CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

One local place is determined by the others for ternary forms

Statement

Let $q = a X^{2} + b Y^{2} + c Z^{2}$ be a nondegenerate ternary diagonal form over $Q$ . If $q$ is isotropic over $Q_{v}$ for every place $v$ except possibly one place $w$ , then $q$ is isotropic over $Q_{w}$ as well.

Facts & Assumptions

Given: A nondegenerate ternary form $q = a X^{2} + b Y^{2} + c Z^{2}$ over $Q$ and a place $w$ .

[L1]

The local isotropy criterion is $q isotropic over Q_{v} ⟺ (- a c, - b c)_{v} = 1$ (Ternary isotropy via the Hilbert symbol).

[L2]

The global reciprocity law is $\prod_{v} (x, y)_{v} = 1$ (Hilbert reciprocity over the rationals).

Proof

technique · direct

1.1

For each place $v \neq w$ , the assumed isotropy and [L1] give $(- a c, - b c)_{v} = 1$ . Multiplying these identities over all $v \neq w$ and using [L2] with $x = - a c$ , $y = - b c$ forces $(- a c, - b c)_{w} = 1$ as well.

L1L2givenalgebra

2.1

Applying [L1] again at the place $w$ shows that $q$ is isotropic over $Q_{w}$ .

L1step 1.1algebra∎

Depends on

Used by

The one-place principle in action Example
Hasse-Minkowski theorem over Q Theorem

Dependency tree · two levels

10 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

Andrew V. Sutherland, 18.782 Lecture 11, Corollary 11.13 (standard reference, not scraped)
Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms, section 4.5 (standard reference, not scraped)