Well-Defined but Not Predetermined Persistent Mathematical Objects, Embedded Observers, and the Provability of P = N P

Abstract

We develop a foundational distinction between \emph{well-definedness} and \emph{predetermination}. A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension. We call such an object \emph{well-defined but non-predetermined}. We next formulate an observer-based semantics. All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history. Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed. A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic--nondeterministic separation under an explicitly history-sensitive semantics. We then derive a conditional result concerning the provability of \(P=NP\). The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).