CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-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.

Using the positive gradient reverses the stable and unstable dimensions

Statement refuted

For $f (u, v) = - ∣ u ∣^{2} + ∣ v ∣^{2}$ , one may use the positive gradient while retaining the descending convention that the unstable dimension equals the Morse index.

Counterexample

Given: The Euclidean quadratic $f (u, v) = - ∣ u ∣^{2} + ∣ v ∣^{2}$ evolved by $+ grad f$ .

1.1

The positive-gradient equations are $\dot{u} = - 2 u$ and $\dot{v} = 2 v$ , so $u (t) = e^{- 2 t} u_{0}$ and $v (t) = e^{2 t} v_{0}$ .

givenalgebra

2.1

Thus the forward-stable space is ${v = 0}$ of index dimension and the backward-unstable space is ${u = 0}$ of coindex dimension. The labels are reversed from the descending convention, refuting the assertion.

step 1.1∎

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.1 (standard reference, not scraped)