LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

Adapted descending field near a compact morse band

Statement

Assume ${A C}_{ω}$ . Suppose the compact closed band of a smooth function on a boundaryless manifold has only finitely many critical points, all nondegenerate. There is a smooth field $X$ with $d f (X) < 0$ at every noncritical point of the band and $X = (2 u, - 2 v)$ in smaller disjoint Morse charts $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ . It can be chosen compactly supported on $M$ and hence complete.

Facts & Assumptions

[F1]

Closed sublevel and level set of a smooth function: Let $f : M \to R$ be smooth on a boundaryless smooth $n$ -manifold. Write $M^{a} = f^{- 1} ((- \infty, a])$ , $M_{a} = f^{- 1} ({a})$ , and $f^{- 1} ([a, b])$ for the closed band. Both endpoints are included. A regular value may have empty fiber. The smooth-manifold convention is def-smooth-manifold.

[F2]

Morse lemma: Let $f : M \to R$ be smooth, let $p$ be a nondegenerate critical point of $f$ , and let $λ$ be the index of $p$ . If $n = \dim M$ , then there are local coordinates $(x^{1}, \dots, x^{n})$ centered at $p$ in which $f = f (p) - \sum_{i = 1}^{λ} (x^{i})^{2} + \sum_{i = λ + 1}^{n} (x^{i})^{2} .$ For $n = 0$ , both sums are empty.

[F3]

The Riemannian gradient is the metric dual of the differential: Let $g$ be a Riemannian metric on a smooth manifold $M$ and let $f : M \to R$ be smooth. The Riemannian gradient of $f$ is the smooth vector field ${grad}_{g} f$ characterized by $g_{x} (({grad}_{g} f)_{x}, v) = d f_{x} (v) for every x \in M and v \in T_{x} M .$ Pointwise, it is the inverse metric-dual of $d f_{x}$ . In a local frame with metric matrix $(g_{i j})$ and inverse $(g^{i j})$ , it is ${grad}_{g} f = \sum_{i, j} g^{i j} \frac{\partial f}{\partial x^{j}} \partial_{x^{i}};$ the displayed coefficients are smooth, so this pointwise definition is a smooth vector field.

[F4]

Assuming countable choice, every smooth manifold admits a Riemannian metric: Assume ${A C}_{ω}$ . Every smooth manifold admits a Riemannian metric.

[F5]

Smooth partitions of unity exist on manifolds with boundary: Assume ${A C}_{ω}$ . Every open cover of a smooth manifold with boundary admits a smooth partition of unity subordinate to it.

[F6]

A manifold bump for a compact set inside an open set: Let $M$ be a smooth manifold, let $K \subseteq M$ be compact, and let $W \subseteq M$ be open with $K \subseteq W$ . Then there exists a smooth function $ρ : M \to [0, 1]$ that equals $1$ on an open neighbourhood of $K$ and satisfies $supp (ρ) \subseteq W$ .

[F7]

Compactly supported smooth vector fields are complete: Every compactly supported smooth vector field on a smooth manifold is complete.

Proof

Given: The objects and hypotheses in the statement.

1.1

Use the closed-band convention. Choose pairwise disjoint Morse neighborhoods of the finitely many critical points, and smaller neighborhoods with compact closure in them. Compactness allows a neighborhood of the band with no other critical points outside these charts. In each chart the field $(2 u, - 2 v)$ has derivative $- 4 (∣ u ∣^{2} + ∣ v ∣^{2})$ .

F1F2algebra

2.1

Choose a metric; away from the critical points the field $- grad f$ has strictly negative derivative. Cover the band neighborhood by the Morse neighborhoods and a regular open set avoiding the closures of the smaller charts. A subordinate partition of unity patches these fields. At a regular point the derivative is a convex combination of strictly negative numbers; on a smaller chart only its local field is present.

F3F4F5step 1.1

3.1

Choose a relatively compact neighborhood of the compact band within the field domain and a bump equal to one near the band. Multiply by it and extend by zero. This leaves the required local formulas intact and gives a complete field. With no critical points the regular field alone is used; with an empty band use zero. In dimension zero each local field is zero and there are no regular points to test.

F6F7step 2.1∎

Depends on

Used by

Unstable disk is the handle core Corollary
Descending flow identifies the local and global attaching regions Lemma
Simultaneous attachment at a morse critical value Proposition

Dependency tree · two levels

27 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)
Benedetti, Lectures on Differential Topology (standard reference, not scraped)