This paper introduces NECR-SPS, an influence-aware shortest path algorithm that unifies exact routing, multi-path influence propagation, attention scoring, and embedding construction in a single mathematically rigorous framework. The method is based on a novel normalized shortest-path restricted resolvent operator derived from a fixed-point formulation of damped path integrals. We establish existence, uniqueness, convergence, and truncation error bounds, and prove exact equivalence between the proposed algorithm and the underlying operator. The framework provides interpretable influence measures and stable embeddings while preserving near-Dijkstra computational complexity.