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.

An incomplete finite trial space can lie strictly above the Hamiltonian ground energy

Example

Let the finite configuration space be $C^{2}$ and the Hermitian Hamiltonian be $H = diag (0, 2)$ . Its ground energy is $0$ , while the one-dimensional trial space $V_{t r i a l} = span {e_{2}}$ has variational energy $2$ . This is the finite-dimensional form of the warning that an incomplete configuration-interaction space need not contain the ground state.

Facts & Assumptions

Given: The Hamiltonian $H = diag (0, 2)$ on $C^{2}$ and trial space $span {e_{2}}$ .

[F1]

A Hermitian matrix has an orthonormal eigenbasis with real eigenvalues (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

Verification

1.1F1givenalgebra

The eigenvalues of $H$ are $0$ and $2$ , with corresponding unit eigenvectors $e_{1}, e_{2}$ by [F1]. For a unit vector $v = (a, b)$ , $v^{*} H v = 2 ∣ b ∣^{2} \geq 0$ , and equality holds at $e_{1}$ , so the full-space ground energy is $0$ .

2.1step 1.1givenalgebra∎

Every unit vector in $V_{t r i a l}$ is $e^{i θ} e_{2}$ for some real $θ$ , and its energy is $(e^{i θ} e_{2})^{*} H (e^{i θ} e_{2}) = 2$ . Thus the restricted minimum is $2 > 0$ , strictly above the full finite-dimensional ground energy.

Depends on

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