DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Downward gradient-like vector fields for a Morse function

Let $f : M \to R$ be Morse. A smooth vector field $X$ is downward gradient-like for $f$ if both conditions hold:

$d f_{x} (X_{x}) < 0$ at every $x \notin Crit (f)$ ; and
for every $p \in Crit (f)$ there are Morse coordinates $(u, v)$ centred at $p$ , with $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ , in which $X = 2 \sum_{i} u_{i} \partial_{u_{i}} - 2 \sum_{j} v_{j} \partial_{v_{j}} .$

Thus the field is the negative Euclidean gradient in the required local Morse model, not merely a strictly descending field off the critical set.

Depends on