The Well-Defined Path Integral: How the UCM Substrate Resolves Five Foundational Problems of Quantum Field Theory (UCM Paper 54)

The Feynman path integral rests on five unresolved mathematical foundations: (1) the measure does not exist in infinite dimensions; (2) the Lorentzian integrand does not converge; (3) rigorous Euclidean constructions exist only in d<4; (4) 4D Yang-Mills existence is an open Millennium Prize problem; (5) Haag's theorem forbids the interaction picture. The UCM substrate combined with the fibre-bundle structure of Papers 52-53 resolves all five: the bulk field contributes a finite Lebesgue measure, the gauge field contributes a finite Haar measure, and gauge-matter unity eliminates the matter integral. The complete partition function is a finite product of well-defined measures.