A compact nonhomogeneous dyadic commutator with divergent symbol truncations

We study the sibling Haar model of the Hilbert transform on L²(μ), where μ is an atomless Radon measure with positive finite dyadic masses, infinite dyadic quadrants, and sibling balance. Its commutator symbol space X combines quadratic mean oscillation with bounded sibling differences β_Q(b) = ⟨b,h_Q²⟩ − ⟨b,h_{Q^s}²⟩. On one fixed absolutely continuous nondoubling measure with sibling balance constant at most eight, we construct a real, mean-zero symbol b supported on [0,1) such that [H,M_b] is compact and b belongs to the finite Haar closure in X modulo quadrant constants. Nevertheless, for every integer N ≥ 1 its conditional expectation satisfies ‖E_N b‖_X ≥ log(N+1), and the truncation errors tend to infinity in X. Slowly tapered finite Haar sums converge to b by preserving the adjacent-coefficient cancellation in β. We also derive the essential norm estimate max{A(b),B(b)} ≤ ‖[H,M_b]‖_ess ≤ 8A(b)+B(b), where A and B are the cofinite tails of quadratic oscillation and |β_Q(b)|, respectively.