The Irreducible Three-Body Congruence Correlation and the Annihilation Lattice on the 6N Skeleton

Extends the two-centre phase-space convolution of Part XX to three twin centres on the 6N skeleton and asks whether the three-body correlation R3(j1,j2) factorizes into pairwise terms. It does not: the irreducible three-body factor U=R3/(R2(j1)R2(j2)R2(j2-j1)) differs from 1 at every lag pair (0 of 870 factorize in the scanned window), so an irreducible three-body congruence term is generic. The extreme case R3=0 (total annihilation) obeys an exact TILING criterion: the joint density vanishes iff the shifted dead-residue sets cover Z/q for some prime q. This corrects a natural dimension-count heuristic (m_A+m_B>=q): tiling requires the hole-sets to be COMPLEMENTARY, not merely numerous. A triplet contains a twin, so their dead-sets always overlap and a Twin-Triplet two-body pair never annihilates (I_5(j) in {1/5,2/5}, never 0); the lowest order at which twin-based annihilation occurs is three-body. Three consecutive twin centres (lags 0,1,2) tile Z/5, so the six members 6N-1,6N+1,6N+5,6N+7,6N+11,6N+13 always contain a multiple of 5 -- recovering the mod-5 inadmissibility of that window from pure phase-space geometry. The annihilation lattice has 216 forbidden lag pairs in the scanned window. The horizontal annihilation here is distinct from the single-centre 'killer prime' (vertical omega-collapse) of Volume I. Closed-form throughout; the integral equals the discrete singular-series ratio at integer lags; no infinitude is claimed.