Oai290-A Typed Boundary Recovery, Complete Complements, and Modular-Band Witnesses A Nonperturbative Certification Framework with Explicit Geometric and Analytic Boundaries

We give a typed certification framework for a declared finite boundary channel and a specified deterministic decoder, without assuming that quantum corrections are perturbatively small. The framework distinguishes recovery of an unknown quantum code, reconstruction of a named bounded observable, and consistency with a separately declared geometric model.

For a complete finite isometry, we prove the constant-complement criterion, its residual-rank consequence, observable-specific certificates, and dimension-explicit approximate recovery bounds. If a confidence event supplies a normalized decoded Choi-overlap interval $F_L\le F_e\le F_U$, then an identified target channel has half-diamond error in

where $d$ is the logical dimension and $b$ is a separately justified half-diamond transfer allowance. Complete-complement and accessible-state witnesses give lower bounds applying to every decoder, whereas the Choi certificate concerns the specified decoder.

A separate operator-algebra module presents, with attribution and a local proof, the bounded modular-band witness statement of the supplied OpenAI manuscript dated 23 September 2026. We derive its one-way consequence for bounded asymptotic defect tests. This existential modular statement is neither a finite-sample channel certificate nor a bulk–boundary correspondence.

Two defects in the proposed synthesis are repaired. Nonconstant complements obstruct exact full-code recovery, but need not obstruct a named observable. Failure of a sufficient certificate is not an obstruction and does not imply failure of a geometric description. The resulting decision procedure has certified, obstructed, and inconclusive outcomes. The inherited characteristic-map obligations remain documentary open gates: no joint map, quantum coefficient, geometric emergence theorem, or gravitational endpoint is claimed.

. Quantum recovery; boundary access; complete complement; observable reconstruction; diamond distance; modular spectrum; bounded witnesses; geometric identification; source discipline.

The motivating question is often phrased as “bulk reconstruction beyond the semiclassical limit.” That phrase contains at least three different mathematical tasks:

1. recovery of a declared quantum input or a named observable through a declared accessible channel;

2. determination of whether an exact quantum-error-correcting description remains valid;

3. justification of a classical geometric interpretation of the recovered data.

This manuscript answers specified versions of the first two tasks and formalizes the additional evidence required for the third. It does not construct a gravitational dictionary.

Exact quantum error correction, information–disturbance relations, observable correction, and quantum-system certification are established subjects KnillLaflamme,KSW,BenyKempfKribs,KlieschRoth. Holographic reconstruction results additionally use their own bulk–boundary and relative-entropy hypotheses JLMS,DongHarlowWall. Those hypotheses are not supplied merely by naming a channel “boundary” or a logical effect “bulk.”

Cheung's supplied manuscripts motivate the separation of complete data, accessible outputs, residual ownership, destructive readout, and physical correspondence CheungI,CheungHawk04A,CheungHawk04B, CheungEn07A,CheungEn07B,CheungEn07C,CheungXVIIA. The finite statements needed here are proved for the objects defined below. No unavailable model-specific result is an implicit premise.

The written mathematical package consists of:

1. a complete finite-channel contract;

2. exact and approximate full-code recovery statements;

3. an observable-specific reconstruction criterion and obstruction;

4. deterministic and finite-confidence certificates for a fixed decoder;

5. a locally proved, separately typed modular bounded-witness module;

6. a conditional endpoint assembly with four explicitly defined operational obligations;

7. geometric-consistency and analytic-admission ledgers that do not convert missing constructions into no-go theorems.

The underlying channel tools and the bounded-witness statement are not claimed as new foundations. The contribution is their normalization-consistent assembly, the explicit separation of claim types, and the repairs needed to make the proposed plan mathematically valid.

1. Full-code recovery is not one-observable recovery. The constant-complement criterion applies to recovery of every input on the full finite code. A proper observable algebra, or one effect, can remain reconstructible when the complete complement is nonconstant.

2. A named observable does not itself have a diamond norm. Its expectation error is an operator-norm quantity. A half-diamond statement about its readout concerns a specified measurement channel, defined in Section sec:observable.

3. The decision is not an exhaustive success/no-go dichotomy. A sufficient upper bound that exceeds a threshold can be inconclusive. A missing calibration, spectral hypothesis, or correspondence map is not a proof that recovery is impossible.

4. Modular boundedness is a separate analytic implication. The bounded-witness theorem has an invariant-mean hypothesis and an existential operator-norm conclusion. It supplies no finite sample count, decoding channel, or gravitational identification.

5. Recovery failure does not establish geometric failure. A declared geometric model must be tested against its own identified observables and prediction laws.

6. Residual rank is an exact constraint, not a stable error parameter. Small nonzero eigenvalues can change rank without giving a uniform positive approximate-recovery lower bound.

7. Open analytic gates stay open. A bounded operator on a standard Hilbert space does not define an unprovided product of distributions or discharge a characteristic wavefront theorem.

All channel results through Section sec:examples are finite dimensional. The modular module has a different ambient algebra and is introduced separately.

Let $C\cong {\mathbb{C}}^d$, $d\ge 2$, be a logical input. Let $A$ be the accessible output and $B$ the complete complement. A declared complete isometry is

Define

If the physical preparation includes a mixed ancillary state, its purifier is included in $B$. Discarded flags, unavailable modes, and unreturned quantum outputs are included according to the declared dilation.

A decoder is a completely positive trace-preserving map, abbreviated CPTP,

The returned quantum channel is $\mathcal{E}=\mathcal{DN}$. A destructive validation analyzer acts afterward and is not included in the returned quantum task.

The following ledgers are separately declared.

1. Complete-data ledger. The code $C$, the preparation class, all reference systems, and the target quantum channel, effect, or bounded observable. If a physical or gravitational interpretation is intended, its identification map and validity class are additional entries.

2. Accessible ledger. The quantum output $A$, collection operations, available instruments, decoder, schedule, and resource restrictions. A classical record is not silently promoted to a surviving quantum output.

3. Complement ledger. The complete $B$ in the same dilation, including preparation purifiers and every unreturned output. A selected residual marginal is not substituted for $B$.

The use of these ledgers follows the documentary interfaces of Cheung's Landmark I, Hawk04-A/B, and En07-A CheungI,CheungHawk04A,CheungHawk04B,CheungEn07A. Their physical models are not identified with (eq:isometry).

For states and channels, set

For a returned logical channel,

The first quantity concerns a specified decoder. The second permits all deterministic CPTP decoders unless a smaller class is explicitly declared.

Every referenced input state obeys

Consequently every joint measurement has total-variation error at most $e(\mathcal{E})$.

If the accessible process is an instrument $\{{\mathcal{I}}_z{\}}_z$, and branch $z$ has a declared CPTP output operation $D_z$, the deterministic returned channel is

It is CPTP because compositions and sums preserve complete positivity and ${\sum }_z{\mathcal{I}}_z$ is trace preserving.

Successful branches alone are not renormalized into a deterministic certificate. Loss can be completed by a declared fixed output, or retained as a flag in a different output type. These are different tasks, not interchangeable conventions CheungEn02B,CheungEn02C,CheungEm05B.

For (eq:isometry)–(eq:channels), the following are equivalent:

1. there is a CPTP decoder $D$ with $D\mathcal{N}={\mathrm{id}}_C$;

2. there is a density operator ${\tau }_B$ such that

   $$\mathcal{C}(X)=\mathrm{Tr}(X){\tau }_B\text{for every }X\in \mathcal{B}(C);$$

3. in one, hence every, orthonormal code basis,

   $$\mathcal{C}(|i⟩⟨j|)={\delta }_{ij}{\tau }_B;$$

4. for the Kraus family $K_a=(I_A\otimes ⟨a|)V$,

   $${K_a}^{†}{K}_b=c_{ab}{I}_C$$

   for scalar coefficients $c_{ab}$.

If these conditions hold and $r=\mathrm{rank}{\tau }_B$, then

The recovered channel preserves every external reference.

Proof. Suppose $D\mathcal{N}={\mathrm{id}}_C$, and dilate $D$ by an isometry $W:A\to C\otimes G$. Then

dilates the identity channel. The Choi matrix of identity has rank one. Every Kraus operator of $T$, whose vectorizations give positive summands of that matrix, is therefore proportional to $I_C$. Consequently

for a single unit vector $\chi$. Tracing $CG$ proves (eq:constantcomplement).

Conversely, diagonalize ${\tau }_B={{\sum }_{a=1}}^r{\lambda }_a|a⟩⟨a|$, with ${\lambda }_a>0$. The isometry has no component in the kernel of ${\tau }_B$. Write

The complementary matrix-unit identities give

Thus $A$ contains $dr$ orthonormal vectors, proving (eq:rankcapacity).

Let $P$ project onto their span, and define the isometry $U:PA\to C\otimes {\mathbb{C}}^r$ by $U{u}_{ia}=|i⟩|a⟩$. For any fixed code state $\sigma$,

is CPTP and satisfies $D\mathcal{N}(X)=X$.

Matrix units span $\mathcal{B}(C)$, proving the equivalence of conditions 2 and 3. Finally,

so constancy is equivalent to condition 4. Equality of channels remains equality after tensoring with any identity map, proving reference preservation.

This is the constant-complement form of the established Knill–Laflamme condition KnillLaflamme. Comparable finite criteria are recorded in Cheung's Hawk04-A, En07-A, Hawk03-A, Hawk01-B, VII-G, and Em05-B CheungHawk04A,CheungEn07A,CheungHawk03A, CheungHawk01B,CheungVIIG,CheungEm05B. The proof above does not import their model-specific channels.

If $\mathcal{C}$ is nonconstant, then ${\epsilon }_{\mathrm{rec}}(\mathcal{N})>0$. Also, if

exact full-code recovery is impossible.

Proof. In finite dimensions the decoder Choi matrices form a compact set: they are positive, satisfy a fixed partial-trace identity, and have fixed trace. Channel distance is continuous, so the infimum in (eq:errors) is attained. If the minimum were zero, Theorem thm:exact would force a constant complement.

For the rank assertion, exact recovery would identify ${\tau }_B=\mathcal{C}(I_C/d)$, and then (eq:rankcapacity) would hold.

The corollary concerns the chosen full code and access interface. It does not say that every named observable has been lost.

If the only accessible output is a finite classical measurement record, every quantum decoder produces an entanglement-breaking channel and has full-code recovery error at least

For a replacement accessible channel,

the optimal recovery error is

Proof. A decoded classical record has the form

On a referenced input its output is a sum of positive reference operators tensored with ${\sigma }_z$, hence is separable. A product state has overlap at most $1/d$ with a maximally entangled state; convexity gives the same bound for separable states. The Bell-projector effect yields (eq:classicalbound).

Every decoded replacement is a replacement with some state ${\tau }_C$. Its normalized Bell output is $I_Q/d\otimes {\tau }_C$, whose Bell overlap is $d^{-2}$, yielding the lower bound $1-d^{-2}$. Preparing $I_C/d$ attains it: the completely depolarizing channel is the uniform average of the $d^2$ Weyl conjugations. Separating identity and using the triangle inequality gives half-diamond distance at most $1-d^{-2}$.

This separates destructive verification from quantum return, as in Cheung's Landmark I and Hawk04-B CheungI,CheungHawk04B. Destruction of an accessible carrier does not establish physical annihilation of the complete complement.

The adjoint ${\Phi }^{\ast }$ of a channel is defined by $\mathrm{Tr}(\Phi (X)Y)=\mathrm{Tr}(X{\Phi }^{\ast }(Y))$. It is unital and completely positive.

Let $F\in \mathcal{B}(C)$ be an effect, $0⪯F⪯I_C$. Its binary measurement channel is

A boundary effect $G\in \mathcal{B}(A)$, $0⪯G⪯I_A$, reconstructs this readout with error

For every boundary effect $G$,

Thus exact reconstruction of this binary readout is equivalent to ${\mathcal{N}}^{\ast }(G)=F$.

For a specified quantum decoder, $G={\mathcal{D}}^{\ast }(F)$ is an effect and

For any Hermitian $O\in \mathcal{B}(C)$, with spectral width $w(O)={\lambda }_{\max }(O)-{\lambda }_{\min }(O)$,

Proof. Put $H={\mathcal{N}}^{\ast }(G)-F$. On a referenced state ${\omega }_{QC}$, the measurement-channel difference has two diagonal blocks $K$ and $-K$, where

For Hermitian $L$ with ${\Vert L\Vert }_{\infty }\le 1$,

Trace-norm duality gives ${\Vert K\Vert }_1\le {\Vert H\Vert }_{\infty }$. The output half-trace norm is ${\Vert K\Vert }_1$. An unreferenced eigenstate attaining the largest absolute eigenvalue of $H$ gives equality. The characterization of channel distance by referenced states then proves (eq:effectdiamond).

Postprocessing contraction proves (eq:effectfromcode). For (eq:observablefromcode), subtract the scalar midpoint $c{I}_C$, where $c=({\lambda }_{\max }(O)+{\lambda }_{\min }(O))/2$. Both channel adjoints fix identity. For every input state, trace duality and the output trace-distance bound give expectation error at most

The difference operator is Hermitian, so its operator norm is the supremum of these absolute expectation errors.

Observable-algebra correction has a broader established theory BenyKempfKribs. We use only the explicitly proved effect and bounded-observable statements above.

For any two code states $\rho ,\sigma$, every boundary effect $G$ satisfies

For full-code recovery, every decoder satisfies

Proof. Each target expectation differs from its reconstructed expectation by at most ${\eta }_F(G)$. The difference between the two boundary-effect expectations is bounded by the boundary trace distance. The triangle inequality gives (eq:effectlower).

For (eq:accesslower), each decoded state is within $e(D\mathcal{N})$ of its ideal input. Apply the triangle inequality and contraction under $D$.

The optimal effect-only error is the finite semidefinite optimization

A verified feasible $G,t$ gives an upper certificate. A favorable unvalidated numerical fit is not a certificate.

Let $C,A,B$ be qubits and $V|i⟩=|i{⟩}_A|i{⟩}_B$, $i=0,1$. Both $\mathcal{N}$ and $\mathcal{C}$ are computational-basis dephasing. For $F=|0⟩⟨0|$, boundary effect $G=|0⟩⟨0|$ gives ${\eta }_F(G)=0$. Nevertheless exact full-code recovery is impossible, and the identity decoder has error $1/2$.

Proof. Dephasing fixes $F$, so ${\mathcal{N}}^{\ast }(G)=F$. The complement distinguishes the two basis inputs and is nonconstant. For the identity decoder,

The triangle inequality gives error at most $1/2$. On a Bell input the output is an equal mixture of two orthogonal Bell states, giving distance $1/2$ and equality.

Cheung's En02-B likewise distinguishes a logical readout or rank-one instrument from subsystem recovery CheungEn02B (:observable). The sharper one-live-qubit streaming bound in En02-C has additional access and scheduling hypotheses CheungEn02C (:C3); it is not a universal statement about every one-qubit channel.

For logical dimension $d$, define

A channel's normalized Choi state has trace one.

For channels with common input dimension $d$,

For a state $\rho$, a rank-one projector $P$, and $F=\mathrm{Tr}(P\rho )$,

Consequently,

Proof. A Bell input proves the first lower bound. For $\Delta =\Phi -\Psi$, write unit referenced vectors as

Then

A singular-value decomposition extends this to all trace-norm-one inputs and all reference dimensions. Divide by two and use the channel-distance cap one.

The effect $P$ proves the pure-target lower bound. Decompose $\rho ={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$. Pure-state trace distance is $\sqrt{1-|⟨\varphi ,{\psi }_k⟩{|}^2}$ for $P=|\varphi ⟩⟨\varphi |$. Trace-norm convexity and square-root concavity give the upper bound. Combining the two inequalities proves (eq:bellbounds).

These standard normalization-sensitive tools agree with FuchsVanDeGraaf,Watrous and the supplied Oai265-B protocol Oai265B. In particular, the mixed-state upper inequality in (eq:purebounds) is not generally an equality.

Suppose the tested returned channel is ${\mathcal{E}}_0$, and the identified target channel is ${\mathcal{E}}_{\star }$, on the same logical interfaces. A proved transfer allowance is

Suppose an event of probability at least $1-\alpha$ jointly gives

Define

On that event,

For every target-code effect $F$,

For every target-code Hermitian $O$,

If ${\mathcal{E}}_{\star }=D_{\star }{\mathcal{N}}_{\star }$ is the specified physical factorization, then also

This last bound does not establish optimality of $D_{\star }$.

Proof. Lemma lem:choipure bounds $e({\mathcal{E}}_0)$. The reverse triangle inequality gives

Theorem thm:effect gives the observable conclusions. Optimal recovery is no worse than the specified decoder or the decoder that ignores its input and prepares $I_C/d$; Proposition prop:classical evaluates the latter.

The transfer allowance is in half-diamond units and is added outside the square root. It is not multiplied by $d$ CheungHawk04B (:certificate).

If

on common identified interfaces, contraction gives $b\le a+k$. Neither comparison follows from a match of selected spectra.

No proof above expands in a gravitational coupling or assumes that $a,k,b$ are small. Large allowances remain legitimate but can make the certificate vacuous. Thus “beyond a small-correction input” means that the inequalities remain valid for the declared channels; it does not mean that an unprovided quantum-gravitational channel has been constructed.

If a physical bounded observable is represented on the code by $O$, and a separately proved dictionary allowance satisfies

for the named preparation class, the complete expectation allowance is

where $a_{\mathrm{readout}}$ is a distinct detector allowance. The dictionary and detector estimates are inputs to this statement. Unbounded observables require an additional domain and energy-constrained analysis absent from this finite contract.

Let $\Delta$ be Hermiticity preserving and $K_{\Delta }=d{J}_{\Delta }$. If a Hermitian $Z$ and $\nu \ge 0$ satisfy

then

Proof. For a pure referenced input $\psi =(A\otimes I){\Phi }_d$, $\mathrm{Tr}(A^{†}A)=d$, its Hermitian output difference $H$ satisfies $-W⪯H⪯W$, where

The first two matrix inequalities imply $Z⪰0$. Testing against the positive and negative spectral projections of $H$ gives

For arbitrary input operators, Hermitian block dilation preserves input and output trace norms for a Hermiticity-preserving map. Jordan decomposition reduces Hermitian inputs to states; convexity reduces states to pure states. This proves the diamond bound.

Semidefinite formulations are standard Watrous. Numerical feasibility, positivity, trace preservation, and rounding must be validated before this lemma is used as a certificate.

Define the complete leakage distance

The infimum is attained because the finite state space is compact.

We use two standard finite-dimensional fidelity facts in the sufficiency direction below:

1. root fidelity $f(\rho ,\sigma )={\Vert \sqrt{\rho }\sqrt{\sigma }\Vert }_1$ satisfies $f(\rho ,\sigma )\ge 1-d_{\mathrm{tr}}(\rho ,\sigma )$ FuchsVanDeGraaf;

2. purifications can be aligned, after enlarging the purifying space, with overlap $f(\rho ,\sigma )$ Uhlmann.

These are explicitly identified standard inputs, not consequences of the supplied modular manuscript.

For the complete finite interface,

The upper bound is an information-theoretic existence statement. It is not a resource certificate for an implemented decoder.

Proof. Fix any decoder $D$ and dilate it to $W:A\to C\otimes G$. Apply $(W\otimes I_B)V$ to one half of a Bell vector, producing a pure vector $\Psi$ on $Q,C,G,B$. Let $F_e$ be the decoded Bell overlap.

For $F_e>0$, project $\Psi$ onto ${\Phi }_d$ in $QC$ and normalize the environmental projection to $\chi$. The pure-state distance from ${\Phi }_d\otimes \chi$ is $\sqrt{1-F_e}$. Tracing $C,G$ gives some ${\tau }_B$ with

By Lemma lem:choipure,

If $F_e=0$, the same final inequality is trivial because $e(D\mathcal{N})=1$ and $\lambda \le 1$. Taking the infimum over $D$ proves the lower bound.

For the upper bound, choose a minimizing ${\tau }_B$ and put $\delta =\lambda (\mathcal{C})$. The Bell-input comparison gives

The second state has a purification ${\Phi }_d{}_{QC}\otimes {\chi }_{BG}$. The first has the purification obtained from $V$, with purifying system $A$. After enlarging $G$, purification alignment supplies an isometry $U:A\to C\otimes G$ whose aligned overlap with this product purification is at least $1-\delta$.

Define $D(Y)={\mathrm{Tr}}_G(UY{U}^{†})$. Its Bell overlap is at least $(1-\delta {)}^2$, because projection onto ${\Phi }_d$ includes the particular environmental vector $\chi$. Therefore

The constant-decoder alternative gives the other cap.

Sharper dimension-independent continuity theorems are available in the information–disturbance literature KSW. The weaker bounds (eq:leakagebounds) suffice here and keep the local derivation explicit.

Suppose states $\rho ,\sigma$ and a complementary effect $H_B$ give

Then every deterministic decoder obeys

A proved lower confidence bound $q_L$ can replace $q$ on its confidence event.

Proof. The complementary output trace distance is at least $q$. For every ${\tau }_B$, its two distances from the corresponding outputs are each at most $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\tau }_B})$. Thus $q\le 2\lambda (\mathcal{C})$. Apply Theorem thm:leakage.

A nonconstant finite complement always has some such positive witness: if it agreed on all states, linearity and the spanning property of states would make it a replacement channel. This establishes exact nonrecoverability, but a small positive $q$ need not obstruct a prescribed positive error threshold.

If $\dim A=m<d$, every decoder has

Proof. A state on $Q\otimes A$ has a pure-state decomposition with Schmidt rank at most $m$. Local Kraus maps on $A$ cannot increase that rank. Every decoded Bell-output state therefore has Schmidt number at most $m$. A unit vector of Schmidt rank at most $m$ has squared Bell overlap at most $m/d$, by Cauchy–Schwarz. Convexity extends this bound to the decoded state. The Bell-projector test proves the claim.

This section supplies one way to instantiate Theorem thm:certificate; it does not report an experiment. The construction follows the same declared-channel discipline as Oai265-B and Cheung's bounded complete-record interfaces Oai265B,CheungEn01C.

For a qubit,

Let $s_x=1,s_y=-1,s_z=1$. Every attempted trial in setting $j$ has a predeclared score $Z_{ji}\in [-1,1]$, including failure records.

Conditional on training and calibration data, assume independent scores within each setting, fixed counts $n_j$, means $m_j$, and a common pre-analyzer channel ${\mathcal{E}}_0$. Set

On a calibration event $G$, assume

Suppose the actual common pre-analyzer state $\sigma$ obeys $d_{\mathrm{tr}}(\sigma ,J_{{\mathcal{E}}_0})\le \delta$. Let the complete score observable $B_j$ satisfy

Then a valid bias allowance is

Proof. Replacing $B_j$ by $A_j$ changes the fidelity combination by at most ${\sum }_j{\mu }_j/4$. The ideal combination on $\sigma$ is $\mathrm{Tr}(P_2\sigma )$. Since $P_2$ is an effect, its difference from $\mathrm{Tr}(P_2{J}_{{\mathcal{E}}_0})$ is at most $\delta$.

For example, independently proved preparation, reference, and joint cross-talk half-distance bounds can give

This requires a fixed environment preparation and an identified channel. A reduced preparation estimate alone does not justify an input-correlated system–environment assignment.

Let ${\alpha }_{\mathrm{s}}>0$ and ${\alpha }_{\mathrm{s}}+\beta \le \alpha <1$. Define

and

With ${\left[u\right]}_{[0,1]}=\max \{0,\min \{1,u\}\}$, set

These endpoints instantiate Theorem thm:certificate with coverage at least $1-{\alpha }_{\mathrm{s}}-\beta$.

Declaring “certified” only when $U\le \epsilon$ gives

Proof. Hoeffding's inequality Hoeffding gives, conditional on the calibration record,

A union bound gives simultaneous score-mean coverage. On that event,

when $G$ also holds. Clipping preserves coverage. Integrating the conditional statistical bound and adding $\mathrm{Pr}(G^c)\le \beta$ proves the result. An invalid acceptance lies in the upper-certificate failure event.

Exact binomial alternatives are available for iid binary pass records, with their own complete-record contract Oai265B. No new execution of that predecessor's script, and no transfer of its simulation results to modular or gravitational data, is claimed.

Fixed-sample formulas must not be applied after unaccounted optional stopping. Calibration, decoder selection, missing readout, reference disturbance, and setting-dependent drift are separate parts of the probability model.

The supplied manuscript Bounded recovery for modular spectral averages, dated 23 September 2026, is attributed to OpenAI OpenAI29002. The supplied dossier labels it 290-02 in Family 290 Dossier. The following theorem reproduces its bounded-witness content with a local argument following the supplied proof. This is not a verification of the paper's bicentralizer application, a machine proof, or authentication of its catalogue metadata.

No theorem in Sections sec:exact–sec:statistics uses Family 290 as a premise. Conversely, no finite decoder theorem is a premise of the modular argument.

Let $M$ be a von Neumann algebra with faithful normal state $\phi$. In standard form, put

Write ${\sigma }_t={{\sigma }_t}^{\phi }$ and let $E_D$ be the spectral resolution of $D$. We use the standard modular identities

These are standard-form modular-theory inputs Takesaki. Inner products in this section are linear in the first variable.

The scalar-centralizer assumption is

It is not assumed for finite logical codes.

Let ${\mu }_k$ be uniform probability measure on $[-k,k]$. Fix once and for all a free ultrafilter $\mathcal{U}$ on $\mathbb{N}$. For bounded continuous scalar $f$, define

Boundary estimates make this mean translation invariant and reflection invariant. It is positive and normalized. For bounded continuous $H$-valued $F$, define

This is a seminorm by the finite-interval triangle inequality and passage to the ultrafilter limit.

If $f(u)=\int g(t)e^{itu}dt$, with $f\in {C_c}^{\infty }(\mathbb{R})$, then

The integral is weak, defined against normal functionals.

With predual conventions

one has

Consequently uniformly bounded $w_n$ whose vectors are supported in bands shrinking to $s$ satisfy ${\Vert \phi w_n-e^s{w}_n\phi \Vert }_{M_{\ast }}\to 0$.

Proof. The inverse Fourier transform of $f$ is integrable. The weak integral has the asserted operator-norm bound. Applying it to $\xi$ and using the spectral theorem proves the vector identity.

For the second claim, put $c=e^{s/2}$ and $\zeta =\xi w-cw\xi$. Modular conjugation gives $J\zeta =w^{\ast }\xi -c\xi w^{\ast }$. For $\Vert y\Vert \le 1$, write

Replacing $w^{\ast }\xi$ by $c\xi w^{\ast }$ costs at most $\Vert \zeta \Vert$. Commutation of left and right actions gives $⟨y\xi ,\xi w^{\ast }⟩=⟨y(\xi w),\xi ⟩$. Replacing $\xi w$ by $cw\xi$ costs at most $c\Vert \zeta \Vert$. The resulting expression is $c^2\phi (yw)$, proving (eq:predualband).

On a band $[s-{\delta }_n,s+{\delta }_n]$, functional calculus bounds the last norm by

which tends to zero.

Under (eq:scalarcentralizer), for every $a\in M$ and $\psi \in M_{\ast }$,

Proof. The unitary mean ergodic theorem gives a Hilbert-norm limit in (eq:Hilbertmean). The operator averages are bounded by $\Vert a\Vert$. Translation by fixed $u$ changes them in norm by at most $|u|\Vert a\Vert /k$ for large $k$. Every weak-star cluster point is therefore fixed by $\sigma$ and hence scalar. Evaluation by $\phi$ identifies it as $\phi (a)1$, which also identifies the vector limit.

The modular action on $M_{\ast }$ is norm continuous on each functional: positive functionals are standard-form vector functionals, and general functionals are linear combinations of positive ones. Thus the averages are Bochner integrals and contractions.

Let $N$ be the norm-closed span of $\chi \circ {\sigma }_u-\chi$, $\chi \in M_{\ast },u\in \mathbb{R}$. Its annihilator in $M$ is the scalar centralizer. Hahn–Banach consequently identifies $N=\{\psi :\psi (1)=0\}$. Interval boundary estimates make the averages tend to zero on the generators of $N$, hence on $N$ by contractivity. They fix $\phi$, proving (eq:predualmean).

For $S\in \mathcal{B}(H)$, define $P(S)$ by

Then $P$ is a positive unital contraction, $P(S)$ commutes with every spectral projection of $D$, and

In particular,

No normality of $P$ is asserted.

Proof. The bounded sesquilinear form defines $P(S)$ with norm at most $\Vert S\Vert$. Positivity and unitality follow from the mean. Translation invariance gives $U_u^{\ast }P(S)U_u=P(S)$, hence spectral commutation.

For $L(a)$, the conjugated operators belong to the weakly closed algebra $L(M)$. Their averaged weak limit belongs to $L(M)$ and commutes with the modular group, so is scalar. Evaluation at $\xi$ gives $\phi (a)I_H$. Equation (eq:averagedform) is the definition for $T^{\ast }T$.

Assume (eq:scalarcentralizer). Fix $T\in \mathcal{B}(H)$, $s\in \mathbb{R}$, and the mean above. Let ${\delta }_n>0$ tend to zero, and let unit vectors $h_n$ have $D$-support in $[s-{\delta }_n,s+{\delta }_n]$. If

then a subsequence has $v_j\in M$ and constants $C_{\ast }<\infty ,\eta >0$ such that

This is the bounded-recovery statement of OpenAI29002 (Theorem 4.1); the proof below follows its supplied construction.

Proof. Choose $f_0\in {C_c}^{\infty }(\mathbb{R})$ with $0\le f_0\le 1$, $f_0=1$ on $[-1,1]$, and support in $[-2,2]$. Write $f_0(u)=\int g_0(t)e^{itu}dt$. For $\rho >0$, set

Then ${\Vert g_{\rho }\Vert }_1=C_1$, and

Pass to a subsequence on which the averages in (eq:positiveaverage) are at least $4\gamma$ for some $\gamma >0$. Density of $M\xi$, filtering by $f_{{\delta }_n}(D)$, and normalization give $y_n\in M$ satisfying

and, by sufficiently close approximation,

Here the averaged-seminorm difference is bounded by $\Vert T\Vert \Vert y_n\xi -h_n\Vert$. Discard finitely many terms so that $d_n\le 1$. The modular identity yields

There is no uniform bound on $\Vert y_n\Vert$.

Fix $n$, write $y=y_n$, and define

The functional $\psi$ is a normal state. By Lemma lem:ergodic,

Although $B_k$ varies with $k$,

Thus, with this $n$ fixed, all sufficiently large $k$ satisfy