D1 Composite Higgs: Discrete Z_4 Crystalline Vacua – Numerical Confirmation and LHC Phenomenology

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