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. The central observable is d_ρ = Var (ρₐb, W), the variance of the windowed cross-correlation treated as a stochastic process. The main result, the Rank-Collapse Theorem, establishes that d_ρ ≈ 0 together with |ρ*| ≈ 1 is sufficient — and under assumptions A1–A5, necessary — for the joint covariance to be effectively rank-1. Both directions are proved: the reverse direction closes via the Popoviciu variance inequality applied to the bounded support established by A5 (sign-consistent ergodic coupling). The document also introduces Conjecture 2 (Complete Recursion Condition), characterizing sustained LOCK through a joint increment condition and three failure modes: exhaustion, cancellation, and non-arrival. Empirical validation spans EEG, financial time series, traffic networks, gravitational wave detection (LIGO), integer factorization, and algebraic number classification from convergence dynamics.