LemmaStatement: 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.

Normalized gradient crosses a compact regular band in controlled time

Statement

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.

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]

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.

[F3]

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

[F4]

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 $supp (ρ) \subseteq W$ .

[F5]

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 and choose a metric. If $K = \emptyset$ , the zero field suffices and all trajectory assertions are vacuous. Otherwise the metric exists under the stated choice axiom.

F1F3given

1.2

The open set ${d f \neq 0}$ contains $K$ . Cover $K$ by finitely many coordinate neighborhoods with compact closures in this open set; their union $W$ is relatively compact. Choose $ρ = 1$ near $K$ with support in $W$ .

2.1

On $W$ set $Y = ρ grad f / ∥ grad f ∥^{2}$ , and set it to zero outside $W$ . The support condition makes this smooth with compact support. Since $d f (grad f) = ∥ grad f ∥^{2}$ , $d f (Y) = 1$ near $K$ .

F2step 1.2algebra

3.1

The field is complete. On every trajectory segment contained in $K$ , differentiation gives $d (f \circ Φ_{t}) / d t = 1$ . Starting at an endpoint the same identity holds on its open neighborhood, so the trajectory enters the band in the required time direction. A first exit before the claimed level would have value strictly between $a$ and $b$ , contradicting continuity. Integrating gives the identity through both endpoints; strict unit speed gives the asserted hitting time.

F5step 2.1algebra∎

Depends on

Used by

Regular sublevels are diffeomorphic Corollary
Deformation lemma for a critical point free slab Proposition
Regular interval diffeomorphism Theorem

Dependency tree · two levels

17 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)
Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)