Let dρ = Var (ρₐb, W) be the relational channel variance of two scalar observers over a rolling window of length W. Three results are established. First, the Rank-Collapse Theorem: under weak stationarity and ergodicity, dρ 0. 45) with finite-sample bias under 4%; bias enters a monotonic negative third-order regime for σₚost > 0. 30, and both boundaries coincide with the operating limit of the AND gate. Third, non-stationarity is self-detectable: empirical dρ > (1−ρ²) ²/W flags structural drift without external ground truth. The N-observer extension yields a corrected rₑff estimator with MAE < 0. 04 (N = 2–10, synthetic) ; real-data validation on METR-LA (207 sensors) shows that the joint entropy-based estimator tracks ground truth across the full range without correction, while the pairwise estimator is N-invariant on heterogeneous data, revealing the intrinsic dimensionality of the sensor network (rₑff ≈ 5. 4, plateau at N ≈ 50). The framework applies to stochastic relational systems including sensor synchronization, coupled oscillators, and correlated financial or biological signals. Cross-domain validation across four real-world systems — electricity transformer load (ETTh1), urban traffic sensors (METR-LA), motor-cortex EEG (10 subjects, 2 task types, 9/10 replication rate), and correlated financial returns — confirms that the instrument operates in the predicted regimes, recovers precursor signals with lead times of seconds to hours, and correctly identifies distinct crisis coupling regimes and discriminates between mechanistically different financial crises: acute homogeneous shocks (COVID-19, 2020), acute heterogeneous contagion (Lehman collapse, 2008), and gradual valuation corrections (dot-com crash, 2000–2002) each produce distinct (ρ, dρ, λ1/λ2, rₑff) signatures. The instrument correctly remains silent on the gradual correction while detecting the two acute crises via separate detection branches. Benchmarking against Granger causality and transfer entropy on ETTh1 establishes the architectural relationship between the methods: dρ is an observability layer for coupling structure — it characterizes when a relational channel is geometrically stable enough for causal attribution to be reliable. Granger causality precedes dρ in the causal chain (directional influence establishes before geometric stabilization; mean lead −4. 0 h), and fires in 38. 7% of rolling windows reflecting persistent background influence rather than discrete events. dρ fires at geometric lock-in with event-level specificity. The methods are not substitutes: dρ tells you when the channel is ready to be read; Granger and transfer entropy tell you what it says.
Jesus David Calderas Cervantes (Wed,) studied this question.