Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Finite-field groupoid cardinality for quiver stack specializations

Definition

Let k=Fq. A finite groupoid is a groupoid with finitely many isomorphism classes of objects and finite automorphism groups. Its groupoid cardinality is ∣G∣=∑[x]∈Iso⁡(G)1∣Aut⁡G(x)∣. For a quotient stack in the finite-quiver construction whose rational-point groupoid is finite, define its point-count value by the groupoid cardinality of that rational-point groupoid. This definition applies, in particular, to Md(Q), the exact-sequence stacks, and the vector-bundle-stack fibers in the examples below. Its compatibility with the relations defining K is established by Finite-field groupoid cardinality respects the motivic stack relations ↗.

Depends on

Used by

Dependency tree · two levels

3 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