Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds
The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces $\mathbb{H}^2$. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold $α_c = \frac{5}{4}Dκ^2$ for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace $α= 2Dκ^2$ where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit ($κ\to 0$) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law $τ\sim |Δν|^{-1/2}$. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.