Emergent Geometry from Mode-Structured Evolution

This paper develops a geometric extension of mode-based analysis in which geometric structures arise from admissible operational evolution rather than being postulated a priori. Within the framework, convergence modes encode structured classes of admissible approximation procedures, while mode-compatible evolutions induce corresponding admissibility structures on asymptotic tangent directions. The paper shows that:- admissible evolution generates constrained structures of realizable directions,- finite-propagation hyperbolic evolution induces cone-type admissibility structures,- invariant geometries arise from admissibility-preserving transformations,- Minkowski geometry emerges as the unique isotropic quadratic invariant associated with hyperbolic admissibility structures. The work proposes an operational reconstruction framework connecting PDE evolution, admissibility constraints, invariance principles, and emergent geometry. The paper also discusses anisotropic, stochastic, and non-conical admissibility geometries, as well as possible extensions toward global admissibility geometry and operational curvature.