Sharp Fixed-Frequency Phase Complexity in Two-Dimensional Linear Systems with Two Independent Delays: A Rank-Controlled 2-4-8 Hierarchy

Abstract

We study real two-dimensional linear systems with two independent discrete delays. At a prescribed frequency, the characteristic equation becomes a bidegree-two phase equation on the two-dimensional torus. We treat this equation as a fiber problem: regular first-phase fibers contain at most one unit-modulus second phase, whereas compatible singular fibers may contain two. We prove that every compatible singular first phase is at least a double zero of the cleared phase polynomial. This multiplicity compensates for the possible fiber doubling and yields a sharp bound for finite phase sets.