LemmaStatement: 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.

The cellular boundary squares to zero

Statement

For every $n \geq 1$ , $d_{n - 1} d_{n} = 0$ .

Facts & Assumptions

Given: The three-skeleton definition of $d$ in Cellular boundary from three consecutive skeleta.

Proof

technique · direct

1.1given

For $n = 1$ the composite is zero because $d_{0} = 0$ by definition.

1.2given

Let $n \geq 2$ . Write $d_{n} = j_{n - 1} \partial_{n}, d_{n - 1} = j_{n - 2} \partial_{n - 1},$ where $j_{n - 1} : H_{n - 1} (X^{n - 1}) \to H_{n - 1} (X^{n - 1}, X^{n - 2})$ is the quotient map. In the long exact sequence of the pair $(X^{n - 1}, X^{n - 2})$ , $j_{n - 1}$ is immediately followed by $\partial_{n - 1}$ .

2.1step 1.1step 1.2∎

Exactness gives $\partial_{n - 1} j_{n - 1} = 0$ , and hence $d_{n - 1} d_{n} = j_{n - 2} (\partial_{n - 1} j_{n - 1}) \partial_{n} = 0.$ Together with step 1.1 this covers every $n \geq 1$ .

Depends on

Used by

Cellular homology Definition
Cohomology of lens spaces from UCT Example
Cellular homology of the one-polygon surface model Lemma
Equivariant p-fold external power and diagonal decomposition Lemma
Incidence curl has zero discrete divergence and gradient curl Lemma
The relative Morse complex of an adapted cobordism Proposition

Dependency tree · two levels

5 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

J. Peter May, A Concise Course in Algebraic Topology, Chapter 13 (standard reference, not scraped)