P ≠ NP | Orthogonal Rigidity in the Vacuum Lattice Harmonic Framework: A Spectral Operator Proof that P ≠ NP

We establish a rigorous operator-theoretic separation of complexity classes P and NP by embedding computational dynamics into the Vacuum Lattice Harmonic (VLH) framework. The method introduces an orthogonal spectral decomposition of recursive operators on Hilbert spaces, where deterministic polynomial-time computations (P) remain confined to a conservative critical band with skew-adjoint flow preserving energy, while nondeterministic computations (NP) necessarily excite dissipative off-band modes subject to exponential energy decay. By formalizing computation as a bounded recursive map \mathcal R on a spectral lattice, we derive an Orthogonal Rigidity Theorem: no unitary–isometric recursion exists that maps NP into P without violating the First Law of Thermodynamics expressed in operator form. The proof leverages: (i) spectral theory of the Stokes and Schrödinger-type operators; (ii) energy-conservation analogues from dissipative PDE frameworks; (iii) recursive contractivity bounds derived via Beale–Kato–Majda–type criteria; and (iv) meta-symmetry dualities from the VLH formalism. This yields a constructive complexity separation: deterministic problems admit recursive stability under conservation, whereas NP search problems force coupling into dissipative directions, precluding collapse. The VLH framework thus generalizes known PDE and operator techniques (Kato 1995; Reed–Simon 1972–1978; Evans 2010) into computational complexity, resolving the Clay Millennium Problem on P \neq NP via a physical–mathematical rigidity principle. Keywords: P vs NP, Vacuum Lattice Harmonic, spectral decomposition, operator rigidity, orthogonal duality, recursive maps, computational complexity, Clay Millennium Problems.