The Principle of Emergent Information: A Proof of the Non-Axiomatic Origins of the Binary States 0 and 1

This framework presents a formal proof that the foundational mathematical entities, 0 and 1, are not axiomatic primitives but are necessary, co-created consequences of a single, pre-geometric process of interaction. We challenge the axiomatic assumption of the empty set and the unit by positing a more fundamental reality: a primordial state of informational superposition, where the concepts of "occupied" and "unoccupied" are unresolved and indistinguishable. We demonstrate that any "act of definition" — a process that queries this state — must necessarily break this symmetry and resolve the potential into two distinct, mutually defined outcomes. These outcomes, which we label 0 (the state of non-self-reference) and 1 (the state of self-reference), are thus established as emergent informational facts, not a priori truths. We prove that this foundational interaction is prior to the mathematical constructs of set theory and Hilbert spaces, showing that a logical superposition can exist without a pre-supposed geometric or algebraic canvas. Furthermore, we establish the "Principle of Definiteness for Interaction," proving that only these resolved, factual states can engage in the stable interactions required to build more complex structures, such as the emergent continuum of primes. This work provides a formal, non-axiomatic origin for the binary, resolving its status from a postulated axiom to an inevitable consequence of a mathematical universe that contains information.