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Vanishing Phase Stiffness and Fluctuation-Dominated Superconductivity: Evidence for Inter-Band Pairing in UTe$_2$

Superconductivity in three dimensions is almost universally governed by Ginzburg-Landau mean field theory, with critical fluctuations typically confined to within a few percent of the transition temperature ($T_{\rm c}$). We report that the heavy-Fermion superconductor UTe$_2$ exhibits a fluctuation regime that extends over a temperature range as wide as $T_{\rm c}$ itself -- the largest observed for any three-dimensional superconductor. Through ultrasound measurements of the elastic moduli and sound attenuation, we find that UTe$_2$ transitions from a mean-field-like state at ambient pressure to a fluctuation dominated state at higher pressures. This regime is marked by elastic softening and an increase in sound attenuation that onsets well above $T_{\rm c}$, with the attenuation remaining anomalously high deep in the superconducting state. Our analysis shows that these features stem from an extremely low superfluid phase stiffness. This results in a kinetic inductance as high as that of granular aluminum, but achieved in the clean limit. We propose that this exotic state is driven by dominant inter-band pairing mediated by ferromagnetic fluctuations, leading to "local" cooper pairs with a coherence length of only a few lattice constants.