Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)
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.

An orientation-reversing cell fails the Jacobian certificate

Example

Take the reference triangle with vertices (0,0),(1,0),(0,1) and the affine map F(x,y)=(y,x). Its Jacobian determinant is −1, so it cannot satisfy the admissibility condition JF≥j0>0.

Facts & Assumptions

Given: The unit reference triangle and the map F(x,y)=(y,x).

[F1]

Admissible isoparametric curved cells: admissibility requires an orientation-preserving diffeomorphism with determinant bounded below by a positive constant.

Verification

1.1givenalgebra

The derivative matrix is DF=(0110), whose determinant is −1.

2.1F1step 1.1∎

Since −1 is not bounded below by any positive j0, this map is rejected by the admissibility certificate.

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