Anti-concentration gain of the coin race

The gain of the coin race. Consider the race of two level chunk systems $B_L, B_R$ (alongside head and closing systems) with $\kappa$ clamped coins, clamp $\varepsilon > 0$, sizes bounded by $c_B$, and both sides carrying a size window $c_{Lo}' \ge \varepsilon$ below all their chunk sizes. Then the expected imbalance of the consumed side masses satisfies

where $B = \kappa\,(c_B+\varepsilon)^2$ and $\gamma = c_B + \varepsilon$. The imbalance is a bounded-increment martingale up to a drift of at most $\varepsilon$ per step (the clamped coin probabilities equalize the two conditional claims), with per-step conditional variance $n_L n_R \ge c_{Lo}'^2$ inside the window; the bound follows from the anti-concentration inequality for discrete martingales with pathwise variance windows. This gain term feeds the expected-total lower bound of the race: it is the mechanism by which the level step gains a $+\sqrt{\kappa}$-order term over the plain tripling, driving the $(\log k)^2$ recursion of the BCR lower bound.