Fixed-Point Theorems and Recursive Sequences: A Universal Pattern of Self-Reference — E8 Intelligence Research

FINDING: Fixed-point theorems (Brouwer, Banach) share a structural, iterative convergence logic with recursive geometric sequences, revealing a universal pattern of self-reference and stability. MATH: - Banach fixed-point theorem: \( d(Tx, Ty) ≤ c · d(x, y) \) for \( 0 ≤ c < 1 \), guarantees unique fixed point \( x^* = T(x^*) \) via iteration \( xₙ₊₁ = T(x_n) \). - Recursive geometric sequence: \( a_n = r · aₙ₋₁ \), with convergence when \( |r| < 1 \). Fixed point \( a^* = r · a^* \) implies \( a^* = 0 \) (trivial) or \( r=1 \). - Brouwer fixed-point theorem: continuous map from closed ball to itself has a fixed point; no simple ratio, but topological invariance. - Key constant: contraction factor \( c \) (analogous to ratio \( r \)) determines convergence speed; no specific numeric constant emerges. CONNECTION: - The iteration \( xₙ₊₁ = T(x_n) \) mirrors geometric sequence recursion \( aₙ₊₁ = r · a_n \). - For linear maps, fixed-point Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com