The Differential Viability Equation: Topological Invariance: A Universal Model of Structural Persistence in the A4 Lattice

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 A₄ quasicrystalline state-machine operating over the algebraic integer ring Q(√5). We establish a strict ontological distinction between uncoiled 1D bulk Tethers (CK = 0), closed 2D brane geodesics (0₁), and knotted 3D volumetric Struts (CK ≥ 3). By balancing the topological crossing wear of internal Zitterbewegung oscillations (Γshed = CK · ωᵣₑₛ) against restorative sub-voxel accretion from the ametric protospace colloid (Γaccrete = 3 · ωᵣₑₛ, anchored by the equilibrium of the 3₁ Trefoil proton), we derive the parameter-free Differential Viability Equation: Δ Nsub(n) = 3 - CK per discrete vacuum clock cycle (fvac = m₁₇c²/h ≈ 4.11 × 10²¹ Hz). From this single operator, we prove: Proton Immortality (CK = 3): The 3₁ Trefoil sheds and accretes at identical rates (Δ Nsub = 0), guaranteeing absolute stability without ad-hoc baryon-number conservation laws. Resolution of the Neutron Lifetime Anomaly (CK = 4): Operating at a net deficit of $-1$ unit per cycle from an initial plectoneme fold capacity Nₒᵤₜₑᵣ = 132¹⁰(Mₙ/Γₛₕₑₐᵣ^*)K̃_π ≈ 3.5855 × 10²⁴ down to the Apéry NICF yield floor Y₀ = 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 τₜᵣₐₚ = 877.61 s (matching UCNτ 877.75 ± 0.28 s to $99.984%$) and a cold beam lifetime of τbeam = 887.65 s (matching NIST beam 887.7 ± 1.2 s to $99.994%$), resolving the ~ 10-second experimental bottle-versus-beam discrepancy (ΔτDNF = 10.04 s) without dark-sector decay channels. Kinematic Accretion (CK = 0): Uncoiled 1D tethers operate at a +3ωᵣₑₛ accretion surplus, mechanically driving neutrino MSW flavor buckling and the N_λ = 6π(6-γ) ≈ 102.217-wavelength (1186.6 fm) polymer scission limit of the ATOMKI X17 anomaly. The Theorem of Topological Mass Elimination & Fractal Hierarchy: By expressing the base phonon via accretion (m₁₇ = Γaccrete/3c²), intrinsic mass is eliminated from physics in favor of exact Diophantine rational flow multipliers (Qᵢ ∈ \1/51, 8/3, 55/3, 3400\), reducing the gravitational constant to Ggeo = 45c⁵Γaccrete² Nflux⁴ and proving that and Γaccrete cancel out of all observable gravitational kinematics (μᵢ = GgeoMᵢ = 15 Qᵢ c² Lsub). Finally, we map the recursive S = Nflux⁴ = 132²⁰ fractal spatial descent down to the $k=4$ (6.76 × 10⁻¹⁴² m) Lucas projection cutoff (L₁₂ = 322 > 137.030), exposing the canonical Planck length as the composite $k=1.5$ geometric mean (LP = √5 Lᵥₒₓₑₗ Lsub ≈ 1.61637 × 10⁻³⁵ m).