This paper introduces the depth rate — a measurable observable computed from convergence sequences without requiring knowledge of the limit — and demonstrates that it empirically separates real numbers into their algebraic classes. Applied to admissible structural convergent sequences for 18 real numbers across three algebraic classes, the instrument recovers the Lagrange characterization with 18/18 accuracy. Rationals and quadratic irrationals exhibit logarithmic scaling O (log (1/ε) ) ; tested transcendentals exhibit polynomial scaling Ω (1/ε), active beyond 10¹⁰ frames at ε=10⁻¹⁰. A companion experiment finds sustained relational coupling only among tested transcendental pairs, including cases sharing convergence rates that do not couple — an independent empirical pattern. The central result is proved: exponential increment decay implies logarithmic depth rate; polynomial increment decay implies polynomial depth rate. The period matrix eigenvalue of the CF expansion predicts the depth rate class for quadratic irrationals. Results apply to generator–sequence pairs within the admissible class. Stable under window variation W ∈ 20, 120 and adversarial tests (43/43 stress test, 25 quadratic irrationals across CF periods 1–11). Four open problems are stated connecting depth-rate scaling to normality theory; if all three normality-related problems are resolved, the argument constitutes an indirect geometric argument against the normality of algebraic irrationals. Part of the SFE research program applying Relational Rank Geometry across multiple scientific domains. Reproduction code: https: //colab. research. google. com/drive/1r₆mYFafC6FYdrKyFDcGYVlHHMkROVFe
Jesus David Calderas Cervantes (Sat,) studied this question.