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.
Being Morse does not by itself force distinct critical values; excellence is a separate generic condition
Remark
The adjective Morse controls only the Hessians at critical points, not the relative heights of those critical points. By Morse functions and excellent Morse functions, repeated critical values are allowed unless one adds the extra adjective excellent.
The point of For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians is that on a compact manifold this extra condition is still generic: repeated critical levels can be broken by local perturbations that do not change the critical Hessians.
Depends on
Used by
Dependency tree · two levels
6 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
- Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)
- Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)