LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-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.

Time translation acts freely on nonconstant trajectories

Statement

The action $s \cdot γ (t) : = γ (t + s)$ of $R$ on $\tilde{M} (p, q)$ has trivial stabilizers.

Facts & Assumptions

Given: A trajectory $γ \in \tilde{M} (p, q)$ and $s \in R$ with $s \cdot γ = γ$ .

[F1]

Along every nonconstant $X$ -orbit, $d f (X) < 0$ (Downward gradient-like vector fields for a Morse function).

[F2]

Distinct endpoints $p \neq q$ make every trajectory in $\tilde{M} (p, q)$ nonconstant (Parametrized Morse trajectory space).

Proof

technique · direct

1.1givenalgebra

Equality $s \cdot γ = γ$ says $γ (t + s) = γ (t)$ for every $t$ , so if $s \neq 0$ then $γ$ is periodic with nonzero period $∣ s ∣$ .

2.1F1F2step 1.1∎

By [F2] the orbit is nonconstant, while [F1] makes $f \circ γ$ strictly decreasing; it therefore cannot be periodic. Hence $s = 0$ , which is exactly freeness.

Depends on

Used by

Dependency tree · two levels

9 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

Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, Proposition 2.2.2 (standard reference, not scraped)