Part 16On the Structural Origin of π and Prime Discreteness from Discrete Rotational Phase Geometry

https: //youtu. be/GNwUB--4RwM? si=sL7vbqOrIT09guIT https: //youtu. be/cKvye6q3M2A? si=jREZGhtffnBTTnpA Revised Zenodo Description (Final) This work is an origin paper addressing a structural question rather than a computational or purely analytic one: Why does the constant appear universally in rotation, waves, probability, and dispersion, and why does prime discreteness persist as an irreducible structure? The manuscript adopts the JS–SH framework, a minimal discrete model composed of spherical rotational phase cells (JS-cells) coupled locally through phase differences. Within this setting, the discrete-to-continuum transition yields two complementary and structurally distinct outcomes. First, continuum imprints arise when symmetry-preserving collective modes survive coarse-graining and manifest as universal constants. In this sense, emerges as an unavoidable imprint of discrete rotational closure under isotropic continuum interpretation. Second, non-closure residues occur when discrete mismatches cannot be factorized, averaged, or eliminated by coarse-graining. Prime numbers are interpreted as irreducible residue objects associated with such non-factorizable phase-closure failures. A central structural result of the paper is that the leading survivor of discrete-to-continuum expansion is generically a quartic-order residue. This quartic structure appears universally across wave, Schrödinger-type, and Euler-type representations derived from discrete nearest-neighbor coupling. The manuscript demonstrates that although this quartic residue may be temporarily suppressed under coarse-graining, it necessarily re-emerges, with its persistence structurally protected by prime-only coordination. Importantly, this work does not claim: a proof of the Riemann Hypothesis, a new definition or computational method for, or a prime-testing or prime-prediction algorithm. Instead, it provides a conceptual and geometric foundation explaining why and prime discreteness arise as complementary consequences of the same discrete rotational structure. Questions involving the Riemann zeta function and critical-line behavior are explicitly deferred to separate analytic investigations. This manuscript is intended to serve as a structural backbone for subsequent analytic work, while remaining self-contained and readable without reliance on advanced number-theoretic machinery.