This monograph presents a complete, constructive resolution of Negami's Planar Cover Conjecture (1986) by shifting the classical combinatorial graph theory into a new paradigm: Dynamic Wave-Point Geometry. Instead of treating finite planar covers as static, discrete structural search problems, this work models the network as a dynamic multi-layered topological space governed by continuous metric flows. We introduce a novel geometric mechanism—the outward-directed tension flow—to untangle overlapping networks. Under this flow, the core of a non-planar crossing is stretched uniformly until its thickness collapses, triggering a discontinuous topological phase transition that tears a vertex into a circular boundary (a puncture) and releases the crossing stress. By applying a gauge-fixed "Pinned Discrete Laplacian System" to the newly generated boundary, the high-dimensional layer-switching connection matrices are uniquely and deterministically inverted. Furthermore, utilizing global wave-action conservation laws, we establish the strict finite boundedness of the required covering index (the number of building layers). This bridges discrete planarity criteria and the qualitative geometric invariants of the projective plane, successfully closing the 40-year-old conjecture.