T₃分形代数框架: 非结合层级结构、黄金标度律与宇宙学应用T₃ Fractal Algebra Framework: Non-Associative Hierarchical Structure, Golden Scaling Law and Cosmological Applications

Abstract

Abstract This paper constructs the primitive T₃ fractal algebra framework—a cross-scale unified mathematical structure based on three-dimensional non-associative commutative algebra, with zero free parameters. The multiplication table of e2 = 1 i2 =− 1 ei = 0 primitive T₃ ( , , ) induces a unique symplectic geometric structure, and its associator field is fully Δ(X, Y, Z ) = ω(X, Z )⋅ (0,tY , δY ) ω characterized by the symplectic area . Within this framework, we rigorously prove that: δ∗ 1.Steady-State Fixed Point Theorem: , establishing the = 2 5 = 4ϕ− 2 algebraic isomorphism between T₃ and the golden ratio ϕ 2.Microscopic Freezing Region Theorem: The RG flow converges exponentially near , δ∗ | εn | ≤ | ε0 |⋅ | 1− cδ∗ |n 3.Golden Energy Scaling Law: , with an average error of 0.26% in En+1/En = ϕ1/4 the microscopic segment 4.Scaling Law Constrained Coupling Theorem: , αup/αdown = ϕ1/4 reconstructing empirical parameters as theorem-derived values 5.Symplectic Area Phase Transition Theorem: The critical symplectic area ωc = 1/δ∗2 = 1/20 δeff , and is fully derived from the associator spectrum 6.Zero Free Parameter Theorem: All 6 parameters of originate from the δeff( | ω | ) primitive T₃ algebraic structure Based on these theorems, we construct a 14-level cosmic architecture ranging 10−35 1025 from the Planck scale ( m) to the Laniakea Supercluster ( m), identifying the Solar System as the critical interface between the microscopic ordered phase and the macroscopic disordered phase, and the Earth as the information center. Recursive expansion of sub-levels within the Solar System improves the scaling law accuracy to 0.001%. The framework also includes calculation rules for the neutrino PMNS mixing matrix, achieving 0% fitting error when . δeff ≥ 2.5New heuristic progress: A structural constraint conjecture for the RG coupling constant from the c multiplication table is proposed, pointing out that c = 3/δ∗2 = 0.15 can be traced back to the joint contribution of three independent multiplication axioms of primitive T₃, with a numerical matching error of 0. This conjecture is currently a heuristic physical picture and has not completed rigorous algebraic derivation. Keywords: non-associative algebra; fractal nesting; golden ratio; renormalization group; symplectic geometry; cosmic hierarchical structure; neutrino physics; zero free parameters; structural constraint conjecture