Median-Extremes Alternation

Given a finite linearly ordered set, we pair positions that are mirror images about its midpoint into bilateral shells, ordered by their distance from the center.

Median-Extremes Alternation (MEA) begins with the central shell and repeatedly selects the unvisited shell having greatest radial contrast with the shell most recently engaged.

We prove that this local rule has a strict unique maximizer at every step and forces the shell order 0, q, 1, q-1, 2, q-2, and so on, for both odd and even cardinalities.

Parity affects only whether the central shell is a singleton or a pair.

A separately supplied global orientation determines the order of the elements within every shell, yielding exactly two mirror traversals for n greater than or equal to 2 before orientation is fixed and exactly one afterward.

Explicit formulas, a linear-time generation algorithm, examples, and scope conditions are given.

The result replaces an earlier fixed-center distance-minimization formulation, which does not generate the canonical MEA traversal.