A Sharp Three-Cycle Bound for Two-Zone Affine Systems with a Virtual Focus

We prove that a planar two-zone piecewise-affine system with a straight switching line, two strict off-line foci, and at least one virtual focus has at most three isolated crossing limit cycles. The estimate includes nonhyperbolic cycles and every switching offset. Known three-cycle examples show sharpness.

The proof uses the Poincare half-map and extended Khovanskii framework of Carmona, Fernandez-Sanchez and Novaes. The additional argument rationally parametrizes contacts of their conic, rules out an exceptional denominator fiber, localizes the contacts by normalized equilibrium parameters, and establishes a trace-sign monotonicity certificate. Choosing between the full opposite-sign half-plane and an energy half-plane gives the intersection bound, with an explicit allowance for two components after clipping. Parameter perturbation treats degenerate conics and multiple isolated cycles.

This is a complete theorem for the stated focus family associated with AMR-046-0017 in UnsolvedMath v1.6.0, not a solution of the combined Gasull Problem 17 table or the general non-focus problem. The half-map methodology, endpoint-sum sign and three-cycle lower examples are credited prior work. The package includes an eight-page English manuscript and exact symbolic and rational regression checks. It is AI-assisted, originating-researcher self-audited and unrefereed. No independent review, proof-assistant verification, exhaustive novelty certification or absolute priority is claimed.