PropositionStatement: AI-adaptedProof: 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.

Deformation lemma for a critical point free slab

Statement

Assume ${A C}_{ω}$ . Under the compact regular closed-band hypothesis with $a < b$ , the formula $H (s, x) = Φ_{- s \max (f (x) - a, 0)} (x)$ , for $(s, x) \in [0, 1] \times M^{b}$ , is a strong deformation retraction onto $M^{a}$ . Here $Φ$ is the complete normalized ascending cutoff flow.

Facts & Assumptions

[F1]

Normalized gradient crosses a compact regular band in controlled time: Assume ${A C}_{ω}$ . Let $f : M \to R$ be smooth on a boundaryless manifold, $a < b$ , and let $K = f^{- 1} ([a, b])$ be compact with $d f \neq 0$ on $K$ . For any Riemannian metric there is a compactly supported smooth field $Y$ agreeing with $grad f / ∥ grad f ∥^{2}$ near $K$ . Its complete flow $Φ$ satisfies $f (Φ_{t} (x)) = f (x) + t$ for $x \in K$ and $a - f (x) \leq t \leq b - f (x)$ . Thus every intervening level is reached in exactly its value difference.

Proof

Given: The objects and hypotheses in the statement.

1.1

For $f (x) \leq a$ the time parameter is zero, so $H (s, x) = x$ . For $a \leq f (x) \leq b$ the controlled-time identity gives $f (H (s, x)) = (1 - s) f (x) + s a \in [a, b]$ . Thus $H$ takes values in $M^{b}$ .

F1givenalgebra

2.1

The maximum function and the complete flow are continuous, so the formula is continuous even at $f = a$ . At $s = 0$ it is the identity; at $s = 1$ its image lies in $M^{a}$ and it fixes that set at every time. This proves the strong retraction, including empty sets. No smoothness across $f = a$ is asserted.

step 1.1algebra∎

Depends on

Normalized gradient crosses a compact regular band in controlled time

Used by

One critical point cell attachment homotopy type Corollary

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)
Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)