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)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Canonical mean and variance for a two-level system

Facts & Assumptions

Given: As stated in the item.

[F1]

Countable canonical model: the finite partition sum and normalized state weights are defined as stated there.

[F2]

Summable canonical differentiation: a finite energy list has the required local summable moment envelope on every compact parameter interval (each of finitely many terms is bounded there), so $- \partial_{β} \log Z$ is the mean and $\partial_{β}^{2} \log Z$ is the variance.

[A1]

$E_{0} = 0$ , $E_{1} = ε > 0$ , $g_{0} = g_{1} = 1$ , and $β > 0$ .

Example

Let the state set be ${0, 1}$ , with energies $E_{0} = 0$ , $E_{1} = ε > 0$ , unit weights, and $β > 0$ . Writing $q = e^{- β ε}$ , the excited-state probability is $q / (1 + q)$ , the mean energy is $ε q / (1 + q)$ , and the energy variance is $ε^{2} q / (1 + q)^{2} > 0$ .

Verification

Given: The finite canonical model [F1] with data [A1].

1.1F1A1algebra

Direct summation gives $Z = 1 + e^{- β ε} = 1 + q$ , hence $p_{1} = q / (1 + q)$ and $p_{0} = 1 / (1 + q)$ .

2.1F1F2A1step 1.1algebra∎

The finite weighted sum gives $\overset{ˉ}{E} = ε p_{1} = ε q / (1 + q)$ . The second centered moment is $p_{0} (0 - \overset{ˉ}{E})^{2} + p_{1} (ε - \overset{ˉ}{E})^{2} = ε^{2} q / (1 + q)^{2}$ , which is positive because $q \in (0, 1)$ . The values also agree with the finite-list derivative identities of [F2].

Canonical mean and variance for a two-level system

Facts & Assumptions

Example

Verification

Depends on

Dependency tree · 2 levels

Sources