Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31
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.

Every diagram in a monoidal category commutes

Statement

False claim: every diagram in a monoidal category commutes.

Facts & Assumptions

Given: The scope warning attached to the coherence theorem.

[L1]

Coherence only covers formal diagrams of canonical morphisms between parenthesised tensor words (Why 'every diagram commutes' is false as stated).

[L2]

A category with finite products is monoidal under its cartesian product (A category with finite products is monoidal).

Refutation

technique · direct
1.1

Let A={0,1} in the cartesian monoidal category of sets supplied by [L2], and let c:A→A be the constant-zero map. Consider the square with all four vertices A, top edge c, and the other three edges 1A.

givenL2construct
2.1

The composite along the top and right edges is c, while the composite along the left and bottom edges is 1A. These maps differ because c(1)=0≠1=1A(1), so the square does not commute.

step 1.1algebra
3.1

This noncanonical square lies outside the scope described in [L1] and is a diagram in a monoidal category that does not commute. Therefore the claim is false.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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