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

Mathematical preprint proving that a prefix-separable order on the standard Boolean cube can require Omega(2^n/n^2) monotone runs under every generic additive sweep. The liminf of n^2 M_n/2^n is at least log 2/(2 log 3), strengthening the v1.0 coefficient by a factor 4 log 2. This linked revision adds two precisely scoped paired-family scan theorems and exact row-lift and conditional-mean obstructions. The proof combines a fixed signed-tree family, a deterministic comparison-multiplicity bound, the established read-k tail theorem and sweep-chamber counts. Existing signed-tree representations, alternating-run statistics and concentration methods are explicitly credited. A positive-density lower bound remains open. Complete analytic proofs and reproducible finite checks are included. Human author of record and responsible depositor: Hongju Liu. Substantial ChatGPT (OpenAI) assistance under human direction is disclosed. Internal review; not externally peer reviewed. Adjacent first-party research, not an amendment of the Trinity Accord.