Arrangement as Relation: A Relation-Calculus Counterpart to Shape Rewriting

This is a preprint of a manuscript under review at Automation in Construction (Elsevier). It is posted for citation and verification; it is not the version of record. It is the companion paper to "Interpretable Relation-Calculus Fitness for Controlled Spatial Layout Search" (submitted concurrently to the same journal). Contemporary formalization of visual design computation diverges into two lineages that share common roots yet follow orthogonal paths: the shape-grammar lineage (Stiny 1975; its contemporary engine Shape Machine, Hong & Economou 2023) computes on shapes via rewriting W'=W-f(u)+f(v), whereas Wang's arrangement lineage (Wang 1987) computes on the relations between elements. They have not, to our knowledge, been juxtaposed as two formalized calculi, implemented as runnable systems, and contrasted through coherent worked examples. We formalize Wang's twelve arrangement moves (5 generic + 7 ordering), make explicit their closed generative structures, and build a three-layer method (validators -> inference -> scoring) re-implementing Wang's Chapter-6 propositional calculus with modern data structures. We validate on Wang's own worked example, the Rosenbaum House, reproducing all 14 core relations of his 1987 step-by-step reconstruction (to our knowledge the first computational realization of his relation calculus as a runnable validator system), then transfer the method to a Taiwanese three-bedroom apartment, where the scoring function is provably order-invariant (monotone from +89.8 to the all-satisfied lower bound -33.0). We provide a prototype open-source library arrm_core (139 tests passing; to be released upon publication) and argue for a clean pipeline division of labor: generation (Shape Machine / Wasp) + interpretation (ArrM).