Geometry and physics intertwine throughout the study of Nature at its fundamental level, their active reciprocating feedback of key importance in string theory. Complex-algebraic methods of geometry in particular have brought about an unrivaled abundance of models, counted well into hundreds and thousands of orders of magnitude — reciprocated by the discoveries such as the wholly unexpected but profound mirror symmetry. Further and very rich possibilities have recently been made possible by generalizations that, at first glance, appear to be non-algebraic. Nevertheless, they are remarkably well-aligned within both overall mirror-symmetric and the gauged linear sigma model framework, and are amenable to almost as comprehensive quantitative analysis, and implying a foundational mirror-symmetric integraton with symplectic geometry.