Comparison Stability for Permutation Deletion Shadows: Closing the 38-Order Condorcet Frontier

Abstract

This paper develops an information-theoretic and order-theoretic method for studying directional deletion shadows of Condorcet domains. For a family D of linear orders, let ∂ᵢD denote the family obtained by deleting alternative i. The directional contraction conjecture asks whether every Condorcet domain satisfies max_i |∂ᵢD| ≥ (3/7) |D|. Building on previous reductions, the paper resolves the first previously open frontier. It proves that every 38-order Condorcet domain on eight alternatives has a deletion shadow of size at least 17: max_i |∂ᵢD| ≥ 17 > (3/7)·38. The argument combines: a weighted stability refinement of Shearer's inequality, exact adjacent-transposition counts, deletion-fiber entropy, star-capacity inequalities, majority-poset structure, cubical isoperimetry, and Sen's value-restriction criterion. The initial seven possible deletion profiles collapse to the unique profile (16^8), and every deletion partition of a hypothetical Condorcet obstruction is forced to have fiber shape (3^6, 2^10). The remaining high-comparison graph is reduced to a matching of size two, three, or four. These three branches are then eliminated using exact rational order-packing duals, saturated-wedge constraints, deletion-incidence budgets, context enumeration, endpoint caps, and directional face-defect arguments. The accompanying reproducibility bundle contains the LaTeX source, exact Python verifiers, rational certificate data, independent finite audits, and an optional A₂-packet sharpness experiment. The terminal computations use exact integer or rational verification; floating-point optimization is used only to discover candidate dual certificates that are subsequently checked exactly. The result closes the first unresolved cardinality of the directional 3/7 program. It does not prove the universal conjecture for all cardinalities and ranks. The paper concludes by formulating a uniform multi-marginal A₂ certificate as a possible scalable route toward the general theorem.