Zero-Closure Factorization of the High-Symmetry Floor: Complete +-i Photon-Pair Covering for Even Systems N=8k and the Minimal p-Wave Condensate for Odd Systems

Paper 4 of the series "Foundations of Self-Consistent Relational-Wave Closed Systems" (10 papers). Reconstructing the floor z_ij=r_d e^{i2pi(i+j)/N} from its defining formula and auditing all pairs/3-waves/4-waves, we prove a cyclic square-wave factorization theorem: the squared wave w=z^2 cycles a regular L-gon with L=ord(q)=N/gcd(N,4), q=e^{i8pi/N}, and the minimal zero-closure block is the smallest prime factor p=spf(L) (minimality by the Lam-Leung vanishing-sum theorem). p=2 iff 8|N iff nu_2(N)>=3 (a 2-adic selection rule from the doubling of the phase step under squaring). For N=8k the minimal unit falls to a 2-wave +-i photon pair covering all M waves; odd systems have no 2-wave pair and a minimal p-wave condensate. The make_parent data floor has no exact +-i pairs (near pairs only). Author: Noriaki Kihara (WF System Co., Ltd.), ORCID 0009-0004-6753-4020.