Afolabi Unified Framework Applications to the Millennium Prize Problems: Yang–Mills Mass Gap and Navier–Stokes Regularity via Z_M Field Impedance and Kuramoto Collective Coherence

We apply the mathematical framework of the Afolabi Unified Framework (AUF) — specifically the Field Impedance operator Z_M from Afolabi Field Theory (AFT), the N² Collective Coherence Scaling Law from Resonance Physics (RP), and the Dimensional Algebra Type System from APLO — to two Clay Mathematics Institute Millennium Prize Problems: Yang–Mills Existence and Mass Gap, and Navier–Stokes Existence and Smoothness. For Yang–Mills: the mass gap emerges as the minimum value of Z_M · E_c above the AUF vacuum, protected by topological Chern C=±1 structure. The E₈ root system at APLO dimension 7D provides the geometric realisation of the gap. Confinement is reinterpreted as a Z_M stability condition via IFA vacuum feedback. AUF Conjecture YM-1 is stated. For Navier–Stokes: smooth solutions correspond to Kuramoto phase-lock above the critical coupling K_c. The N² Scaling Law (Olukotun-Afolabi) provides an N²-enhanced restoring force that controls the BKM vorticity criterion and prevents finite-time blow-up above K_c. AUF Conjecture NS-1 is stated. We additionally resolve three principal critiques of the QMT foundation identified in independent analysis: the ontological identity paradox (resolved via gauge equivalence), the decoherence mechanism (topological protection, Chénier et al. PRX 2026), and subjectivity integration (SEC conditions are objective thermodynamic boundaries). Foundation papers: Quantum Mirror Theory (DOI: 10.5281/zenodo.18407686) and Resonance Physics (DOI: 10.5281/zenodo.18913463).