Scale-Adaptive Cut-and-Flow for Stationary-Point Search on Periodic Domains

Abstract

On a periodic domain, recursive localization becomes cheaper when each separator is sampled at the scale of the region that remains. For a function f: 𝕋ᵈ → ℝ with a 1-periodic C¹ lift and 1-Lipschitz gradient, we turn this observation into a deterministic exact value–gradient algorithm that finds a queried point x with ‖∇f(x)‖ ≤ ε in O_d(ε^{-2(d-1)/(d+1)}) queries; in dimension three the bound is O(1/ε). At spatial scale R, sampling a separator and descending below its true minimum cost (R/δ)^{d-1} + δ²/ε². Balancing at the current scale makes these costs geometrically summable. A face-complete net retains quadratic accuracy at lower-dimensional face minima. As a finite stress test, the same principle gives an O(√K n) coefficient-query algorithm for periodic integer potentials whose second differences are bounded by K. As a boundary test, we show why the argument does not directly compactify ℝᵈ: a bounded smooth function may admit no bounded exact trap, and fixed full-range additive walls face a radius–cancellation dichotomy. Thus the unrestricted three-dimensional problem remains open.