Operational Calculus: A Meta-Framework for Distinguishing Existence from Survivability in Mathematical Processes

We introduce Operational Calculus, a meta-framework that sits above classical foundations to distinguish between mathematical existence and survivability. While classical mathematics establishes what objects exist, Operational Calculus characterizes which structures persist under truncation, projection, and irreversible limiting processes. We formalize the notion of operational coherence as a primitive concept, establish axioms governing information flow in directed systems, and prove a Fundamental Theorem characterizing which mathematical statements admit operational proofs without requiring reconstruction from collapsed data. This framework provides a unified explanation for why certain major conjectures (Birch–Swinnerton-Dyer, Riemann Hypothesis, Navier–Stokes regularity) resist classical proof techniques: they require reconstruction of high-dimensional information from collapsed representations, violating the irreversibility axiom. We demonstrate that Operational Calculus strictly contains classical mathematics while forbidding operationally illposed inverse operations, and we show how classical results like the Fundamental Theorem of Calculus exemplify information loss under differentiation.