  We characterize the earliest galaxy population in the JADES Origins Field
(JOF), the deepest imaging field observed with JWST. We make use of the
ancillary Hubble optical images (5 filters spanning $0.4-0.9\mu\mathrm{m}$) and
novel JWST images with 14 filters spanning $0.8-5\mu\mathrm{m}$, including 7
medium-band filters, and reaching total exposure times of up to 46 hours per
filter. We combine all our data at $>2.3\mu\mathrm{m}$ to construct an
ultradeep image, reaching as deep as $\approx31.4$ AB mag in the stack and
30.3-31.0 AB mag ($5\sigma$, $r=0.1"$ circular aperture) in individual filters.
We measure photometric redshifts and use robust selection criteria to identify
a sample of eight galaxy candidates at redshifts $z=11.5-15$. These objects
show compact half-light radii of $R_{1/2}\sim50-200$pc, stellar masses of
$M_{\star}\sim10^7-10^8 M_{\odot}$, and star-formation rates of
$\mathrm{SFR}\sim0.1-1\,M_{\odot}\,\mathrm{yr}^{-1}$. Our search finds no
candidates at $15<z<20$, placing upper limits at these redshifts. We develop a
forward modeling approach to infer the properties of the evolving luminosity
function without binning in redshift or luminosity that marginalizes over the
photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a $z=12$ luminosity function in good
agreement with prior results, and that the luminosity function normalization
and UV luminosity density decline by a factor of $\sim2.5$ from $z=12$ to
$z=14$. We discuss the possible implications of our results in the context of
theoretical models for evolution of the dark matter halo mass function.
