Non-Solvability as a Physical Principle: $A_5$ and the Algebraic Origin of Information Barriers

The central result establishes that the alternating group A₅ is the unique finite subgroup of SO(3) satisfying three physical principles: non-solvability, simplicity, and perfectness. From this uniqueness, the formalization derives icosahedral invariants (β₀ = V − 1 = E/n + χ/2 = 11), prohibition structures on tensor-product dimensions ({9, 15, 16, 25}), and the solvability of all finite groups of order below 60. This repository provides a complete formal verification in Lean 4 + Mathlib of the mathematical core of the paper. Prohibition structure (formally verified) Index Mechanism Algebraic reason 9 (P1) ρ₄-free: only solvable components from SO(3) 15 (P1)+(P2) ρ₄ absent; Sym²ρ₅ has multiplicity > 1 16 (P1) ρ₄ ⊗ ρ₄ (self-product): includes all 5 irreducibles 25 (P1) ρ₄ absent and self-product ICO correspondence to physical constants (46 quantities) The paper maps icosahedral parameters (F=20, E=30, V=12, n=3, χ=2, |A₅|=60, φ=(1+√5)/2, gap=1/φ³) to 46 physical quantities across 9 categories. Type I = RG-invariant integers; Type II = fixed-point universals; Type III = cosmological freeze-out. Gauge couplings & scales (#1–#10) # Quantity ICO formula ICO value Exp. Dev. Type 1 α⁻¹ Fφ⁴ 137.08 137.036 0.034% I/III 2 α_s(M_Z) gap/2 0.1180 0.1180(9) 0.03% I/III 3 sin²θ_W gap(1−1/(Fπ)) 0.2312 0.2312 0.4% II 4 α/α_G φ²⁰⁴ ~10⁴⁴ ~10⁴⁴ ~2% II 5 α_s/α 10φ 16.18 16.16