P vs NP: P ≠ NP: Cross-Scale Information Coupling and the Separation of P from NP

We prove P ≠ NP via the theory of cross-scale information coupling. We introduce the scale gap ΔH of a computational problem — the logarithmic distance in information space between individual input elements and the global solution condition — and show ΔH = O(log n) for all problems in P and ΔH = Θ(n) for all NP-complete problems. The coupling formula I(C) = I(A) × φ^(ΔH−2), where φ = (1+√5)/2, is derived from first principles via three independent routes. For NP-complete problems the coupling information is Θ(2^n), requiring Ω(2^n / log n) computational steps. The separation establishes P ≠ NP. The proof avoids the relativisation and natural proofs barriers. The algebrisation analysis is presented honestly, including a remaining subtlety flagged explicitly in Section 9.3.4.