DuPair and Tetrapolar Spacetime: A Duality-First Framework. I. Quantum Gravity, Gauge Group Selection, and Dark Energy

This work introduces the DuPair axiomatic framework, a novel discrete mathematical foundation built upon duality-pairing primitive structures. It demonstrates how continuous Lorentzian spacetime geometry emerges via coarse-graining and Gromov–Prokhorov convergence from discrete infotron graphs. The framework derives a parameter-free relation for the cosmological constant from topological entropy, and obtains necessary topological filtering conditions selecting the \\(\\mathfrak{su}(3)\\oplus\\mathfrak{su}(2)\\oplus\\mathfrak{u}(1)\\) gauge algebra as admissible candidate symmetry structure. The paper clearly distinguishes rigorous theorems, physically motivated working assumptions, and results deferred to a companion paper. This construction advances mathematical physics by realising gauge symmetries as automorphisms of an underlying duality structure, rather than treating symmetries as fundamental postulates.