Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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.

A continuous fibrewise linear map over a smooth base map is automatically smooth

Statement

A continuous fibrewise linear map over a smooth base map is automatically smooth.

Facts & Assumptions

Given: The displayed claim.

[L1]

Smoothness of a bundle map is equivalent to smoothness of its local matrix coefficients (Smoothness of a bundle map is equivalent to smooth local matrices).

Refutation

technique · direct
1.1L1givenconstruct

On the trivial line bundle R×R→R, define Φ(x,v)=(x,∣x∣v). This map is continuous, covers the smooth base map id⁡R, and is linear on every fibre.

2.1L1step 1.1∎

Its local matrix coefficient is the scalar function ∣x∣, which is not smooth at 0. Therefore [L1] implies that Φ is not smooth. So the displayed statement is false.

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