ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

A degenerate local ground eigenspace has a deterministic tie choice

Example

For two spin- $1 / 2$ sites let $Z = diag (1, - 1)$ and $H = Z \otimes Z$ . Use the initial state $∣ 00 ⟩$ with rank-one open-boundary TT cores $G_{k} (1, i_{k}, 1) = 1_{{i_{k} = 1}}$ for $k = 1, 2$ , and run one width-two sweep without rank caps. The two-site ground eigenspace is $span {∣ 01 ⟩, ∣ 10 ⟩}$ . The deterministic standard-basis projection rule in Finite Hermitian ground vectors and a deterministic tie rule selects $∣ 01 ⟩$ .

Facts & Assumptions

Given: The ordered two-site computational basis $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ , the initial $∣ 00 ⟩$ core list and width-two no-cap sweep specified above, and the Hamiltonian $Z \otimes Z$ .

[F1]

Finite matrix-product states and Hermitian Hamiltonians: a finite Hermitian Hamiltonian acts on the finite tensor-array coefficient space.

[F2]

Finite Hermitian ground vectors and a deterministic tie rule: among lowest eigenvectors, the rule projects standard basis vectors in order and takes the first nonzero projection.

[F3]

Finite alternating tensor optimization with effective eigenproblems: the exact local solve uses that deterministic ground-vector rule.

Verification

1.1givenF1algebra

In the ordered basis, $H = diag (1, - 1, - 1, 1)$ . Its least eigenvalue is $- 1$ and its eigenspace is exactly $span {∣ 01 ⟩, ∣ 10 ⟩}$ .

2.1F2F3step 1.1algebra∎

The orthogonal projection onto that eigenspace sends $∣ 00 ⟩$ to zero and sends $∣ 01 ⟩$ to itself. Hence the first standard basis vector with nonzero projection is $∣ 01 ⟩$ ; it is already unit length and has positive first nonzero coordinate. The local solve therefore returns $∣ 01 ⟩$ deterministically, with energy $- 1$ .

Depends on

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