CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 simplex has zero reduced simplicial homology

If $Δ^{n}$ is a simplex, then ${\tilde{H}}_{k}^{s i m p} (Δ^{n}) = 0$ for every $k$ .

Given: A simplex $Δ^{n}$ .

1.1given

The previous lemma gives a contracting homotopy for the augmented simplicial chain complex of $Δ^{n}$ .

2.1step 1.1∎

A contractible chain complex is acyclic, so every reduced simplicial homology group of $Δ^{n}$ vanishes.

Depends on