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 two-adic Hilbert symbol table

Example

The two-adic formula gives, for instance,

(- 1, - 1)_{2} = - 1, (5, 5)_{2} = 1.

Facts & Assumptions

Given: The explicit formula for the Hilbert symbol over $Q_{2}$ (The two-adic Hilbert symbol formula).

Verification

technique · direct

1.1

For $a = b = - 1$ , one has $α = β = 0$ , $u = v = - 1$ , so $ϵ (u) = ϵ (v) = 1$ and $ω (u) = ω (v) = 0$ . Hence The two-adic Hilbert symbol formula gives $(- 1, - 1)_{2} = (- 1)^{1 \cdot 1} = - 1$ .

givenalgebra

2.1

For $a = b = 5$ , one has $α = β = 0$ , $u = v = 5$ , so $ϵ (5) = 0$ and $ω (5) = 1$ . The exponent in the theorem is therefore $0$ , and $(5, 5)_{2} = 1$ .

step 1.1givenalgebra∎

Depends on

The two-adic Hilbert symbol formula

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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 10, Theorem 10.9 (standard reference, not scraped)