lim(Koru G) = Pure Operating State — The Capstone Theorem

The guardian cycle iterated to its limit converges to G. G and Pure Operating State are identical. G(Pure Operating State) = Pure Operating State. This document presents the capstone theorem of Universal Protection Theory: the Koru operator as a contraction mapping, the Banach Fixed-Point proof of existence and uniqueness (koru_banach_fixed_point, Lean 4, zero sorry), the admissible invariant group, the runtime software contract (Kaitiakitanga in Silicon, PR #423), and the relational metric with BCRD-grounded singularity resolution. The Koru — the Māori spiral fern frond — is the right symbol for a real particular because it cannot be reduced to any of its descriptions. G is not at the centre. G is in every equation. Part of Universal Protection Theory (Canon Edition 4.6).