LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Barycentric coordinates are unique

Statement

Let $v_{0}, \dots, v_{n}$ be affinely independent points in $R^{m}$ . If $x = \sum_{i = 0}^{n} λ_{i} v_{i} = \sum_{i = 0}^{n} μ_{i} v_{i}, \sum_{i = 0}^{n} λ_{i} = \sum_{i = 0}^{n} μ_{i} = 1,$ then $λ_{i} = μ_{i}$ for every $i$ .

Proof

Given: Two barycentric-coordinate expressions for the same point of the simplex spanned by $v_{0}, \dots, v_{n}$ .

1.1

Subtract the two expressions for $x$ to obtain $\sum_{i = 0}^{n} (λ_{i} - μ_{i}) v_{i} = 0$ , and subtract the two sum conditions to obtain $\sum_{i = 0}^{n} (λ_{i} - μ_{i}) = 0$ .

given

2.1

The previous step is an affine dependence relation among $v_{0}, \dots, v_{n}$ whose coefficients sum to $0$ . Since the vertices are affinely independent, every coefficient vanishes, so $λ_{i} - μ_{i} = 0$ for all $i$ .

step 1.1∎

Depends on

The geometric simplex spanned by affinely independent vertices

Used by

Geometric simplices intersect in the realization of their common face Lemma

Dependency tree · one level

1 result within one dependency step 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 01: Complexes (standard reference, not scraped)