LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Spectral flattening preserves gaps and symmetry relations

Statement

For a finite-dimensional self-adjoint gapped Hamiltonian $H$ , define $sign (H)$ by its spectral decomposition: it is $+ 1$ on positive eigenspaces and $- 1$ on negative eigenspaces. The path $H_{t} = (1 - t) H + t sign (H)$ is continuous, remains gapped, and preserves every unitary or antiunitary symmetry relation $S H S^{- 1} = σ H$ with $σ \in {+ 1, - 1}$ . At $t = 1$ it is a self-adjoint involution.

Facts & Assumptions

Given: A finite-dimensional complex Hilbert space, a self-adjoint endomorphism $H$ with $0 \notin spec (H)$ , and a unitary or antiunitary operator $S$ satisfying $S H S^{- 1} = σ H$ for a real sign $σ = \pm 1$ .

[F1]

A finite-dimensional self-adjoint endomorphism is normal (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F2]

Every normal endomorphism of a finite-dimensional complex inner-product space has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F3]

The Hamiltonian and symmetry conventions are those in Finite-dimensional gapped free Hamiltonians and stable equivalence.

Proof

technique · diagonalize and check each eigenspace

1.1givenF1F2algebra

If the Hilbert space is zero-dimensional, the unique path is the unique endomorphism path and the assertion is vacuous. Otherwise, by [F1] and [F2], choose an orthonormal eigenbasis of $H$ . Every eigenvalue $λ$ is real because self-adjointness gives $⟨ H v, v ⟩ = ⟨ v, H v ⟩$ , while $H v = λ v$ makes the two sides conjugates; since $H$ is gapped, $λ \neq 0$ . On its eigenspace $H_{t}$ acts by $(1 - t) λ + t sgn (λ)$ , which is positive for $λ > 0$ and negative for $λ < 0$ for every $t \in [0, 1]$ . Thus $H_{t}$ has no zero eigenvalue.

2.1step 1.1algebra

Each eigenvalue expression is continuous in $t$ , so the diagonal operator path $H_{t}$ is continuous. At $t = 1$ , its eigenvalues are $\pm 1$ , giving $H_{1}^{*} = H_{1}$ and $H_{1}^{2} = I$ .

3.1givenF3step 1.1algebra∎

The symmetry relation implies $S sign (H) S^{- 1} = sign (σ H) = σ sign (H)$ : $S$ sends a $λ$ -eigenvector to a $σ λ$ -eigenvector, and the scalar values involved are real, so the statement holds for antiunitary as well as unitary $S$ . Therefore $S H_{t} S^{- 1} = (1 - t) σ H + t σ sign (H) = σ H_{t}$ for every $t$ .

Depends on

Used by

Antiunitary AZ symmetry classes give Clifford extension problems Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)