Standing-Wave Coherence from Isotropy: A Pre-Dynamical Admissibility Derivation of Minimal Structure

Modern science explains how structures evolve once space, time, and dynamical laws are assumed. We ask a prior question: which geometric forms are admissible as the first coherent differentiations of a maximally symmetric (isotropic), undifferentiated state? The analysis is deliberately pre-dynamical and pre-physical: no temporal evolution, field equations, energetics, or mechanism are assumed.“Zero” is interpreted operationally as nondifferentiation (maximal isotropy), not emptiness and not a physically extant point. The sphere appears only as a symmetry object encoding “all directions are equivalent” once differentiation is contemplated. Coherence is a closure-compatibility constraint (standing-wave–like only in the sense of global consistency under closure).Under isotropy-preserving closure and minimality, continuous differentiation is disfavored and a finite set of extrema is forced. The smallest non-degenerate configuration requires four extrema; imposing single-scale maximal symmetry uniquely selects the regular tetrahedron (up to rotation). Minimal conjugate completion yields the star tetrahedron, and the cube/octahedron arise as induced envelopes. We record admissible extension pathways toward packing and Voronoi structure (preview only), deferring substrate selection to later work. The results are admissibility claims conditional on explicit postulates, not assertions of physical necessity.