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At fixed particle numbers, the perturbations satisfy $\int \delta n_i{d}^{3}x=0$. Expanding the uniform interaction energy, its linear term is $\int ({\mu }_1\delta n_1+{\mu }_2\delta n_2)d^{3}x=0$, and its exact quadratic change in this contact-interaction approximation isCompleting the square givesHence the miscibility criterion for a two-component condensate isThese are precisely strict positive definiteness of the interaction matrix, and make every nonzero admissible long-wavelength density perturbation cost energy. With both equilibrium densities positive, a negative eigenvalue would instead allow a small sinusoidal perturbation along its eigenvector, with zero mean and negative energy change. The omitted positive gradient term cannot cure arbitrarily long wavelengths in an unbounded system.

If cross-repulsion exceeds the geometric mean of the self-repulsions, the unstable direction favors separation of species; sufficiently strong cross-attraction favors a joint density increase and collapse. Equality gives a zero-cost direction at this order, so it is marginal rather than strictly stable. Constraints near an absent component or finite-system wavelength restrictions require a separate boundary analysis.