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.

Repulsive Fermi covariance and zero-temperature Bose occupation step

Physical scope

A symmetric-parameter instance of the finite Fermi atom and a single Bose zero-temperature boundary in the isolated onsite models. No lattice or material-specific conclusion is drawn.

Example

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the inherited Bose occupation-space construction.

For the Fermi atom of Exact onsite Bose and Fermi interacting atoms, set $h = 0$ , $μ = ϵ$ , and $U > 0$ . Then $Ξ_{F} = 3 + e^{- β U}$ and $Cov (n_{↑}, n_{↓}) = (e^{- β U} - 1) / (3 + e^{- β U})^{2} < 0$ . For the Bose site, set $μ = ϵ + U$ with $U > 0$ ; the two zero-temperature minimizing occupations are $1$ and $2$ , each with limiting Gibbs probability $1 / 2$ , and the entropy tends to $k_{B} \log 2$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $β > 0$ , $U > 0$ , and the two exact onsite model statements.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every at most countable family of nonempty sets has a choice function; the Bose conclusion retains this premise from the theorem.

[F1]

Exact onsite Bose and Fermi interacting atoms: at $h = 0$ , $μ = ϵ$ , the Fermi covariance has the displayed formula; for the Bose site the boundary $μ = ϵ + U r$ has exactly the adjacent minimizing occupations $r, r + 1$ with equal zero-temperature limiting probabilities.

Verification

1.1givenA1F1algebra

Substituting $h = 0$ and $μ = ϵ$ into the Fermi weights gives $w_{↑} = w_{↓} = 1$ , $w_{2} = e^{- β U}$ , and $Ξ_{F} = 3 + e^{- β U}$ . Since $U, β > 0$ , $e^{- β U} < 1$ , so [F1] gives the strictly negative covariance stated in the Example.

2.1givenA1F1algebra∎

In the Bose result of [F1], set $r = 1$ and $μ = ϵ + U$ . The adjacent minimizing occupations are $1, 2$ , their limiting probabilities are $1 / 2, 1 / 2$ , and the binary entropy is $- k_{B} (2) (1 / 2) \log (1 / 2) = k_{B} \log 2$ .

Depends on

Dependency tree · 2 levels

Sources

Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger, Many-body physics with ultracold gases, §§IV.A and IV.E