Universality and recognition barriers for the(2,1)-gapped consecutive-ones property

We prove a representation theorem and a sharp barrier to structural recognition for the one hole consecutive-ones problem. Every upward-closed Boolean predicate with a polynomial time verifier can be compiled, with polynomial explicit size, into actual binary-matrix rows of total internal-zero budget one and strictly nested prefix interfaces. Conversely, every uni formly polynomial-size family of such modules defines an upward-closed NP predicate. The construction preserves witnesses and realizes arbitrary finite intersection complexes of zero defect domains. The principal consequence is a second-level completeness theorem stated entirely in terms of binary matrices. For every fixed r≥2, given a feasible base matrix and strictly nested ad ditional unit-budget rows, deciding whether every infeasible subfamily contains an infeasible subfamily of at most r rows is Π₂ᴾ-complete. Hardness persists when the base and every individual extension have explicit feasible orders, and all compatibility information through order r follows a fixed pattern. This goes beyond proving that pairwise consistency can fail: it characterizes the difficulty of certifying that bounded-order consistency is sufficient. The construction rests on an independently audited assignment-faithful NP-completeness re duction. Four rows force every slot of size q≥3, optimally within the stated guard-omitting class, reducing the scaffold from 5n+21m to 5n+10m rows without changing its 3n+8m+1 columns. An exact endpoint-charge identity compiles certified block orders into weighted finite-domain constraint systems for decision, counting, and minimum additions. A sepa rate open-row dynamic program works without a supplied scaffold. We also establish #P completeness of counting valid labeled-column orders under polynomial-time Turing reduc tions, correcting for unequal clause multiplicities by interpolation. The release includes complete proofs, executable compilers and exact recognizers, an inde pendent direct-zero checker, fresh global reduction tests, portable certificates for selected infeasible cases, a reproducible 24-instance benchmark, and two exact scientific examples. Prior Helly and endpoint methods are attributed to Patterson and to Chepoi, Creignou, Her mann and Salzer; standard CSP machinery is not claimed as new. Computational validation and external review are distinguished from mathematical proof.