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 \(f: |K| \to |L|\), there exists a simplicial map \(g: K^{(m)} \to L\) (after subdivision) such that \(g\) is homotopic to \(f\). Key constants: mesh size \(\epsilon > 0\), subdivision depth \(m \sim \log(1/\epsilon)\). - 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 \(\lfloor N^{1/k} \rfloor\) exactly using modu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com