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 lower energy bound does not ensure a countable partition function

Facts & Assumptions

Given: As stated in the item.

[F1]

Countable canonical model: $Z$ is the extended nonnegative sum and a canonical probability law is defined only when $0 < Z < \infty$ .

[A1]

The state set is $N$ and every state has zero energy and unit weight.

Example

Take $I = N$ , $E_{n} = 0$ , and $g_{n} = 1$ for every $n$ . The energy list is bounded below, but $Z (β) = + \infty$ for every real $β$ , so this model has no normalized canonical law.

Verification

Given: The countable model in [F1] with the data in [A1].

1.1F1A1algebra

Every summand is $g_{n} e^{- β E_{n}} = 1$ , so the first $N$ terms sum to $N$ for every $N$ ; their supremum is infinite.

2.1F1step 1.1∎

Thus $Z (β) = + \infty$ , which violates the finite-normalization condition in [F1]; no normalized canonical probabilities $p_{n}$ are defined.

Depends on

Countable canonical model

A lower energy bound does not ensure a countable partition function

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources