Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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.

Boone sharp is not word inversion

Example

Choose the distinct tape symbols a=s0 and b=h in Sˉ. Then (ab)#=a−1b−1, whereas (ab)−1=b−1a−1. These have different spellings.

Facts & Assumptions

Given: Set a=s0 and b=h.

[F1]

The machine tape alphabet S contains its blank s0, while h is a new symbol distinct from every member of S; the subsequent tape alphabet is Sˉ=S∪{h}. (Boone machine semigroup and augmented configurations)

[F2]

Sharp reverses each tape-letter sign in the same order; an inverse word reverses the whole order as well. (Boone group presentation and special word)

Verification

1.1

By [F1], a,b are distinct letters of Sˉ. Applying the defining letter substitution to the two positions gives (ab)#=a#b#=a−1b−1.

F1F2givenalgebra
1.2

The word b−1a−1 is the inverse spelling: (ab)(b−1a−1) cancels first bb−1 and then aa−1; the product in the other order cancels first a−1a and then b−1b. Thus (ab)−1=b−1a−1.

F2givenalgebra
2.1

The two outputs start with different symbols and neither has adjacent inverse pairs, so their reduced spellings differ. For a single tape letter the two operations agree, and both send the empty word to the empty word. The length-two instance displays why sharp cannot be replaced by word inversion; no inequality in an arbitrary quotient group is inferred from spelling alone.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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