Strong CP from Z₃ Trisection: A Topological Resolution of the Strong CP Problem without an Axion, and the Near-Term Neutron-EDM Falsification Test

The strong CP problem is among the most durable puzzles of the Standard Model. Quantum chromodynamics admits a CP-violating term in its Lagrangian proportional to a parameter θQCD that could a priori take any value in [0, 2π) ; neutron electric-dipole-moment measurements constrain it to |θQCD| < 10^ (-10) (Abel et al. 2020; PDG 2024). The Standard Model offers no explanation for the extreme smallness of this parameter. The conventional resolution is the Peccei–Quinn mechanism (Peccei the parameter θQCD is the relative phase between them; a non-zero value would break the equivalence. Because Z3 is forced by the Euler–Poincaré constraint on a hex-pentagon tiling of S² (which gives exactly 12 pentagonal defects in a cuboctahedral arrangement, forcing a trisection with Nc (Nc+1) = 12 and therefore Nc = 3), the symmetry is topological, not dynamical. No Peccei–Quinn mechanism is needed; no axion is predicted to exist. The framework makes two firm empirical commitments. First, it predicts that no axion will ever be detected at any mass and any coupling consistent with the Peccei–Quinn relation mₐ proportional to 1/fₐ. Every null result from ADMX, HAYSTAC, CAPP, CASPEr, IAXO, and future axion searches is consistent with the framework. An axion discovery at any experiment falsifies the topological resolution. Second, it predicts that the neutron electric dipole moment receives only weak-sector contributions, dₙ ~ 10^ (-31) e. cm, far below the current PSI 2020 bound of 1. 8e-26 e. cm but within reach of the n2EDM and SNS nEDM experiments in the late 2020s. A measured non-zero EDM above ~10^ (-29) e. cm with confirmed strong-CP origin would falsify the topological resolution. This paper is self-contained. It recapitulates the topological chain from the Euler characteristic of S² to the Z3 trisection in full, presents the QCD-vacuum-as-flat-connections identification, derives θQCD = 0 from the Z3 symmetry, explains why the Peccei–Quinn mechanism becomes unnecessary, and discusses the wider elegance claim that the same Z3 symmetry solves four long-standing Standard Model problems simultaneously: Nc = 3 colours, three fermion generations, the lepton CP phase δPMNS, and θQCD = 0. One topological symmetry, four puzzles dissolved.