Lab for Latent Actors: Hiding Segments, Tempo and the Value of a Probe.

Two kinds of patient, invisible in the data, can explain every trial result. The first question

is which part of their structure the evidence actually reveals. The second is how a decision

maker can force the rest into view, especially when someone with an interest in the outcome

chooses what is revealed. This companion to the note Partial Gluing and Level Two [1] builds

a synthetic laboratory for these questions. It is run as an observe–orient–decide–act loop, and

it proves three theorems.

Theorem A (hiding segment). Pair tables never identify two latent types. They leave

a segment of explanations, on which the probability of every event on k variables is a

polynomial of degree at most k−2 in one unseen parameter. A small subgroup that is

very different is indistinguishable from a large one that is mildly different.

Theorem B (tempo). Under pair-only disclosure the weight is never learned (tempo zero).

After one informative triple table everything converges at n−1/2. Agreement on a single

variable is a regular relation. Only the merge of the two types slows to n−1/4

.

Theorem C (stratified value of information). A probe is worthless when it touches a

variable on which the types agree, and collapses the segment otherwise. Its worst-case

value is zero whenever the decision has a saddle point on the segment, as a treat-or-not

choice does, while its minimax-regret value stays positive whenever the best act changes.

Its graded value rises with an explicit signal, and the best probe can be named from the

pair tables alone. With verifiable disclosure and a skeptical regulator, sponsors unravel.

A robust decision maker can lose from new evidence only when that evidence makes the

evidence as a whole vacuous.