ratwolf
Bronze Coder
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.
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, strictly increasing bijection of the real line; the family of maps forms a one-parameter group; and as time tends to infinity every open cell between consecutive half-integers collapses onto the integer it contains. "Integer-valued" therefore appears as the endpoint of a continuous process with a single, exactly composable resolution parameter, the time. The integers are not generated by the flow: they enter through the period of the potential. What the flow provides is an exactly solvable, exactly composable route to that prescribed discrete target. We state the closed form, derive the main properties, give a geometric interpretation as a hyperbolic Möbius dilation of the circle, and give the exact time it takes to resolve a point to a given accuracy.
The full derivations and proofs, the verification script and the visualization code are in the GitHub repo:
https://github.com/ratwolfzero/continuous-discreteness.
Abstract
We study the gradient flow of the periodic potentialIntegers 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, strictly increasing bijection of the real line; the family of maps forms a one-parameter group; and as time tends to infinity every open cell between consecutive half-integers collapses onto the integer it contains. "Integer-valued" therefore appears as the endpoint of a continuous process with a single, exactly composable resolution parameter, the time. The integers are not generated by the flow: they enter through the period of the potential. What the flow provides is an exactly solvable, exactly composable route to that prescribed discrete target. We state the closed form, derive the main properties, give a geometric interpretation as a hyperbolic Möbius dilation of the circle, and give the exact time it takes to resolve a point to a given accuracy.
The full derivations and proofs, the verification script and the visualization code are in the GitHub repo:
https://github.com/ratwolfzero/continuous-discreteness.
