Ontic Field Theory: A Rigorous Framework for Quantified Existence in Physics and Mathematics

We introduce a mathematically rigorous formalism for encoding ontological states—actuality, potentiality, and nullity—within physical theories. Defining the Ontic Field as a real scalar field over spacetime, we establish its integration into field theory, quantum mechanics, and category-theoretic logic. Our work provides axiomatic foundations, proves formal theorems relating existence amplitudes to operator spectra and sheaf cohomology, and clarifies the epistemological status of mathematical entities via ontological weights. This framework bridges formal ontology, logic, and physics, offering new tools for quantum gravity, foundational logic, and modal metaphysics.