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 \(f:S → S\), a fixed-point-free involution \(d\) (e.g., \(d(x) = x\) in Boolean logic, or Cantor's \(d(n) = 1 - aₙₙ\) for binary sequences) yields \(f(x) ≠ x\) for all \(x\) — the essence of Gödel's undecidability and Turing's halting problem. - Fixed-point combinator: \(Y = λ f.(λ x. f(xx))(λ x. f(xx))\) — self-reference as a fixed point in lambda calculus. - Wigner-Eckart corrections: For broken symmetry \(G → H\), matrix elements \( α' j' m' | T^k_q | α j m \) acquire corrections proportional to \( φ | φ \) (order parameter) — the leading correction scales as \(~ φ / Λ\) (spontaneous breaking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com