THE FINITE DATA PRINCIPLE A Unified Classification Framework for Mathematical Problems Formal Development in ZFC with Applications to the Millennium Problems

AbstractWe present a rigorous formalization of the Finite Data Principle (FDP) within Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). The FDP provides a classificationcriterion for mathematical problems: a problem is solvable if finite algebraic or analyticdata forces the conclusion via a theorem provable in ZFC; a problem is independent if itsresolution requires quantification over infinite structures without finite closure, rendering itundecidable in ZFC. We develop the formal apparatus of forcing theorems, kernel conditions,and oracle barriers. We then apply this framework to classify the seven MillenniumPrize Problems, demonstrating that four (Poincar´e, Riemann, Yang-Mills, BSD) are solvablevia finite spectral/kernel data, while three (P vs NP, Hodge, Navier-Stokes) are independentdue to inherent infinite quantification. The classification is exhaustive and predictive.This paper synthesizes and formalizes results from the Caravan’s complete resolution of theMillennium collection.