Topology

Non-Closure of Doubling Vesica Chains: A Constructive Proof of Irreducible Remainder in Self-Similar Geometric Expansion

Authors: Patrick Nod Glavin
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We present a constructive geometric proof demonstrating that a chain of Vesica Piscis figures following a doubling rule (R, 2R, 4R, u2026, 2n—1R) cannot be enclosed by a Vesica of radius 2nR that closes at the chain’s origin. The linear extent of n doubling iterations sums to 2n — 1, producing an irreducible remainder of exactly one base unit R. This remainder is invariant across all scales of the construction and cannot be eliminated by further iteration. We show that this non-closure property originates at the unit construction itself: the radius R, which generates the Vesica, cannot be consumed by the figure it produces. The result is established through direct lattice construction without appeal to limiting processes or infinite series. We discuss connections to self-similar fractal structures, aperiodic tilings, and renormalizationgroup theory. The construction arises from Geosectometry, a geometric framework developedby the author for investigating natural-state properties of constructive geometry.
Category: Topology