ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 trivial character has Schur index one

Example

For every finite group $G$ , the trivial character $1_{G}$ has character field $Q$ and Schur index $m_{Q} (1_{G}) = 1$ .

Facts & Assumptions

Given: A finite group $G$ and its one-dimensional trivial complex representation.

[L1]

The Schur index is the least positive multiplicity with a model over the character field (Schur index as minimal realization multiplicity).

Verification

technique · direct

1.1

Every value of $1_{G}$ is $1$ , so its character field is $Q$ . The one-dimensional representation on $Q$ in which every element acts as $1$ is a $Q$ -model.

L1algebra

2.1

Thus multiplicity $1$ is attainable, and [L1] makes the Schur index $1$ .

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Gabor Wiese, Galois Representations, Corollary 2.5.10 (standard reference, not scraped)