Emergent Geometry from Mode-Structured Evolution

This paper develops an operational approach to emergent geometry within the framework of mode-based analysis. Rather than treating metric and causal structures as primitive entities, the paper investigates how geometric structures may arise from admissible classes of operational evolution. Convergence modes encode admissible approximation procedures, while mode-compatible evolutions induce corresponding admissibility structures on asymptotic tangent directions. The central claim is that causal cones and Lorentzian geometry need not be postulated a priori. Instead, they may emerge as invariant structures associated with isotropic hyperbolic admissibility regimes. The paper develops: - mode tangent structures, - admissibility geometries, - admissibility-preserving transformations, - invariant quadratic reconstruction, - operational interpretations of isotropy and finite propagation. Within this framework, Minkowski geometry appears as the unique isotropic quadratic invariant compatible with hyperbolic admissibility structures. More generally, the work proposes that geometry should be understood as an invariant organization of admissible operational evolution rather than as a fixed background structure.