Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-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.

Distinct reduced forms can represent the same integers

Statement refuted

Two binary quadratic forms can represent exactly the same integers and still fail to be properly equivalent. The forms

f=(2,1,3)andg=(2,−1,3)

have this property.

Facts & Assumptions

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

[L1]

Reducedness is defined by ∣b∣≤a≤c together with the boundary sign condition (Reduced positive-definite binary quadratic forms).

[L2]

Each proper-equivalence class of positive-definite forms contains exactly one reduced form (Each proper-equivalence class of positive-definite integral binary quadratic forms contains exactly one reduced form).

[L3]

A form represents an integer when it takes that value at some integer pair (Integers represented, and primitively represented, by a binary quadratic form).

Counterexample

technique · direct
1.1

Both forms are reduced: for each one, ∣ ±1 ∣≤2≤3, and no boundary clause is violated because ∣b∣<a<c.

L1givenalgebra
1.2

For every integers x,y, one has g(x,y)=2x2−xy+3y2=f(x,−y). Thus g represents exactly the integers that f represents, and conversely.

L3givenalgebra
2.1

The reduced triples are distinct because (2,1,3)≠(2,−1,3). Therefore [L2] forbids proper equivalence between f and g.

L2step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

9 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