A Meta‑Dynamical PDE on the Coefficient Space of D‑Operators (One Possible Realization of a "Theory about Theories'')

We construct a model partial differential equation (PDE) whose dependent variables are the coefficient functions of differential operators and whose solutions are continuous families of free kinetic operators in quantum field theory. The construction builds on the rigorous framework of the D‑operator space \ (D\) and the Global Time‑Shared Object (GTSO) hierarchy. The configuration space is an affine space of lower‑order coefficients of admissible operators on a fixed compact manifold, equipped with a chosen flat \ (L²\) metric. Geodesics of this metric recover the affine interpolation paths in the fixed‑principal‑symbol coefficient model, mirroring one part of the SOT connectivity constructions. To bias deformations toward selected admissible regions, we introduce a penalty functional that discourages violations of positivity, hyperbolicity, and gauge invariance. The Euler–Lagrange equation for the modified action is a semilinear wave equation on the infinite‑dimensional coefficient space. We provide a detailed derivation, explicit examples for scalar fields with varying mass, and a functional‑analytic discussion of well‑posedness. This PDE provides one concrete model of the landscape of free field theories and demonstrates how the recursive GTSO program can generate analysable meta‑equations.