ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 nonmaximal quadratic order Z[sqrt 5] inside O_Q(sqrt 5)

Example

For $K = Q (\sqrt{5})$ , the subring $A = Z [\sqrt{5}]$ is an order in $O_{K} = Z [(1 + \sqrt{5}) / 2]$ . It has index $2$ and $disc (A) = 20 = 2^{2} \cdot 5$ .

Facts & Assumptions

Given: $K = Q (\sqrt{5})$ and $A = Z [\sqrt{5}]$ .

[F1]

An order is a unital full-rank subring of $O_{K}$ (Order in a number field).

[F2]

The integral basis $(1, ω)$ with $ω = (1 + \sqrt{5}) / 2$ has field discriminant $5$ (The integral basis and discriminant of Q(sqrt 5)).

[F3]

The order-index discriminant formula applies to $A \subseteq O_{K}$ (Order-index discriminant formula).

Verification

technique · direct

1.1

Since $\sqrt{5} = 2 ω - 1$ , $A$ is the subring generated by an integral element and has full-rank basis $(1, \sqrt{5})$ ; hence [F1] makes it an order.

F1F2given

2.1

Relative to $(1, ω)$ , the basis $(1, \sqrt{5})$ has determinant $2$ . Thus $[O_{K} : A] = 2$ , and [F3] with [F2] gives $disc (A) = 2^{2} \cdot 5 = 20$ .

F2F3step 1.1algebra∎

Depends on

Used by

Index 2 obstructs reading factorisation from Z[sqrt 5] modulo 2 Example

Dependency tree · two levels

10 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

J. S. Milne, Algebraic Number Theory, Proposition 2.9 (standard reference, not scraped)