Why Identical Coils Attract:- A Resonant Curvature--Universality Framework Beyond Ampere's Law

I revisit the classical problem of pairwise attraction between identical current-carrying coils andpresent a variational framework that extends the Maxwell action with two controlled higher-ordercontributions: a resonant modal coupling term and a curvature-equivalence functional. The extendedaction recovers the conventional mutual-inductance (Ampère) force as the leading-order limit whilepredicting explicit, experimentally accessible corrections that depend on modal overlap, relativephase, and field-geometry mismatch. I introduce the Resonant–Curvature Index (RCI) as adimensionless diagnostic of topology–resonance coupling and provide concrete protocols to calibratethe two effective couplings (denoted $\alpha$ and $\beta$) from force measurements. Using finite-element eigenmode extraction I demonstrate numerically that resonant corrections of order$\sim$ a few percent of the Ampère baseline and curvature corrections of order $10^{-6}$--$10^{-5}\,$Nare achievable with laboratory-scale coils; both are within reach of modern torsion-balancesensitivities. I provide clear experimental designs, uncertainty analysis, and control measurementsto falsify the extended model. The resulting theory is intentionally conservative in its claims: it is aclassical, gauge-consistent augmentation of Maxwellian electrodynamics that makes falsifiablepredictions and supplies practical methods for parameter estimation and replication. Speculativeextensions to plasma, condensed-matter, and quantum circuits are indicated as avenues for futurework but are separated from the core experimental claims.<p></p>