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

Abstract 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 [8] builds a synthetic laboratory for these questions. It is run as an observe–orient–decide–act loop. Part I proves four 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. Theorem D (the decision on the segment). An act is safe on the whole segment iff it is safe at both walls, and the set-rule choice depends on the pair tables alone, so no undetectable perturbation changes it. Info-gap prefers acts least exposed to the hidden moment, but before the probe its nominal point is a convention. After a probe the decision settles on the oracle act at median sample size n∗ = (zτ/d)2, where d is a computable radius, and never settles when the truth sits on a boundary. Part II turns to an actor who adds records to tilt the decision, with no relation between injected and genuine records. Theorem E (slopes). A decision’s margin has a price in injected records: the margin divided by the steepest single-record influence, attained by copies of one record. Theorem F (the hidden channel). Injections that leave every pair table unchanged form a segment of their own. They move the estimate of the unseen parameter by εs/vT while 1the regulator’s segment stays put, so there is always a window in which the decision is taken over and nothing is flagged. The best probe is also the hardest to poison. Theorem G (stealth with equations). With four variables the tetrads can detect injection, except along their tangent directions. Those directions multiply on the strata, but there the Andr´e–Quillen obstruction stops them from being extended. Theorem H (one geometry). The adversary’s segment and the regulator’s are the positivity part and the model part of one line, read in units of vT . The regulator’s margin is the distance to the decision boundary and the adversary’s room the distance to the detection boundary, so Part II completes the decision lab rather than adding to it. Theorem J (sources and bands). When the decision maker sees which source and batch each table came from, a trusted source that is adversarial in some batches can move the pooled estimate by at most cs π/(1−π) standard errors without being exposed, at every sample size. Gluing tests on two covers turn every undetected tilt into a bounded delay and localise concentrated attacks without writing the source off. Theorem K (lifting). Each singularity has a lift: decision chambers and plateaus on the fibre, mixtures of acts, the blow-up of the merge, the adversarial lift whose block size is the poisoning budget, and an orbifold atlas. Around it sit propositions on dilation, on the value of a probe in its size (a Radner–Stiglitz flat start, and a negative Wald value), on backtracking, and on rectangularity as gluing. Theorem L (one picture). Settling, the value of a probe, every robustness number, the decision-relevance of a gluing failure and the stealth window are all readings of one chamber arrangement on the fibre. Each robustness number is the same margin to a wall divided by the capacity of a gauge. Four pre-registered studies test the quantitative claims, and one example is worked through every step by hand (section 17). In Study 11, twenty-one of the twenty-four registered predictions held, and the three that failed are reported together with what they show. In Study 11b, all four predictions about settling held. In Study 12, all four predictions about injection held: about 6% of injected records flips a mandated-triple decision without a single replicate being flagged. In Study 13, all five predictions about a sometimes-adversarial source held: pooling is tilted in 88% of replicates, while the gluing filter never settles wrongly more than 2% of the time and is only delayed. The paper is written for three readerships: decision theory under severe uncertainty, Bayesian persuasion and verifiable disclosure, and machine learning with reinforcement learning and adversarial training.