TheoremStatement: Literature-sourcedProof: 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.

Every positive-definite integral binary quadratic form is properly equivalent to a reduced form

Statement

Every positive-definite integral binary quadratic form is properly equivalent to a reduced form.

Facts & Assumptions

Given: A positive-definite integral binary quadratic form $f$ .

[F1]

A positive-definite form is reduced exactly when it satisfies the inequalities and boundary sign condition of Reduced positive-definite binary quadratic forms.

[L1]

If a positive-definite form is not reduced, then some properly equivalent positive-definite form has strictly smaller reduction measure (A non-reduced positive-definite form admits an equivalent positive-definite form with smaller reduction measure).

[L2]

Every nonempty subset of $N$ has a least element (The well-ordering principle).

Proof

technique · direct

1.1

Let $S$ be the set of reduction measures of the positive-definite forms properly equivalent to $f$ . The set $S$ is nonempty because it contains the measure of $f$ , and $S \subseteq N$ .

givenconstruct

2.1

By [L2], the set $S$ has a least element. Choose a positive-definite form $g$ properly equivalent to $f$ whose reduction measure is that least element.

step 1.1L2choose

3.1

If $g$ were not reduced, [L1] would produce a properly equivalent positive-definite form $h$ with strictly smaller reduction measure. Since $h$ is properly equivalent to $g$ and $g$ is properly equivalent to $f$ , the form $h$ is also properly equivalent to $f$ , so its measure lies in $S$ , contradicting the choice of $g$ .

step 2.1L1algebra

4.1

Therefore $g$ is reduced, and it is properly equivalent to $f$ by construction.

F1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

15 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

William Stein, Elementary Number Theory and Elliptic Curves, Theorem 9.3.2 (standard reference, not scraped)
Andrew Granville, Binary Quadratic Forms, Exercise 4.1e (standard reference, not scraped)