Principia Boundary Calculus: A Spectral Conservation Principle for Saturated Fields
Year: 2026
Type: preprint
Abstract: Boundary Calculus – A Spectral Conservation Principle for Saturated Fields (Preprint, v2.0) 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 $\{\pi, 6\}$ with zero free algebraic parameters. Fundamental Law VI – Spectral‑Phase Unification Law VI (Complex Keplerian Manifold).In the superradiant large‑$N$ limit of the Master $\Omega$-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 now closed. Numerical verification (10+ tests) is included in the full document. Key Results - Spectral Action $\mathcal{A} = \sum_n (1/n^2)(\dot v_n^2 + \lambda_n v_n^2)$ – conserved in the linearised superradiant limit - Dynamical Phase Action $\mathcal{P} = \sum_n (1/n^2)(\dot\theta_n^2 + \omega_n^2 \theta_n^2)$ – independently conserved - Complex Action $\mathcal{Z} = \mathcal{A} + i\mathcal{P}$ – conserved; level sets define the Complex Keplerian Manifold $\mathcal{M}_\Omega$ - Real projection of $\mathcal{M}_\Omega$ is the Keplerian body $K_\Omega$ with critical mixing $t_{\text{crit}} = 6/(\pi+6) \approx 0.656$ - Nonlinear bound: $\left|\frac{d\mathcal{A}}{dt}\right| \le \frac{\pi^3}{36}\frac{\mathcal{A}_0}{N^2}$ – conservation up to $1/N^2$ corrections - Scaling exponents: $\alpha_{\text{coh}} = (6-2\pi)/(\pi+6) \approx -0.031$, $\alpha_{\text{frag}} = -4/\pi \approx -1.273$ – match A100 simulations to $0.3\%$ - Phase torus quantization: $\mathcal{P}_n = \pi^2/36$ per mode, winding numbers $n\cdot\pi/6$ - Berry phase per particle: $-\pi$ Constants derived from $\{\pi,6\}$ Symbol Expression Numerical value $\Omega$ $6/\pi$ $1.9098593171$ $t_{\text{crit}}$ $6/(\pi+6)$ $0.6563407742$ $P^*$ $\pi/(\pi+6)$ $0.3436592258$ $\Psi_{\max}$ $\pi^3/[6(\pi+6)]$ $0.5652967678$ $\zeta(2)$ $\pi^2/6$ $1.6449340668$ $\eta_{\text{HOUT}}$ $\pi^6/36864$ $0.0260793510$ $\kappa_{\text{EM}}$ $(e^{\pi/6}-1)\pi/6$ $0.3602840213$ Companion documents - Hu Tao / Walnut Report v37 (H.O.U.T. Framework) – DOI: 10.5281/zenodo.20272324 - Omega‑Calculus v3.0 – DOI: 10.5281/zenodo.20408702 - Boundary Calculus v15.0 (Complete Edition – rigorous proofs) – same DOI as this record - Gap Closure Compendium v3 (definitive master synthesis) – same DOI Version information - Version: 2.0 (Preprint – Simplified Edition) - Complete rigorous edition: Boundary Calculus v15.0 (included in this Zenodo record) - License: CC BY‑NC‑ND 4.0 - Version Date: 1 June 2026 - Publish Date: 30 May 2026 (original); version update 1 June 2026 Preprint version. The complete rigorous edition (v15.0) and the Gap Closure Compendium v3 are available as additional files in this Zenodo record.
Author: Omandac, Clarence
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162860059
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