OPAQUE AND UNBOUNDED : Two Words That Carry an Argument, Cashed Out in Examples — On the Arithmetic of the First, the Missing Record of the Second, and Why Neither Is Technobabble

A stochastic model is called "unbounded" and "opaque." Both words are used across thecorpus as if their meaning were obvious. It is not. Unbounded is an arithmetic claim: the space of possible inputs has no ceiling, so coveragecannot be computed. This paper states the arithmetic at four scales, gives worked examples from deployed systems, and shows why "we tested a million inputs" is not a coverage statement. Opaque is a record-keeping claim: when the system fails, there is no trace from the failure to a rule, a source, a signer, or a decision point. This paper lists the questions a post-incidentreview normally asks, and shows that none of them can be answered. It then does the harder thing: it states what bounded and attributable mean, using systems that already work — an ATM, a pacemaker, an airbag, a chess engine, a trafic light. Bounded does not mean small; chess is bounded and its game tree is about 10^120. Bounded means the space is finite and declarable. The paper concludes that the two words are not technobabble. They are the names of twoabsences, and each absence has a demonstrable consequence: an unbounded system cannot be verified, and an opaque system cannot be corrected. A system with both cannot be trusted in a domain where failure matters. The paper also states what it does not prove. It does not prove that no stochastic systemcan ever be bounded. It does not prove that bounded systems are safe. It proves that thetwo words mean something specific, and that the something is checkable.