SFE: Rank Geometry of the Relational Channel — Finite-Sample Calibration and Operating Envelope

Let d_ρ = Var(ρ_ab, 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, d_ρ < δ implies E[r_eff] = 2/(1+ρ²) + O(δ) and Var(r_eff) ≤ 16ρ²δ/(1+ρ²)⁴, so near-zero d_ρ together with |ρ| ≈ 1 is equivalent to the joint covariance being approximately rank 1. Second, the rolling estimator is reliable for σ_post < 0.20 (ρ* > 0.45) with finite-sample bias under 4%; bias enters a monotonic negative third-order regime for σ_post > 0.30, and both boundaries coincide with the operating limit of the AND gate. Third, non-stationarity is self-detectable: empirical d_ρ > (1−ρ*²)²/W flags structural drift without external ground truth. The N-observer extension yields a corrected r_eff estimator with MAE < 0.04 (N = 2–10) and an empirically characterized band structure with analytic motivation deferred to subsequent work. The framework applies to stochastic relational systems including sensor synchronization, coupled oscillators, and correlated financial or biological signals.