<![CDATA[This paper proposes a computationally verifiable integrate fixed point framework on the integrated metric space , where combines a continuous component endowed with the matrix induced metric with invertible and a discrete component defined by the shortest-path metric of a finite weighted graph. The objective is to obtain verifiable conditions that guarantee existence, uniqueness, and predictable convergence of fixed points for coupled continuous–discrete dynamics, while embedding the graph geometry directly into the metric via the scaling parameter . Our method studies the coupled operator and derives explicit sufficient inequalities ensuring that satisfies a Chatterjea-type contraction on , yielding an effective contraction factor . In particular, the threshold implies that admits a unique fixed point and that the hybrid Picard iteration converges geometrically in . Numerical experiments support these findings and clarify the integrate mechanism, when maps every vertex to a fixed node, the discrete mode stabilizes after the first iterate, and the successive iterate error decays exponentially at a rate consistent with , with numerical and analytic fixed points agreeing up to floating-point tolerance. Practically, the bound provides an a priori, computable convergence for implementations of matrix graph iterations relevant to graph structured computing and networked models. Future work includes reducing conservatism in the sufficient bounds, exploring richer couplings, and extending the analysis to broader graph classes.]]>
