Analytic Limits of the Teixido Envelope and Domination Root Monotonicity for Trees and Sparse Graphs

We investigate the location of roots for the domination polynomial D(G,x) within families of sparse graphs. While it is known that roots for general graphs are dense in the complex plane, we identify a strict geometric boundary governing the roots of trees, which we term the Teixido Envelope. Using the Beraha-Kahane-Weiss theorem on the limits of recursive polynomials, we derive the analytic path limit curve Lpath and formally define the Teixido Constant ( τ=−0.5 ) as the real-part accumulation limit for star graphs. Through extensive computational enumeration of 50,000 random trees and formal verification in the Rocq Prover, we establish the Monotonicity Principle: increasing the branching factor of a graph induces a geometric contraction of roots toward the stable interior of the envelope. These bounds provide a new metric for structural stability in networked systems. Comments / Notes (Crucial for IP): This work establishes the mathematical foundation for "Topological Analytical Homeostasis" (TAH). Machine-verified proofs for the Teixido Constant (τ ) and the domination stability bounds are included in the supplementary dataset (DOI: Topological Analytical Homeostasis: A Unified Geometric Paradigm for Antifragile AI, Zero-Multiplication Hardware, and Quantum Interconnects).