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

We test whether the local occupation variance Fi = <n_i^2> - <n_i>^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 (mu_max >= 1.0) shows that Fi is the top selector, ranking above geometric centrality (geo), mean occupation (maxn/minn), disorder amplitude (dis_amp), 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, mu_max <= 0.3) show that Fi beats all within-shell permutations by +0.002 to +0.006 at tau = 3, J/U >= 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 (gamma_extra in {0.1,0.2,0.5,1.0,2.0}) shows the advantage peaks near gamma_extra = 0.5 to 1.0 and collapses at very high rates, confirming that the paper's choice gamma_extra = 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.