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.

Sublevels of height on the sphere

Example

Assume ${A C}_{ω}$ . For height $f (x) = x_{n + 1}$ on $S^{n} \subset R^{n + 1}$ , $n \geq 1$ , the sublevel is empty for $a < - 1$ , a point at $a = - 1$ , a closed $n$ -disk for $- 1 < a < 1$ , and all of $S^{n}$ for $a \geq 1$ . The regular-sublevel changes use one $0$ -handle and one $n$ -handle.

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.

[F2]

Index zero handles create components: A $0$ -handle on a smooth $n$ -manifold with boundary attaches along the empty set and adds one disjoint $n$ -disk component. This includes an empty starting manifold and $n = 0$ .

[F3]

Index n handles cap boundary spheres: An $n$ -handle attaches along its whole $S^{n - 1}$ boundary. For $n \geq 2$ it fills a boundary component diffeomorphic to $S^{n - 1}$ . For $n = 1$ its attaching $S^{0}$ is a pair of boundary points, possibly in different components. For $n = 0$ it is the same disjoint point attachment as a $0$ -handle.

Verification

Given: The objects and hypotheses in the example.

1.1givenalgebra

A critical point has the vertical vector normal to the sphere, so the only critical points are the two poles. In horizontal coordinates $z$ at those poles, height is respectively $- \sqrt{1 - ∣ z ∣^{2}}$ and $\sqrt{1 - ∣ z ∣^{2}}$ ; their Hessians at zero are $+ I$ and $- I$ . The indices are $0$ and $n$ , and their values are $- 1$ and $1$ .

2.1step 1.1algebra

Stereographic coordinates from the north pole identify $S^{n} ∖ {north}$ with $R^{n}$ and give height $(∣ w ∣^{2} - 1) / (∣ w ∣^{2} + 1)$ . For $- 1 < a < 1$ the sublevel is therefore $∣ w ∣^{2} \leq (1 + a) / (1 - a)$ , a closed disk. The values at and beyond the poles give the point, empty set and whole sphere stated above.

3.1F1F2F3step 1.1∎

The sphere is compact, and each band crossing only one pole satisfies the handle theorem. The lower change adds a disjoint disk; the upper change caps its boundary by the whole-boundary attachment. At $n = 1$ the cap attaches along two endpoints, as required by the endpoint qualification.

Depends on

Used by

Nothing in the library uses this result yet.

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

Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)
Benedetti, Lectures on Differential Topology (standard reference, not scraped)