Directional Extremal Geometry of Nonlinear Contraction Orbits from One Observed Transition

Abstract

Given one observed transition of a contraction, we study how far later iterates can move in a prescribed direction. We consider globally defined Euclidean contractions in dimension at least two that reproduce the transition and have Lipschitz constant at most q<1. A single map must generate every state, so nonconsecutive iterates constrain one another as well as successive ones. Under these constraints, scaled planar rotations attain the exact directional support at step two for q≥1/2 and step three for q≥4/5, at every angle. In noncollinear directions, only the rotation trajectory attains the bound, and the support gap bounds deviations from it. For the second-step displacement opposite the observed step, nonlinear maps exceed the affine optimum for q<1/2, whereas planar maps exceed the scalar optimum for q>1/2. An exact backward threshold determines when the first-step bound controls the entire future. Its extension to a restricted range of directions uses a computer-assisted proof of sufficiency in exact arithmetic. For 0.8≤q≤0.83 and 0.995≤c≤1, where c is the cosine of the angle between the prescribed direction and the reverse of the observed step, the exact all-future support is the largest of the first three supports; each can be uniquely largest as the direction varies. In a double-precision replay of a released network, the test for keeping the current class unchanged over the next three iterates first certified a window 0.9 iterations earlier on average than its classical comparator. The all-future support beyond the proved ranges and any runtime benefit of the window test remain open. The attained bounds give exact uniform positivity tests for affine functions on the proved ranges.