Precise asymptotics at the tip of the Mandelbrot set

What the math gives to ML

The paper develops a transferable mechanism for dynamics approaching a non-hyperbolic degeneracy: induce the map at progressively deeper return scales, separate nearly horizontal branches from transverse branches, and use pressure with uniform exponential-tail control to quantify the resulting complexity increase. Its key quantitative result is a square-root law: transverse branch size and pressure grow proportionally to the square root of the distance from the critical parameter. A neural analogue is a multiscale induced-pressure monitor and controller for recurrent or state-space networks, estimating return-map Jacobian pressures and reducing recurrent gain before the system enters a marginal regime. The square-root scaling is a falsifiable hypothesis for a deliberately engineered recurrent bifurcation, rather than a theorem for arbitrary neural networks.