Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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.

Regular-level slices for unparametrized trajectories

Example

For f(θ)=cos⁡θ on S1, take p=0, q=π, and regular value c=0. The level f−1(0)={π/2,3π/2} has one point on each of the two downward orbit classes, so it realizes M(p,q) as a two-point slice.

Facts & Assumptions

Given: The circle height flow and the regular value 0.

[F1]

A regular intervening level identifies an unparametrized moduli space with its trajectory slice (A regular level identifies unparametrized trajectories).

Verification

technique · direct
1.1

The two arcs from 0 to π cross f−1(0) respectively at π/2 and 3π/2, and strict descent prevents a second crossing.

given
2.1

By [F1], these two slice points are exactly the two unparametrized trajectory classes.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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