ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Finite analyticity versus the global-model cusp and shell inequivalence

Physical scope

The N=2 member and positive transition parameter of the explicitly defined finite global three-sector quantum model. This is not an example of a local or short-range quantum lattice.

Example

For the global three-sector model of Commuting quantum Ising and nonadditive phase countermodel, take $N = 2$ and $β_{c} = (\log 2) / ϵ$ . Its finite Gibbs probability of the one-dimensional middle sector is exactly $1 / 9$ , while a uniform spectral shell with half-width $0 \leq Δ < 2 ϵ$ assigns that sector probability $1$ . The finite partition function is still analytic at $β_{c}$ ; the cusp belongs to the limiting normalized log partition as $N \to \infty$ .

Facts & Assumptions

Given: The three-sector Hamiltonian and finite canonical preparation of Commuting quantum Ising and nonadditive phase countermodel, with $ϵ > 0$ joules.

[F1]

Commuting quantum Ising and nonadditive phase countermodel: for each $N \geq 1$ , $Z_{N} = 1 + e^{- β N ϵ} + 4^{N} e^{- 2 β N ϵ}$ , and a shell with $0 \leq Δ < N ϵ$ contains only the middle sector.

Verification

1.1givenF1algebra

At $N = 2$ and $β = β_{c}$ , $e^{- β N ϵ} = e^{- 2 \log 2} = 1 / 4$ and $4^{N} e^{- 2 β N ϵ} = 16 e^{- 4 \log 2} = 1$ . Hence $Z_{2} = 9 / 4$ and the middle-sector Gibbs probability is $(1 / 4) / (9 / 4) = 1 / 9$ . The finite partition is a sum of positive exponentials, so it is analytic at this parameter.

2.1givenF1step 1.1algebra∎

Since $2 ϵ$ is the distance from the middle energy to either outer energy at $N = 2$ , every shell with $0 \leq Δ < 2 ϵ$ selects exactly the one-dimensional middle sector and gives it uniform probability $1$ . Thus the two specified finite preparations disagree on this observable, while the analytic finite sum and limiting cusp remain distinct claims.

Finite analyticity versus the global-model cusp and shell inequivalence

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources