Geometric Resolution of the Strong CP Problem without an Axion

The strong CP problem—why the QCD vacuum angle satisfies |¯θQCD|<10−10 despite receiving O(1) contributions—has resisted solution for nearly fifty years. The Peccei–Quinn axion remains undetected after four decades of experimental searches. We present a geometric resolution within the BCT Superfluid Lattice Model 9, in which the quantum vacuum crystallises into a D4 root lattice with tetrahedral void symmetry group Td. Three independent mechanisms enforce θQCD = 0 exactly: (i) the Z2 centre of the D4 Weyl group maps θ→−θ, requiring θbare = 0; (ii) Td reflection symmetry forces all BCT Yukawa couplings to be real, giving arg det Mq = 0 to all orders; (iii) D4 modular self-duality (D∗4 = D4) makes the partition function satisfy Z(θ) = Z(−θ). The result is radiatively stable: the one-loop correction δθ∼JBCT/(16π2) = 2.4×10−9 is 23 orders of magnitude below the residual θres = (ΛQCD/MR)2α0 = 8.0×10−34. BCT predicts dn = 1.0×10−32 e·cm (CKM contribution only; no QCD θ-term), testable at n2EDM and next-generation facilities. No axion is required or predicted.