Tarski Fixed Points: PPAD-Complete Yet Categorically Expressible — E8 Intelligence Research

FINDING: Tarski fixed-point theorem for monotone functions on complete lattices is computationally hard (PPAD-complete), yet the category-theoretic exponential object (function space) provides the structural framework for such fixed points; the 1D Brouwer case and the cosine iteration converge to the golden-ratio-adjacent Dottie number. | MATH: Tarski: every monotone f: L→L on a complete lattice L has a least fixed point, lfp(f) = ⋂{x | f(x) ≤ x}. Computational complexity: finding a Tarski fixed point is PPAD-complete (Etessami et al.). Brouwer 1D: continuous f:[0,1]→[0,1] ⇒ ∃c: f(c)=c. Cosine iteration: xₙ₊₁ = cos(xₙ) ⇒ x* ≈ 0.7390851332 (Dottie number). Category theory: exponential object B^A satisfies Hom(C×A, B) ≅ Hom(C, B^A) — the adjunction that makes function spaces internal. | CONNECTION: The Dottie number 0.739085… is not a simple ratio, but its complement 1 − 0.739085 = 0.260915 and its reciprocal 1/0.739085 ≈ 1.353 — no direct golden ratio. However, the *structure* of fixed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com