Topological Fixed Points and Discrete Approximations in Computational Mathematics — E8 Intelligence Research

FINDING: Simplicial approximation bridges continuous maps to discrete combinatorial structures, while fixed-point theorems on metric spaces reveal constraints on surjective self-mappings; exact integer root extraction via dimension reduction addresses computational limits. MATH: - Simplicial approximation theorem: For continuous \(f: |K| → |L|\), there exists a simplicial map \(g: K⁽ᵐ⁾ → L\) (after subdivision) such that \(g\) is homotopic to \(f\). Key constants: mesh size \(ε > 0\), subdivision depth \(m ~ log(1/ε)\). - Fixed-point theorem (arXiv:1511.04069): No nontrivial surjective uniformly asymptotically regular mapping on a metric space; for firmly nonexpansive semigroups, existence of fixed points requires boundedness or compactness — no explicit constants, but the obstruction is topological (non-contractibility). - Exact integer roots: Dimension reduction via arithmetic transforms — e.g., computing \( N1/k \) exactly using modu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com