This paper formulates Thermo-Optical Guidance (TOG) as a stochastic field theory whose fundamental object is a non-normal propagator rather than an unstable eigenvalue spectrum. The linearized operator takes the form L(k)=Dk2I+AJL(k)=Dk2I+AJ, where JJ is nilpotent (J2=0J2=0), producing transient amplification ∥etL∥∼te−Dk2t∥etL∥∼te−Dk2t while both eigenvalues remain strictly negative. Key results include: Exact quadratic propagator: The Martin–Siggia–Rose action yields Gμeff(k,ω)=(−iω+Dμk2−A2/(−iω+Dκk2))−1Gμeff(k,ω)=(−iω+Dμk2−A2/(−iω+Dκk2))−1. Exact εε-pseudospectrum: The pseudospectral boundary is computed analytically; it extends into the right half-plane for any A≠0A=0, confirming amplification without threshold. Exact semigroup norm: ∥etL∥∥etL∥ is obtained in closed form, with optimal wavenumber migration kopt∼T−1/2kopt∼T−1/2. Two-coupling RG flow: Coupled beta functions βg=Cg2(1+c1a)βg=Cg2(1+c1a), βa=−2a+C3ag2βa=−2a+C3ag2 show that the non-normal coupling aa is dangerously irrelevant, and the k−1k−1 regime is proved to be a finite-scale crossover, not an asymptotic fixed point. Exact response spectrum: S(k)=(2Dk2)−1∣1−A2/(D2k4)∣−1S(k)=(2Dk2)−1∣1−A2/(D2k4)∣−1 exhibits a cusp at kx=(A2/D2)1/4kx=(A2/D2)1/4 with discontinuous logarithmic derivative. Operator geometry: The nilpotent structure J2=0J2=0 is preserved under RG; transient growth is identified as curvature in the operator manifold. The theory makes falsifiable predictions for transient amplification, optimal wavenumber migration, and spectral cusp structure, testable in active nematics, superfluid films, bacterial suspensions, and thermo-optical cavities.
Petar Dryanovski (Sat,) studied this question.