Matrix-element selection rule for sublattice-selective leakage suppression in flat-band Lieb mechanical lattices

Flat-band lattices support compact localized states with sublattice-selective amplitude patterns, but it is less clear whether this geometry can protect nearby boundary resonances from disorder-induced leakage into the bulk. Here we identify a matrix-element selection rule for sublattice-selective leakage suppression in a gyroscopic Lieb mechanical lattice. The lattice hosts a nearly A-sublattice-dark quasi-flat band and a boundary resonance in the adjacent low-density spectral window. Under sublattice-resolved onsite disorder averaged over 48 realizations with bootstrap 95% confidence intervals, the small-W leakage prefactors obey a BC /a A = 58 (CI [49, 71]) for W ≤ 0.12; a full-range quartic fit and a fit-free integrated disorder response give more conservative ratios of 35 and 8.8, respectively. Bond-stiffness disorder shows a matching bond-resolved selectivity a AC /a AB = 34 with near-perfect variance additivity (0.997), confirming that the mechanism is not tied to onsite disorder. A Fermi-golden-rule analysis on the strip eigenmodes attributes the asymmetry to a suppressed matrix element ⟨ψ b |V A |ψ e ⟩ whose isotropic scalar-site overlap-density evaluation supports the observed order-of-magnitude asymmetry (details in Sec. 4.3). The selection rule remains present under spring anisotropy k y /k x ∈ [0.5, 2.0] at every anisotropy where the perturbative fit converges above the twelve-realization ensemble noise floor. The mechanism is geometric rather than topological and provides a route to reducing selected bulk-leakage channels in multi-sublattice mechanical, phononic and photonic flat-band platforms.