  While standard solar model (SSM) predictions depend on approximately 20 input
parameters, SSM neutrino flux predictions are strongly correlated with a single
model output parameter, the core temperature $T_c$. Consequently, one can
extract physics from solar neutrino flux measurements while minimizing the
consequences of SSM uncertainties, by studying flux ratios with appropriate
power-law weightings tuned to cancel this $T_c$ dependence. We re-examine an
idea for constraining the primordial C+N content of the solar core from a ratio
of CN-cycle $^{15}$O to pp-chain $^8$B neutrino fluxes, showing that nonnuclear
SSM uncertainties in the ratio are small and effectively governed by a single
parameter, the diffusion coefficient. We point out that measurements of both
CN-I cycle neutrino branches -- $^{15}$O and $^{13}$N $\beta$-decay -- could in
principle lead to separate determinations of the core C and N abundances, due
to out-of-equilibrium CN-cycle burning in the cooler outer layers of the solar
core. Finally, we show that the strategy of constructing "minimum uncertainty"
neutrino flux ratios can also test other properties of the SSM. In particular,
we demonstrate that a weighted ratio of $^7$Be and $^8$B fluxes constrains a
product of S-factors to the same precision currently possible with laboratory
data.
