ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Uniform onsite energy and exact pressure

Example

Fix $d \geq 1$ , a finite site dimension $q \geq 1$ , and $c \in R$ . On $Γ = Z^{d}$ , the Hilbert space of a finite region $Λ$ is the tensor product $H_{Λ} = ⨂_{x \in Λ} C^{q}$ , with sites ordered lexicographically, dimension $q^{∣ Λ ∣}$ , and identity $I_{Λ}$ . Here ${Tr}_{Λ}$ is the ordinary finite-dimensional matrix trace. Define a translation-invariant interaction by $Φ ({x}) = c I_{{x}}$ for each site $x$ , and $Φ (Z) = 0$ for every other finite nonempty support $Z$ . Its free-boundary Hamiltonian and partition function are $H_{Λ} = \sum_{\emptyset \neq Z \subseteq Λ} Φ (Z)$ and $Z_{Λ} (β) = {Tr}_{Λ} (e^{- β H_{Λ}})$ for $β > 0$ . If $c$ is interpreted as an energy, use units of joules for $c$ and inverse joules for $β$ .

Facts & Assumptions

Given: The self-contained finite tensor-product Hilbert spaces, interaction, Hamiltonian, and partition-function definitions in this Example section.

[A1]

The scalar $c$ is real, $d$ and $q$ are positive integers, and every finite region used below is nonempty.

Verification

technique · direct evaluation of the finite-volume trace

Given: [A1] and the definitions in the Example section.

1.1givenA1algebra

At each site, the only possibly nonzero interaction term containing it is its singleton term, and that term has operator norm $∣ c ∣$ . Every possibly nonzero support is a singleton, so this interaction is translation invariant, has support-size bound $M = 1$ , range $R = 0$ , and uniform per-site norm bound $κ = ∣ c ∣$ .

2.1givenA1step 1.1algebra∎

Each term $c I_{{x}}$ embeds into $A_{Λ}$ as $c I_{Λ}$ . Therefore $H_{Λ} = c ∣ Λ ∣ I_{Λ}$ , and the dimension formula gives $Z_{Λ} (β) = q^{∣ Λ ∣} e^{- β c ∣ Λ ∣}$ . Thus $∣ Λ ∣^{- 1} \log Z_{Λ} (β) = \log q - β c$ for every finite nonempty $Λ$ ; this same value is the limit along every van Hove sequence. For $c = 0$ it is $\log q$ , and for $q = 1$ it is $- β c$ .

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.