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 ($\theta_M \approx 57.56^\circ$) anchored around hydrostatic compression nodes of unruptured $C_3$-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 $\mathbb{R}^3$ ($Z=2$ tetrahedron with $\Delta\theta < 2.82^\circ$; $Z=8$ stella octangula with $\sum \mathbf{J} \equiv 0$; $Z=20$ icosahedral face triangulation with 10 $\alpha$-clusters). (2) Coordination Shells ($28, 50, 82, 126$): qualified as a structured combinatoric model. (3) Stability Threshold at $N=126$ ($^{208}\text{Pb}$): accounting for substrate screening within the dense void composite via the linear Voigt approximation ($\mu_{\text{eff}} = \mu_0(1 - \alpha_p)$), the effective transverse wave speed drops to $c_{\text{eff}} \approx 0.835\,c_0$. The critical shear coherence length is derived as $\lambda_{\text{crit}}^{\text{eff}} \approx 6.945\text{ fm}$, matching the experimental charge radius of lead-208 ($R \approx 7.11\text{--}7.15\text{ fm}$) within $\sim 2.4\text{--}2.9\%$ (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.