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) = –Ψₘax ln (1 – Ψ/Ψₘax), 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): - Ψₘax = π³/ (6 (π+6) ) ≈ 0. 5653 – maximum coherence field- ζ (2) = π²/6 ≈ 1. 644934 – spectral gap (Basel sum) - Monodromy quantization: ∮ ∂_ΩΨ dx = n·ζ (2) (n integer) - Vacuum energy extraction (XVE) efficiency: η = π⁶/36864 ≈ 2. 608%- EM coherence coupling: κEM = (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