Plane-Decomposition Readout in N-Body Fixed-Generator Systems

Fourth paper of the Dimensional Generation Structure series. Under a fixed antisymmetric generator, each rotation plane of the N-body complete pairwise relational-wave system is an invariant subspace carrying a single-frequency rotation, kinematically isomorphic to the two-body AB system. Five theorems are established: (1) constant-angle, constant-amplitude plane rotation with angle 2 arctan(gamma sigma_j), orientation fixed by the generator (K p = sigma q, residual gauge SO(2)); (2) under the in-plane SO(2) gauge the absolute phase is unreadable while signed progression, frequency, and amplitude are readable - Axiom 5 reproduced at the level of derived quantities; (3) exact energy decomposition H = sum H_j + H_ker with individual conservation; (4) the static kernel admits no progression-based readouts, while the projector, kernel squared quantity, and state projection are covariant (dynamical realization of Consequence 16.3); (5) the squared closure also decomposes, sum Q_j + Q_ker = R^2, with each term individually conserved - a double conservation register per plane, yielding a no-go: under a fixed generator neither closure nor energy-like quantities move between planes, so spontaneous state splitting requires dynamics beyond the fixed generator. The zero-order harmonic second-order structure Delta^2 c = -4 sin^2(theta/2) c holds on every plane at machine precision; the exponent -2 in the continuous-spectrum re-expression is a definitional consequence, explicitly distinguished from integer-harmonic closure. The plane counterpart of the AB same-time positional readout D_AB and the construction of the Axiom-17 observed acceleration on adopted planes are stated as connection problems. Experiments F-I (N = 3-9, 32 trials x 720 steps, fixed generator, no sequential renormalization) verify all claims, including frequency nondegeneracy as an independent general-position assumption. Japanese and English texts, TeX/PDF, four figures, experiment code, and reproduction data are included.