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.

Torus from one 0-handle, two 1-handles and one 2-handle

Example

Assume ${A C}_{ω}$ . On $T^{2} = S^{1} \times S^{1}$ , let $f (θ, ϕ) = - \cos θ - 2 \cos ϕ$ . It yields one $0$ -handle, two $1$ -handles and one $2$ -handle, with critical values $- 3, - 1, 1, 3$ .

Facts & Assumptions

[F1]

One critical point handle attachment: Assume ${A C}_{ω}$ . Let $f : M \to R$ be smooth on a boundaryless $n$ -manifold and let $a < b$ be regular values. If $f^{- 1} ([a, b])$ is compact and has exactly one critical point $p$ , nondegenerate of index $k$ , then $M^{b}$ is diffeomorphic to $M^{a}$ with one $k$ -handle attached and corners rounded. No orientation or Morse–Smale hypothesis is required.

Verification

Given: The objects and hypotheses in the example.

1.1

The critical equations are $\sin θ = 0$ and $2 \sin ϕ = 0$ . Modulo $2 π$ there are four solutions: $(0, 0), (π, 0), (0, π), (π, π)$ . The Hessian is $diag (\cos θ, 2 \cos ϕ)$ , giving respectively indices $0, 1, 1, 2$ and values $- 3, - 1, 1, 3$ .

givenalgebra

2.1

The torus is compact. Choose successive regular levels, for example $- 4, - 2, 0, 2, 4$ . Each intervening closed band contains exactly one of these nondegenerate points. Applying the handle theorem to each band gives the stated counts from the empty sublevel to the full torus.

F1step 1.1∎

Depends on

One critical point handle attachment

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Benedetti, Lectures on Differential Topology (standard reference, not scraped)