ExampleConstruction: AI-adaptedVerification: 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.

A local obstruction to a rational conic

Example

The conic

x^{2} + y^{2} = 3 z^{2}

has no $Q_{3}$ -point, and therefore no rational point.

Facts & Assumptions

Given: The ternary Hilbert-symbol criterion (Ternary isotropy via the Hilbert symbol).

Verification

technique · direct

1.1

The equation is the isotropy problem for $X^{2} + Y^{2} - 3 Z^{2}$ , so Ternary isotropy via the Hilbert symbol says it is locally soluble over $Q_{3}$ exactly when $(3, 3)_{3} = 1$ .

givenalgebra

2.1

Write $3 = 3^{1} \cdot 1$ twice in the odd-prime formula. Then $(3, 3)_{3} = (- 1)^{(3 - 1) / 2} = - 1$ , so the form is not isotropic over $Q_{3}$ . Therefore the conic has no $Q_{3}$ -point and hence no rational point.

step 1.1givenalgebra∎

Depends on

Ternary isotropy via the Hilbert symbol

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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 (standard reference, not scraped)