LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Reduction of lifted traces recovers modular traces

Statement

In the fixed splitting system, $\overline{φ_{V} (g)} = tr (g ∣ V)$ for every finite-dimensional $V$ and every $p$ -regular $g$ .

Facts & Assumptions

Given: V,g and the splitting system in the Statement.

[F1]

The lifted trace is the sum of the unique lifts of the eigenvalues with multiplicity (Lifted modular trace on p-regular elements).

Proof

1.1

Reduction $O \to k$ is a ring homomorphism. Therefore $\overline{φ_{V} (g)} = \sum_{j = 1}^{\dim V} \overline{\hat{λ_{j}}} = \sum_{j} λ_{j}$ . This uses additivity of reduction, not additivity of root lifting.

F1algebra

2.1

In an eigenbasis of the p-regular operator the diagonal entries are precisely the $λ_{j}$ , so their sum is its trace. The definition supplies that eigenbasis decomposition. If $V = 0$ both sides are zero; if $g = 1$ , both reduce to $(\dim V) 1_{k}$ .

F1step 1.1algebra∎

Depends on

Lifted modular trace on p-regular elements

Used by

Irreducible Brauer characters are independent on p-regular elements Theorem

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

Halle, Galois actions on Neron models of Jacobians, Section 5.3, p.875 (standard reference, not scraped)