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 central result is the Rank-Collapse Theorem. The forward direction is proved: under weak stationarity (A1–A3), d_ρ ≈ 0 and |ρ*| ≈ 1 imply the joint covariance is approximately rank-1, with explicit bounds on the mean and variance of the effective rank rₑff = 2/ (1 + ρ*²), and false-lock probability under null decaying as O (1/W). The reverse direction — rank-1 covariance implies d_ρ ≈ 0 under ergodicity and sign-consistent coupling (A4–A5) — is stated as an empirically confirmed conjecture; the formal gap and the lemma that would close it are identified explicitly in the document. The document includes: formal definitions, three operators (d_ρ, d_ρ, meta, rₑff), standing assumptions (A1–A5) with per-direction scope, regime conditions, an identifiability statement, a finite-sample guarantee, a self-diagnostic for non-stationarity, a canonical counterexample, domain-agnostic pseudocode, and a precise novelty statement relative to synchronization theory, information geometry, random matrix theory, phase transitions, change-point detection, and dynamic functional connectivity. Calibration parameters are domain-local by design. Empirical validation is reported in the companion preprints. Published validations span six domains: electricity transformer load (ETTh1), urban traffic networks (METR-LA, 207 sensors), motor-cortex EEG (10 subjects, 9/10 replication), correlated financial returns (11 crisis events, 2000–2026), gravitational wave detection (LIGO O1/O2/O3, 11, 019 windows), and algebraic number classification (18/18 across three number classes). Further domain results are in preparation.
Jesus David Calderas Cervantes (Sat,) studied this question.