  We present a novel analytic extraction of high-order post-Newtonian (pN)
parameters that govern quasi-circular binary systems. Coefficients in the pN
expansion of the energy of a binary system can be found from corresponding
coefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change
$\Delta U$ in the redshift factor of a circular orbit at fixed angular
velocity. Remarkably, by computing this essentially gauge-invariant quantity to
accuracy greater than one part in $10^{225}$, and by assuming that a subset of
pN coefficients are rational numbers or products of $\pi$ and a rational, we
obtain the exact analytic coefficients. We find the previously unexpected
result that the post-Newtonian expansion of $\Delta U$ (and of the change
$\Delta\Omega$ in the angular velocity at fixed redshift factor) have
conservative terms at half-integral pN order beginning with a 5.5 pN term. This
implies the existence of a corresponding 5.5 pN term in the expansion of the
energy of a binary system.
  Coefficients in the pN series that do not belong to the subset just described
are obtained to accuracy better than 1 part in $10^{265-23n}$ at $n$th pN
order. We work in a radiation gauge, finding the radiative part of the metric
perturbation from the gauge-invariant Weyl scalar $\psi_0$ via a Hertz
potential. We use mode-sum renormalization, and find high-order renormalization
coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of
the expression for $\Delta U$. The non-radiative parts of the perturbed metric
associated with changes in mass and angular momentum are calculated in the
Schwarzschild gauge.
