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 compact two-generator mixture attains the d+1 support bound

Physical scope

A constructed finite two-generator mixture with a continuous concave entropy functional; its values are illustrative and are not observations.

Example

Let $K = {0, 1} \subset R$ be the compact generator domain, with dimensionless coordinate $x$ . On the convex ambient domain $[0, 1]$ , define the affine extension $\tilde{s} (x) = s_{0} + a x$ and restrict it to $s = \tilde{s} ∣_{K}$ , where fixed $s_{0}, a > 0$ have units $J m o l^{- 1} K^{- 1}$ . For $x, y \in [0, 1]$ and $t \in [0, 1]$ , $\tilde{s} ((1 - t) x + t y) = (1 - t) \tilde{s} (x) + t \tilde{s} (y)$ , so the extension is concave. Take the barycenter to be $\overset{ˉ}{x} = \frac{1}{2}$ and declare every Borel probability mixture on $K$ with that barycenter feasible.

Facts & Assumptions

Given: The compact domain, entropy map, normalization, and feasible fixed-barycenter mixtures in the Example.

[F1]

Attained entropy-maximizing mixtures on compact phase domains: Under these compactness, continuity, barycenter, and feasibility hypotheses, an entropy-maximizing mixture exists with at most $d + 1$ atoms, here $d = 1$ .

[F2]

The Borel sigma-algebra of a topological space, Probability measures and probability spaces, and Measure space and integral: A Borel probability measure on finite $K$ assigns nonnegative weights summing to one, and the integral of a simple function is the finite weighted sum of its values. The supplier Attained entropy-maximizing mixtures on compact phase domains defines $δ_{x} (A) = 1$ when $x \in A$ and $0$ otherwise for Borel $A \subseteq K$ .

Verification

1.1givenF2algebra

Write $w_{0} = P ({0})$ and $w_{1} = P ({1})$ . By [F2], $w_{0} + w_{1} = 1$ and the barycenter equation is $w_{1} = \frac{1}{2}$ . Hence the only feasible measure is $P_{*} = \frac{1}{2} δ_{0} + \frac{1}{2} δ_{1}$ .

2.1givenF1F2step 1.1algebra∎

Its entropy per mole is the finite simple-function integral

\int_{K} s d P_{*} = \frac{1}{2} s (0) + \frac{1}{2} s (1) = \frac{1}{2} s_{0} + \frac{1}{2} (s_{0} + a) = s_{0} + \frac{a}{2} .

Since it is the only feasible mixture, it attains the maximum. Its two support points equal $d + 1$ for $d = 1$ , as in [F1].

A compact two-generator mixture attains the d+1 support bound

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources