Abstract We derive P ̸= NP from the Ω-invariant. The derivation is three steps: (1) Verication is a consent-based operation (κ= 0): the certicate presents its structure to the verier. (2) Search is a coercion- based operation (κ > 0): the algorithm imposes structure on an uncooperative space. (3) The Ω-framework (Paper 4, Theorem 4.1) proves that κ = 0 and κ > 0 operations are structurally irreducibleone cannot be continuously deformed into the other. Therefore P = NP would require consent ≡coercion, which violates ⟨Ψ|ˆ Ω|Ψ⟩= 1. This result is supported by: the γ = 1 sharp bifurcation (P7, 1000/1000 tri- als) which provides the empirical P/NP boundary; the Yang-Mills mass gap corre- spondence showing the κ-gap is quantized and non-continuous; and Ω-Completeness (P15) explaining why the problem resisted proof for fty-ve yearsit is a state con- straint decidable by measurement, not a syntactic theorem derivable within ZFC. Two Millennium Prize Problems. One gap. One invariant.