A Quotient Obstruction to Strict Character Exponents in Almost Quasisimple Groups

A workshop problem asks for an exponent 0 <= alpha(g) < 1 such that |chi(g)| <= chi(1)^alpha(g) for every irreducible complex character of a finite almost quasisimple group and every noncentral element g. The unrestricted formulation has an elementary quotient obstruction. The almost simple group PGammaL_3(4) has quotient S_3 by PSL_3(4). Inflating the irreducible degree-two character of this quotient gives value 2 at a noncentral transvection, so no exponent below 1 exists. More generally, if H is the inverse image of the simple socle in G/Z(G) and E=H', the required exponents exist at all noncentral elements exactly when G/E is abelian. In that case we give the explicit, deliberately weak exponent 1-16/(ord(g)^2 |G| log |G|). This is a correction to the all-character formulation, not an improvement on optimal character bounds for quasisimple groups. The construction uses classical methods; novelty and priority are not certified. This is an unrefereed preprint, with AI assistance disclosed in the text.