Global distinction, local derivatives, and its limit

Suppose $D_{\mathrm{f}}\in [d_{-},d_{+}]$ and $\lambda A_{r_{\mathrm{c}}}\ge {A_{\min }}^{\mathrm{add}}>0$ over the tested alternative family. For known $\mathcal{R}$, the untruncated mean ranges have gap

A positive value gives global mean separation, not just local rank.

If sensor or timing calibration makes $\mathcal{R}$ uncertain, use the joint-family extrema

with any certified mean departures subtracted. Different free-time experiments cannot be assigned independent arbitrary preparation covariances within these families.

For complete-record density or mass function $p_i$, differentiation of (eq:fourtime) yields, when differentiation under the moment integral is justified,

Thus gravity is transverse to the support-amplitude derivative at this observation boundary when controls and competing diffusion are fixed. This calculation supplies a separating functional for those two derivatives; it does not remove other nuisance columns automatically.