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.

A diffusive time step and a four-state entropy reversal

Physical scope

Exact finite specializations of the stated walk, rotor, and permutation models; no observations or collision-derived diffusion law are asserted.

Example

This example evaluates one finite walk step size, one rotor period, and the first three states of the exact finite permutation model in Diffusion-scale walk and recurrence/coarse-entropy counterexamples.

Facts & Assumptions

Given: The three explicitly defined models and their units and conventions from Diffusion-scale walk and recurrence/coarse-entropy counterexamples.

[F1]

Diffusion-scale walk and recurrence/coarse-entropy counterexamples: under $ℓ^{2} = 2 D δ t$ , the walk has variance $n ℓ^{2}$ ; the rotor period is $2 π I / ∣ p_{0} ∣$ ; the finite permutation has coarse entropies $0, \log 2, 0$ and fine entropy $\log 2$ .

Verification

technique · direct

Given: $t = 1$ s, $D = 1$ m $^{2}$ s $^{- 1}$ , $δ t = 1 / 8$ s, $I = 1$ kg m $^{2}$ , and $p_{0} = 1$ kg m $^{2}$ s $^{- 1}$ .

1.1givenF1algebra

The scaling gives $ℓ = \sqrt{2 D δ t} = 1 / 2$ m and $n = ⌊ t / δ t ⌋ = 8$ , so $Var (S_{n}) = 8 (1 / 2 m)^{2} = 2$ m $^{2}$ and $φ_{S_{n}} (k) = \cos ((1 / 2 m) k)^{8}$ . This is a finite model marginal; its Gaussian limit is the separate $δ t ↓ 0$ conclusion of [F1].

2.1givenF1algebra∎

The rotor return period is $2 π I / ∣ p_{0} ∣ = 2 π$ s. The finite permutation calculation has coarse entropy $0$ at times zero and two and $\log 2$ at time one, while the fine entropy remains $\log 2$ at all three times.

Depends on

Diffusion-scale walk and recurrence/coarse-entropy counterexamples

A diffusive time step and a four-state entropy reversal

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources