Discreteness as the Late-Time Limit of a Smooth Flow
An exactly solvable gradient flow whose time-evolution maps are smooth bijections of the real line at every finite time, and whose infinite-time limit is rounding to the nearest integer. Abstract We study the gradient flow of the periodic potential Integers are its stable fixed points and half-integers its unstable ones. The flow can be solved in closed form. At every finite time it is a smooth,…
A gradient flow, characterized by smooth bijections of the real line at every finite time, exhibits rounding towards the nearest integer as it approaches infinite time. This flow is governed by a periodic potential, with integer values serving as stable fixed points and half-integers as unstable ones. The flow can be solved in closed form, forming a one-parameter group of maps that smoothly transform the real line.
As time progresses towards infinity, every open interval between consecutive half-integers collapses onto the integer it encompasses. Despite the flow providing an exact route to discrete targets, integers are not generated by the flow itself but are introduced through the periodicity of the potential. The flow's closed-form solution, geometric interpretation as a hyperbolic Möbius dilation of the circle, and exact resolution time for desired accuracy are detailed in the associated GitHub repository.
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