Second–Order Closure and the Structural Necessity of the Gaussian Kernel

Second--Order Closure and the Structural Necessity of the Gaussian Kernel, Version 2, presents a structural derivation of the Gaussian kernel from positive second-order quadratic data. The paper starts from a finite-dimensional real quadratic structure and asks which positive scalar kernels can be constructed from the quadratic datum alone while remaining invariant under linear re-description, compatible with orthogonal direct sums, locally bounded, stabilizing, and canonically normalizable. The main result is that these requirements force the Gaussian form. Re-description invariance forces radial dependence on the quadratic value. Orthogonal direct-sum compatibility forces the exponential form. Scalar stabilization selects the negative exponential sign in the positive quadratic class. Finally, the normalized Gaussian density is obtained by separating the intrinsic kernel form from the density induced by the quadratic structure, and by fixing the remaining scalar normalization through the canonical two-dimensional rotational scalar. A central point of the paper is the distinction between the Gaussian kernel and the normalized Gaussian density. The determinant factor appearing in arbitrary coordinates is not part of the primitive coordinate-free kernel; it is the coordinate expression of the volume density canonically induced by the positive quadratic form. This separation clarifies why the Gaussian expression is structurally forced before any coordinate representation is chosen. Version 2 strengthens the formulation by making the admissible positive quadratic class explicit, separating kernel form from density form, clarifying the role of the two-dimensional rotational scalar, and removing possible ambiguity between structural derivation and analytic normalization. The argument does not rely on probability, Fourier analysis, the heat equation, or the central limit theorem. Instead, it treats the Gaussian as the unique kernel compatible with positive second-order closure, re-description invariance, direct-sum multiplicativity, stabilization, and canonical normalization. The paper is intended as a structural and axiomatic account of why the Gaussian kernel arises from positive quadratic data, rather than as a new analytic proof of standard Gaussian integral formulas.