ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

A Superlinear Count Test Is Not Local-Tame

Example

For the unit cell $C_{0} = [0, 1)^{d}$ , the observable $f (ω) = N_{C_{0}} (ω)^{2}$ is local but is not one of the tests defining the local-tame topology.

Facts & Assumptions

Given: The unit-cell configuration space and the definition of local-tame observables.

[F1]

Every cell configuration is finite, and $N_{C_{0}}$ is measurable with respect to the restriction to $C_{0}$ (Poisson Configuration Reference).

[F2]

A local-tame observable supported in $C_{0}$ must satisfy $∣ f (ω) ∣ \leq b (1 + N_{C_{0}} (ω))$ for one finite constant $b$ (Local-Tame Topology).

Verification

Given: $f (ω) = N_{C_{0}} (ω)^{2}$ .

1.1F1algebra

By [F1], $N_{C_{0}}$ is finite and measurable from the restriction to $C_{0}$ , so its square is a measurable local observable.

2.1givenF1F2constructalgebra∎

Fix any finite $b \geq 0$ and choose an integer $n > b + 1$ . The cell configuration space contains configurations with exactly $n$ distinct points in $C_{0}$ . On each such configuration, $f = n^{2} > b (1 + n)$ , so the tame growth bound in [F2] fails for this $b$ . Since $b$ was arbitrary, $f$ is not tame.

A Superlinear Count Test Is Not Local-Tame

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels