An Explicit Cox-Graded Complex with Chamber-Dependent Homology on Six Quotient Stacks

This preprint constructs an explicit four-term Cox-graded complex over an eight-variable Cox ring equipped with a rank-three torus grading and studies its restrictions to six quotient stacks arising from prescribed monomial semistable opens. The same complex has different homological behavior across the six chambers. In two chambers it reduces to an ordinary ideal resolution, while in the other four it carries a codimension-two conormal contribution in negative degree. Despite this chamber dependence, the reflexive hull of the degree-zero homology is the same line bundle in every chamber. The accompanying computation archive verifies the matrix identities, Cox degrees, monomial saturations, colon ideals, chamber support pattern, and relevant coordinate intersections through exact finite calculations using only the Python standard library. 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.