We introduce Boundary Algebra, a new mathematical foundation based on the primitive distinction between boundary and content. From five axioms (Enclosure and Boundary Theory, EBT), we derive a four-valued logic, a four-mode proof theory with Gödel incompleteness as a structural feature, and Zermelo-Fraenkel set theory as a special case. Concrete models (Boolean matrix, graph, free algebra) prove consistency. A topos-theoretic interpretation (the Boundary Topos) yields a subobject classifier Ω = {0,V,1,∞} and a Lawvere-Tierney topology that recovers classical logic externally. The ontological square embeds into a diamond lattice whose Hasse graph Laplacian has spectrum {0, 2, 3, 4}, giving provisional masses for four fields in a unified Lagrangian. The precise correspondence to known physical theories depends on the complete metric graph spectrum and is reserved for future investigation. We outline quantization, generalization to n dimensions, and a theory of discrete dynamical systems with active boundary that generalizes Collatz convergence.