Topological Mapping of NP-Complete Complexity Classes onto Fractal Manifolds: A Framework for Direct P vs NP Resolution

We introduce a geometric program that transports the combinatorial structure of Boolean satisfiability into the language of dynamical systems, fractal geometry, spectral theory, and topological invariants. To each conjunctive normal form formula on n variables we associate three objects: an arithmetization polynomial on Complex space, a contractive iterated function system whose attractor is a Cantor-type fractal encoding the satisfying assignments, and a family of Hermitian operators on a Hilbert space of dimension 2^n, parametrized by a torus, whose spectral statistics and Chern-type invariants encode satisfiability. We prove a sequence of unconditional results: the exactness of the arithmetization identity, a Moran-type formula for the Hausdorff dimension of the attractor, the identification of the Julia set of the Boolean Newton map, and the integrality of the Chern number of a gapped Hermitian family. We then isolate exactly which additional statements would be needed for the program to yield a separation or collapse of P and NP, and prove a no-go proposition showing that any poly-time computable, faithful topological invariant of satisfiability would itself imply P=NP.