We ask whether the Plancherel-saturation structure that fixes the renormalized mass at c = 7/45 for the 3x +1 Syracuse map (Humphrey, 2026b) generalizes to the qx + 1 family for odd q ≥ 3, and answer at three rigor tiers. First, the full-period Fourier transform Fp satisfies a family-level magnitude formula | Fp (ξ) | = p (r+3) /2 on a support of size pr−1; its residual equidistribution lemma, previously empirical at q = 3, is now established by a machine-checked second-moment identity formalized in Lean 4 and verified to machine precision at small p, leaving a single cited standard fact (a binomial lift) as the only undischarged hypothesis. Second, the short-window character sum Kp exhibits universal √N saturation across p ∈ 3,. . . , 23, though the bilinear bound that would explain it is not produced; we locate the Burgess-type wall precisely. Third, the renormalized mass follows a candidate ˜cq = (q − 3) /q governed by a universal q/3 ratio, with q = 5, 7 deviations reported transparently. We state three negative claims upfront—chiefly that none of this dissolves the conditional dependence on Tao (2022) Proposition 1. 17, since Fp is the wrong object to substitute for µn. The work was pre-registered.
Nathan Humphrey (Sat,) studied this question.