ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Direct cancellation in the boundary squared of a two-simplex

Example

For the identity singular $2$ -simplex $ι_{2} : Δ^{2} \to Δ^{2}$ , $\partial^{2} ι_{2} = (v_{2} - v_{1}) - (v_{2} - v_{0}) + (v_{1} - v_{0}) = 0.$ Each vertex appears exactly twice with opposite signs.

Facts & Assumptions

Given: The identity singular $2$ -simplex $ι_{2} : Δ^{2} \to Δ^{2}$ .

[L1]

The boundary of $ι_{2}$ is the alternating sum of its three oriented edges (Boundaries of the standard one- and two-simplices).

[L2]

Singular boundaries square to zero (The singular boundary squares to zero).

Verification

technique · direct

1.1

Expanding [L1] once more gives $\partial^{2} ι_{2} = \partial (v_{1} v_{2}) - \partial (v_{0} v_{2}) + \partial (v_{0} v_{1}) = (v_{2} - v_{1}) - (v_{2} - v_{0}) + (v_{1} - v_{0}),$ where each edge boundary is the degree-one formula from the previous example.

L1givenalgebra

2.1

The three positive and three negative vertex terms cancel pairwise, so the sum is $0$ , exactly as predicted by [L2].

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)