Local number variance as a causal handle in the one-dimensional Bose–Hubbard chain

We test whether the local occupation variance Fi = - ² identifies causally effective intervention sites in a one-dimensional open Bose-Hubbard chain under Lindblad dephasing. Additional dephasing at the top-k highest-Fi sites is compared against matched-budget random targeting across 100 independent trials per condition, using exact Lindblad evolution in the fixed-particle-number sector (L in 6, 7, 8, tau in 1, 2, 3). The primary sweep reveals three regimes: at J/U = 0. 12 targeted intervention is harmful (95% CI strictly below zero at all tested L and tau) ; near J/U ~ 0. 20 there is a crossover that is sensitive to system size and horizon; at J/U >= 0. 30 targeted intervention is reliably beneficial (CI above zero at every tested (L, tau) combination), with best effects at J/U = 0. 40, tau = 3: +0. 047 (L = 6), +0. 068 (L = 7), +0. 072 (L = 8). To disentangle variance information from geometry, we perform four additional experiments. A disorder selector sweep (L = 6, 50 realizations, 9 selectors) at strong disorder (muₘax >= 1. 0) shows that Fi is the top selector, ranking above geometric centrality (geo), mean occupation (maxn/minn), disorder amplitude (disₐmp), and a generator-action selector (gen) ; the Fi-geo gap grows monotonically with disorder strength and J/U, confirming that Fi carries information beyond raw spatial position. Shell-matched permutation controls (L = 6, muₘax = 0. 30, establishing that the advantage is not reducible to shell geometry. A deterministic inhomogeneous tilt chain (L = 6) where Fi != geo by construction gives Fi > geo with a gap of +0. 032 to +0. 054 at J/U = 0. 40, tau = 3, confirming the result under designed symmetry breaking. A scan of the extra dephasing rate (gammaₑxtra in 0. 1, 0. 2, 0. 5, 1. 0, 2. 0) shows the advantage peaks near gammaₑxtra = 0. 5 to 1. 0 and collapses at very high rates, confirming that the paper's choice gammaₑxtra = 0. 5 is near-optimal. Taken together, local number variance identifies causally effective intervention sites in the positive pocket (J/U >= 0. 30), and this advantage is not reducible to geometric centrality, shell position, mean occupation, or disorder amplitude, while surviving the tested deterministic symmetry-breaking controls.