MMP.16:4.3 - Find what a design can distinguish

First seek a simple consequential contrast. For a set-valued model, take the union of R(h,theta,d) over the remaining possible theta. Disjoint unions for two alternatives give a separating observation under those assumptions. If the unions overlap, identify records that would settle the needed distinction and records that would leave it open.

Equal sets alone do not establish equal statistical information: probability laws can weight the same possible records differently. In a probabilistic model, compare the full record laws or a statistic whose retained information suffices for the requested result. Distinct means are one possible contrast, not a universal criterion.

An impossibility conclusion needs its scope. Two alternatives that give the same record law under every currently feasible design, yet different required answers, demonstrate a distinction unavailable through those designs. More repetitions under the same uninformative access do not resolve it. Changing the observation type, access or target may do so. Failure of a numerical search to find a good design establishes only that search result.

A narrower target can remain obtainable. Suppose all feasible records leave the individual parameters unresolved, but every compatible parameter pair gives the same total response. Return the total when it answers the work question. Recovering each parameter is then unnecessary.