# Verification of the character exponent counterexample

The source-checked written argument proves that PGammaL_3(4) is almost simple and has quotient S_3 by PSL_3(4). Inflation of the irreducible degree-two quotient representation gives character value two at the noncentral order-two element [I+E_12]. Hence no exponent alpha<1 satisfies the source's inequality for every irreducible character. The proof includes a self-contained primitive-action proof of simplicity, an injective semilinear conjugation action, and the explicit quotient matrices.

The general criterion uses E=H', where H is the preimage of the simple socle of G/Z(G). The desired exponents exist at every noncentral element if and only if G/E is abelian. The proof uses the projective kernel and strict trace inequality. A roots-of-unity estimate supplies the weak explicit exponent 1-16/(ord(g)^2 |G| log |G|). It is not claimed to be optimal or to improve the deep quasisimple bounds.

## Exact computational checks

The supplied standard-library checker performs 1,222,248 explicit checks in each mode. It enumerates 60,480 projective invertible matrices over F_4, generates all 20,160 determinant-one classes, verifies all matrix inverses, the determinant quotient, scalar invariance, and double transitivity on 21 projective points. It verifies all 36 products in the six-element quotient representation, exact character norms, and the noncentral witness. Finite eigenvalue-multiplicity tests supplement the analytic bound. No floating-point computation is used.

Normal and optimized receipts agree except for their mode flag and identify the identical checker source hash. Explicit exceptions enforce checks even under `python -O`. The finite computations do not prove the arbitrary-group theorem; its proof is in the manuscript.

## Source and scope

The two complete report pages containing Problem 1, its neighboring theorems and references were inspected. The current Hugging Face original statement agrees with the frozen original. The source does not require faithful characters. Its quasisimple theorems and the problem of optimal bounds for representations nontrivial on E are unaffected.

The proof is accepted on the originating researcher's detailed self-audit. No external expert review, proof-assistant verification, exhaustive literature search, or absolute priority certification is claimed. Classical ingredients are attributed. This is an unrefereed formulation correction, not a claim to have obtained a new deep character-bound theorem.