Derivability Classes of Optimal Reduction: The Derivability Spectrum from Proof to Measurement

A sorting algorithm has a provable minimum. No comparison-based sort can order N elements in fewer than ⌈log₂(N!)⌉ comparisons. This is not an empirical finding. Nobody timed a thousand sorting algorithms and reported the fastest. It is a mathematical theorem derived from the task's information-theoretic structure: N! possible orderings exist, each comparison eliminates at most half, and therefore log₂(N!) comparisons are necessary regardless of how clever the algorithm is. The floor is known before anyone sorts anything. An emergency physician diagnosing chest pain has no such theorem. The best physicians reach a correct diagnosis in five to eight operations — a focused history question, a targeted examination maneuver, a key lab result, a pattern recognition, a confirmation. The worst take forty to sixty operations, methodically checking every possibility. The best observed performance is the only estimate of the floor. Maybe four operations suffice. Maybe three. Maybe the current best is already optimal. Nobody can prove it either way because no structural argument connects the task's properties to a minimum operation count. These two cases represent opposite ends of a spectrum. At one end, the minimum number of operations for a task is derivable from the task's structure — you can prove the floor before anyone performs the task. At the other end, the minimum is known only from observation — you measure the best performers and call that the estimate. Between these ends lies territory where partial structural arguments constrain the minimum without determining it exactly. This paper classifies tasks by where on this spectrum their minimum operation count lives, identifies the structural properties that determine placement, and examines whether the classification has a hierarchy analogous to the complexity classes of theoretical computer science. The vocabulary is small and builds in order. Processing is what any system does when it must act on information — a CPU executing instructions, a physician diagnosing, a pilot navigating, a developer debugging. The unit of processing is the **op**: one irreducible transformation by one processor. Processing entropy is the op count a specific processor requires for a specific task. Through repetition under consistent conditions, processing entropy decreases toward zero as the processing chain dissolves into structure that produces correct results without consuming the processor's scarce sequential pipeline. Before dissolution, there is a floor: the **optimal reduction** R*, the minimum number of correct ops any competent processor requires for reliable execution of a given task. Below R*, the processor is not dissolved — it is operating without sufficient verification. Above R*, there is measurable inefficiency that practice can eliminate. R* is the boundary between competence and waste, and knowing its value is fundamental to training design, performance assessment, and efficiency engineering. Prior work in this series defined R*, used it to measure dissolution progress, and noted that it is provable in some domains and only empirical in others. This paper asks the structural question: what determines which?