A Kazhdan–Lusztig interaction formula for nested (non-split) matroids

This record contains the preprint “A Kazhdan-Lusztig interaction formula for nested (non-split) matroids”, together with its LaTeX source (main.tex, refs.bib) and an ancillary Python verification engine (kl_engine.py). The Kazhdan-Lusztig (KL) polynomial and inverse KL polynomial of a matroid are valuative invariants, and closed formulas are known when the nontrivial cyclic flats form an antichain (uniform, paving, sparse paving, elementary split, and corank-two matroids). In all of these the naive inclusion-exclusion over the individual stressed subsets already reproduces the invariant. This paper treats the complementary chain (nested, or Schubert) case, where it need not, and measures the discrepancy by the interaction I := P_M - PM1 - PM2 + P_U relative to the single-stress companions. Main results (chains of arbitrary length, arbitrary corank). For a nested matroid whose nontrivial cyclic flats form a chain of any length k, we prove that the 2^k-fold inclusion-exclusion interaction of the constant inverse-KL coefficients is given by an explicit alternating multiple-binomial formula, valid at every rank; and we give a flat-count recursion, valid for all k and all nullity orderings, for the linear KL interaction. Two-flat case (in full). We prove closed forms for the linear KL coefficient and the constant inverse-KL coefficient, a general degree law expressing the interaction at any corank as an explicit binomial combination of hyperplane values, and a single-binomial closed form (for nullity rho_1 >= 2) for the remainder in the next inverse-KL coefficient. Relation to prior work. The nearest existing formulas, for corank-two matroids via the inverse-KL deletion formula of Braden, Ferroni, Matherne and Nepal (arXiv:2510.01086), lie on the antichain side, where the interaction vanishes identically. The matroids studied here are on the chain side and of unbounded corank, so they are not covered by those formulas; the new content is at corank three and above. We also locate the results relative to the valuative framework of Ferroni and Schroeter, and determine the precise threshold at which the single-binomial structure of the low inverse-KL coefficients breaks down. Reproducibility. All computational claims are reproducible from the ancillary file kl_engine.py (Python standard library only), which combines a block-symmetric KL engine with an independent brute-force cross-check computed directly from the lattice of flats; running the file executes a self-test. Versioning. Version 2 supersedes version 1 (2026-07-04). The concept DOI 10.5281/zenodo.21198033 always resolves to the latest version. AI use disclosure. Generative AI tools (Claude, Anthropic) were used to assist with reference verification, exact symbolic computation and its verification, and proofreading. The author designed the research, independently verified all proofs, computations, and literature attributions, and accepts full accountability for the final content.