Alphabeta MathHow 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 nonzero-Chern band has no global smooth periodic frame
Statement refuted
Every smooth isolated Bloch band on the two-torus admits a global smooth periodic normalized eigenvector.
Statement references
Facts & Assumptions
Given: Axiom of Choice, Countable Choice, the smooth-frame obstruction theorem, and the band in Gap closings and Chern regions in the two-band lattice model.
Axiom of Choice and Countable Choice are the principles in The Axiom of Choice and The Axiom of Countable Choice ().
Gap closings and Chern regions in the two-band lattice model computes at and verifies the uniform gap.
Gapped band curvature and Chern invariance proves that a global periodic orthonormal frame forces the Chern number to be zero.
Counterexample
Given: [A1], [F1], [F2], and the lower band of the stated two-band Hamiltonian at . Its spectral gap is open and its Chern number is .
By [F1], this is a smooth rank-one orthogonal band projector with . Suppose it had a global smooth periodic normalized eigenvector . Since its range is one-dimensional, would be a global orthonormal frame of the line bundle.
Apply [F2] to that frame: it would force , contradicting step 1.1's value . Thus this smooth isolated band has no global smooth periodic normalized eigenvector, refuting the universal statement.