Odd Primes from Statistics: How the centre of SU(3), Gentile modes and anyonic exchange put the Euler factor at 3 into thermal sums, and why even centres lose the factor at 2 (Paper P16)

In thermal physics the prime 2 is everywhere: the step from bosons to fermions multiplies zeta values by the Euler factor 1 − 2^(1−s). The present series showed this for blackbody, Fermi, neutrino and heat-channel constants and noted that a twist of order q > 2 of the thermal boundary condition would bring in odd primes, but proposed no physical system for it. Here we give two. The first is quantum chromodynamics (QCD) at imaginary chemical potential, where the centre ℤ₃ of the gauge group SU(3) acts as a twist of order three (Roberge and Weiss). The second is the ideal gas of anyons in two dimensions, where the exchange phase is a continuous twist. Key results: (i) the sum over the three colours in a confining background is the distribution relation of the polylogarithm and keeps only world lines whose winding number is a multiple of 3; (ii) a bound state of N fermions keeps the Euler factor at 2 exactly when N is odd, the arithmetic form of the rule that baryons of SU(N) are fermions for odd N; (iii) at the Roberge–Weiss point θ = π/3 a quark–antiquark pair has, mode by mode, the partition function 1 + y + y² of a Gentile mode with at most two quanta, so the Euler factor at 3 appears; (iv) the Fourier coefficients of the free-quark baryon density known from lattice studies are exactly those of the Roberge–Weiss envelope, and their slow decay 16/(27k) is the signature of its cusp; (v) the second virial coefficient of ideal anyons is B₂/λ_T² = 1/12 − 2B̄₂(α/2), a periodic Bernoulli polynomial, with both Euler factors at 2 and 3 at the Laughlin value α = 1/3; (vi) a Gentile mode of order q is, up to a factor, the character of the q-dimensional representation of SU(2), so the Roberge–Weiss point is the spin-1 Brillouin sum of the paramagnet; the paramagnet removes the multiples of a modulus from the zeta sum, the colour centre keeps exactly these multiples, and the two Euler factors 1 − m^(−s) and 1 − m^(1−s) are exchanged by a finite Fourier transform and the reflection s ↔ 1 − s; (vii) even centres, as for SU(2) and SU(4), lose the Euler factor at 2 at every step of the ladder N = 2, 4, 8, …, and a twist of order four puts the character χ₋₄ of ℚ(i) into the charge density but not into the pressure. All statements are elementary and proved; the physics is classical. What is new is one reading: particles are waves, their thermal world lines wind around the time circle, and the Euler factor of a prime p is the result of the interference of windings with a phase of order p. Two primers make the paper readable for physicists and for number theorists. The companion code is included as P16_companion_code.zip. It contains six Python check scripts (mpmath and sympy) and the figure script. One of the check scripts independently recomputes every printed number and identity, with 269 identity checks and 6 negative controls. The archive also holds the pre-registration of that revision check with its SHA-256 file, reference logs for runs at 50 and 100 digits, all failed runs, and a runner that repeats every run and compares it with its reference log. The figures used by the LaTeX source are in P16_figures.zip.