  Some of the most remarkable phenomena---and greatest theoretical
challenges---in condensed matter physics arise when $d$ or $f$ electrons are
neither fully localized around their host nuclei, nor fully itinerant. This
localized/itinerant "duality" underlies the correlated electronic states of the
high-$T_c$ cuprate superconductors and the heavy-fermion intermetallics, and is
nowhere more apparent than in the $5f$ valence electrons of plutonium. Here we
report the full set of symmetry-resolved elastic moduli of $PuCoGa_5$---the
highest $T_c$ superconductor of the heavy fermions ($T_c$=18.5 K)---and find
that the bulk modulus softens anomalously over a wide range in temperature
above $T_c$. Because the bulk modulus is known to couple strongly to the
valence state, we propose that plutonium valence fluctuations drive this
elastic softening. This elastic softening is observed to disappear when the
superconducting gap opens at $T_c$, suggesting that plutonium valence
fluctuations have a strong footprint on the Fermi surface, and that $PuCoGa_5$
avoids a valence-transition by entering the superconducting state. These
measurements provide direct evidence of a valence instability in a plutonium
compound, and suggest that the unusually high-$T_c$ in this system is driven by
valence fluctuations.
