Spectral Ontology: A Theory of Coherent Identity

Spectral Ontology presents a complete and irreducible framework for mathematical existence grounded in the spectral resolution of self-adjoint operators. It proposes a generative law of identity:

∑O=R+E+RE=I

in which structure arises not from axioms, but from the coherence of spectral participation. This foundational equation defines the conditions under which mathematical, physical, and logical identity may persist.

Through the tripartite decomposition into Relevance (R), Existence (E), and Interaction (RE), the framework unifies number theory, quantum mechanics, formal logic, and entropy as consequences of a single spectral law. The Riemann Hypothesis is reinterpreted as a spectral admissibility condition; Gödel’s incompleteness theorems emerge naturally as signatures of sub-coherent fields.

In a final act of closure, the theory formalizes the undifferentiated coherence field FΩ—the ontic origin of identity prior to spectral decomposition—thus resolving not only the structure of being, but the birth of structure itself. With this, Spectral Ontology does not propose a new foundation. It defines the condition for all possible foundations.

Note: This paper is retained as part of the developmental archive leading toward Zero-State Axioms. It contains provisional formulations and should not be read as the final ZSA framework.