Insertion Fibers, Deletion Loss, and Cylinder Rigidity for Ulam Anticodes

How large can a family of permutations be if every pair becomes identical after deleting at most k labels? The Ulam anticode conjecture predicts the cylinder bound n!/(n − k)!, with cylinder equality when n ≥ k + 3. Results Three related working papers develop sharp insertion bounds, cylinder-shadow rigidity, exact deletion identities, and joint capacity methods. Shared-core three-row capacity has a self-contained all-order proof. Three row types inside one host have sharp capacity with 32 exact finite Hall inputs. A hypothetical noncylinder extremizer therefore has at most two full penultimate rows. Separated and boundary pairs have sharp capacity; an interior pair reduces to three explicit fibers. Maximal near-threshold families can nevertheless have a full-cylinder dimension gap of order √n. Methods and scope The arguments combine comparison graphs, vertex covers, Hall expansion, insertion matchings, explicit word comparisons, and exact finite verification where required. Diameter-three cardinality and equality hold for n ≥ 155 with the stated dependencies; fixed-diameter eventual classifications are treated separately from fixed-corank limits. Open questions The arbitrary-diameter conjecture remains open. Families with zero, one, or two full rows require further control. The remaining targets include exact-threshold extraction, the adjacent and interior one-gap joint capacities, and the weighted global deletion selector.