LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-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.

A regular level identifies unparametrized trajectories

Statement

Let $f (q) < c < f (p)$ and suppose that $c$ is a regular value. Each class in $M (p, q)$ has exactly one representative whose value at $0$ lies in $f^{- 1} (c)$ . Thus evaluation induces a bijection

$M (p, q) ≅ \tilde{M} (p, q) \cap f^{- 1} (c) .$

Facts & Assumptions

Given: A Morse trajectory from $p$ to $q$ and a regular $c$ strictly between its endpoint values.

[F1]

Along a nonconstant trajectory, $\frac{d}{d t} (f \circ γ) = d f (X) < 0$ (Downward gradient-like vector fields for a Morse function).

[F2]

A regular level is an embedded hypersurface (A regular level set is an embedded submanifold).

Proof

technique · direct

1.1

Continuity and the endpoint limits give a time $t_{0}$ with $f (γ (t_{0})) = c$ ; strict decrease in [F1] makes $t_{0}$ unique. Translating by $t_{0}$ therefore gives exactly one representative in the stated slice.

F1given

2.1

Conversely, two slice representatives in one time orbit differ by a translation, and step 1.1 forces that translation to be zero. The slice lies in the embedded hypersurface of [F2], giving the asserted identification.

F2step 1.1∎

Depends on

Used by

Dependency tree · two levels

16 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

Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, Remark 2.2.3 (standard reference, not scraped)