Same Function, Different Geometry: Choosing Optimization Coordinates for Gaussian Representations

Optimization coordinates are part of a learning algorithm, even when they leave its represented functions unchanged. We investigate their practical value in Gaussian representations, where appearance is linear at fixed geometry but joint fitting is nonlinear. Exact appearance absorption and transported Adam/AdamW controls isolate coordinate effects from capacity. We construct energy-normalized coordinates that compensate appearance as shape changes, cancelling continuous single-atom shape–appearance coupling. With the same five-profile iRprop+ calibration allowance for peak and energy coordinates, a prospective study on 64 fresh images yields mean RGB PSNR gains of 1.242 dB over a frozen historical incumbent and 0.939 dB over independently calibrated peak coordinates, averaged across attained 2, 5 and 10-second output budgets. Both medians are positive and mean LPIPS decreases; a complete repaired-arithmetic replay retains these gains. Benefits extend to three of four domains; Aircraft and arithmetic-mean error against calibrated peak remain limitations. Timings measure warm output delivery with monitored host activity. Separately, fixed output measurements yield a sharp, separation-independent angle law and first-order minimax certificate for moving one-dimensional Gaussian clusters. Prospective selection and optimizer-payoff bounds distinguish useful geometry from useful computation. Together, these results make coordinate choice a testable design decision rather than an inherited implementation detail.