Alphabeta Math
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
1.1

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.

givenalgebra
2.1

Iterating the estimate yields (n/(n+1))rd. Since 0<n/(n+1)<1, this tends to 0 as r tends to infinity.

step 1.1algebra∎

Depends on

Used by

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