Non-Pseudo-Arithmetic Hyperbolic 4-Manifolds

Abstract

We give three explicit finite-volume hyperbolic Coxeter lattices in dimension four whose adjoint trace fields are not totally real. They are the reflection groups of the Ma--Zheng polytopes $P_{11,6}$, $P_{11,7}$, $P_{11,8}$. The obstruction is contained in two cyclic products of the doubled unit-normal Gram matrix $A=2G$. Those products recover elements $s=\sqrt5$ and $u$ satisfying $$ (s-1)u^2+2(s-3)u+s-7=0. $$ After the embedding $s\mapsto-\sqrt5$, this quadratic has discriminant $8(1-\sqrt5)<0$. The cyclic-product field therefore has a complex place. Vinberg&#39;s identification of that field with the adjoint trace field and the Emery--Mila total-reality obstruction imply that the three lattices are not pseudo-arithmetic. Torsion-free orientation-preserving finite-index subgroups yield orientable noncompact finite-volume hyperbolic $4$-manifolds with the same obstruction. Exact computations check the census data and provide independent field and finite-volume certificates. A pinned Lean~4 development verifies the radical identity and the number-field obstruction. It also checks the lattice and manifold implications conditional on explicit geometric and literature hypotheses.