Optimal galaxy magnification: generalized fundamental-plane residuals and number counts are all you need

Weak-lensing magnification complements cosmic shear through its effects on galaxy fluxes, sizes, and number counts. We derive the optimal magnification statistics from galaxy properties and show that the optimal measurement can be decomposed into an optimal mean statistic and total number counts. The optimal mean statistic can be constructed from the residual distribution of a generalized fundamental plane (GFP), F(i,), which predicts galaxy size or, equivalently, flux from surface brightness i and additional lensing-invariant properties. The function F can take any form that reduces the intrinsic scatter and thereby improves magnification precision. For a Gaussian residual distribution, the optimal mean statistic reduces to the inverse-variance-weighted residual, or simply the residual for constant scatter. Together with total number counts, this statistic captures all the magnification Fisher information under Poisson statistics, achieving the same precision as a full-likelihood analysis. This provides a simple and physically intuitive optimal compression of the full magnification likelihood, avoiding the need to model the potentially high-dimensional galaxy-property distribution, particularly when additional observables are incorporated to reduce the GFP scatter. Our results demonstrate that GFP residuals and number counts provide an optimal combination for weak-lensing magnification measurements in galaxy surveys.