BLACK AND GOLD IV Swarm Coherence, Geometric Observability, and the Closed Architecture under the Common Integrity Gate

BLACK AND GOLD IV Swarm Coherence, Geometric Observability, and the Closed Architecture under the Common Integrity Gate C. J. Tully Independent Researcher, Engineer — Tamworth, NSW, Australia ORCID: 0009-0007-5661-7332 October 2026 ABSTRACT This paper is the fourth of the black-and-gold series. It places empirical dual-swarm coherence results and geometric observability criteria under the common Quasi-Equivalence Index gate established in Paper I. Over 500 timesteps a convergent swarm architecture achieves mean S5 coherence 0.734 and sequestration efficiency 79.8%, while an oscillatory control remains at 0.180 and 7.7%. The coherence–efficiency correlation is substantially stronger in the convergent case. A surrounding beacon array supplies an independent geometric observability criterion whose Fisher information must be well-conditioned; an external measurement is admitted only when the updated state continues to satisfy the hard gate. The McKay placement of tetrahedral and octahedral peaks is recorded as a reading of form, not as a derivation from the corpus. The architecture remains closed: every admitted state, command, measurement, and scaling has already been shown to leave the common certificate intact.