On the Geometry of Fractal Prime Structures: Adèlic Spectral Regularization, Multi-Scale Coercivity, and Physical Manifestations

Abstract

We construct an operator-theoretic framework demonstrating that the distribution of prime-gap fluctuations and scale-invariant geometric networks can be analyzed as dual manifestations of an underlying adèlic spectral geometry. By formulating dilation flows over the adèle class space \mathbb{X} = \mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^\times, we introduce a \theta-twisted Fredholm regularizer that resolves the ultraviolet domain and convergence difficulties historically associated with unregularized prime-zeta summation formulas. We derive a universal dimensionless stabilizer \theta_* \approx 0.23708503 as the unique root of a unitary feedback functional balancing Hamiltonian perturbation theory, diffusion entropy, and binary state branching. This invariant establishes a contractive stability margin \kappa = 1 - \theta_*^2 \approx 0.94379 and determines an effective capacity dimension D_f \approx 2.225 for the prime-gap difference lattice under the regularized metric. Through a generalized Feshbach–Schur complement, we prove that high-frequency arithmetic fluctuations decouple stably from macroscopic low-frequency geometric sectors, guaranteeing uniform operator coercivity. Finally, we investigate the projection of this spectral architecture into scale-invariant physical models: formulating a spectral analogue of the Kolmogorov -5/3 turbulent cascade from prime-gap logarithmic density, modeling an effective ultraviolet dissipation scale for fluid cascades, and analyzing the filamentary scaffolding and void morphology of cosmic web topologies through an exact 76.3\% \,/\, 23.7\% Pareto-modular partition on the modular curve X_0(11).