Depth Calibrates the Effective Bandwidth of Sinusoidal Neural Fields

The frequency-scale hyperparameter of a sinusoidal neural field, ${\omega }_0$ or $f_{\max }$, is commonly treated as the main control on what the network represents. The spectrum a trained network realizes departs from that nominal scale through weight rescaling and through the harmonics that layer composition generates, and how the two relate as the network deepens has not been characterized. We measure the realized spectrum directly, through an effective bandwidth $B_{\mathrm{eff}}$ defined as the 95% spectral-energy radius of the trained output on a dense grid, over roughly 7,000 fits across two datasets, two architectures and four depths. Near-optimal nominal values span 4-10.7x across depths while the realized $B_{\mathrm{eff}}$ spans only 1.5-2.3x. Depth amplifies spectral expansion. At a fixed knob the realized bandwidth grows from 18 to 727, and the operational expansion boundary shifts down with depth, [32, 16, 12, 12] on both datasets under a fixed 2x criterion. A refined sweep resolves that boundary into a steep, localized crossover, and near-optimal nominal scales lie no farther than its near edge. The geometry supports a target-spectrum-conditioned selector. It uses one ratio, fit on two shallow depths of one dataset, with the signal bandwidth $B_{\mathrm{sig}}$ estimated from training coordinates only. Over the completed candidate grid its mean regret is 0.20 dB on held-out depths, 0.59 dB on a held-out dataset without fitting a target-side bandwidth map, 0.52 dB transferring to an architecture-specific SIREN map and 0.12 dB on a 32-image held-out cohort. Its worst case is 2.35 dB, at one frequency just past the expansion boundary. Across depth the effective bandwidth is the more stable quantity, and it is the one to calibrate. All experiments concern controlled 2D image regression with sinusoidal-activation networks.