Analytic Exact Solution of the 90-Degree High-Symmetry Floor: The sigma_max Eigenmode of the Integer Antisymmetric Generator Defined by Z4 Phase Labels, and the Closed-Form Spectrum

Paper 3 of the series "Foundations of Self-Consistent Relational-Wave Closed Systems" (10 papers). Quantizing the phase to the Z4 label k_e=(i+j) mod 4 makes the generator K_ef=A_ef sin(90deg (k_f-k_e)) an integer matrix (0,+-1), so the floor is algebraically exact as the sigma_max eigenmode (residual ~1e-15). We give closed forms for sigma^2 and the squared amplitudes by parity class (sympy-exact), the parity bipartite structure W_ef=A_ef 1[p_e != p_f] with a reduced Perron problem (2x2 even / 3x3 odd) valid for arbitrary N, the automatic square-closure sum z^2=0 from the bipartite eigenmodes, connectivity of W for all N>=3, the sign-gauge family 2^(M-1), and the rank/kernel of K (rank 2(N-1), ker (N-1)(N-4)/2 for N>=6, with the low-dimensional exceptions N=3,4,5). Author: Noriaki Kihara (WF System Co., Ltd.), ORCID 0009-0004-6753-4020.