Exact Selection and Deletion of Balanced Pairs

How can a balanced comparison be selected in a uniform linear extension of a finite poset, and when does it survive deletion? This working paper develops exact mechanisms toward the 1/3–2/3 conjecture. Principal results Geometric deletion-count selectors and extremal correlation give quantitative balanced pairs and complete retaining deletion chains in several structural classes, including every width-two nonchain. A maximal antichain of size w contains a balanced pair when w ≥ 5k, where each member has at most k exterior incomparabilities. The proof retains its structural inputs and 65 exact arithmetic certificates. For a maximal element whose incomparable neighbours form a chain, a balanced partner exists exactly when 2e(P) ≥ 3e(P−z). This also gives sharp preservation of half-probability card pairs. An unconditioned ideal-cut selection criterion is proved for every width-two nonchain. Exact named-core and matching-fibre decompositions identify the exterior weights and currents needed beyond conditional selection. Methods and scope The methods combine order-polytope geometry, entropy, exact insertion counts, correlation inequalities, ideal conductances, swap currents, and rational certificates. Explicit uniform-poset examples and nonuniform-law obstructions delimit localized cuts, prescribed-card menus, and correlation or moment relaxations. Open questions The unrestricted 1/3–2/3 conjecture, universal deletion lifting, and a common comparison-charge construction of total coefficient at most three remain open. The project contains both manuscripts, editable sources, complete code and certificates, proof reviews, and the preceding material.