Worlds Next Door. IV. Mapping the Late Stages of Giant Planet Evolution with a Precise Dynamical Mass and Luminosity for ε Ind Ab

We present new JWST/NIRCam 4-5 $\mu$m (F410M, F430M) and JWST/MIRI 18-25 $\mu$m (F1800W, F2100W, F2550W) imaging detections of the nearby (3.6 pc) cold (275 K) gas giant exoplanet $\epsilon$ Ind Ab. The F2550W detection of $\epsilon$ Ind Ab constitutes the longest wavelength image of an exoplanet acquired to date. Combining three decades of radial velocity monitoring, Gaia-Hipparcos absolute astrometry, and relative astrometry from direct imaging (including the new NIRCam astrometry), we conduct a comprehensive re-analysis of $\epsilon$ Ind Ab's orbit and obtain a dynamical mass $M_{\rm Ab} = 6.5^{+0.7}_{-0.6}\;M_{\rm Jup}$. Using $\epsilon$ Ind Ab's NIRCam and MIRI photometry, we assemble the first 4-25 $\mu$m spectral energy distribution (SED) of a cold gas giant outside the Solar System. The NIRCam photometry supports a metal-enriched atmosphere for $\epsilon$ Ind Ab based on analysis with atmospheric model grids, consistent with predictions from the giant planet mass-metallicity relation. While the current data do not provide definitive evidence for or against the presence of water ice clouds, we tentatively find that the H$_2$O vapor absorption-dominated F2550W photometry is systematically brighter ($>1\sigma$, but $<2\sigma$) than predictions from cloud-free/rainout chemistry models and better explained by a cloudy model. We calculate a bolometric luminosity of $\log L_{\rm bol}/L_\odot = -7.23 \pm 0.03$ dex by directly integrating $\epsilon$ Ind Ab's SED. Combining this with the planet's dynamical mass and age ($3.5 \pm 1.0$ Gyr), we demonstrate excellent agreement with evolutionary model predictions in a new regime of low luminosities, low masses, and old ages. Our results establish $\epsilon$ Ind Ab as a benchmark system for planetary evolution studies and set the stage for the detailed atmospheric characterization of this temperate extrasolar world.