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.

Hamiltonian functions for one vector field differ by one global constant on a disconnected manifold

Statement refuted

Hamiltonian functions for one vector field differ by one global constant even when the manifold is disconnected.

Facts & Assumptions

Given: The proposed claim.

[F1]

Such Hamiltonians differ only by a locally constant function, which may take different values on different components. Hamiltonians for a fixed vector field differ by a locally constant function.

Refutation

technique · direct
1.1

Let M be the disjoint union of two copies of the standard symplectic plane. The zero function H generates the zero vector field. Let K equal zero on the first component and one on the second; then dK=0, so K generates the same field.

F1construct
2.1

But K−H takes both values zero and one and is not one global constant. It is locally constant exactly as [F1] predicts.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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