The Equation of Motion of a Single Wave in a Relational-Wave Closed System: Kuramoto-Type Phase Coupling on L(K_N) and the Condition for a Self-Consistent Floor on the 90-Degree Phase-Difference Skeleton

Paper 1 of the series "Foundations of Self-Consistent Relational-Wave Closed Systems" (10 papers). For the closed-system map z -> exp(dtau K) z on the line graph L(K_N) with a phase-only real antisymmetric generator (defined in the prior work, Concept DOI 10.5281/zenodo.22112008), we decompose the one-step map into amplitude and phase equations: dr_e/dtau = (1/2) sum_f r_f sin 2(phi_f - phi_e) (amplitude, driven by the second harmonic) and r_e dphi_e/dtau = sum_f r_f sin^2(phi_f - phi_e) (phase). Taking the empirically observed 90-degree two-axis phase lattice as given, we show that on the 90-degree skeleton the identity sin(psi) e^{i psi} = i sin^2(psi) yields D^{-1} K D = i W with W = A_ef sin^2(psi_ef) real symmetric, non-negative and 0/1, so the complex self-consistent floor problem Kz = i omega z is exactly equivalent to the real symmetric eigenproblem Wr = omega r. When W is irreducible, Perron-Frobenius guarantees a unique positive-amplitude principal eigenvector, i.e. a unique self-consistent floor. Author: Noriaki Kihara (WF System Co., Ltd.), ORCID 0009-0004-6753-4020.