Tetrapolar Spacetime and π-Circular Topology:A Discrete Framework for Spacetime Emergence and Gauge Symmetry

We present a complete,mathematically rigorous discrete spacetime framework:Tetrapolar Space-time(TPS)based on π-circular compact topology.The fundamental substrate consists of discreteinformation elements carrying four independent microscopic components:temporal-causal(T),Z₂dual(C),chiral-ordered (L),and compact-phase(D).The background tensor Kμu is uniquelyinduced by coarse-graining the discrete tetrapolar structure.Under compact support,exponentialcorrelation decay,uniform ellipticity,and ergodicity,the discrete system converges uniquely to emer-gent 4D Lorentzian spacetime.The full Standard Model gauge group SU(3)c×SU(2)L×U(1)Demerges naturally from topological properties of the four axes.The L-axis provides an intrinscglobal orientation that couples to the fermion axial current,inducing a negative effective chemicalpotential for right-handed fermiors and dynamically explaining cosmic chiral asymmetry.The cos-mological consLant A is uniquely delermined by Lhe topological vacuum energy of Lhe L-axis andthe ground-state information entropy In2,without fine-tuning or ad hoc inputs.Dark matter isidentified as topologically inert W=0 states,with cosmic abundance naturally matching observa-tions.This framework is ultraviolet-complete,singularity-free,and provides a unified foundation forspacetime emergence,gauge symmetry,and cosmic structure.