DuPair: A Binary Dual Pairing Framework and Its Tetrapolar Spacetime Realization for Quantum Gravity, Gauge Symmetry, and Dark Energy

This paper introduces DuPair (Binary Dual Pairing Calculus), a constraint-based mathematical physics framework that takes binary duality pairing as its primitive organizational principle—rather than symmetry. DuPair is positioned as a meta-tool: it does not replace existing mathematical tools such as Lie group classification or index theorems, but imposes pairing-compatibility constraints on their combinations. From three pairing constraints—the tripartite direct-sum decomposition, the single-vertex winding bound |W_i| <= 1, and the cardinality #O = 3 of the pairing-subspace index set—DuPair uniquely selects SU(3)_C x SU(2)_L x U(1)_Y as the admissible gauge group, with explicit generators and no phenomenological input. In the physical realization Tetrapolar Spacetime (TPS), four independent discrete components (T,D,C,P) on a locally finite directed graph generate a unique continuum limit: the graph ensemble converges exponentially to a smooth Lorentzian manifold via Gromov-Prokhorov convergence, with Lorentzian signature emerging from the Schur complement of the correlation tensor. The same topological data yield the parameter-free relation Lambda L0^2 = 3 ln 2, fixing the cosmological constant to within 0.2% of the Planck 2018 observed value, and the scaling relation H0 = (c / l_P) sqrt(pi / (N_tot ln 2)). The bound |W_i| <= 1 provides a first-principles UV cutoff at the Planck scale, rendering all loop integrals convergent without renormalization. The framework yields falsifiable predictions including Lambda L0^2 = 3 ln 2, gravitational-wave echoes from black-hole cores, linear Lorentz-violating dispersion, and exactly three fermion generations. The binary structure of DuPair is structurally isomorphic to the system-environment duality in Qian Xuesen's Engineering Cybernetics, suggesting that the same pairing-first logic may unify physical law and control theory. This work presents the complete mathematical foundation of the framework, including the algebraic axioms, measure theory, differential geometry, and spectral theory, together with the physical derivation of the Standard Model gauge group, the cosmological constant, and the ultraviolet completion of quantum gravity.