Alphabeta Math
CorollaryStatement: 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.

No Morse--Smale trajectories for nonpositive index drop

Statement

If (f,X) is Morse--Smale, p≠q, and λ(p)≤λ(q), then M(p,q)=∅.

Facts & Assumptions

Given: Distinct critical points p,q in a Morse--Smale pair with λ(p)≤λ(q).

[F1]

The unparametrized trajectory space, when nonempty, is a smooth manifold of dimension λ(p)−λ(q)−1 (The unparametrized trajectory space is a smooth manifold).

Proof

technique · direct
1.1

If M(p,q) were nonempty, [F1] would give it dimension λ(p)−λ(q)−1≤−1.

F1givenassume-contraalgebra
2.1

A nonempty smooth manifold has a nonnegative integer dimension, contradicting step 1.1. Thus the moduli space is empty.

step 1.1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

6 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