Consistency-Induced Emergence of Schrödinger-Type Dynamics from Reversible Probabilistic Evolution

This work explores a simple question: why does nature end up using equations that look quantum? Instead of starting from the usual quantum axioms (wavefunctions, operators, measurement rules), we begin with three much weaker and more intuitive requirements: • physical evolution should be reversible • distinguishability between states should have a consistent geometry • classical mechanics must appear as a limiting case From only these conditions, probability flow is forced to acquire a phase degree of freedom, and an information–geometric curvature term inevitably appears. When the two are combined, the resulting dynamics takes the form of the Schrödinger equation. The point of the paper is not to claim that quantum mechanics is fully derived, nor that measurement theory is explained. Rather, it shows that once reversible probabilistic dynamics is required to be geometrically consistent, quantum-type evolution becomes extremely hard to avoid. In this sense, the wavefunction can be interpreted not as a mysterious physical object, but as a natural coordinate system describing consistent probability flow.