The Algebraic Framework of Reality in YUCT — A Self-Consistent System of Twenty Closed Loops

This document, Appendix AF of the Yakushev Unified Coordination Theory (YUCT), presents the complete mathematical skeleton of the theory: a self-consistent system of twenty-one interlocking algebraic loops. Derived from first principles, these loops are anchored by three exact rational constants: the universal scaling exponent β = 2/3, and the coordination invariants Sₒdd = 6/5 = 1. 2 and Sₑven = 4/5 = 0. 8. These constants are not fitted parameters but necessary consequences of hierarchical coordination and fractal geometry. The twenty-one loops are organized into two classes: - Hard Loops (I–VI, XIV, XVI): Exact mathematical identities that function as universal invariants. There are eight such loops in total. Any empirical deviation would immediately falsify the core framework. - Soft Loops (VII–XIII, XV, XVII–XXI): Universal scaling laws applied to specific systems, providing quantitative engineering blueprints across domains. This class includes the newly identified asymptotic relation bridging YUCT and Feigenbaum chaos theory (Loop XXI). The loops collectively govern phenomena spanning more than 50 orders of magnitude: - Particle Physics: Half-integer quantization of fermion masses, the weak scale hierarchy (mW ≈ MPl (2/3) ⁹6) - Cosmology: The baryonic mass of the Universe, the cosmological constant, quantization of time- Geometry: An approximation of π from Sₒdd and the golden ratio (error ~10^-5) - Chaos Theory: The Feigenbaum constants δ and α are algebraically linked to YUCT via the geometry of the circle- Molecular Biology: Genome structure, mutation rates, metabolic scaling, codon usage- Information Theory: Coordination efficiency of quantum entanglement and two-factor authentication- Economics & Sociology: Critical group sizes, economic volatility, evolutionary singularities The framework is rigid and falsifiable. It contains no free continuous parameters. A single, high-confidence experimental failure in any Hard Loop would shatter the entire algebraic skeleton. This maximal vulnerability to empirical refutation is the defining strength of the theory. This document serves as the central reference for the algebraic core of YUCT. All subsequent applications—from alloy design (PLCM) to gravitational wave predictions—derive from the invariant structure presented here. Contents: 1. Foundations: Coordination, D+I·R Triad, Universal Scaling Law (β=2/3, κc=1/3) 2. The Primary Algebraic Loop: Derivation of Sₒdd, Sₑven, and β3. Loops I–VI: The Six Core Physical Loops (Constants, Fermion Masses, Weak Scale, Geometry, Baryonic Mass, Time Quantization) 4. Loops VII–XXI: Extended Inventory Across Chemistry, Biology, Economics, Sociology, Information Theory, and Chaos Theory5. Hard Loops vs. Soft Loops: A Crucial Distinction6. Falsifiability: The Rigidity Principle and Unified Falsification Criteria7. Epistemological Implications: A Pythagorean-Platonic Physics on a Rigorous Foundation This is the "algebraic skeleton of reality"—a precise, testable, and ultimately disposable hypothesis about the invariant structure underlying all coordinated systems. Relation: Is part of DOI: 10. 5281/zenodo. 18444598