The Theorems of the Universal Cascade Law: Complete Statement of Results

Description/Abstract: Complete indexed reference for the mathematical results of the Universal Cascade Law (UCL) research program, established across six companion papers (the Math Sextet). Twenty-three theorems, three corollaries, and five testable predictions are presented in a unified numbering system (L1-L23, C1-C3, P1-P5) with theorem statements, source papers, proof status, and DOI references. The results are organized in three parts: Part I (L1-L6) establishes the foundation — the UCL axiom, the geometric derivation of the Feigenbaum constants, the Decay Bounce self-grounding mechanism, the Randolph Constant λᵣ = δ/α = 1. 86551077, the four-layer universality hierarchy, and the closed loop within classical dynamics. Part II (L7-L19) builds the quantum bridge — the null result confirming the law's diagnostic (C₂ absent → no cascade), the classical cascade in nonlinear systems (δ₂ = 5. 09 converging), topology universality (coupling order selects the Feigenbaum family member), the quantum phase transition (γq = 1. 1838, Δγ = 0. 738), cascade encoding in the Wigner function topology, the breathtaking comparison (classical cascade versus quantum silence in the same Hamiltonian), the whisper in quantum fluctuation structure, the central result — the Quantum Emergence Theorem (self-grounding across the quantum-classical boundary, loop closes), N-convergence to the infinite-dimensional limit (0. 000% variation), semiclassical emergence (cascade crystallizing from quantum fog), and self-grounding in the quantum noise floor (whisper scaling exponent −4. 657 matches δ = 4. 669 to 0. 26%). Part III (L20-L23) extends to gravity — the Lovelock-Lucian correspondence (EFE as the unique equations the law permits for 4D metric gravity), two competing universality classes in general relativity (Feigenbaum orbital and Gauss map curvature), α-survival under universality class interference, and universal merger onset in 10/10 SXS numerical relativity waveforms (p < 0. 001). Corollaries address observability scaling (detection thresholds as δ⁻ⁿ), quantum observability, and a correction to the published Rössler bifurcation point. Predictions target modified gravity theories (f (R), Lanczos-Lovelock, Hořava-Lifshitz with specific δ values), LIGO subcycle ratios (α = 2. 503 ±15%), and derivation of the Kolmogorov −5/3 turbulence exponent from cascade architecture. The complete self-grounding chain is presented: Prerequisites → Constants → Mechanism → Detectability → Quantum Emergence → Quantum Fingerprint → Gravitational Uniqueness → Prerequisites. One closed loop. Every level governed by the same δ. Keywords: Universal Cascade Law, theorem index, Feigenbaum cascade, self-grounding, Randolph Constant, universality hierarchy, quantum emergence, decay bounce, Lovelock correspondence, universality class, observability scaling, reference standard, mathematical proof, canonical numbering, complete results License: CC BY 4. 0 (All Rights Reserved until public release) Communities: Mathematics, Mathematical Physics, Nonlinear Dynamics Access Right: Restricted Notes: Canonical reference document for the Universal Cascade Law theorem set. Intended as the primary citation point for individual theorems (e. g. , "Theorem L15, Randolph 2026"). Contains the complete self-grounding chain from classical dynamics through quantum emergence through gravitational uniqueness. All twenty-three theorems are either proved analytically, proved computationally to stated precision, or follow by deduction from established results (Lanford 1982, Feigenbaum 1978/1979, Sard 1942).