Nuclear Stereometry in the Elastic Continuum: Topological Derivation of Magic Numbers (2, 8, 20, 28, 50, 82, 126) from Mach Cone Interference and the Shear Coherence Limit (Paper 2)

Description / Abstract: This paper presents a deterministic stereometric formulation of nuclear structure that eliminates empirical, ad-hoc potential wells (such as Woods--Saxon parameterizations with artificially amplified spin-orbit coupling). The nucleus is modeled as a close-packed, laminar assembly of interfering nucleonic Cherenkov--Mach acoustic shock cones (θM ≈ 57.56^∘) anchored around hydrostatic compression nodes of unruptured C₃-symmetric neutron tri-blades. We demonstrate that the sequence of nuclear magic numbers partitions naturally into two distinct regimes: (1) Rigorous Geometric Core ($2, 8, 20$): exact space-filling closures of regular Platonic and Keplerian polyhedra in R³ ($Z=2$ tetrahedron with Δθ < 2.82^∘; $Z=8$ stella octangula with ∑ J ≡ 0; $Z=20$ icosahedral face triangulation with 10 α-clusters). (2) Coordination Shells ($28, 50, 82, 126$): qualified as a structured combinatoric model. (3) Stability Threshold at $N=126$ (²⁰⁸Pb): accounting for substrate screening within the dense void composite via the linear Voigt approximation (μeff = μ₀(1 - αₚ)), the effective transverse wave speed drops to ceff ≈ 0.835\,c₀. The critical shear coherence length is derived as λcritᵉᶠᶠ ≈ 6.945 fm, matching the experimental charge radius of lead-208 (R ≈ 7.11--7.15 fm) within ~ 2.4--2.9% (less than $3%$). For $N > 126$, the nuclear radius exceeds the coherence length, triggering boundary phase slip, alpha instability, and spontaneous fission, precluding a superheavy island of stability.