Computable diagonal loading attaining a target spatial white noise gain in spherical-harmonic minimum variance distortionless response beamforming

With few snapshots, minimum-variance distortionless-response beamforming on spherical microphone arrays loses white noise gain (WNG) and cancels its target. A Kantorovich bound that sets diagonal loading for a target WNG, applied to the spherical-harmonic modal covariance, can miss the spatial WNG by up to the squared modal-transform condition number. A computable rule searches the identity-loaded modal family for the least loading meeting a spatial-WNG floor and reports infeasibility. The exact spatial-WNG-constrained solution loads the capsule covariance instead and meets the tested floors at every frequency. On a simulated Eigenmike at 4.5 kHz, the rule meets the floor, with equality when loading is needed, where the transferred bound realizes 0.44-0.80 for a requested 0.1, and at equal targets from 0.1 to 0.5 keeps 0.3-0.7 dB more signal-to-interference-plus-noise ratio and 0.6-1.4 dB more directivity than the exact loading. The lead grows at lower frequencies, vanishes from about 5.5 kHz, and ends at an interferer-dependent target where the loading collapses. On measured room responses, the rule meets the target at every snapshot count while the unloaded weight self-cancels under deficiency and residual steering mismatch.