ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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 quaternary Hasse-Minkowski calculation

Example

The quaternary form

q (X, Y, Z, W) = X^{2} + 2 Y^{2} - 3 Z^{2} - 6 W^{2}

is isotropic over $Q$ .

Facts & Assumptions

Given: The global square-class approximation lemma and the Hasse-Minkowski theorem (Global approximation of finitely many square classes, Hasse-Minkowski theorem over Q).

Verification

technique · direct

1.1

The two binary subforms $X^{2} + 2 Y^{2}$ and $3 Z^{2} + 6 W^{2}$ both represent the common value $3$ : indeed $1^{2} + 2 \cdot 1^{2} = 3$ and $3 \cdot 1^{2} + 6 \cdot 0^{2} = 3$ . No approximation lemma is needed in this concrete instance because the common rational value is already explicit.

givenalgebra

2.1

Substituting these representations gives $q (1, 1, 1, 0) = 3 - 3 = 0$ , so $(1, 1, 1, 0)$ is a rational isotropic vector. This concrete calculation is exactly what Hasse-Minkowski theorem over Q guarantees once the local square classes have been matched.

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Theorem 11.17 (standard reference, not scraped)