Exact Selection and Deletion of Balanced Pairs

A finite poset unfolds into many linear extensions: total orders faithful to its comparisons. Motivated by the 1/3–2/3 conjecture, this working-paper project asks how a balanced pair can be selected exactly, how its comparison law changes under deletion and reinsertion, and when a single pair can persist through an entire deletion sequence. The main manuscript, Exact Selection and Deletion of Balanced Pairs, develops structural selection and deletion theorems; the companion, Balanced Pairs: Exact Certificates and Algebraic Complements, supplies complementary proofs, finite certificates, and obstructions. The project keeps three questions distinct: existence of some balanced pair, exact identification of a specified comparison probability, and hereditary persistence of one chosen pair. Principal results include exact completion and joint-rank laws; a hereditary characterization of balance through obstruction sets of size at most one; fixed-pair deletion chains in width-two and related classes; extremal deletion correlations; antichain and moment-based selection criteria; endpoint inequalities and envelope-contained transport; and local Catalan kernels with exact certificates through min⁡(m,n)≤8min(m,n)≤8. The methods combine fibrewise counting, controlled exchanges, lattice-path switching, monotone coupling, rational duality, and finite-language reachability, with reproducible exact certificates. The project also records precise counterexamples and open boundaries: the unrestricted conjecture, universal charge coverage, general deletion lifting, arbitrary multiple coordinate bounds, full transport, and higher-width selection remain open.