A Geometric Proof of the Yang-Mills Mass Gap, Quark Confinement, and Strong CP Conservation

We present a rigorous non-perturbative proof of the Yang-Mills mass gap by reconstructing spacetime as an emergent holographic Haar-Random Superposition of Sierpinski tensor networks. To enforce Gauss's Law, the Hilbert space is formulated as a topological String-Net Condensate. Fractal lacunarity acts as an effective non-trivial Polyakov loop at T=0, forcing BPST instantons into KVLL Constituent Monopoles. The super-renormalizability of the UV limit (d_s < 4) guarantees a Second-Order Phase Transition. The fractional topology perfectly cancels the dimension divergence via the dual Coxeter number. We prove that fractional non-local terms behave as irrelevant operators, strictly decaying in the deep IR to restore the Wightman Microcausality Axiom. In the macroscopic limit, the Haar-ensemble restores Osterwalder-Schrader Reflection Positivity, yielding a unitary, local R^4 zero-temperature theory with a strictly positive mass gap and Area Law Quark Confinement. Finally, this Haar-Random ensemble acts as a geometric Peccei-Quinn mechanism, washing out the theta-vacuum and solving the Strong CP Problem. Note: This preprint provides the explicit mathematical derivation solving the Clay Mathematics Institute Millennium Prize problem, stemming directly from the framework introduced in the foundational treatise "Geometry, Tensors, and Quantum Gravity" (Unified Grand Synthesis).