Cohesive ideals without universal Tukey representatives

We give a negative answer to Question 3.17 of Tom Benhamou’s Scales in the point spectrum, in the formulation of arXiv:2603.00305v1. For nonempty linear orders K and L without greatest elements, let I be the ideal of noncofinal subsets of the coordinatewise product K × L. A directed order Q is I-cohesive exactly when K ≤_T Q or L ≤_T Q. This class has a universal Tukey representative if and only if K and L are Tukey comparable. Distinct infinite regular cardinals yield counterexamples; the case (ω, ω₁) suffices. The English preprint includes complete written proofs and states the exact scope of a companion Lean 4 formalization. The decisive (ω, ω₁) counterexample and the general classification and representability criterion have recorded kernel checks in the fixed public repository version linked below. General regular-cardinal observations are presented as written proofs. Formal verification does not establish historical priority or independent expert peer review. Files: five-page PDF and editable standalone LaTeX source ZIP with compilation instructions. Version v1, 28 September 2026. AI assistance disclosure: OpenAI Codex assisted mathematical exploration, drafting, checking, formalization and manuscript preparation. See the manuscript for the full disclosure and verification scope. Fixed formal companion: https://github.com/EgoFakeFantasy/OpenProblemFormalizations/tree/bfc7696ef2f053c964a02dd22497215cf718c685