Exponential Run Complexity of Prefix-Separable Orders on the Boolean Cube

TA-TR-2026-22 · Version 1.0 · 2 October 2026

Human author of record and responsible depositor: Hongju Liu.

Published DOI: 10.5281/zenodo.23103274

Status: Published English mathematical preprint; not externally peer reviewed. All 13 deposited files passed anonymous exact-byte SHA-256 readback, and DOI resolution passed.

Every prefix of the constructed Boolean-cube order has an explicit strict affine separator. Nevertheless, every generic additive sweep produces exponentially many monotone runs: the paper proves an order-of-growth lower bound of 2^n/n^2, with liminf constant 1/(8 log 3). The main manuscript contains the complete analytic proof. The supplement documents exhaustive small-dimensional checks, sampled higher-dimensional prefix checks, and the precise relationship to prior permutation results.

The result is an extremal statement about rankings and linear sweeps. It does not establish a statistical-learning guarantee or an empirical biological-landscape result. A constant-density lower bound, an optimal constant and deterministic explicit constructions achieving the general lower bound remain open here. The paper credits known signed-tree constructions, alternating-run statistics and concentration methods; exhaustive historical priority is not certified.

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