D1 Composite Higgs: Discrete Z_4 Crystalline

In the D1 composite Higgs model, the SO(9)/SO(8) coset is deformed by a scalar operator thatΦD1transforms the usual Mexican hat potential into a characteristic four-toothed clover shape with exactZ4symmetry. This deformation naturally lifts the flat directions and produces precisely fourdegenerate discrete minima---termed crystalline vacua---without fine-tuning or additional ad-hocsymmetries.We provide numerical confirmation of this structure through toy lattice relaxation simulations,which show that random initial field configurations dynamically cluster into exactly four sharpZ4-related points. The renormalisation group flow of the deformation coupling exhibits a robustΦD1𝒪(1)plateau at the TeV scale, ensuring naturalness.Full phenomenological scans of the modified Higgs couplings and demonstrate excellentκV κfagreement with the latest 2025 LHC Run 3 constraints. The distinctive shape of the effectivepotential and the resulting pattern of fermion coupling deviations provide clear experimentalsignatures that distinguish the D1 framework from conventional continuous coset models.