Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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 divisor and class of a fractional ideal

Example

Assume the Axiom of Choice.

In R=Z, the fractional ideal

I:=409Z=(2)3(5)(3)−2

has divisor

div⁡(40/9)=3[(2)]+[(5)]−2[(3)],

and its ideal class is trivial.

Facts & Assumptions

Given: The Axiom of Choice, the Dedekind domain R=Z, and the fractional ideal I=(40/9)Z.

[L1]

For x∈Q×, the principal-divisor sequence sends x to the valuation vector of (x) and then sends that divisor to the trivial class of (x) (The principal-divisor exact sequence for a Dedekind domain).

Verification

technique · direct
1.1

The generator 40/9 contributes prime exponents 3 at (2), 1 at (5), and −2 at (3), so div⁡(40/9) is the displayed valuation vector.

L1givenalgebra
2.1

Because I is principal, its class is zero in the class group by [L1].

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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