Geometric Hammocks: Surface Topology, Laminations, and the Ziegler Spectrum of String Algebras

We establish a dictionary between the representation theory of string and special biserial algebras and the model theory of the Ziegler spectrum, over an algebraically closed field of arbitrary cardinality. Associating to a string algebra a marked surface with forbidden boundary segments and a finite-area hyperbolic metric, we prove Schr"oer's hammock posets are strictly order-isomorphic to geometric fans of pointed admissible arcs, and that Cantor--Bendixson limit points of the Ziegler spectrum correspond to pointed complete geodesics and $\mathbb{K}^\times$-decorated closed geodesics. Three advances follow. First, the string closure and higher Cantor--Bendixson strata are homeomorphic to the corridor space of admissible surface geodesics; the pushout map is a homeomorphism if and only if the algebra is domestic; and the Krull--Gabriel dimension equals the topological accumulation depth, governing radical filtration stabilization. Second, via prefix-free dyadic blocks in the geometric perfect core, we verify Puninski's cardinality-free two-chain criterion, resolving the open problem of superdecomposable pure-injectives over uncountable fields for all non-domestic string and special biserial algebras. Third, on the perfect core we construct purely algebraic invariants: localized pp-shape entropy, a $K_0$-weighted pressure function whose Legendre--Fenchel dual reconstructs the ergodic dimension simplex and separates iso-entropic families, and a pp-bifurcation index giving a Gauss--Bonnet obstruction to self-intersection. We further show an arational pure-injective module generates a minimal definable subcategory if and only if its lamination is minimal arational, in which case the inverse limit of the quotient pp-lattice reconstructs the transversal Cantor cross-section.