LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Mesh tends to zero under iterated subdivision

Statement

For an affine $n$ -simplex of finite diameter and $n > 0$ , every simplex in its $r$ -fold barycentric subdivision has diameter at most $(n / (n + 1))^{r}$ times the original diameter. Hence the mesh tends to zero.

Facts & Assumptions

Given: An affine $n$ -simplex with $n > 0$ and finite diameter $d$ .

Proof

technique · direct

1.1givenalgebra

Each barycentric subsimplex has vertices which are barycenters of nested faces; their convex-coordinate differences have diameter at most $n / (n + 1)$ times that of the parent.

2.1step 1.1algebra∎

Iterating the estimate yields $(n / (n + 1))^{r} d$ . Since $0 < n / (n + 1) < 1$ , this tends to $0$ as $r$ tends to infinity.

Mesh tends to zero under iterated subdivision

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources