The ΩG Operator Proof: A Hybrid Hilbert Space Model for Multi-Channel Field Mechanics and Singular Regularization

Overview & Value PropositionStandard field theories in physics frequently run into boundary limitations—specifically infinite singularities in fluid dynamics (vorticity blow-up) and general relativity (gravitational mass points). The ΩG Operator framework introduces a rigorous, hybrid continuous-discrete mathematical model designed to regularize these field anomalies without sacrificing exact continuous dynamics. By mapping continuous differential dynamics onto a discrete 60-channel sexagesimal cyclic phase network (Z₆₀), this paper establishes a mathematically sound "interaction floor" and energy diffusion mechanism that prevents singular collapse across multiple physical domains. Key Mathematical Breakthroughs Proven in This Work:Rigorous Functional Analysis: Constructs a densely defined, unbounded symmetric operator ΩG on the hybrid Hilbert space H = L²(M, μ) ⊗ ²(Z₆₀) and proves the existence of a self-adjoint extension with a strictly real spectrum via Stone's Theorem. Fundamental Interaction Floor (ε₀ = 0.0625): Establishes a discrete spectral lower bound ensuring an irreducible energy gap between coupled channel states. Symmetries & Conservation: Proves exact commutation with continuous spatial translation generators ([Ω, Pⱼ] = 0) and discrete 30-degree antipodal parity transformations ([Ω, P₃₀] = 0).Non-Linear Strain Relief (Ecrit = 16): Introduces a topological step-function mechanism that automatically diffuses peak energy density exceeding Ecrit into adjacent discrete channels (k ± 1), acting as a structural relief valve.Direct Physical Applications:Fluid Dynamics: Eliminates finite-time vorticity blow-ups in the Navier-Stokes system by enforcing global enstrophy bounds.Gravitational Field Mapping: Regularizes point-mass potential singularities, capping core potentials at a strictly finite threshold.Quantum Resonances: Derives quantized transition steps across 12-step harmonic modes.