Non-Solvability as a Physical Principle: A₅ and the Algebraic Origin of Information Barriers

We consider a restricted model class in which microscopic three-dimensional geometry admits a finite rotation holonomy group \(H \leq {{SO}{(3)}}\), and ask whether \(H\) can be uniquely determined from physical postulates alone. Three postulates—Finiteness (H1), Irreducibility (H2), and Perfectness (H3)—motivated respectively by Planck-scale discreteness, the absence of privileged directions, and the absence of universal abelian charges, select \(A_{5}\) (the icosahedral rotation group of order 60) as the unique admissible holonomy under Klein’s classification. Five logically independent sufficient conditions converge on the same conclusion, and the entire derivation chain is formally verified in Lean 4 (sorry = 0, axiom = 0). The central result is not a derivation of the Standard Model but a specification of falsifiable structural constraints. The representation theory of \(A_{5}\) yields a threefold prohibition structure defined by three independent mechanisms: a tensor-product selection rule for the four-dimensional irreducible representation \(\rho_{4}\) (P1), a multiplicity-free exclusion principle (P2), and a Coxeter-exponent filter derived from the exceptional Lie algebra \(E_{8}\) via the McKay correspondence (P3)—that pre-specifies the forbidden \(\varphi\)-power exponents \(\{ 9,15,16,25\}\) with explicit algebraic reasons. As a proof of concept, the pure \({SU}{(3)}\) Yang–Mills one-loop coefficient \(\beta_{0} = 11\) is reconstructed from icosahedral cell data as the identity \({V - 1} \equiv {{E/n} + {\chi/2}}\), a Type I (renormalization-group-invariant, hereafter RG-invariant) quantity immune to the scale problem. The scale problem is addressed by classifying RG survival mechanisms into Types I–III, with main-text claims restricted to Type I.