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 v2, 30 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 This revision adds acknowledgements to the Rethlas development team (Haocheng Ju, Jiedong Jiang, Shurui Liu, Guoxiong Gao, Yuefeng Wang, Zeming Sun, Leheng Chen and Bin Wu; project leaders Liang Xiao and Bin Dong). See https://github.com/frenzymath/Rethlas and https://web.stanford.edu/~srliu/homepage/blog/rethlas-guide/ . Mathematical statements, proofs and formalization scope are unchanged. Rethlas assisted exploratory proof refinement and internal AI checking of the rectangular counterexample; later general classification and Lean formalization belong to the subsequent Codex-assisted work. AI checking is not independent peer review or kernel verification.