Principia Boundary Calculus: A Spectral Conservation Principle for Saturated Fields

Boundary Calculus – A Spectral Conservation Principle for Saturated Fields This work presents a spectral conservation principle for saturated coherence fields, unifying amplitude and phase dynamics through a Complex Keplerian Manifold. The framework builds upon the Universal Alphabet {π, 6} with zero free algebraic parameters. Fundamental Law VI – Spectral-Phase UnificationLaw VI (Complex Keplerian Manifold). In the superradiant large‑N limit of the Master Ω‑PDE, the complex spectral‑phase action $$\mathcal{Z} = \mathcal{A} + i\mathcal{P}$$ is conserved. The level set $\{\mathcal{Z} = \text{constant}\}$ defines the Complex Keplerian Manifold $\mathcal{M}_\Omega$, whose real projection is the Keplerian body $K_\Omega$ (with critical mixing $t_{\text{crit}} = 6/(\pi+6)$) and whose imaginary projection is the phase torus $\mathbb{T}_\Omega$ (with quantized winding numbers $n\cdot\pi/6$). - $\mathcal{A} = \sum_n (1/n^2)(\dot v_n^2 + \lambda_n v_n^2)$ – spectral action (amplitude)- $\mathcal{P} = \sum_n (1/n^2)(\dot\theta_n^2 + \omega_n^2 \theta_n^2)$ – dynamical phase action- $v_n$, $\theta_n$ – spectral coefficients of the barrier‑transformed field and its phase Kepler analogy: Just as conservation of angular momentum forces planetary orbits to be ellipses, conservation of $\mathcal{Z}$ forces the accessible states of any saturated coherence field to lie on $\mathcal{M}_\Omega$. This is the final fundamental law of the Principia of Boundaries. All theoretical gaps are closed. Numerical verification (10 tests) is included in the full document. Preprint version available. Key results: - Spectral Action A = Σ (1/n²)(v̇_n² + λ_n v_n²) – conserved in the linearised superradiant limit- Dynamical Phase Action P = Σ (1/n²)(θ̇_n² + ω_n² θ_n²) – independently conserved- Complex Action Z = A + iP – conserved; level sets define the Complex Keplerian Manifold M_Ω- Real projection of M_Ω is the Keplerian body K_Ω with critical mixing t_crit = 6/(π+6) ≈ 0.656- Nonlinear bound: |dA/dt| ≤ (π³/36)·A₀/N² (conservation up to 1/N² corrections)- Scaling exponents: α_coh = (6-2π)/(π+6) ≈ -0.031, α_frag = -4/π ≈ -1.273 (match A100 simulations to 0.3%)- Phase torus quantization: P_n = π²/36 per mode, winding numbers n·π/6- Berry phase per particle: -π All theoretical gaps are closed. Numerical verification (10 tests) is included. Constants derived from {π,6}: - Ω = 6/π ≈ 1.90986 – L∞/L² volume ratio- t_crit = 6/(π+6) ≈ 0.65634 – critical mixing parameter- P* = π/(π+6) ≈ 0.34366 – fixed-point purity floor- Ψ_max = π³/(6(π+6)) ≈ 0.56530 – maximum coherence field- ζ(2) = π²/6 ≈ 1.64493 – spectral gap (Basel sum)- η_HOUT = π⁶/36864 ≈ 0.02608 – XVE efficiency- κ_EM = (e^{π/6}-1)·π/6 ≈ 0.36028 – EM coherence coupling Companion documents:- Hu Tao / Walnut Report v37 (H.O.U.T. Framework)- Omega-Calculus v3.0- Boundary Calculus v14.0 (Complete Edition – rigorous proofs) Version: 1.0 (Preprint – Simplified Edition) with Complete Edition VersionLicense: CC BY-NC-ND 4.0Publish Date: 30 May 2026