Principia Boundary Calculus: A Spectral Conservation Principle for Saturated Fields

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 \, 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 -PDE, the complex spectral‑phase action = A + iP\ is conserved. The level set \Z = constant\ defines the Complex Keplerian Manifold M_, whose real projection is the Keplerian body K_ (with critical mixing t₂ₑ₈ₓ = 6/ (+6) ) and whose imaginary projection is the phase torus T_ (with quantized winding numbers n/6). - A = ₙ (1/n²) (vₙ² + ₙ vₙ²) – spectral action (amplitude) - P = ₙ (1/n²) (ₙ² + ₙ² ₙ²) – dynamical phase action - vₙ, 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 Z forces the accessible states of any saturated coherence field to lie on M_. 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 A = ₙ (1/n²) (vₙ² + ₙ vₙ²) – conserved in the linearised superradiant limit - Dynamical Phase Action P = ₙ (1/n²) (ₙ² + ₙ² ₙ²) – independently conserved - Complex Action Z = A + iP – conserved; level sets define the Complex Keplerian Manifold M_ - Real projection of M_ is the Keplerian body K_ with critical mixing t₂ₑ₈ₓ = 6/ (+6) 0. 656 - Nonlinear bound: |dAdt| ³36A₀N² – conservation up to 1/N² corrections - Scaling exponents: ₂₎₇ = (6-2) / (+6) -0. 031, ₅ₑ₀₆ = -4/ -1. 273 – match A100 simulations to 0. 3\% - Phase torus quantization: Pₙ = ²/36 per mode, winding numbers n/6 - Berry phase per particle: - Constants derived from \, 6\ Symbol Expression Numerical value 6/ 1. 9098593171 t₂ₑ₈ₓ 6/ (+6) 0. 6563407742 P^* / (+6) 0. 3436592258 _ ³/6 (+6) 0. 5652967678 (2) ²/6 1. 6449340668 ₇₎ₔₓ ⁶/36864 0. 0260793510 ₄₌ (e^/6-1) /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.