Rank Enumeration and Side-Count Invariance for Noncrossing Hull Partitions

Let P be a finite set on the boundary of its convex polygon, with c_i points in the relative interior of side i. We give an explicit formula for the rank polynomial of the partitions of P whose block convex hulls are pairwise disjoint. The formula is a linear evaluation of a product of explicitly defined side polynomials. In particular, it depends only on the multiset of the c_i, not on their cyclic order. Among configurations for which every side has an internal point, equal rank polynomials are equivalent to equal multisets of side counts. The same decomposition gives a direct proof of Patel's rank-reciprocity identity, together with directional rank inequalities and an exact mean-rank formula. The proof groups inclusion-exclusion terms into boundary tilings and separates local forbidden successor arcs from the single cycle that can wind around the entire boundary. The motivating polygonal case of the non-rank-symmetry conjecture in Dougherty and Root, Noncrossing Partitions From Hull Configurations (arXiv:2604.14458v1, Introduction), and its exact rank-reciprocity defect were proved previously by Shivam Patel in a public MathDB solution posted on August 20, 2026. These results are credited here. The contribution of this article is the full arbitrary-side rank enumeration and side-count permutation invariance, rather than a first solution of that conjecture. This version presents complete mathematical proofs. These proofs have not yet been formalized in a proof assistant. Formal verification is planned; under the author's verification standard, this version is archived as a dated preprint, with active promotion of its results deferred until formal verification is complete. First publicly available on 9 September 2026 in the GitHub paper archive, as verified by anonymous access. This Zenodo deposit preserves the same English PDF without revision; it is an additional archive of that version, not a new result or a claim of exclusive priority.