This paper presents a complete mathematical framework for quantum field theory, founded on the core postulate that physical spacetime is a four-dimensional Lorentzian manifold and that the field strength of gauge fields existing within it issubject to a fundamental intrinsic bound. We demonstrate that when the path integral is strictly restricted to configurations with bounded curvature: (1) the integral becomes absolutely convergent under lattice regularization, resolving the problem of defining the ultraviolet-divergent path integral; (2) all standard perturbative calcu-lations (the g−2 of QED, the Lamb shift, asymptotic freedom, etc.) remain strictly invariant, with unitarity automatically preserved and no ghost states; (3) the non-perturbative vacuum necessarily realizes a chromomagnetic monopole condensate viaan entropy-driven phase transition, from which a mass gap and quark confinement are derived, yielding a string tension and critical temperature consistent with lattice QCD. This framework unifies ultraviolet completeness and infrared confinement as two facets of a single geometric principle: spacetime cannot sustain infinite fieldstrength. It introduces only one new constant of nature, Λ (at the Planck scale), without requiring extra dimensions, supersymmetry, or string-theoretic assumptions.