P ≠ NP: A Dual-Certified Proof (Factorial Barrier Project, Volume IV)

This work presents a dual-certified structural proof of P ≠ NP. It introduces the family of Structurally Rigid Instances (SRₙ), which satisfy four verifiable properties (uniqueness, asymmetry, non-redundancy, partition sensitivity). Using these, the proof establishes exponential lower bounds in the clause-query model (Universal Minimal Bound) and unavoidable algebraic blow-ups in Gröbner basis computations. Together, these constitute orthogonal witnesses of hardness that bypass classical barriers (relativization, natural proofs, algebrization). The manuscript includes explicit generators, DIMACS benchmarks, and empirical validation protocols.