The Categorical Mixing Problem: A Foundational Diagnosis of P vs NP (The Categorical Mixing Problem: A Research Programme, Paper II of III)

This paper applies the Categorical Mixing framework, developed in Paper I of this research programme (DOI: 10.5281/zenodo.18945326), to the P vs NP problem. It argues that the problem's fifty-year resistance to resolution is not a consequence of its depth but of a specific and identifiable structural defect in its formulation: the implicit treatment of two distinct axiomatic systems as one. Two unwritten axioms are identified. The first is the unproven assumption that solving and verifying are commensurable operations of the same fundamental type. The second is the conflation of the mathematical register — in which truths are timeless and scale-independent — with the computational register, in which process, sequence, and directionality are native concepts. The paper further demonstrates that the three major barrier results in complexity theory — relativization, natural proofs, and algebrization — are not evidence of the problem's irreducible difficulty but predictable symptoms of categorical mixing, marking precisely the points at which standard proof techniques collide with unacknowledged axiom boundaries. The paper concludes with a proposal for what a correctly reformulated version of the question would require, and argues that such a reformulation is the necessary precondition for any resolution of the underlying question about the relationship between solving and verification. This is the second paper in a three-part research programme. Paper III develops the formal logical treatment of axiomatic collision across mathematical foundations.