Certified Two-Step Consequence-Minimality of the Nine-Dimensional ACRF-4 Host-Boundary Quotient: A Reproducible Closure of the Finite-Amplitude Observability Problem

This theorem note closes an explicit open problem left by the previously published finite-amplitude permutation-quotient analysis of canonical ACRF-4 (DOI: 10.5281/zenodo.22184552). The prior paper proved that the exact two-host quotient by internal relabelings has generic dimension nine and admits the chart W_S=(m_S,p_S,D_S), W_C=(m_C,p_C,D_C), B_SC=(A,B,C), but it did not prove that all nine quotient directions are required for future host consequence. It reported a numerical rank-nine observability audit and stated that an analytic nonzero 9x9 minor or certified interval evaluation remained necessary before promoting consequence-minimality to theorem status. The present note reconstructs the canonical ACRF-4 dynamics independently and studies the future-only two-step host readout O_2(q)=(H(Fq),H(F^2q)). At an explicit admissible generic witness, the 12x9 Jacobian has numerical rank nine with smallest singular value approximately 0.10846. A selected 9x9 minor has high-precision determinant 0.0013227091156746834762072035769499..., and interval automatic differentiation on a radius-1e-12 box around the witness gives the enclosure [0.001322709112668265, 0.001322709118681102], which excludes zero. Therefore the future-only two-step host-consequence Jacobian has rank nine throughout that explicit neighborhood. By the rank theorem, no smooth local state representation of dimension below nine can reproduce the stipulated two-step host consequence there. Because the selected minor is real analytic and not identically zero on the regular chart component, full rank and local consequence-minimality hold generically away from an analytic exceptional set. Within canonical finite ACRF-4, this establishes that the six intrinsic host coordinates alone are generically insufficient for complete two-step future host consequence; the retained relational boundary contributes irreducible consequence-bearing degrees of freedom. The result is model-specific and does not establish a universal Registration Theory cross-registration law, multi-host composition, redundant accessibility, decoherence, spacetime reconstruction, Lorentz structure, or a universal propagation constant. The publication package includes the PDF, LaTeX source, a reproducibility script, and the complete numerical/interval certificate output.