Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (
TK). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces (
Hm) circumvent this via constant negative curvature (
K<0) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce \textbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (
R1,m). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish \textbf{MAN-2D} (
R1,1→H1) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into
D/2 independent two-dimensional Minkowski blocks. We further formulate \textbf{MAN-4D} (
R1,3→H3) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of
SO+(1,3) to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.