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 $v \in T_{p} M$ .

[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 $1 \cdot 1$ gives $v (1) = 1 v (1) + 1 v (1) = 2 v (1)$ , so $v (1) = 0$ .

2.1F1step 1.1

Every constant germ is $c \cdot 1$ for some $c \in R$ , so linearity from [F1] gives $v (c \cdot 1) = c v (1) = 0$ .

3.1step 2.1∎

Thus every constant germ is annihilated by $v$ .

Depends on

Derivations at a point and the tangent space

A derivation annihilates constant germs

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources