Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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 derivation annihilates constant germs

Statement

Every derivation at p sends every constant germ to 0.

Facts & Assumptions

Given: A derivation v∈TpM.

[F1]

A derivation is linear and satisfies the Leibniz rule on germs (Derivations at a point and the tangent space).

Proof

technique · direct
1.1F1given

Let 1 denote the constant germ with value 1. Applying the Leibniz rule from [F1] to 1⋅1 gives v(1)=1 v(1)+1 v(1)=2v(1), so v(1)=0.

2.1F1step 1.1

Every constant germ is c⋅1 for some c∈R, so linearity from [F1] gives v(c⋅1)=c v(1)=0.

3.1step 2.1∎

Thus every constant germ is annihilated by v.

Depends on

Used by

Dependency tree · two levels

3 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