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.

The one-place principle in action

Example

For the ternary form

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

the unknown $2$ -adic local value is forced by the values at the other places.

Facts & Assumptions

Given: The one-place principle for ternary forms (One local place is determined by the others for ternary forms).

Verification

technique · direct

1.1

The criterion for $q$ is the symbol $(2, 2)_{v}$ . At the real place this is $1$ because both arguments are positive. At every odd prime $p$ , both arguments are units, so the odd-prime formula gives $(2, 2)_{p} = 1$ .

givenalgebra

2.1

Since every place except possibly $2$ gives value $1$ , One local place is determined by the others for ternary forms forces the remaining $2$ -adic value also to be $1$ . Hence $q$ is isotropic over $Q_{2}$ without a separate direct computation there; globally, $(1, 1, 1)$ is the corresponding visible isotropic vector.

step 1.1givenalgebra∎

Depends on

One local place is determined by the others for ternary forms

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)