Certifying Output-Space Separation of Conditional Location-Scale Mixtures

Analyses of conditional generative models routinely assume that conditions far apart are mapped to output distributions far apart in Wasserstein distance. We ask when this separation can instead be certified for a learned conditional location-scale mixture. For each pair of conditions, our decision procedure returns either an explicit lower bound on output distance or an honest refusal. Three certificates cover complementary regimes. A spectral moment certificate combines mean displacement with the full Bures distance between output covariances, using exact covariance addition under convolution. It has constant one, is dimension-free, and detects anisotropic changes that a trace-only certificate misses. A heat-semigroup certificate converts an integrated absolute density gap into an output-transport lower bound with a three-halves power law for a fixed Laplace kernel in every dimension; scale interpolation extends it to different kernels with an additive Bures slack. A latent inverse certificate feeds these output bounds. Our direct proof extends to every kernel admitting finite-order PDE inversion and yields an exponent twice the inversion order, exactly matching the detour through the first Wasserstein distance. Thus direct inversion does not improve the exponent within this proof template. We also make two normally implicit side conditions checkable. The globally Lipschitz gradient needed by the quadratic refinement reduces to strong convexity of the dual potential; it holds for single-point targets and in the classical smooth-density regime but fails for multi-cell atomic transports. A mollifier uncertainty principle forces spatial moment and derivative size to grow jointly at least at the square-root-of-dimension scale. Consequently, the best latent constant within this proof architecture deteriorates like the reciprocal of dimension to the three-halves power, regardless of mollifier choice. Numerically, we verify the heat bridge in dimensions two, four, and eight without empirical optimal transport, exhibit an anisotropic pair that defeats the trace certificate, and certify a trained mixture density network. Throughout, we distinguish certification from refusal: the exponents are dimension-free, the latent constants are not, and the composed guarantee is polynomial rather than a linear coarse embedding.