Why Nature Minimizes: The Many-to-One Root of Time, Action, and Emergence

The Principle of Least Action, the arrow of time, and the emergence of smooth physics from discrete rules are treated as separate results requiring separate explanations. This paper argues they are three instances of one structural fact: many-to-one mappings. Computation is many-to-one: 1+1=2 is deterministic, 2=x+y is not. The inverse does not exist. This is the arrow of computation (Bernstein, 2026h). Feynman's path integral is many-to-one: all quantum paths exist, but paths far from the extremal cancel under phase interference. What survives is the classical trajectory. The Principle of Least Action is what remains when everything else cancels. Coarse-graining (averaging over fine details) is many-to-one: many microstates map to one macrostate. When discrete rules are averaged at large scales, only extremal macro behavior survives, for the same structural reason. Variational emergent physics is forced by coarse-graining regardless of whether the fundamental rule is variational. A forcing argument completes the picture. Observers require stable structures. Stable structures require conservation laws. Noether's theorem proves conservation laws require symmetries of a variational formulation. The chain from observers to variational physics runs one direction with no exit. Under the Simplicity Dominance Principle (Bernstein, 2026d), the simplest observer-supporting frameworks dominate. If all such frameworks produce variational physics through the Noether chain, SDP derives the Principle of Least Action for any universe we could inhabit.