This document presents two companion papers that substantially strengthen and refine the auxiliary mathematical proofs outlined in the main Decad preprint. The work develops a three-regulator completion motivated by the finite-resolution structure of the chiral spacetime fabric lattice (CSFL), including the free-particle--CSFL interface, and demonstrates how the same core physical principles yield constructive results across two long-standing problems in mathematical physics.Paper A develops an unconditional Decad-completed constructive formulation of (SU(3)) Yang--Mills theory with mass gap and stochastic quantization. The paper separates the finite-lattice core, the continuum-reduction argument, and the full three-regulator Decad completion. Within the completed formulation, the simultaneous ultraviolet/infrared removal yields a reflection-positive continuum Euclidean theory, Osterwalder--Schrader/Wightman reconstruction in four-dimensional Minkowski space, and a strictly positive mass gap.Paper B establishes unconditional global existence, smoothness, uniqueness, and classical recovery for a Decad-completed Navier--Stokes-type vorticity model. In the saturated minimal formulation, the braided coefficient is bounded, reflecting finite local topological capacity of the CSFL. The resulting completed equations remain globally regular while recovering the classical constant-viscosity incompressible Navier--Stokes equations in the high-threshold ordinary-scale limit.The regulators employed in both papers arise naturally from the asymmetric, dual-sector, bi-chiral structure of the CSFL. The braided damping term, chiral asymmetry floor, and collective strain/deformation-cost regulator express the same finite-resolution topological principles in gauge-field and fluid settings. While the classical equations encounter serious mathematical difficulties at extreme curvature, vorticity, or strain concentration, the Decad-completed versions supply a finite-resolution completion that preserves the well-tested classical limits at accessible scales. 

Version update This version substantially revises the earlier combined companion-paper package. Paper A has been strengthened from a conditional constructive programme to an unconditional Decad-completed Yang--Mills construction, separating the finite-lattice core, continuum reduction, and full three-regulator completion. Paper B has been reorganised to distinguish fixed-threshold Decad regularity from high-threshold classical recovery, and now states the recovery of classical constant-viscosity Navier--Stokes as the ordinary-scale continuum limit of the completed theory. Both papers now include clearer mathematical-formulation subsections, explicit Decad-admissibility hypotheses, and expanded discussion of the shared CSFL regulator architecture.