Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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.

Symplectic manifolds can have odd dimension

Statement refuted

A symplectic manifold can have odd dimension.

Facts & Assumptions

Given: A symplectic manifold (M,ω).

[F1]

Every finite-dimensional symplectic vector space has even dimension. Symplectic vector spaces have even dimension.

Refutation

technique · direct
1.1

At every p∈M, nondegeneracy makes (TpM,ωp) a symplectic vector space.

given
2.1

By [F1], dim⁡TpM is even; this is dim⁡M on the component containing p. Hence no odd-dimensional component is symplectic, contrary to the claim.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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