Rigorous Implementation of CDUFD Axioms A1 and A4 on Fixed Background Geometry: Unconditional Proof of Non-Empty Solution Space

Within the CDUFD 4+1 axiom system, we focus on axioms A1 and A4. On a flat 4‑torus we construct an explicit discrete network: vertices are lattice points, the kernel is a Gaussian, and we define Wilson gauge fields and Wilson fermions on this fixed background. Using the positive definiteness of the Gaussian kernel and Gromov–Hausdorff convergence theory, together with standard lattice gauge theory techniques, we rigorously prove: 1. The kernel is strictly positive definite and the intrinsic distance metric space converges to a flat torus (A1). 2. In the continuum limit, the Wilson gauge and fermion actions converge to the Yang–Mills action and a vector‑like Dirac action. 3. By explicit construction of U (1) vortices and SU (2) instantons we verify topological defects with quantised charges (A4). The construction does not contain dynamical geometry, but a trivial Langevin dynamics (_=0) can be assigned to satisfy the formal requirement of A2. Axiom A3 is not directly relevant here. Hence this construction is a legitimate element of the solution space S₀₁-₀₄, unconditionally proving that S₀₁-₀₄ is non‑empty. We call it a “mathematical survivor”.