Fractal-Spectral Field Theory: A √2-Recursive Framework with Natural UV Regularization

We construct a mathematically rigorous quantum field theory framework based on a fractal-spectral operator with scaling exponent √2. The central object is a Hamiltonian H = −Δ + Vfractal on a fractal-spectral Hilbert space, where the potential encodes a tower of texture modes with kinetic weights (√2) ^−2n = 2^−n per level — derived from the fractal-temporal Lagrangian, not postulated. The framework rests on three analytic pillars: a Mourre estimate (ensuring absence of singular continuous spectrum), trace-class bounds (connecting eigenvalues to geometric data), and a parametrix expansion (controlling perturbative corrections). Together they produce a Master Trace Formula linking particle masses to spectral data and a natural UV regularization through the fractal product ∏ (1 + k²/Λₙ²) ⁻¹, which yields Gaussian suppression without artificial cutoffs. The hierarchy problem receives a structural resolution: quantum corrections to the Higgs mass from each fractal level are suppressed by 2^−n, producing a convergent geometric series (∑2^−n = 2) instead of a quadratically divergent integral. No supersymmetry, extra dimensions, or fine-tuning is required — the suppression is a direct consequence of the √2 kinetic weights. The particle mass spectrum exhibits a structural pattern with ratios (√2) ^kₙ between levels. Consistency checks include the electroweak mixing angle sin²θW (one fitted parameter), the W/Z mass ratio (follows from sin²θW), and the muon-to-electron mass ratio at leading order. These are presented as structural matches, not exact predictions — the derivation of precise Standard Model parameters from spectral data remains an open problem. Testable predictions beyond the Standard Model include: log-periodic modulations in running coupling constants with universal frequency ωf = 2π/ln√2 ≈ 18. 1, atomic spectroscopy corrections at order ~10⁻¹⁵, and CMB signatures from the fractal texture tower. The theory reduces to the Standard Model at low energies and connects to the fractal-temporal Lagrangian for gravity, dark energy, and the cosmological bounce. Open problems are stated explicitly: Borel summability of the perturbative series, derivation of exact masses from spectral data, first-principles derivation of the gauge group SU (3) ×SU (2) ×U (1), and modular structure of the partition function.