Isotope-Engineered Remote Entanglement in Diamond NV Centers via Non-Equilibrium Unified Theory: Analytic Expression for the Complexity Parameter D(h) and Scalable Qubit Design

We derive an exact closed-form expression for the complexity parameter D(h) that en- forces F = 0 (spatial symmetry maintenance equilibrium) in the Non-Equilibrium Unified Equation. Calibrated at the carbon bond length h0 = 1.42 Å with D = 1 for 12C-12C, the analytic form Dopt(h) ≈ 3(h/h0) 2 − 2 holds with < 0.01% error for h ≳ 10 nm. This enables deterministic design of remote entanglement distances in 12C-enriched diamond NV-center arrays. For experimentally realized spacings (10 nm arrays, 40–100 nm scalable registers, up to 3 m heralded entanglement), we predict D values from 1.5 × 104 to 1021, yielding 4–30% enhancement in coherence time T2 via isotope tuning (13C mixing ∆D = +0.7–+4%). The framework unifies wave-particle phase transition with F = 0 remote binding, providing a fully analytic, isotope-engineered blueprint for scalable quantum processors.