A.3.3.PI:4.2 - Compare situations merged by the description

Construct two admitted states or histories with the same retained description. Keep the rule, input conditions and future question fixed. Derive their relevant continuations. A difference in the requested result identifies information lost by that description.

For a deterministic discrete model, let x be its state, u its input, F(x,u) the next state and z = r(x) the retained description. An update z’ = G(z,u) is well-defined when r(F(x,u)) has the same value for every admitted x having the same z, for each input covered by the claim. This condition says how to construct G: choose any compatible x, apply F and retain r of the result. All compatible choices must give the same answer. C.29.1 supplies the general comparison of operations under a representation change.

A prediction of one selected output at a fixed horizon may need less information than an update of the entire retained state. Compare the output that the question asks for. If the result will be used repeatedly, also determine how to update the information from which the next prediction is made.

For a probabilistic model, compare the distributions of the relevant future result under the merged states or histories. The current readout can conceal different next-event probabilities even when the full model is Markov, as :5.2 shows.

One divergent pair refutes sufficiency for a claim covering both cases. Matching a few pairs supports only those comparisons. A general sufficiency claim needs an argument or subject Method covering its stated class; a local counterexample or conditional forecast may already be enough for the present work.