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.

Boundaries of the standard one- and two-simplices

Example

Let $ι_{1} : Δ^{1} \to Δ^{1}$ and $ι_{2} : Δ^{2} \to Δ^{2}$ be the identity singular simplices.

Their singular boundaries are $\partial ι_{1} = ι_{1} δ_{0} - ι_{1} δ_{1} = v_{1} - v_{0}$ and $\partial ι_{2} = ι_{2} δ_{0} - ι_{2} δ_{1} + ι_{2} δ_{2},$ the alternating sum of the three oriented edges of the standard triangle.

Facts & Assumptions

Given: The identity singular simplices $ι_{1} : Δ^{1} \to Δ^{1}$ and $ι_{2} : Δ^{2} \to Δ^{2}$ .

[L1]

The affine face maps $δ_{i}$ insert a zero in the $i$ th slot (The standard topological simplex and its affine face maps).

[L2]

The singular boundary is the alternating sum of the face restrictions (The singular boundary operator).

Verification

technique · direct

1.1

By [L1], the two face maps of $Δ^{1}$ pick out the terminal and initial vertices of the interval. Therefore [L2] gives $\partial ι_{1} = ι_{1} δ_{0} - ι_{1} δ_{1} = v_{1} - v_{0} .$

L1L2given

2.1

Again by [L1], the three face maps of $Δ^{2}$ are the affine inclusions of the three edges opposite $v_{0}$ , $v_{1}$ , and $v_{2}$ . Applying [L2] gives the displayed alternating sum $\partial ι_{2} = ι_{2} δ_{0} - ι_{2} δ_{1} + ι_{2} δ_{2} .$

L1L2step 1.1∎

Depends on

Used by

Direct cancellation in the boundary squared of a two-simplex Example

Dependency tree · two levels

3 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)