FRCMFD Equations — Current Version-Locked Form (May 21, 2026)
FRCMFD Equations — Current Version-Locked Form (May 21, 2026) Below are the complete, version-locked equations as they stand after Gate 1 verification. All components are now mathematically consistent and numerically validated. 1. Core Field Equation (v2 Spectral-Operator Formulation) The fundamental field equation for the substrate excitation field Ψ ( x , t ) Ψ(x,t): ∂ t 2 Ψ − v 2 ∇ P s i + μ Ψ + λ ∣ Ψ ∣ 2 Ψ = κ S ^ Ψ ∂ t 2 Ψ−v 2 ∇ P si+μΨ+λ∣Ψ∣ 2 Ψ=κ S ^ Ψ where: Symbol Meaning Value (Current) Ψ Ψ Complex scalar substrate excitation field Variable v v Propagation speed (assumed c c) 1.0 1.0 μ μ Restoring coefficient − 1.0 −1.0 λ λ Nonlinear saturation coefficient 1.0 1.0 κ κ Source coupling coefficient 1.0 1.0 S ^ S ^ Spectral operator − i ∂ ϕ −i∂ ϕ 2. Spectral Operator (Angular Momentum) For axisymmetric analysis with winding number m m: S ^ = − i ∂ ∂ ϕ S ^ =−i ∂ϕ ∂ Under the separable ansatz Ψ = Φ ( r , z ) e i ( m ϕ − ω t ) Ψ=Φ(r,z)e ...
