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.

Finite bad places for a ternary form

Example

For the ternary form

q (X, Y, Z) = X^{2} + 2 Y^{2} - 15 Z^{2},

the only primes that can require an explicit local check are $2, 3, 5$ .

Facts & Assumptions

Given: The ternary Hilbert-symbol criterion and the almost-all-primes isotropy theorem (Ternary isotropy via the Hilbert symbol, Almost all local completions are isotropic in dimension at least three).

Verification

technique · direct

1.1givenalgebra

The diagonal coefficients of $q$ are $1, 2, - 15$ , so the only primes dividing $2 a b c$ are $2, 3, 5$ . The theorem Almost all local completions are isotropic in dimension at least three guarantees that only finitely many primes can be bad; the exact finite list for this form is determined in the next step.

2.1step 1.1givenalgebra∎

The criterion Ternary isotropy via the Hilbert symbol reduces the local question to the single symbol $(15, 30)_{p}$ . For $p \notin {2, 3, 5}$ both arguments are $p$ -adic units, so the odd-prime formula gives value $1$ . Thus every prime outside ${2, 3, 5}$ is good, and only $2, 3, 5$ need separate local computation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)