DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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.

The local algebra of smooth function germs

Definition

Let $M$ be a smooth manifold and let $p \in M$ . Let $C_{p}^{\infty} (M)$ denote the set of smooth germs at $p$ . For germs represented by smooth functions $f$ and $g$ near $p$ , define $[f] + [g] : = [f + g], [f] [g] : = [f g], c [f] : = [c f] for c \in R .$ These operations make $C_{p}^{\infty} (M)$ the local algebra of smooth function germs at $p$ , once representative independence is checked.

Depends on

The germ of a smooth function at a point

The local algebra of smooth function germs

Definition

Depends on

Dependency tree · 2 levels

Sources