Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Time-dependent pullback differentiation for a translation

Statement

For Xt=∂x, Φt,s(x)=x+t−s, and ωt=t dx, the time-dependent pullback identity holds.

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that If Φt,s is the local evolution of Xt and ωt is a smooth time-dependent form, then ddtΦt,s∗ωt=Φt,s∗(ω˙t+LXtωt). (Differentiation of a pulled-back form along a time-dependent flow).

Verification

technique · direct
1.1

Here Φt,s∗(t dx)=t dx, so the left side is dx.

F1given
2.1

Since ω˙t=dx and L∂x(t dx)=0, the right side is also dx.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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