Condorcet Domains in Spatial Models

Majority coherence is obtained classically in two ways. The spatialroute, associated with Black's median-voter theorem, restrictspreferences through low-dimensional geometry. The combinatorial routerestricts the admissible linear orders directly, giving a Condorcetdomain on which majority is always acyclic. The two are often discussedtogether, but their quantitative relation has not been explicit: howlarge a Condorcet domain can one fixed spatial model contain? Let P_d(n) denote the largest Condorcet domain contained in ad-dimensional weighted-affine ranking model, and likewise for theordinary Euclidean ideal-point model, on n alternatives. The centralresult is that for every fixed dimension d both quantities grow asTheta_d(n^{2d}), whereas unrestricted maximum Condorcet domains growexponentially. Fixed-dimensional spatial coherence and combinatorialmajority coherence are therefore asymptotically distinct restrictions.In the plane the paper pins down exact values: 4, 9, and 19 alternativesat n = 3, 4, 5, where 19 falls strictly below the combinatorial maximumof 20, and certifies at least 38 at n = 6. The small-case upper boundsrest on a balancing-space obstruction and an exhaustive classification ofmaximum five-alternative domains. The paper also supplies an exact prefix-sum, root-cone, and Galecriterion for spatial realizability, and uses pair-comparison shatteringto define and bound the spatial dimension of a preference domain: adomain of size 2^(n-1) has exact spatial dimension n-1. Families ofexponential size force dimension Omega(n / log n), and the paperconjectures that linear dimension is necessary. All computer-assistedclaims use deterministic exact-arithmetic certificates and reproduciblescripts.