Computational Directional Asymmetry: A Distribu- tional Reformulation of P vs NP

We define computational directional asymmetry AM(x) = log2 TM(x)−log2 TV(x,M(x)), measuring the log-ratio of solving time to verification time for a total candidate solver M on instance x. We prove that a language LR defined by an NP relation is in P if and only if there exists a polynomial-time solver M with sup|x|=nAM(x) = O(log n), establishing a precise equivalence between the polynomial-time boundary and a logarithmic bound on directional asymmetry. We introduce the asymmetry spectrum ΣNP, the collection of asymptotic asymmetry exponents across all NP relations, and show that P= NP corresponds to its complete degeneracy: ΣNP = {0}. Within this framework, existing complexity concepts — worst-case, average-case, parameterized, and cryptographic hardness — appear as different statistics of a single distributional object, and Impagliazzo’s Five Worlds become continuous tail regimes. We analyze the local mechanism generating asymmetry through a search-space structure distinct from solution-space topology, and show that global asymmetry accumulates as compound local information deficits. We formulate two conjectures — a structural-to- computational bridge and a meaning-dimension inexhaustibility claim — whose conjunction implies P = NP, and analyze the framework against the relativization, natural proofs, and algebrization barriers. The definitions and main theorem have been verified in Lean 4 with Mathlib.