ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

Zero-point energy and ground-degeneracy entropy

Physical scope

A direct low-temperature comparison of the one-dimensional harmonic oscillator and the zero-field two-state spin model covered by the linked theorem. It records a model result, not a universal third-law statement or observation.

Example

For an oscillator with gap $ϵ = ℏ ω > 0$ , the $T ↓ 0$ canonical state approaches the nondegenerate ground level, with energy $ϵ / 2$ and entropy zero. For the separate spin Hamiltonian $H_{s} = - h σ_{z}$ , choosing $h = 0$ leaves two exactly degenerate states equally occupied at every temperature, with entropy $k_{B} \log 2$ . These model calculations show why ground degeneracy must be included when stating a low-temperature entropy limit.

Verification

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $ϵ > 0$ , the oscillator and spin models in Exact quantum oscillator and spin ensembles with their adopted canonical Gibbs preparations, and $k_{B} > 0$ .

[F1] Exact quantum oscillator and spin ensembles, Statement: oscillator energy tends to $ϵ / 2$ and entropy to zero; the spin entropy is $k_{B} \log 2$ when $h = 0$ .

1.1givenF1algebra

Since $e^{- β ϵ} \to 0$ as $T ↓ 0$ , the oscillator formulas give $\overset{ˉ}{n} \to 0$ , $U = ϵ (1 / 2 + \overset{ˉ}{n}) \to ϵ / 2$ , and the entropy function in [F1] tends zero.

2.1givenF1algebra∎

With $h = 0$ , the spin Gibbs weights are both $1 / 2$ , so $S = - k_{B} (2) (1 / 2) \log (1 / 2) = k_{B} \log 2$ independently of $T$ . Both conclusions follow from these Hamiltonians together with the adopted canonical Gibbs preparations.

Zero-point energy and ground-degeneracy entropy

Example

Verification

Depends on

Dependency tree · 2 levels

Sources