CDUFD Framework: A Conditional Construction within the A1–A4 Solution Space (A Non-emptiness Example under the A2 Conjecture)

Within the CDUFD 4+1 axiom system, we focus on the solution space S₀₁-₀₄ defined by the first four mathematical axioms (A1–A4). Under the A2 conjecture—which is strongly supported by numerical Causal Dynamical Triangulations (CDT) simulations showing a second-order geometric phase transition—we explicitly construct a discrete network solution based on dynamical Regge calculus and lattice gauge theory. We verify that the construction satisfies A1 (positive definite kernel and convergence to a smooth manifold) and A4 (topological defects with quantized charges, exemplified by a U (1) vortex construction). Under the condition that the system is in the non-critical regime (finite correlation length), we prove that the continuum limit of the geometric action converges to the Einstein–Hilbert action, and the matter action converges to the Standard Model gauge theory with vector-like fermions (Wilson fermions). The chiral nature of the electroweak interaction and the three-generation mass pattern are not directly encoded in this vector-like limit; they are proposed to emerge from critical lag dynamics at the electroweak critical point, as developed in CDUFD Phenomenological Series II. This shows that S₀₁-₀₄ is non-empty (conditionally on A2). The construction does not address the external anchoring axiom A5; it provides a mathematical existence proof for the qualitative framework. Open problems, including the critical regime convergence and chiral fermions on fluctuating lattices, are identified.