The G34 Reflection Representation and Its Finite Orthogonal Geometries

We investigate the finite orthogonal and incidence geometries associated with the exceptional complex reflection group G34. Starting from an Eisenstein Hermitian projector and computational reflection data, we construct finite orthogonal models, explicit intertwiners, and generalized quadrangle incidence structures. The resulting geometry includes 40 distinguished two-ovoids of GQ(4,2). An exhaustive computation enumerates exactly 16,120 labeled two-ovoids, not isomorphism classes. Independent computational verification was performed by Stein 87 (4 November 2026), with additional successful local reproduction checks. This work is presented as a preprint. The complete computational supplement is not yet publicly available, and some original input datasets require separate redistribution authorization. One cross-signing verification remains outstanding. No claim concerning the Hodge conjecture is established by this manuscript.