A Finite Complete-Relation Closed System with Time Evolution as Vertices: Exact Minimal Closure at N=1, K=4 under U^K=I and Complete-Network Analysis of Even and Odd Harmonics

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We analyze the minimal exact closed system N=1, K=4 in which distinct phase states are treated as ordinary state vertices and all their relations form a complete graph K4, closed by U^K=I with the primitive root U=exp(2 pi i/4)=i. For two-wave states S_k=(U^k, Uᵐᵏ) we compare the minimal nontrivial even harmonic m=2 and odd harmonic m=3, and derive exactly: the internal order reduction ord(U^2)=2 versus ord(U^3)=4; distinct edge-length multisets of the complete state network under the C^2 Euclidean readout; and the parity selection rule Q_loop(m)=4 for even m and 0 for odd m, obtained as a corollary of the general theorem Q_loop(m;K)=K [K|2] + K [K|2m]. A Gram-structure analysis distinguishes m=1 from m=3, which share identical distance multisets. Mathematical results are strictly separated from physical identifications (photon, physical time, spatial distance), which are stated as hypotheses. All figures, data, and reproduction programs are included.