The Uncomputable Fixed Point: Topological Limits of Algorithmic Self-Reference — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Brouwer's fixed-point theorem and its combinatorial proof via Sperner's lemma reveal a deep topological obstruction to computable self-reference — any continuous map from a compact convex set to itself must have a fixed point, but the constructive/computable analogue fails, exposing a fundamental limit on algorithmic knowledge. | MATH: Brouwer fixed-point theorem: ∀f: Dⁿ→Dⁿ continuous, ∃x∈Dⁿ: f(x)=x. Sperner's lemma: any Sperner labeling of a triangulated n-simplex contains an odd number of fully labeled (rainbow) subsimplices — combinatorial parity invariant. Cohomological reformulation: the map f induces H⁰(Dⁿ)→H⁰(Dⁿ) with Lefschetz number L(f)=1≠0, forcing a fixed point. Computable analogue: there exists a computable continuous function on the unit interval with no computable fixed point (e.g., f(x)=x/2 + 1/4 on [0,1] — fixed point at 1/2 is computable, but a diagonal construction yields a computable f with fixed point at a non-computable real). | CONNECTION: The parity inv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212279866
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