Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26
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.

Improper equivalence can merge two distinct proper classes

Example

If one allows determinant −1 substitutions as well as determinant 1 substitutions, then the two distinct reduced forms

(2,1,3)and(2,−1,3)

become equivalent.

Facts & Assumptions

Given: The forms f=(2,1,3) and g=(2,−1,3).

[F1]

Proper equivalence uses determinant-one integer matrices (Proper equivalence of binary quadratic forms).

[L1]

The previous counterexample shows that f and g are not properly equivalent (Distinct reduced forms can represent the same integers).

Verification

technique · direct
1.1

The matrix N=(100−1) has determinant −1, and direct substitution gives f∣N=(2,−1,3)=g.

F1givenalgebra
2.1

Thus allowing determinant −1 merges the two distinct proper classes from [L1]: the forms are equivalent under a unimodular substitution, but not under a determinant-one unimodular substitution.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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