Relational Rank Geometry (RRG): Formal Core — A Geometric Certificate for Rank Collapse

Relational Rank Geometry (RRG) is a model-light geometric instrument for detecting when two scalar observers sharing an unknown coupling mechanism transition between geometric regimes — emergence, lock, and exit — certified by a rank condition on their joint covariance, without access to the coupling mechanism itself. At its core is the observable dᵣho = Var (rhoₐb, W), the variance of the windowed cross-correlation treated as a stochastic process. Its main result, the Rank-Collapse Theorem, establishes that dᵣho ≈ 0 together with |rho*| ≈ 1 is sufficient—and under A1–A5 necessary—for the joint covariance to become effectively rank-1. Both directions are proved: the reverse direction closes via the Popoviciu variance inequality applied to the bounded support established by A5. The document also introduces Conjecture 2 (Complete Recursion Condition), which characterizes sustained LOCK through a joint increment condition and three distinct failure modes: exhaustion, cancellation, and non-arrival. Empirical validation spans multiple domains, including EEG, financial time series, traffic networks, gravitational wave detection (LIGO), and algebraic number classification from convergence dynamics.