TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

The simplicial boundary squares to zero

Statement

For every simplicial complex $K$ and every $n \geq 1$ , one has $\partial_{n - 1} \partial_{n} = 0 : C_{n} (K) \to C_{n - 2} (K) .$

Proof

Given: An integer $n \geq 1$ and an oriented $n$ -simplex $[v_{0}, \dots, v_{n}]$ .

1.1

Expanding $\partial_{n - 1} \partial_{n} [v_{0}, \dots, v_{n}]$ produces the sum of all codimension-two faces obtained by deleting two vertices, once by deleting $v_{i}$ then $v_{j}$ and once by deleting $v_{j}$ then $v_{i}$ .

given

2.1

The two appearances of the same codimension-two face have opposite signs because the exponents differ by $1$ . Hence every codimension-two face cancels with its partner, and the full sum is $0$ .

step 1.1

3.1

The boundary maps are homomorphisms, so vanishing on every oriented simplex implies $\partial_{n - 1} \partial_{n} = 0$ on all of $C_{n} (K)$ .

step 2.1∎

Depends on

Used by

Simplicial cycles, boundaries, and homology Definition

Dependency tree · two levels

4 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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)