What Stands Between and What Is Required: A theory of the forms, the requirement, and the shape of arrangements

Anything that stands between two things — a measurement, a category scheme, a summary, a filter, a copy sent elsewhere — carries some differences and merges the rest. Grain theory is a statement of what that costs, and of what follows from it. Its subject is a gap between two questions that are usually run together: what a thing in the middle lets through, and what needed to get through. The first is a matter of what can reach what; the second is not, and nothing here derives it from the first. What the theory states is what happens where the two come apart — where something that had to be told apart was not. The book states the theory as a relation with two clauses, builds only the apparatus those clauses force (partitions and the refinement order on them, and nothing stronger), and works out five ways of standing between: a system's discriminations, an aggregation over many sources, a set of distinctions fixed before the cases arrive, a filter that discards, and an artifact carried to a second site. It then treats two mechanisms lying beneath those, what holds of several positions at once, and what accumulates across populations of them. Two chapters account for large cases on those results: natural selection, read as three mediating positions in a chain, and the shape and direction of arrangements at the scale of institutions. Every proposition carries a mark saying what it rests on, and a premise list written with it. Of 274 propositions, 227 are derived — 198 with a derivation printed in the appendix, 31 immediate from a named result — alongside 34 asserted, 5 argued, 3 measured against a record, and one standing result of another field. The book also states what it does not say (112 non-claims), what would refute each result (146 falsifiers), and where it does not close (89 declared gaps, each carrying what it costs). Nothing in it establishes prevalence: it says what is unavailable where its conditions hold, not how often they hold.