Deterministic Registration-Cycle Realization of Quadratic Projective Weights in Registration Theory: An Explicit Born-Form Existence Construction and the Remaining Linear-Coupling Problem

This timestamp records a bounded milestone in Registration Theory (Physical Foundations): an explicit deterministic preparation-measurement-reset model that reproduces the binary quadratic projective weight νP,I(Bi) = ||Pi v||² / ||v||² in a controlled tangent/refinement limit, while isolating the remaining unproved physical bridge. The construction combines an already-derived isotropic rank-three H4 relational sector, a finite fine-registration fiber, a deterministic cyclic successor that visits each fine state exactly once per period, and the existing retained-registration detector result that post-threshold outcome is fixed by the sign of the incoming imbalance and reset returns the detector to readiness. Uniform finite-cycle visitation gives an exact empirical measure on the fine fiber. In the directional refinement limit, full-rank lattice counting in the isotropic H4 tangent geometry gives uniform spherical area. With the explicit linear seed coupling δ = ε(λ̂·nI + mP,I), the detector basins become spherical caps with relative area (1 + m)/2, and the projective contrast identity m = ⟨v,(2Pi−I)v⟩/||v||² yields the quadratic projective weight. The paper does not claim that canonical ACRF-4 has derived the Born rule. The decisive linear particle-apparatus coupling is an additional model assumption, not a result already forced by ACRF-4 or by the detector sign-preservation theorem. A sensitivity theorem shows that replacing the linear offset m by a general function f(m) changes the limiting frequency law to (1 + f(m))/2 on the unsaturated interval. Thus deterministic cycling and isotropic geometry alone do not select the Born form. The contribution is therefore an explicit deterministic RT-compatible Born-form existence construction with a precise claim boundary. The named next proof obligation is H-B/U8: derive the linear tangent particle-apparatus coupling, including its relative coefficient, directly from canonical ACRF-4 dynamics rather than choosing it phenomenologically. Canonical theory name: Registration Theory (Physical Foundations). Version: 1.0.