  We present new cosmological parameter constraints from the eBOSS
Lyman-$\alpha$ forest survey. We use a new theoretical model and likelihood
based on the PRIYA simulation suite. PRIYA is the first suite to resolve the
Lyman-$\alpha$ forest in a ($120$ Mpc/h)$^3$ volume, using a multi-fidelity
emulation technique. We use PRIYA to predict Lyman-$\alpha$ forest observables
with $\lesssim 1\%$ interpolation error over an $11$ dimensional ($9$
simulated, $2$ in post-processing) parameter space. We identify an internal
tension within the flux power spectrum data. Once the discrepant data is
removed, we find the scalar spectral index at $k = 0.78$ h/Mpc to be $n_P =
0.97 - 0.995$ at $68\%$ confidence from the Lyman-$\alpha$ forest flux power
spectrum alone, in good agreement with Planck. The amplitude of matter
fluctuations is $\sigma_8 = 0.733 \pm 0.026$ at $68\%$ confidence, in agreement
with Dark Energy Survey weak lensing measurements and other small-scale
structure probes and in tension with CMB measurements from Planck and ACT. The
effective optical depth to Lyman-$\alpha$ photons from our pipeline is in good
agreement with earlier measurements. We add measurements of the mean
temperature of the intergalactic gas from $z=3.8 - 2.2$ and use them to
constrain the duration and heating rate of helium reionization, finding a
preference for an early, hot, helium reionization event, as suggested by
measurements from the helium Lyman-$\alpha$ forest. Adding the mean IGM
temperature data also increases the significance of the $\sigma_8$ tension. In
the near future we will use our pipeline to infer cosmological parameters from
the DESI Lyman-$\alpha$ data.

Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies