Higher-Order Smoothing and Spectral Dominance in a One-Dimensional Biharmonic Stabilizer

We present a constrained five-dimensional Effective Field Theory (EFT) in which Standard Model flavor hierarchies emerge naturally from the geometry of localized fermion zero modes on a scalar background governed by a biharmonic stabilizer. We demonstrate that the fourth-order stabilizer operator provides a robust "spectral shielding" mechanism, suppressing high-frequency radiative corrections to the background geometry by factors of O(n-4). This mechanism ensures that the localization centers remain stable against one-loop drift (Condition K4), justifying the rigidity of the geometric ansatz. We calibrate the model to a "Snap-Point" configuration characterized by a shared Left-Handed lattice and a single width parameter per sector. This minimal structure reproduces the observed quark masses and CKM mixing angles to within O(10%) accuracy at tree level. Furthermore, we explicitly verify that a single natural bulk phase gradient generates a Jarlskog invariant of the observed magnitude (J ~ 10-5) without fine-tuning (Condition K5), while a "leakage" regime for right-handed neutrinos reproduces the anarchic mixing observed in the lepton sector. The framework is defined by a rigorous "Kill-Condition Ledger" of falsifiable audits, identifying the coupled scalar-sector mixing (Condition K6) as the final critical test for the model's viability.