Sharp Degree Bounds for Multipartite Lorentzian Coxeter Forms

Partition at least four vertices into at least two nonempty parts. Assign Coxeter exponent \(2\) within each part, a fixed exponent \(m\geq5\) between parts, and replace selected cross-part exponents by \(3\). Suppose each vertex has at most one replacement neighbor in any other part. If the replacement graph has maximum degree \(\Delta\) and \((\cos(\pi/m)-1/2)\Delta<1\), the normalized Coxeter form has exactly one negative eigenvalue and no zero eigenvalue, with an explicit lower bound for its positive eigenvalues.

The uniform degree caps are \(3\) for \(m=5\) and \(2\) for every finite \(m\geq6\); both are sharp. A paired-part construction gives the entire obstruction spectrum. The diagrams avoid \(A_3\) and \(B_3\), so the established rewriting theorem of Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees supplies a quadratic word algorithm for each fixed Artin presentation. Nonstar examples have unboundedly many exponent-3 pairs.

This is a complete spectral construction and an application of an existing algorithm, not a new general Artin word algorithm or a resolution of the small-type problem. The manuscript is AI-assisted, self-audited and unrefereed; no absolute priority claim is made.