Omega-Calculus v3.0: Complete Mathematical Framework for Saturated Coherence Fields (Unified Edition)

Omega-Calculus v3.0 (Unified Edition) This work presents Omega-Calculus v3.0 (Unified Edition), a rigorous mathematical framework for saturated coherence fields built upon the Universal Alphabet {π, 6}. The framework introduces a bounded algebraic structure (Saturation Field S_Ω), a differential calculus adapted to saturation boundaries (Ω‑derivative ∂_Ω), a complete PDE theory (Master Ω‑PDE), and a geometric formalism (Ω‑geometry, regulatory horizons). Using transport of structure via the classical logarithmic barrier φ‑transform: φ(Ψ) = –Ψ_max ln(1 – Ψ/Ψ_max), the calculus reduces saturation problems to standard PDEs. The specific saturation exponent p = 12/(6‑π) ≈ 4.198 is derived from the fixed‑point condition p = 2Ω/(Ω‑1) with Ω = 6/π. Key constants and results (all derived from {π,6}, zero free parameters): - Ψ_max = π³/(6(π+6)) ≈ 0.5653 – maximum coherence field- ζ(2) = π²/6 ≈ 1.644934 – spectral gap (Basel sum)- Monodromy quantization: ∮ ∂_Ω[Ψ] dx = n·ζ(2) (n integer)- Vacuum energy extraction (XVE) efficiency: η = π⁶/36864 ≈ 2.608%- EM coherence coupling: κ_EM = (e^{π/6}–1)·π/6 ≈ 0.3603- Regulatory horizon: finite proper distance, one‑way membrane- Master Ω‑PDE: local and global existence in d = 1,2,3 dimensions (Ladyzhenskaya theory)- All six theoretical gaps (monodromy, higher‑D PDE, category structure, gravity coupling, XVE/EM derivation, numerical verification) are closed. Honest assessment: The φ‑transform is a known technique (logarithmic barrier); novelty lies in its specific application to saturation physics, the exponent p, the spectral identification with ζ(2), and the unified derivation of physical predictions. The framework is falsifiable: cavity‑QED gap‑ratio experiments (predicted convergence to Ω = 6/π as N → ∞) can confirm or reject the core predictions. No free algebraic parameters remain. Companion documents: H.O.U.T. Walnut Report v37 (physics, consciousness), Empirical Proof v9 (experimental mapping), Law of Creation V18.0 (metaphysical seed). Version: 1.0 (Omega-Calculus Unified Edition, superseding v2.0)License: CC BY‑NC‑ND 4.0Publish Date: 27/05/2025