Alphabeta Math
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)=X2+2Y2−15Z2,

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
1.1

The diagonal coefficients of q are 1,2,−15, so the only primes dividing 2abc 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.

givenalgebra
2.1

The criterion Ternary isotropy via the Hilbert symbol reduces the local question to the single symbol (15,30)p. For p∉{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.

step 1.1givenalgebra∎

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