Rigidity of Mather's $β$-function on a KAM set for analytic billiards-like maps and unique quasi-analytic continuation

What the math gives to ML

The paper's transferable mechanism is quasi-analytic rigidity: for analytic twist maps, Mather's beta function has a highly regular extension, so equality on a positive-measure subset of KAM rotation numbers forces equality on all Diophantine rotation numbers. The quantitative asset is not billiard geometry itself, but a uniqueness principle for parameterized functions whose derivatives obey a quasi-analytic growth bound. A neural implementation could represent a parameter-dependent response curve with an analyticity-constrained head and use sparse positive-measure anchor observations to test whether the entire curve is identified, providing a principled alternative to unconstrained interpolation.