Quantitative realizer stability and copula estimation from unlabelled two-dimensional orders

An unlabelled poset of dimension at most two need not determine the permutation that realizes it. This preprint bounds the lower-left empirical-copula discrepancy between arbitrary realizers by the largest smaller side of a proper strong module, proves an explicit strong-module tail for samples from copulas with density bounded below, and combines these results with classical algorithms and concentration inequalities to obtain a polynomial-time estimator modulo coordinate transpose. It also proves that absence of a positive-mass interval block suffices for universal arbitrary-realizer consistency and identifiability, while explicit diagonal, ordinal-sum, and blockwise-transpose examples show that the natural converses fail. The work does not claim an optimal deterministic constant, an optimal or minimax stochastic rate, a complete classification of equal-order-law permutons, physical isometry, or established priority. AI-assistance disclosure: the author used OpenAI ChatGPT and Codex and Anthropic Claude for research planning, literature searches, conjecture and counterexample exploration, drafting and LaTeX preparation, formalization planning, and adversarial mathematical and bibliographic review. Their outputs were checked and revised by the author, who takes full responsibility for the work; the systems are not authors and their output is not treated as independent evidence.