Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

The word category on words of length three

Example

Among the unit-free binary words of length three, there are exactly two objects: ((−□−)□−),(−□(−□−)).

Facts & Assumptions

Given: The binary-word category W.

[L1]

Binary words are built recursively from e0, (−), and □, with length determined additively (The category of binary words).

[L2]

The category W is monoidal and therefore has the unique structural arrows between equal-length bracketings (The category of binary words is monoidal).

Verification

technique · direct
1.1

A unit-free length-three word must be obtained by combining exactly three copies of (−) with two binary operations. Up to the placement of brackets, the only possibilities are ((−□−)□−) and (−□(−□−)). Words of length three containing one or more copies of e0 also exist, but are outside this unit-free subcollection.

givenL1
1.2

These two words have the same length, so there is exactly one morphism between them in each direction. In the monoidal structure of [L2], those are the associator and its inverse at the generator.

L1L2
2.1

Thus the unit-free part of the length-three slice of W is the basic coherence picture: two bracketings and one canonical comparison each way.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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