ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 degree-two Jordan-quiver Hall product

Example

Let $k = F_{q}$ , let $S = I_{1}$ be the simple Jordan-quiver representation, and let $I_{2}$ be the indecomposable length-two Jordan block. Then $[S] * [S] = (q + 1) [I_{(1, 1)}] + [I_{2}] .$

Facts & Assumptions

Given: A finite field $k = F_{q}$ , the nilpotent Jordan-quiver category $J_{k}$ , and its untwisted Hall product.

[F1]

The middle isomorphism classes are indexed by partitions and $I_{λ} = ⨁_{j} I_{λ_{j}}$ (Jordan-quiver isomorphism classes are indexed by partitions).

[F2]

$[M] * [N]$ counts subobjects isomorphic to $N$ with quotient isomorphic to $M$ (The untwisted Hall algebra of a finitary abelian category).

[F3]

The Jordan-quiver category over a finite field is finitary, and its subobjects are invariant subspaces (Jordan-quiver categories are abelian; finite-field models are finitary).

Verification

technique · direct

1.1F1F2F3

The category is finitary by [F3], so its Hall product is defined. Any middle object contributing to $[S] * [S]$ has vector-space dimension two. By [F1], its type is either $(1, 1)$ or $(2)$ .

1.2F1F2F3

In $I_{(1, 1)} = S \oplus S$ , the nilpotent operator is zero, so every one-dimensional subspace is an invariant subobject by [F3], and its quotient is $S$ . There are $(q^{2} - 1) / (q - 1) = q + 1$ such lines.

1.3F1F2F3

In $I_{2}$ , an invariant one-dimensional subspace must be annihilated by the nilpotent operator, so it is contained in $\ker x$ . This kernel is one-dimensional, hence it is the unique such subobject by [F3], and the quotient is $S$ .

2.1step 1.1step 1.2step 1.3∎

The two possible middle types and their counts from steps 1.1–1.3 yield the displayed Hall product.

Depends on

Used by

Jordan Hall–Littlewood basis and pairing in degrees zero, one and two Example

Dependency tree · two levels

25 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

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §2.2, Example 2.2 (standard reference, not scraped)