $F=|⟨\Omega ,\varphi ⟩|$, then

On $L^2$ sites, the corresponding sufficient energy-density tolerance is $\Delta /(2L^4)$. A fixed energy-density accuracy in a thermodynamic-first optimization problem is not that guarantee.

Accordingly:

• Family 281's QAOA/SK energy-approximation record has a different model and order of limits;

• Family 283's supplied-oracle synthesis record, and the conditional Oai283-A certification interface, do not find the ground vector or its PEPS tensors;

• En02-B's exact streaming decoder for a declared code is not a many-body ground-state approximation algorithm;

• destructive Enter is a measurement interface, not a lattice Hamiltonian or a preparation circuit.

These are scope comparisons, not reductions OpenAIOverview,CheungEn02B,CheungI,Oai283Draft.

There are qubit Hamiltonians on even open $L\times L$ squares with $R=0$, $J=1$, and gap one above the ground space, containing ground vectors with

Thus the extension to every vector of an unrestricted degenerate ground space is false.

Proof. Let $A$ be the left half of the square. Pin one site $x_0\in A^c$ with

The spectrum is $\{0,1\}$, with a highly degenerate ground space.

Choose $m=L^2/2-1$ disjoint pairs joining distinct sites of $A$ to distinct sites of $A^c\setminus \{x_0\}$. Put a Bell pair on each pair, and put $|0⟩$ on the two unpaired sites, including $x_0$. This is a ground vector. Each Bell pair contributes $\log 2$ across the vertical cut.

Here the correct degenerate gap inequality is

not (eq:gap) with one selected vector. The latter already excludes any second orthogonal ground vector.

A topologically motivated degenerate extension must add real structure, such as a restricted ground-space class and relevant local indistinguishability or boundary conditions. A universal topological correction does not repair the volume-law example.

On a two-qubit induced edge, let

These Hamiltonians have a unique ground vector and full gap at least $3/4$. The restricted Hamiltonian consisting of terms supported entirely in $\{a\}$ has gap $ϵ$.

Proof. The first operator has eigenvalues $0,1,1,1$, and the perturbation is positive with norm $ϵ$. By the min–max principle, the second eigenvalue of $H_{ϵ}$ is at least one, while the first is at most $ϵ$. They are separated by at least $1-ϵ\ge 3/4$, which also proves uniqueness. The restriction is $ϵ|1⟩⟨1{|}_a$.

Adding a subsystem-gap assumption would change the frozen theorem. It is not a harmless audit convenience.

There is a unique, nearest-neighbour, unit-gap ground state satisfying

for every $A$, but no constants $\alpha ,\gamma$ satisfy

for all its interior one-site regions.

Proof. Choose a sufficiently large open square, two neighbouring interior sites $a,b$, and another interior site $c\notin \{a,b\}$. Let the ground state be a Bell pair on $a,b$ and $|0⟩$ elsewhere, with Hamiltonian

The disjoint projectors give a unique ground state and gap one. Only cuts separating $a,b$ have nonzero entropy, at most $\log 2$, and every such cut crosses their edge.

Both $\{a\}$ and $\{c\}$ have boundary size four, but their entropies are $\log 2$ and zero.

The strict-area-law theorem of Kim–Lin–Ranard–Shi uses exact entropic conditions in its own geometric setting and constructs a commuting parent under those hypotheses KimLinRanardShi. An upper inequality does not supply its hypotheses. Literal equalities also require care about empty regions, topology, and boundary conventions; a topological term is not defined by fitting an upper allowance.

The plan's blanket exclusion of all Rényi and mutual-information consequences is stronger than necessary.

If a pure state satisfies $S(A)\le C|{\partial }_{\Lambda }A|$, then:

1. for every $\alpha >1$, including the limiting min-entropy case,

   $$S_{\alpha }(A)\le S(A)\le C|{\partial }_{\Lambda }A|;$$
2. $$I(A:A^c)=2S(A)\le 2C|{\partial }_{\Lambda }A|;$$

3. for disjoint $X,Y$,

   $$I(X:Y)\le 2\min \{S(X),S(Y)\}.$$

   These do not give a distance-decay theorem, an $\alpha <1$ Rényi bound, or a topological correction.

Proof. For eigenvalues $p_i$ and $\alpha >1$, Jensen's inequality gives

Taking logarithms and dividing by $1-\alpha <0$ proves the Rényi assertion. The limiting case follows by continuity. Purity proves the complementary mutual-information identity. The last assertion follows from the Araki–Lieb inequality.

For $N\ge 3$, let ${\delta }_N=1/\log N$, and consider

Then

For fixed $0<\alpha <1$,

These are Schmidt spectra of finite bipartite pure states. The example disproves a spectrum-only implication; it is not claimed to be a counterexample in the frozen gapped-local class.

On a connected square lattice, the GHZ vector has entropy $\log 2$ for every nonempty proper subset, so it satisfies an area allowance with coefficient $\log 2$. Two separated nonempty subsystems have mutual information $\log 2$ whenever their union has nonempty complement. This example is not asserted to be a unique uniformly gapped local ground state. It shows why the collar estimates cannot be recovered from an entropy allowance alone.

For comparison, the maximally mixed product state has zero connected correlations and zero mutual information across disjoint regions, yet $S(A)=|A|\log q$. This is a mixed-state example, not an instance of the pure-ground-state theorem. Correlation decay and pure-state entanglement entropy are different inputs.

The family descriptions are documentary labels from the supplied catalogue, not verified premises OpenAIOverview. No theorem from those families is needed for the explicit counterexamples above.

For states and channels, use

The diamond norm includes arbitrary untouched reference systems Watrous.

Let $C\simeq {\mathbb{C}}^d$, $d\ge 2$, and let $V:C\to R\otimes B$ be an isometry. Its accessible and complete complementary channels are

Define information-theoretic recovery error and complete leakage by

A resource-constrained decoder is a further, separate optimization.

Let ${\tau }_C=I_C/d$. Then

is CPTP and $\mathcal{L}{\mathrm{Ad}}_V={\mathrm{id}}_C$. Also, for every reference operator $X$,

Proof. Compression and effect-measurement followed by fixed-state preparation are CP. Their traces sum to $\mathrm{Tr}Y$. On the encoded range the second term vanishes and $V^{†}V=I$. Isometric embeddings preserve nonzero singular values, proving the norm identity.

The domain of $\mathcal{L}$ is $R\otimes B$, not $R$. This is the complete-output distinction in En07-B CheungEn07B ({thm:global}).

The following are equivalent:

1. a CPTP map $\mathcal{D}:R\to C$ satisfies $\mathcal{DN}={\mathrm{id}}_C$;

2. $\mathcal{E}(X)=\mathrm{Tr}(X){\sigma }_B$ for one fixed state ${\sigma }_B$;

3. $\mathcal{E}(E_{ij})={\delta }_{ij}{\sigma }_B$ on all logical matrix units.

If $\mathrm{rank}{\sigma }_B=r$, then $\dim R\ge dr$. All identities preserve arbitrary untouched references.

Proof. The equivalence of (ii) and (iii) is linearity.

Assume (iii), and diagonalize

Write

The matrix-unit identity implies

Thus $dr$ orthonormal vectors occur in $R$. Let $K\simeq {\mathbb{C}}^r$, put

Then

An explicit deterministic decoder is

It is CPTP by the same trace argument as in Lemma lem:global, and it exactly recovers $X$.

Conversely, dilate an exact decoder as an isometry $L:R\to C\otimes G$. The composite $Z=(L\otimes I_B)V$ dilates the identity. Since its decoded marginal on input $|i⟩$ is pure,

Recovery of $E_{ij}$ gives $⟨{\xi }_j,{\xi }_i⟩=1$; hence all normalized ${\xi }_i$ are the same vector. Tracing $C,G$ gives (iii). Every assertion is an operator identity, so reference extension preserves it.

This is a standard exact-correction criterion KnillLaflamme, proved here in the finite form compared in En07-B/C CheungEn07B,CheungEn07C. The dimension count $\dim R\ge dr$ is not a lattice area law.

For $d\ge 2$,

The maximally mixed replacement attains the infimum. Consequently every constant accessible channel has optimal recovery error $1-d^{-2}$.

Proof. On a maximally entangled input, the identity output is $|{\Omega }_d⟩⟨{\Omega }_d|$, while a replacement output is $I_d/d\otimes \sigma$. Their overlap against the ideal Bell projector is $d^{-2}$, independently of $\sigma$. The effect test therefore gives the lower bound.

For the upper bound, let $(W_{ab}{)}_{a,b=0}^{d-1}$ be the shift-phase Weyl unitaries. Their twirl is

Hence

The diamond norm is at most $2(1-d^{-2})$. Division by two gives equality. A decoder after a constant channel is still constant and can prepare $I/d$, proving the final assertion.

With $R\simeq B\simeq C\simeq {\mathbb{C}}^d$, define

For every pure logical input, both accessible states have entropy zero and the same spectrum. Nevertheless,

whereas

Proof. The accessible channels are identity and replacement; the complements are replacement and identity. Apply Lemma lem:constant.

For an explicit reference-sensitive distinction, on a Bell input the good accessible output is maximally entangled with the reference, with mutual information $2\log d$. The bad accessible output is uncorrelated with it, with mutual information zero. This information is absent from the selected pure-input accessible entropy.

Let ${\sigma }_K$ be any finite state, with purification $|\chi {⟩}_{KP}$. Set $R=C_R\otimes K$, $B=P\otimes L$, with $L\simeq C$, and define

For every pure input, the accessible spectra coincide and both entropies equal $S({\sigma }_K)$. If $H_R=I_{C_R}\otimes H_K$, all accessible energy-measurement distributions coincide as well. The optimal recovery errors are still $0$ and $1-d^{-2}$.

Proof. The accessible channels are

A pure logical factor changes neither the nonzero spectrum nor the entropy of ${\sigma }_K$. Spectral effects of $H_R$ act only on $K$. Tracing $K$ decodes the first channel, and the second is constant.

Taking ${\sigma }_K$ to be Gibbs gives the precise selected-thermal-data comparison in En07-C CheungEn07C ({prop:entropy}). It does not identify the whole channel from its occupation data.

For

the ground state is unique, the gap is one, all subsystem entropies vanish, and the state is an exact bond-dimension-one PEPS. Its Schrödinger evolution under $H_0$ is stationary.

Proof. The on-site commuting projectors have a common one-dimensional kernel and smallest positive eigenvalue one. The vector is a product and has energy zero.

A stationary choice of a radiation subsystem also has entropy zero. There is no radiation history until radiation factors and their time-dependent access are specified.

Choose adjacent qubit sites $a,b$, and let

Define

Every $K_{\theta }$ has a unique ground state ${\Omega }_{\theta }$, gap one, and local interaction norms at most one. For every cut,

The states have exact PEPS representations of bond dimension at most two.

A cut separating $a,b$ has entropy

Thus the constant path $\theta (t)=0$ and a path increasing from $0$ to $\pi /2$ have the same uniform gap and spatial allowance but different entropy histories.

Proof. The pair projector has a one-dimensional kernel and gap one. All other projectors act on disjoint sites. If a cut does not separate $a,b$, its reduction is pure. Otherwise the only mixed factor has eigenvalues ${\cos }^2\theta ,{\sin }^2\theta$, with entropy at most $\log 2$. The edge $\{a,b\}$ crosses every such cut. A Schmidt expansion on that one edge gives bond dimension two. The entropy is zero at the endpoints and $\log 2$ at $\theta =\pi /4$.

This corrects corrupted transcriptions in which the eigenvalues or $\log 2$ lost their trigonometric or logarithmic symbols. The result is the finite path comparison of the supplied Oai265-A source, proved locally here CheungOai265A ({prop:gappedpath}).

Let

Then $G$ is Hermitian and $e^{-i\theta G}|00⟩=|{\chi }_{\theta }⟩$. A differentiable path is prepared by

In contrast, $K_{\theta (t)}{\Omega }_{\theta (t)}=0$. Unless the vector is constant, it does not solve the Schrödinger equation generated by that instantaneous parent Hamiltonian. The ground-state family and its preparation dynamics are different data.

There are finite sequential isometric processes with the same logical dimension, the same emission-register dimensions, and exact final logical recovery, whose pure-input radiation entropies are, respectively,

after three emissions, for any $k\ge 2$. Both can be appended to the same independent $\mathsf{AL}$-record.

Proof. Use outgoing registers $R_1,R_2\simeq {\mathbb{C}}^k$ and $R_3\simeq C$. In the first process, emit $|0⟩$ into each of $R_1,R_2$, then emit the logical state into $R_3$. In the second process, first create a $k$-dimensional Bell pair between $R_1$ and a retained register, emit its retained half as $R_2$, then emit the logical state as $R_3$. Every step is isometric on the admitted input with fresh pure ancillas. The first radiation prefix is mixed only in the second process. After the second emission the pair is pure inside the radiation, and final recovery is tracing $R_1,R_2$.

This is an engineered finite entropy turnover, not a derivation of a black-hole Page curve.

Let $\Lambda :C^2\to C^2$ be one implemented deterministic decoder channel, including all its branches. Define

The Bell projector obeys

Measure the three oriented observables

on separate independent preparations. If their ideal means are $r_s$, then

Before collecting records, specify:

1. the channel, its preparation, every branch and failure outcome, and the measurement settings;

2. $N$ attempted observations per setting, with independent fresh runs and no unpriced postselection or stopping rule;

3. a bias allowance $b_{\mathrm{cal}}$ for each oriented mean;

4. a half-diamond transfer allowance $b$ between the tested channel and any separately named target implementation.

Independence, calibration, and transfer are assumptions to be established for the experiment, not consequences of simulation.

Set

Under Specification spec:stat, with probability at least $1-\alpha$, the separately compared target channel has half-diamond error from identity in

Proof. Hoeffding's inequality for observations in $[-1,1]$ gives