The Tetrahedral Kernel Field Theory: A Variational Formulation on a Tetrahedral Manifold with Corners

We formulate a variational field theory on a tetrahedral domain treated simultaneously as a compact domain with piecewise smooth boundary and as an oriented simplicial complex. The construction is motivated by a kernel-first viewpoint in which invariant structure is primary and geometric realization is derived.

We introduce a kernel projection on Sobolev fields, define a distinguished identity boundary patch, and derive the Euler–Lagrange equations for a scalar field coupled to a smooth Riemannian metric. A key conceptual contribution is the separation of simplicial topological closure (∂² = 0) from analytic flux balance via the divergence theorem.

Under standard coercivity and semicontinuity assumptions, existence of minimizers is established using the direct method of the calculus of variations. The framework provides a mathematically rigorous bridge between invariant kernel structure, simplicial topology, and continuum variational analysis.