CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 CW complex with no cells in adjacent dimensions has zero cellular boundary

If $X$ has no cells in adjacent dimensions, every cellular differential is zero; consequently $H_{n} (X; G) ≅ C_{n}^{c e l l} (X; G)$ .

Given: A CW complex with no two nonzero cellular chain groups in consecutive degrees.

technique · direct

1.1given

For every $n$ , either $C_{n}^{c e l l}$ or $C_{n - 1}^{c e l l}$ is zero, so $d_{n} = 0$ .

2.1step 1.1∎

Thus cellular homology equals the chain group in each degree; Cellular homology computes singular homology transfers this calculation to singular homology.

Depends on