TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Consistent finite classical laws define a quasilocal C-star state

Statement

Let $q \geq 1$ and, for every $N \geq 1$ , let $μ_{N} : {1, \dots, q}^{N} \to [0, 1]$ satisfy $\sum_{x \in {1, \dots, q}^{N}} μ_{N} (x) = 1.$ Assume consistency under deleting the last coordinate in the explicit sense that, for every $N \geq 1$ and $x \in {1, \dots, q}^{N}$ , $μ_{N} (x) = \sum_{j = 1}^{q} μ_{N + 1} (x, j) .$ For a local matrix $A \in A_{N}$ , $N \geq 1$ , define $ω_{N} (A) = \sum_{x \in {1, \dots, q}^{N}} μ_{N} (x) A_{x x} .$ For each $N \geq 1$ , if $f : {1, \dots, q}^{N} \to C$ and $D_{f}$ is the diagonal matrix with entries $(D_{f})_{x x} = f (x)$ , then $ω_{N} (D_{f}) = \sum_{x} μ_{N} (x) f (x)$ . This rule extends uniquely from the local algebra $A_{l o c}$ to a state on its norm completion $A$ of Quasilocal completion of finite matrix registers. The theorem constructs a classical diagonal state; it does not say that every state on $A$ is diagonal or classical.

Facts & Assumptions

Given: Consistent probability laws $μ_{N}$ and the local embeddings in Quasilocal completion of finite matrix registers.

[F1]

Quasilocal completion of finite matrix registers: each local matrix has a well-defined operator norm, the embeddings preserve it, and $A_{l o c}$ is dense in its C*-completion.

[F2]

States and positive functionals on a C star algebra: a state is a positive bounded linear functional of norm one.

[F3]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: the inner-product pairing obeys $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ .

Proof

1.1givenF1algebra

Under the inclusion $A \mapsto A \otimes I_{q}$ , the diagonal entry at $(x, j)$ equals $A_{x x}$ . Consistency of $μ_{N + 1}$ says $\sum_{j} μ_{N + 1} (x, j) = μ_{N} (x)$ ; substituting this identity into $ω_{N + 1} (A \otimes I_{q})$ gives $ω_{N} (A)$ . Hence the rule is well-defined on the union of positive-length local matrix algebras. Define $ω_{0} (c I) = c$ on $A_{0} = C I$ . It agrees with the $N = 1$ formula because $\sum_{x} μ_{1} (x) = 1$ . Each $ω_{N}$ is linear, and these identities show the rule on the full union is linear and satisfies $ω (1) = 1$ .

2.1givenF1F3step 1.1algebra

For a local matrix $A$ , positivity of the weights gives $∣ ω_{N} (A) ∣ \leq \sum_{x} μ_{N} (x) ∣ A_{x x} ∣ \leq ∥ A ∥$ because $∣ ⟨ A e_{x}, e_{x} ⟩ ∣ \leq ∥ A ∥$ by [F3], and the same marginal calculation makes the functionals consistent. Thus $∣ ω (A) ∣ \leq ∥ A ∥$ on $A_{l o c}$ . If $A_{n}$ is a norm-Cauchy sequence of local matrices, $(ω (A_{n}))$ is Cauchy; define its limit to extend $ω$ . The bound proves independence of approximating sequence, linearity and continuity, and density makes the extension unique.

3.1givenF1F2step 2.1algebra∎

If $A_{n} \to A$ in norm, then $A_{n}^{*} A_{n} \to A^{*} A$ by continuity of multiplication and adjoint in the C*-norm. For local $A_{n}$ , its diagonal entries obey $⟨ A_{n}^{*} A_{n} e_{x}, e_{x} ⟩ = ∥ A_{n} e_{x} ∥^{2} \geq 0$ , so $ω (A_{n}^{*} A_{n}) \geq 0$ . Passing to the limit shows positivity of the extension. Its norm is at most one by step 2.1 and at least one because $ω (1) = 1$ and $∥ 1 ∥ = 1$ . Hence it is a state by [F2].

Consistent finite classical laws define a quasilocal C-star state

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources