Constrained Form: A Rank Theory of Architectural Selection — Theoretical Core

This deposit contains the compact theoretical presentation of a rank theory of architectural form. The central claim is that architectural form can emerge as the global state of a locally regulated constructive process. The process is modelled through five operative components — a generative rule, a metric apparatus, a kinematics of the pace, a dynamics of emplacement, and an observable that measures the built with the same instruments that built it. A form-state is the joint configuration the process closes: a planimetric relation, a dimensional class, and a volumetric mode. The rank of the process measures its residual formal freedom and bounds the admissible space of layout, dimension and volumetric type before any form is generated. This selection is structural: a third explanatory category, irreducible both to design intent and to material constraint. Held at rank one (one slope rule, additive measurement, no global coordinates), the Egyptian apparatus of cord, cubit and seked selects the right square pyramid as its closed volumetric class and admits exactly three irreducible form-states. The three pyramids of Giza realise all three, with no dimension adjusted to fit. Across the wider pyramid corpus the same rule leaves realisations that are coherent but partial, exactly as the theorem requires. Saturation — one complex displaying every admissible form under one constant generator — is therefore predicted to be rare; Giza is its first attested case rather than its only possible one. This file is the theoretical core. The extended edition with fuller apparatus is available at: https://doi.org/10.17605/OSF.IO/D538Y The earlier preprint “Constrained Form: A Rank-One Constructive Regime at Giza” is superseded.