Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Finite-sector quantum Gibbs preparation minimizes the declared free-energy functional

Physical scope

The finite-dimensional fixed-particle-number quantum sector at assigned positive temperature, conditional on the adopted Gibbs-preparation assumption and the exact NRQM entropy/free-energy definitions; no equilibration or macroscopic thermodynamic-limit claim is made.

Statement

Work in the nonzero finite-dimensional fixed- $N$ sector of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, with self-adjoint energy $H_{N}^{σ}$ in joules and assigned $T > 0$ in kelvins. Adopt the Gibbs preparation of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature and its unit convention $β = (k_{B} T)^{- 1}$ . For $F (ρ) = tr (ρ H_{N}^{σ}) - T S (ρ)$ over the declared sector density operators, the canonical preparation is the unique minimizer and $\min_{ρ} F (ρ) = F_{e q} = - k_{B} T \log tr (e^{- β H_{N}^{σ}}) .$ This quantum statistical functional is not identified here with a macroscopic Helmholtz potential without an additional model bridge; its minimum does not establish equilibration.

Facts & Assumptions

Given: A nonzero finite-dimensional fixed- $N$ sector, its self-adjoint energy, an assigned $T > 0$ , and the adopted canonical preparation.

[F1]

Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model: on the finite-dimensional sector, $S (ρ) = k_{B} s (ρ)$ and $F (ρ) = tr (ρ H_{N}^{σ}) - k_{B} T s (ρ)$ , with $β = (k_{B} T)^{- 1}$ .

[F2]

The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature and The canonical preparation as the unique free-energy minimiser of the declared finite-sector model (both current carrier statuses say published): the former adopts the finite-sector Gibbs preparation under its exact AC/Countable Choice and finite positive-trace hypotheses; the latter proves that this preparation uniquely minimizes the declared finite-sector functional with minimum $- k_{B} T \log Z_{N}$ . Their current statements and the theorem proof were inspected, but their published records lack the required audit/source-review evidence, so this use remains escalated.

[F3]

Dimensions, SI unit conventions and quantity types of the framework and SI quantities and unit conventions: $H_{N}^{σ}$ and $F$ are in joules, $T$ in kelvins, and $k_{B}$ in $J K^{- 1}$ .

[A1]

The Axiom of Choice and Countable Choice are assumed as required by [F2]; the sector dimension is finite and at least one.

Proof

Given: The declared finite-sector quantum model, the entropy/free-energy functional, and the canonical Gibbs assignment.

1.1F1F2F3A1

The finite-sector class, assigned positive temperature, canonical preparation, and entropy functional in the Statement are exactly the hypotheses and definitions of [F1]–[F2]. Countable Choice is stated explicitly in [A1], as required by [F2].

2.1F1F2F3step 1.1∎

By [F2], every declared sector preparation obeys $F (ρ) \geq - k_{B} T \log Z_{N}$ , with equality only for the canonical preparation. Replacing $β^{- 1}$ by $k_{B} T$ using [F1]–[F3] gives the displayed minimum and units. This is a conditional preparation-model deduction, not a claim that dynamics reaches the minimizer.

Finite-sector quantum Gibbs preparation minimizes the declared free-energy functional

Statement

Facts & Assumptions

Proof

Depends on

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Sources