Emergent Quantum Dynamics From Fixed-Point Geometry

This work introduces a minimal variational framework in which localized energy basins and their fixed points form the foundation for emergent quantum dynamics. Rather than imposing quantization axiomatically, discrete spectra arise naturally from geometric regularity, topological invariants, and adjacency-driven transport within a continuous relational configuration space. Key contributions include: Emergent Schrödinger Dynamics: Oscillatory modes near basin fixed points admit a complex structure, leading to a Schrödinger-type evolution derived from a quadratic variational principle rather than canonical quantization. Topology-Induced Quantization: Integer-valued invariants such as winding and linking numbers enforce discrete excitation spectra without probabilistic postulates. Hydrogenic Consistency: The framework reproduces the hydrogen spectrum with emergent parameters, matching experimental wavelengths to within 0.1 nm. Conceptual Foundations: Spacetime is treated as an emergent descriptor of basin adjacency and deformation, not as a pre-imposed manifold. This approach provides a conservative and mathematically rigorous route to quantum-like behavior grounded in geometry and topology, while remaining compatible with classical correspondence in the coarse-grained limit. Extensions to relativistic regimes, gauge structures, and gravitational coupling are deferred to future work. Keywords: Emergent quantum dynamics, fixed-point geometry, basin framework, topology-induced quantization, Schrödinger evolution, variational principles.