A spectral approach to approximate symmetry: Defining "Structural Entropy" via the log-determinant of a near-commutant

I am looking for feedback, literature precedents, or potential spectral pitfalls regarding a framework that uses operator near-kernels to track the continuous breaking of global geometric symmetry under metric deformation.

​1. Setup & The Near-Kernel

​Let (M, g) be a compact Riemannian manifold and Delta_g its Laplace-Beltrami operator. For a given operator A (such as a Killing vector field generating an exact symmetry on an unperturbed metric g_0), define the derivation delta = [Delta_g, A].

​Under a smooth metric deformation g(lambda) = g_0 + lambda * h, the exact symmetry breaks, and A shifts into the near-kernel of the derivation, where delta_lambda(A) scales on the order of lambda.

​2. Entropy & Log-Determinant Representation

​To measure the volume of this approximate commutant space, we define a scalar "Structural Entropy" (S) using a regularized functional log-determinant of the positive-definite operator (delta^dagger * delta):

​S proportional to: -ln det(delta^dagger * delta + epsilon * I) = -Tr(ln(delta^dagger * delta + epsilon * I))

​Here, epsilon > 0 serves as a scale-dependent regularizer defining the boundary threshold of the near-kernel.

​3. Quadratic Decay

​Because the maximum-symmetry metric sits at a local maximum of this volume measure, the first-order variation vanishes globally over the unperturbed state. Applying standard perturbation theory, a generic symmetry-breaking deformation forces a quadratic decay:

​S(lambda) = S(0) - C * lambda^2 + O(lambda^3)