The Shannon-Lebesgue Barrier A Measure-Theoretic and Information-Theoretic Resolution to P vs NP via the Banach-Tarski Obstruction

We present a rigorous cross-domain impossibility proof for P = NP, establish ing that polynomial-time computation lacks the descriptive capability to separate complex geometric structures without invoking non-measurable sets. By formalizing a Turing Machine as a finite-capacity Shannon channel, we demonstrate that the backbone information rate of NP-complete problems (R = Ω(n)) strictly exceeds the channel capacity CM = O(poly(n)). By Shannon’s Sphere Packing Bound, this bandwidth deficiency topologically guarantees overlap between decision regions. Be cause Turing-computable functions are strictly Borel-measurable, they cannot per form the non-measurable decompositions (necessitating the Axiom of Choice, akin to the Banach-Tarski paradox) required to cleanly separate overlapping spheres. Any attempt to reconstruct these spheres via geometric flows (e.g., Ricci flow) induces unavoidable topological information damage. Consequently, ρ > 0 is an irreducible property of polynomial computation