Sharp moments of nonnegative gamma sums on the Euclidean sphere

Unrefereed research candidate. Add up n independent gamma (for example exponential) random variables with nonnegative weights whose squares sum to one, so that the variance of the total is fixed. This paper determines exactly how small and how large the q-th moment of the total can be, for every real q > 0 and every common gamma shape α > 0. Minimum (Theorem 1). The minimum equals min1≤m≤n Γ(mα+q)/(Γ(mα) mq/2) and is attained only at vectors with m equal nonzero weights. For exponential variables one variable is best up to q = 5.3197…, then two equal weights, then three, the optimal number growing by one at a time and roughly like 0.40 q. For α = 1 this proves Conjecture 4.8 of Brazitikos and Pandis (arXiv:2512.12254), previously known for integer q (Brazitikos and Pandis), for q ≤ 4 (Brazitikos, Tang and Tkocz, arXiv:2602.03058) and, in an unrefereed AI-generated dossier, for q ≤ 5. Maximum (Theorem 3). The maximum is attained either at the flat vector or at a vector with one larger weight and n−1 equal smaller ones, located by the unique interior local maximum of an explicit hypergeometric function. The flat vector is the unique maximiser below a switch exponent q*; for n = 2, q* = 2α+2 and the switch is continuous; for n ≥ 3, 2+2α√(n−1) < q* < nα+2 and the maximiser jumps. Also: large-q asymptotics of the minimum, negative exponents, and consequences for positive Gaussian quadratic forms (α = 1/2) and symmetric Dirichlet averages. The archive contains the paper (PDF and Markdown source), the prior-art and credit record, four verification scripts (computer algebra, exact rational comparisons, 30-digit hypergeometric sign checks and global optimisation; 26,147 explicit checks that remain active under python -O) with recorded outputs, a deliberately broken negative-control run, the figures with their data, and the raw Crossref and arXiv records from which every reference was copied. Assurance: written proofs, not machine-verified; computations are finite checks at sample parameters; the numerical switch values q* are numerics; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded, dated literature search. Creator: Anonymous. Text CC0-1.0; code MIT.