ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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A zero-dimensional class-A flattened Hamiltonian

Example

On $C^{2}$ , let $H = diag (- 1, + 1)$ . Define the $d = 0$ rank-difference index by $ν (H) = \dim E_{< 0} (H) - \dim E_{< 0} (H_{r e f})$ , where $H_{r e f}$ is the fixed reference system; choose the zero-band reference, whose negative eigenspace has dimension zero. Then $H$ is already flattened, with occupied projection $P = diag (1, 0)$ , and the example has $ν (H) = 1$ .

Facts & Assumptions

Given: A finite-dimensional class-A Hamiltonian on $C^{2}$ and the zero-band reference.

[F1]

A finite-dimensional free Hamiltonian is self-adjoint, is gapped at zero when zero is outside its spectrum, and stable equivalence is by gap-preserving homotopy after adding fixed reference bands (Finite-dimensional gapped free Hamiltonians and stable equivalence).

Verification

1.1givenF1algebra

The eigenvalues of $H$ are $- 1$ and $+ 1$ , so $H$ is self-adjoint and gapped. The projection onto its negative eigenspace is $P = (I - H) / 2 = diag (1, 0)$ . Direct multiplication gives $P^{2} = P = P^{*}$ ; its image is the one-dimensional span of $(1, 0)$ .

2.1step 1.1algebra∎

Therefore $\dim E_{< 0} (H) = 1$ , while the stated zero-band reference has negative eigenspace dimension zero. Substitution in the defined rank difference gives $ν (H) = 1 - 0 = 1$ .

Depends on

Finite-dimensional gapped free Hamiltonians and stable equivalence

Used by

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Sources

Alexei Kitaev, Periodic table for topological insulators and superconductors (2009) (standard reference, not scraped)
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford Modules, Topology 3, Supplement 1 (1964) (standard reference, not scraped)
M. F. Atiyah, K-Theory (lectures, notes by D. W. Anderson, 1967), with K-Theory and Reality reprint (standard reference, not scraped)