Residual Galactic binary foreground in LISA stochastic gravitational-wave background inference: source power concentration and spectral degeneracy

Galactic compact binaries are expected to form a dominant foreground in the millihertz band of the Laser Interferometer Space Antenna (LISA). Residual power from injected sources that do not meet the adopted recovery criteria can bias stochastic gravitational-wave background (SGWB) inference or increase its uncertainty. We use LISA Data Challenge 2A Sangria injections and the Erebor comparison table to construct a catalog residual spectrum between 0.4 and 6.0 mHz with orbit-averaged long-wavelength Michelson $X$ source powers. The source power concentration in each frequency bin determines the excess kurtosis of a random-phase source sum; instrumental noise and fiducial SGWB power strongly reduce the resulting excess kurtosis in most bins. The residual spectrum also overlaps an isotropic power-law SGWB in the mean binned power. We use a fixed covariance obtained by summing independent Fourier-mode power variances. For a frequency-independent SGWB with fiducial amplitude $\Omega_0=10^{-11}$, marginalizing over the dimensionless residual-power factor $\beta$ increases the $\Omega_0$ uncertainty by $13.6\%$ when the residual power is distributed uniformly over the Fourier frequencies in each bin. The largest Gaussian prior standard deviation on $\beta$ that limits this increase to $10\%$ is $0.0073$. More concentrated distributions of the residual power among Fourier frequencies reduce the increase, reflecting unresolved frequency structure. Omitting the fiducial residual with the covariance held fixed shifts the best-fitting $\Omega_0$ by $119.5$ times the uncertainty obtained with $\beta$ fixed. This projection of the residual spectrum onto the SGWB spectrum is not a posterior detection significance. The numerical values are conditional on the catalog-level scalar power model, fixed instrumental noise, and independent mode-power covariance.