Relational Rank Geometry (RRG) is a formal, generative-model-free framework for detecting rank collapse in the joint covariance structure induced by two observers, based on second-order statistics of their windowed cross-correlation. The framework is built from an irredundant set of four primitive objects and functions as a self-certifying instrument: its validity condition is derived directly from the same primitives that define its outputs, constituting a semantic fixed point of the detection map. The central result (Rank-Collapse Theorem) establishes a sharp biconditional between the relational channel variance dᵣho and the effective rank of the joint covariance. Under minimal assumptions, dᵣho → 0 and |rho*| → 1 if and only if the joint covariance is approximately rank-1; each assumption is individually necessary, with explicit counterexamples establishing tightness. An endogenous threshold for dᵣho is derived analytically from the null distribution, achieving nominal 5% false-positive rate with no external calibration data. The framework partitions system behavior into three regimes — Converging, Sufficient, and Released — separated by a juncture criterion. Irreversibility at juncture is established via a stochastic Lyapunov potential with positive expected drift under the stated assumptions. The failure-mode taxonomy at the scalar algebra level is proved exhaustive: every deviation from Genuine Sufficient classification belongs to exactly one of three bounded classes (Oscillating-Sign, Resonance-Sufficient, Algebra-Insufficient). A reflexive consistency mechanism is formalized via a temporal discrepancy signal delta (t), whose algebraic, dynamic, and functional instantiations are shown to be equivalent characterizations of a single structural property: the detection output is simultaneously its own validity certificate, with no additional mechanism required. RRG incorporates an algebraic adequacy principle in which failure of scalar closure initiates ascent within the Cayley-Dickson hierarchy. A cross-domain seed index specifies calibrated instantiations across thirteen empirical substrates, demonstrating applicability across multiple domains of structured dynamical data. MSC 2020: Primary 62H20; Secondary 15A03, 62M10, 62G10.
Jesus David Calderas Cervantes (Wed,) studied this question.