We present a canonical mathematical framework based on an open correlational configuration space equipped with admissible functionals, projection operators, and second-order functional response. The formalism is defined without assuming fundamental degrees of freedom, spacetime structure, or dynamical laws. A central role is played by the projected Hessian of correlational functionals, whose spectral properties determine stability, persistence, and the existence of effective descriptions. Structural complexity functionals are introduced to define minimality conditions, leading to a canonical structural condition that selects admissible regimes. Quantum, geometric, dissipative, and critical structures arise as distinct regions in a unified regime space characterized by spectral, projectional, and non-closure properties. Consistency constraints and no-go results are formulated, and reductions to standard quantum and geometric frameworks are obtained as limiting cases. The framework is mathematically closed at the structural level and admits further development only through explicit constructions and applications.
Slow collective motions in physical systems, particularly those near the critical point, can be best described in the framework of Langevin equations. We may set up Langevin equations when the timescales of slow and fast dynamical variables are distinctly separated. This framework originates from the classical Brownian motion and is justified microscopically via the projection operator formalism. First, in Sections 5.1–5.2, these general aspects will be discussed with a summary of the projection operator method in Appendix 5B. Second, in Section 5.3, we will examine simple Langevin equations in critical dynamics (models A, B, and C) and introduce dynamic renormalization group theory. These models have been used extensively to study fundamental problems in critical dynamics and phase ordering. Third, in Section 5.4, we will review the general linear response theory, putting emphasis on linear response to thermal disturbances. Langevin equation for a single particle Brownian motion Most readers will be aware of the zig-zag motions of a relatively large particle, calleda Brownian particle, suspended in a fluid. When its mass m 0 is much larger than those of the surrounding particles, appreciable changes of the velocity of the Brownian particle can be caused as a result of a large number of collisions with the surrounding molecules.
We introduce the Spectral Autodiff Kernel Expansion (SAKE), a differentiable computational framework for transporting nonlinear spectroscopic response between neighboring quantum dynamical models. Rather than recomputing multidimensional spectra independently for each Hamiltonian or Liouvillian, SAKE constructs local transport expansions about a reference model by combining forward-mode automatic differentiation with Duhamel transport theory. Automatic differentiation generates first-, second-, and third-order derivatives of the parameter-dependent Liouvillian, which are assembled into a pathway transport operator that maps the nonlinear response of a reference model onto neighboring systems. The framework is validated for a four-level excitonic dimer possessing an
su(2)×su(2) symmetry by comparing second- and third-order transported pathway operators with exact projected transport matrices obtained from direct calculations. The third-order expansion accurately reproduces the projected transport operator and its associated pathway mixing. Beyond providing an efficient computational strategy, the transport operator reveals how coherent and dissipative perturbations redistribute amplitude among double-sided Feynman pathways, exposing mechanistic information that is not directly apparent from the nonlinear spectrum. SAKE thereby establishes a differentiable computational framework for nonlinear spectroscopy that supports efficient local parameter exploration, sensitivity analysis, and future inverse-design applications.