Diagonalization as Symmetry Breaking in Computation and Infinite Systems — E8 Intelligence Research

FINDING: Diagonalization is the core fixed-point-free involution underlying incompleteness, self-reference, and symmetry breaking in computation; spontaneous symmetry breaking corrects Wigner-Eckart relations in infinite systems. MATH: - Diagonalization: For a set \\(S\\) and function \\(f:S \\to S\\), a fixed-point-free involution \\(d\\) (e.g., \\(d(x) = \\neg x\\) in Boolean logic, or Cantor's \\(d(n) = 1 - a_{nn}\\) for binary sequences) yields \\(f(x) \\neq x\\) for all \\(x\\) — the essence of Gödel's undecidability and Turing's halting problem. - Fixed-point combinator: \\(Y = \\lambda f.(\\lambda x. f(xx))(\\lambda x. f(xx))\\) — self-reference as a fixed point in lambda calculus. - Wigner-Eckart corrections: For broken symmetry \\(G \\to H\\), matrix elements \\(\\langle \\alpha' j' m' | T^k_q | \\alpha j m \\rangle\\) acquire corrections proportional to \\(\\langle \\phi | \\phi \\rangle\\) (order parameter) — the leading correction scales as \\(\\sim \\langle \\phi \\rangle / \\Lambda\\) (spontaneous breaking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com