Intrinsic restriction norms for BMO

For every absolutely continuous probability measure on Euclidean space, we construct a single BMO function whose pair-difference tail controls, with dimensional constants, the distributional envelope of the classical BMO unit ball at all thresholds. This yields sharp restriction-norm estimates, uniform in the finite source and target exponents, and a common test for arithmetic-center rearrangement-invariant Banach norms and best-constant rearrangement-invariant quasi-Banach norms. A second argument compares oscillations on arbitrary measurable profiles occupying a fixed positive fraction of a containing ellipsoid, independently of their finer fragmentation. The resulting affine growth order is sharp. Weighted atomic models realize the restriction norms by duality and give exact reverse capacities and universal large-exponent limits. Stability, simultaneous smooth approximation, and composition estimates are consequences and applications of these common estimates.