ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

The middle sector at N=4 and the transition point

Physical scope

Exact N=4 specialization of the stated global three-sector finite-state model; no short-range material system is asserted.

Example

Specialize the finite-sector model of Explicit nonadditive transition and ensemble inequivalence to $N = 4$ at its positive transition parameter.

Facts & Assumptions

Given: The three sectors, energies, and finite exponential-tilt law from Explicit nonadditive transition and ensemble inequivalence.

[F1]

Explicit nonadditive transition and ensemble inequivalence: at $β_{c} = (\log 2) / ϵ$ , the sector weights are $1$ , $2^{- N}$ , and $1$ ; every shell half-width $0 \leq Δ < N ϵ$ isolates $B_{N}$ , and its uniform counting probability is $Q_{N, Δ} (ω) = ∣ S_{N, Δ} ∣^{- 1}$ for $ω \in S_{N, Δ}$ , so the shell probability of $B_{N}$ is $1$ .

Verification

technique · direct

Given: $N = 4$ , $ϵ > 0$ , $β = β_{c}$ , and $0 \leq Δ < 4 ϵ$ .

1.1givenF1algebra

Since $β_{c} N ϵ = 4 \log 2$ , the three aggregate canonical sector weights are $1$ , $e^{- 4 \log 2} = 1 / 16$ , and $4^{4} e^{- 8 \log 2} = 1$ . Thus $Z_{4} = 33 / 16$ and $P_{β_{c}, 4} (B_{4}) = (1 / 16) / (33 / 16) = 1 / 33$ .

2.1givenF1algebra∎

The shell centered at $4 ϵ$ with half-width $Δ < 4 ϵ$ excludes the neighboring energy levels $0$ and $8 ϵ$ , so it contains exactly the single state in $B_{4}$ . Its uniform probability is therefore one on $B_{4}$ , compared with the canonical value $1 / 33$ .

Depends on

Explicit nonadditive transition and ensemble inequivalence

The middle sector at N=4 and the transition point

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources