  We derive multidimensional bosonization directly from the electron gas in a
low-energy, low momentum regime where $\omega\gg \frac{k^2}{k_F}$, such that
the dispersion can be linearized. To reach this limit, the Fermi momentum and
the number of patches are scaled simultaneously keeping the width of each patch
finite. We apply this to obtain an exact low-energy solution of the problem of
a Fermi surface coupled to a gapless boson, free of disorder and
electron-electron scattering. Contrary to claims in the literature, we show
that the bosonized theory exactly reproduces the $\omega^{2/3}$ of electrons,
previously obtained in large-$N$ theories. We argue that correction to the
self-energy due to tangential dispersion are subdominant at sufficiently low
energies such that $v_F k\gg \left(\frac{g^4 v_F}{k_F}\right)^{1/3}
\omega^{2/3}$, where $g$ is the coupling constant.
