Counterexamples to Park's Coherent-Sheaf Conjecture and a Bounded-Edge MCM Extension

This paper disproves Park’s conjectural extension of asymptotic syzygy nonvanishing from line bundles to arbitrary coherent sheaves with full support. Explicit counterexamples are constructed on projective space using ideal sheaves of arithmetically Cohen–Macaulay subschemes. Linear subspaces yield rank-one torsion-free examples in every lower weight, together with exact multiplicity formulas and a single coefficient sheaf that violates the predicted behavior in all lower weights simultaneously. A second family is obtained from the Koszul resolution of a point. These examples show that the conjecture can fail for reflexive sheaves and for sheaves satisfying any prescribed Serre condition below the maximal Cohen–Macaulay condition. The failure is explained through higher deficiency sheaves, which contribute additional strands to the duality governing the right edge of the syzygy table. The paper also establishes a positive replacement. For full-support maximal Cohen–Macaulay sheaves on Cohen–Macaulay integral projective varieties, the nonvanishing groups in each weight form a single interval. The expected endpoint growth is recovered whenever the corresponding exponent is positive, while the two exponent-zero edges are shown to remain uniformly bounded. The accompanying computation package verifies the principal finite examples and the relevant Koszul-complex calculations. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.