ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Simultaneous handles at a repeated critical value

Example

Assume ${A C}_{ω}$ . On $T^{2}$ let $f (θ, ϕ) = - \cos θ - \cos ϕ$ . Its two index-one critical points both have value zero. For every $0 < ε < 2$ , crossing the closed band $[- ε, ε]$ attaches two disjoint $1$ -handles simultaneously.

Facts & Assumptions

[F1]

Simultaneous attachment at a morse critical value: Assume ${A C}_{ω}$ . Let $f$ be smooth on a boundaryless manifold and let $a < b$ be regular values. Suppose the closed band is compact and its critical points are finitely many nondegenerate points $p_{1}, \dots, p_{m}$ , all at the same value $c \in (a, b)$ . Then $M^{b}$ is obtained from $M^{a}$ , up to diffeomorphism and corner rounding, by attaching disjoint handles of indices $ind (p_{j})$ . If $m = 0$ , no handles are attached and the regular-band conclusion applies.

Verification

Given: The objects and hypotheses in the example.

1.1

The critical equations are $\sin θ = \sin ϕ = 0$ . The Hessian is $diag (\cos θ, \cos ϕ)$ . At $(0, 0)$ it is positive definite with value $- 2$ ; at $(π, π)$ it is negative definite with value $2$ . At $(π, 0)$ and $(0, π)$ it has one negative entry and value zero.

givenalgebra

2.1

For $0 < ε < 2$ , both endpoints are regular and the compact band contains precisely the two saddle points. The simultaneous-attachment proposition gives two disjoint index-one handles, with no need to perturb their equal values. The restrictions on epsilon exclude both the collapsed band and endpoints through the extrema.

F1step 1.1∎

Depends on

Simultaneous attachment at a morse critical value

Used by

Nothing in the library uses this result yet.

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

Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)