Registration Geometry II: Topological Writing, Sector Locking, and the Amplitude-Completed Hopf Gate

This paper develops a topological writing mechanism for Registration Geometry using an amplitude-completed Hopf field. It begins from the fact that the Hopf invariant cannot change during a continuous evolution of a regular S²-valued field with fixed boundary conditions. A transition between topological sectors, therefore, requires a temporary opening of the configuration space, such as a zero of an underlying unconstrained amplitude field.

An explicit polynomial energy functional is introduced for a three-component field Φ taking values in R³. Because the functional contains no inverse powers of |Φ|, it remains finite when Φ passes through zero. The paper then constructs a closed-form finite-energy path from the trivial Hopf sector to a charge-one sector. At the midpoint of this path, the amplitude vanishes on a resolved gate circle, allowing the normalized field to change its Hopf charge. Once the amplitude is again bounded away from zero and the boundary data remain fixed, the resulting topological sector is conditionally locked under continuous regular evolution.

The analytical results are supported by a numerical audit using a stereographic charge-one Hopf map. The computation resolves the amplitude-zero gate exactly, verifies finite energy throughout the path, and recovers a Hopf charge close to one after the transition. The accompanying source archive contains the implementation, tests, numerical audit script, data, and reproducibility materials.

The paper does not claim dynamical controllability, permanent gate closure from the potential alone, a universal writing-energy threshold, global smoothness, or convergence to a unique minimizer. Its contribution is the explicit amplitude-completed construction, the finite-energy topology-changing path, and the gate-and-lock interpretation of stable topological records.