arXiv:2611.12931v1 Announce Type: new Abstract: We give the first complete topological, geometric, and ergodic classification of the Ziegler spectrum $\Zg(\Lambda)$ of a non-domestic string algebra $\Lambda$ over an algebraically closed field of arbitrary cardinality, via a dictionary with Thurston's theory of geodesic laminations on a cusped hyperbolic surface. This is a model-theoretic counterpart of Haiden--Katzarkov--Kontsevich's realization (\emph{Publ.\ Math.\ IHES} 2017) of stability conditions on the bounded derived category of a gentle algebra via flat surfaces: pure-injective modules correspond to admissible geodesics, and the Ziegler topology to a corridor quotient of Chabauty space. Three consequences follow. The Krull--Gabriel dimension equals the surface's topological accumulation depth, and the string closure with all higher Cantor--Bendixson strata is globally homeomorphic to an explicit corridor space, with the pushout map to $\Zg(\Lambda)$ a homeomorphism if and only if $\Lambda$ is domestic. Prefix-free dyadic blocks in the geometric perfect core 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. Three purely algebraic invariants on the perfect core---localized pp-shape entropy, a $K_0$-weighted pressure function satisfying an algebraic Ruelle--Walters Variational Principle, and a pp-bifurcation index with a Gauss--Bonnet simplicity obstruction---strictly separate uncountable iso-entropic and iso-pressure families, and the inverse limit of the quotient pp-lattice recovers the transversal Cantor cross-section.
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