Central Projection of a Vacuum Universe and Particle-like States with a Spread Phase (Observational Note)

Observational note (thought-experiment style; not a proof or claim paper). Taking an exterior-free vacuum universe as a single intrinsic geometric system, we observe that particle-like states (phase-bearing local modes) carrying a spread phase can be placed on the subjective geometry produced by its central projection. Starting from the fifth frequency nu_5 and composite radius R_U = 1/nu_5 of the preceding note, and from the central (gnomonic/radial) projection, we observe: (i) the subjective geometry is defined as the homogeneous 4-sphere S^4(R_U) in R^5 (sectional curvature 1/R_U^2), with the central projection regarded as a gnomonic chart covering one hemisphere; no Lorentzian metric or causal structure is introduced, so it is not called physical spacetime; (ii) because the geometry is homogeneous with no origin or reference axis, no absolute position phase is defined for a single particle; (iii) a particle-like state is represented minimally by P_a = (nu_a, R_a); (iv) under the model convention that reads a frequency as the winding number of phase and the spread as an effective diameter, the spread (full width) is read as twice the curvature radius, W_a = 2 R_a = 2/nuR,a, with the spread frequency identified with the curvature frequency (nu_w = nuR,a). The scope is limited to defining the single-particle geometric stage; relative phase, interference, and statistics of multiple particles are deferred to a subsequent note. This record contains Japanese and English versions (Markdown, LaTeX, PDF).