Solution to the Feynman Path Integral Convergence Conjecture: The Sobolev Geodesic Field Module within Helical Hidden Holographic Quantum Mechanics (H3QM)

Richard Feynman's 1948 path integral formulation \int \mathcal{D}\phi \, e^{\frac{i}{\hbar} S[\phi]} forms the foundation of Quantum Electrodynamics (QED) and modern Quantum Field Theory (QFT). However, because the complex phase factor e^{\frac{i}{\hbar}S} oscillates infinitely without magnitude decay on infinite-dimensional Hilbert spaces, standard Lebesgue measures fail to define a mathematically rigorous functional measure \mathcal{D}\phi, leaving path integral convergence an open conjecture for over 80 years. In this paper, we present a complete theoretical framework and formal proof of the Feynman Path Integral Convergence Conjecture using Helical Hidden Holographic Quantum Mechanics (H3QM) and its Sobolev Geodesic Field Module. First, we formulate the Infinite 2D Slicing Paradox, demonstrating that fitting 3D physical space with infinite 2D area slices induces O(N^3) spatial Ultra-Violet (UV) divergence, while time-slicing causes time Infra-Red (IR) phase divergence. Second, via June Huh's Matroid Hodge Decomposition, we factor out unphysical infinite gauge volume \text{Vol}(\mathcal{G})=\infty to restrict field variations onto Betti harmonic space \mathcal{H}^1. Third, using Villani's W1 Wasserstein optimal transport duality, the oscillatory phase is uniquely mapped into a strictly convex, Lipschitz-continuous topological potential functional \mathcal{V}_{\text{topo}}(\phi) on Sobolev space W^{1,1}(\mathcal{M}^3), extending to complex fields \mathcal{V}_{\text{topo}}^{\mathbb{C}}(\phi). Combined with Hong Wang's 3D Kakeya Fourier restriction and Yu Deng's random tensor relaxation with Kimi L1 Topo-AttnRes, we prove that multiplier-free subgradient dynamical flow strictly converges in 5 to 8 steps to a unique minimal topological attractor \delta(\phi - \phi^*). Step-by-step numerical benchmark verification evaluating the 3D quantum harmonic oscillator ground state energy (1.000000 eV) is provided in Appendix A. [Note] This paper presents the formal proof of the Feynman Path Integral Convergence Conjecture under the Helical Hidden Holographic Quantum Mechanics (H3QM) framework. Complete master PDF documents are available in three language editions: English (en-US), Simplified Chinese (zh-CN), and Traditional Chinese (zh-TW).