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 macrocell count and a nonuniform microscopic distribution

Physical scope

A stipulated two-microstate finite model with a nonuniform probability vector; it is not an equilibrium preparation, measurement or empirical claim.

Example

Let $Ω = {ω_{1}, ω_{2}}$ with $ω_{1} \neq ω_{2}$ , and declare the finite numerical energy map $ϵ : Ω \to R$ by $ϵ (ω_{1}) = ϵ (ω_{2}) = 0$ , so the assigned physical energies are $E (ω_{i}) = 0 J$ . This supplies the finite microstate pair $(Ω, ϵ)$ required in Macroscopic and microscopic entropy quantities; no mechanical internal-energy or equilibrium-macrostate identification is assumed. Choose $M = {m}$ and the constant coarse-graining map $π : Ω \to M$ , so both states belong to the macrocell $M_{m} = Ω$ , and assign $p_{1} = 1, p_{2} = 0$ . The probability vector is deliberately nonuniform. The example shows that the macrocell-count definition and the Gibbs–Shannon definition can take different values on separately specified inputs.

Facts & Assumptions

Given: The finite model below and the entropy definitions.

[F1]

Macroscopic and microscopic entropy quantities: $S_{B} (m) = k_{B} \log N_{m}$ , while $S_{G} (p) = - k_{B} \sum_{i} p_{i} \log p_{i}$ , with the zero-probability summand defined as zero.

[F2]

SI quantities and unit conventions: $k_{B} = 1.380649 \times 10^{- 23} J K^{- 1} > 0$ .

[L1]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is strictly increasing on $(0, \infty)$ and $\log 1 = 0$ .

[A1]

The two states are distinct, $ϵ : Ω \to R$ is identically zero, and $π : Ω \to {m}$ is constant. Thus $M_{m} = Ω$ has $N_{m} = 2$ , and $p = (1, 0)$ is a probability vector.

Verification

Given: [F1]–[F2], [L1], and [A1].

1.1F1A1algebra

Since $N_{m} = 2$ , the macrocell entropy is $S_{B} (m) = k_{B} \log 2$ .

1.2F1L1A1algebra

The only nonzero probability term is $1 \log 1 = 0$ ; the $p_{2} = 0$ term is zero by definition. Hence $S_{G} (p) = 0$ .

2.1F2L1step 1.1step 1.2algebra∎

Since $2 > 1$ , [L1] gives $\log 2 > \log 1 = 0$ . With $k_{B} > 0$ from [F2], step 1.1 is strictly positive while step 1.2 is zero, so these two defined quantities differ in this model. This example asserts no general inequality or thermodynamic identification.

A macrocell count and a nonuniform microscopic distribution

Example

Facts & Assumptions

Verification

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