OAI.BoxTransport.Routing.balanced_box_routing

The theorem states that for every natural number n, rational centers a_i, b_i in ℚ³ and rational positive half-widths h_i, k_i in ℚ³ (all coordinates strictly positive) indexed by i in Fin n, the following holds. Suppose the closed boxes {x ∈ ℝ³ : |x_j − (a_i)_j| ≤ (h_i)_j for all j} are pairwise disjoint over distinct indices, the target boxes with centers b_i and half-widths k_i are likewise pairwise disjoint, and each box pair is volume balanced, meaning the product over the three coordinates of k_i,j / h_i,j equals 1. Then there exist a vector field U on space-time ℝ × ℝ³ with values in ℝ³ and a map Φ : ℝ → ℝ³ → ℝ³ such that U is admissible, Φ is a global flow of U, and Φ delivers every box. Admissible means U is C^∞, compactly supported, spatially divergence-free at every time (the sum of the jth spatial derivatives of the jth component is zero), vanishes for all times t outside [1/4, 3/4], and is effective: there are computable procedures giving rational approximations, to any requested precision at rational space-time points, of every mixed partial derivative of U (taken along time or coordinate directions, in any finite list of axes), together with computable moduli of continuity and computable uniform bounds on these derivatives. Global flow means Φ(0,x) = x, each t ↦ Φ(t,x) is differentiable with derivative U(t, Φ(t,x)), and every differentiable curve γ starting at x with γ'(t) = U(t, γ(t)) coincides with Φ(·,x) for all t. Delivers means that for each i there is an open set containing the ith source box on which Φ(1,·) agrees with the coordinatewise affine map x_j ↦ (b_i)_j + (k_i,j / h_i,j)(x_j − (a_i)_j), which carries the source box onto the target box.