Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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.

Hilbert symbols over the real numbers

Example

Over R, the Hilbert symbol depends only on the signs of its two arguments:

(1,1)∞=(1,−1)∞=(−1,1)∞=1,(−1,−1)∞=−1.

Facts & Assumptions

Given: The real Hilbert-symbol formula and the equivalent solvability/norm interpretation (The real Hilbert symbol formula, Equivalent formulations of the Hilbert symbol).

Verification

technique · direct
1.1

The theorem The real Hilbert symbol formula gives value 1 whenever at least one argument is positive, so (1,1)∞=(1,−1)∞=(−1,1)∞=1.

givenalgebra
2.1

The same theorem gives (−1,−1)∞=−1. Equivalently, the norm form from C=R(−1) is u2+v2, which never represents −1 over R, matching Equivalent formulations of the Hilbert symbol.

step 1.1givenalgebra∎

Depends on

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