The Differential Viability Equation: Topological Invariance: A Universal Model of Structural Persistence in the A4 Lattice
Year: 2026
Type: preprint
Abstract: Standard Quantum Field Theory treats radioactive decay and neutrino flavor oscillation as inherently probabilistic phenomena governed by empirical half-lives, weak-mixing parameters, and virtual particle loops. Within the Dynamic Needle Framework (DNF), this manuscript eliminates fundamental quantum probability by modeling the physical vacuum as a discrete 5-dimensional $A_4$ quasicrystalline state-machine operating over the algebraic integer ring $\mathbb{Q}(\sqrt{5})$. We establish a strict ontological distinction between uncoiled 1D bulk Tethers ($C_K = 0$), closed 2D brane geodesics ($0_1$), and knotted 3D volumetric Struts ($C_K \ge 3$). By balancing the topological crossing wear of internal Zitterbewegung oscillations ($\Gamma_{shed} = C_K \cdot \omega_{res}$) against restorative sub-voxel accretion from the ametric protospace colloid ($\Gamma_{accrete} = 3 \cdot \omega_{res}$, anchored by the equilibrium of the $3_1$ Trefoil proton), we derive the parameter-free Differential Viability Equation: $\Delta N_{sub}(n) = 3 - C_K$ per discrete vacuum clock cycle ($f_{vac} = m_{17}c^2/h \approx 4.11 \times 10^{21}\text{ Hz}$). From this single operator, we prove: Proton Immortality ($C_K = 3$): The $3_1$ Trefoil sheds and accretes at identical rates ($\Delta N_{sub} = 0$), guaranteeing absolute stability without ad-hoc baryon-number conservation laws. Resolution of the Neutron Lifetime Anomaly ($C_K = 4$): Operating at a net deficit of $-1$ unit per cycle from an initial plectoneme fold capacity $N_{outer} = 132^{10}(M_n/\Gamma_{shear}^*)\tilde{K}_\pi \approx 3.5855 \times 10^{24}$ down to the Apéry NICF yield floor $Y_0 = 6$, the free neutron is governed by a deterministic geometric countdown timer. Evaluating isotropic ($v = 0$) versus directed ($v > 0$) codimension-2 meniscus drag across the Topological Colloid predicts an ultracold trap lifetime of $\tau_{trap} = 877.61\text{ s}$ (matching UCN$\tau$ $877.75 \pm 0.28\text{ s}$ to $99.984\%$) and a cold beam lifetime of $\tau_{beam} = 887.65\text{ s}$ (matching NIST beam $887.7 \pm 1.2\text{ s}$ to $99.994\%$), resolving the $\sim 10$-second experimental bottle-versus-beam discrepancy ($\Delta\tau_{DNF} = 10.04\text{ s}$) without dark-sector decay channels. Kinematic Accretion ($C_K = 0$): Uncoiled 1D tethers operate at a $+3\omega_{res}$ accretion surplus, mechanically driving neutrino MSW flavor buckling and the $N_\lambda = 6\pi(6-\gamma) \approx 102.217$-wavelength ($1186.6\text{ fm}$) polymer scission limit of the ATOMKI X17 anomaly. The Theorem of Topological Mass Elimination & Fractal Hierarchy: By expressing the base phonon via accretion ($m_{17} = \hbar\Gamma_{accrete}/3c^2$), intrinsic mass is eliminated from physics in favor of exact Diophantine rational flow multipliers ($Q_i \in \{\frac{1}{51}, \frac{8}{3}, \frac{55}{3}, 3400\}$), reducing the gravitational constant to $G_{geo} = \frac{45c^5}{\hbar\Gamma_{accrete}^2 N_{flux}^4}$ and proving that $\hbar$ and $\Gamma_{accrete}$ cancel out of all observable gravitational kinematics ($\mu_i = G_{geo}M_i = 15 Q_i c^2 L_{sub}$). Finally, we map the recursive $S = N_{flux}^4 = 132^{20}$ fractal spatial descent down to the $k=4$ ($6.76 \times 10^{-142}\text{ m}$) Lucas projection cutoff ($L_{12} = 322 > 137.030$), exposing the canonical Planck length as the composite $k=1.5$ geometric mean ($L_P = \sqrt{5 L_{voxel} L_{sub}} \approx 1.61637 \times 10^{-35}\text{ m}$).
Author: Hansen, William
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225376860