Condorcet Domains in Spatial Models
Year: 2026
Type: article
Abstract: A Condorcet domain is a family of rankings whose every finite voting profile has an acyclic strict majority relation. This project studies the largest such family contained in one fixed d-dimensional weighted-affine or ordinary Euclidean spatial model on n alternatives. Principal analytical results Metric-projection anchors supported on at most 2d alternatives yield a shifted restriction-sampling inequality. It gives positive fixed-dimensional leading constants κd with Pd(n) asymptotic to κd times the binomial coefficient n choose 2d, and the uniform sparse-dimensional law log Pd(n) = 2d log(n/d) + O(d). Spatial products and coordinate recycling show that weighted and ordinary models share a concave spatial entropy and a common exponential rate Λ of their normalized leading coefficients. The entropy has the expansion hsp(δ) = 2δ log(1/δ) + γδ + o(δ). Exact iterated-limit formulas establish γ ≥ log Λ. Full-star stabilization and simultaneous dilute limits Let Td(n) be the largest number of full d-dimensional coordinate cubes in a spatial Condorcet domain. Exact balanced blow-up proves uniform stabilization of its normalized count in the weighted model, with multiplicative error at most exp(4d²/n) for n ≥ 4d. Products and coordinate recycling give a common capacity Λtop and convergence along every integer sequence with d → infinity and n/d → infinity, in both models. Exact variational formulas express Λtop as a supremum of finite normalized cube counts. At fixed integer ratio M ≥ 2, the residual ray entropy lies between log Λtop − E(M) and log Λtop, where E(M) = 2 + (M−2) log(1−2/M) for M > 2, E(2)=2, and E(M)=2/M+O(M⁻²). A shared-direction construction gives 0.598153547253148… ≤ Λtop ≤ Λ. The unrestricted cardinality equality γ = log Λ remains open. A subexponential comparison of Pd(Md) with Td(Md), uniform as M tends to infinity, would prove the stronger identity γ = log Λ = log Λtop. Planar classifications The common weighted and ordinary planar optima on three through six alternatives are 4, 9, 19, and 38. On five alternatives, an algebraic classification of six balancing families, followed by small exact proof graphs, gives the sharp upper bound and precisely two relabeling classes of extremizers, exchanged by reversal. Both are ordinarily realizable. On six alternatives, a portable proof graph for two local obstructions leaves 72 classes; six explicit prefix-cone covers of a single pair-sum hyperplane exclude 68, and exact ordinary witnesses realize all four survivors. Each cover has a short analytical proof, valid also on its boundary. The affine classification therefore needs no resonance-chamber enumeration or connectivity calculation. The finite local class exhaustion and recorded pattern containments remain exact certificate dependencies. The independent chamber criterion and its wall-closure certificate are retained. These global classifications remain computer-assisted. The stronger full-balancing classification is retained with its separate certificate. Further structural proofs A four-order root-cone argument, a seven-order saturation core, an ideal-lattice and corner-deletion proof, a six-order non-Lorentzian obstruction, and small interleaving tables replace several larger computations. Stars and bars gives the 224- and 492-order construction counts. The 492-order value reproduces a published lower bound; optimality and novelty as a record are not claimed. An exact deletion identity and a sum-of-squares clone-allocation argument clarify obstacles to unrestricted dilute stabilization. A comparison-entropy argument shows why an unlocalized deletion deficit does not provide the missing spatial estimate. An explicit ordinary Euclidean grid family shows that shattered matching sets can overcount full cubes by an exponential factor even in simultaneous dilute limits, while its own cardinality and full-cube counts are exponentially equivalent. This rules out a general shattering shortcut and is not a counterexample to the conjecture about global extrema. Scope and remaining questions The concise analytical manuscript does not use finite planar classifications as premises. The comprehensive companion distinguishes analytical results, exact finite certificates, and conditional implications. The unrestricted identity γ = log Λ, equality of Λtop and Λ, exact values of the capacities, the general entropy curve, and possible differences between weighted and ordinary finite optima remain unresolved. Fully computation-free global planar classifications also remain open. Contents The deposit contains the analytical manuscript, a comprehensive companion, editable LaTeX sources, exact certificate programs and data, independent audit reports, corrections and open issues, and reproducibility instructions.
Author: Fathi, Kevin
Language: English
Subfield: Economics and Econometrics
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7170078103
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