For mean embeddings perturbed by at most \(\varepsilon _i\) and \(\varepsilon _j\), the inner product moves by at most \(\varepsilon _i\lVert \mu _j\rVert +\varepsilon _j\lVert \mu _i\rVert +\varepsilon _i\varepsilon _j\).

Under the same radii, \(\lvert \operatorname {dist}(\mu _i',\mu _j')-\operatorname {dist}(\mu _i,\mu _j)\rvert \le \varepsilon _i+\varepsilon _j\). Additivity on the metric, rather than on the squared quantity, is the same structural fact that makes the metric-level bracket of ??cor:random-w2-metric sharper than its squared counterpart.

Squaring the metric bracket gives an asymmetric interval, with the lower end truncated at zero. The asymmetry is not cosmetic: it is why the squared-cost formalization cannot simply inherit the metric statement.

Under bi-Lipschitz constants stated in the true, unobserved distance and per-asset embedding radii, the covariance inner product is bracketed by polarization expressions evaluated at the radius-inflated and radius-deflated measured distance. The hypothesis that the bi-Lipschitz relation holds in the unobserved distance is the load-bearing one and is not testable from the measured distances.