The chunk-combining lemma: uniform windows from conditional hitting times

The chunk-combining lemma (the repaired form of Lemma 10 of Bubeck–Coester–Rabani, STOC 2023). Let $C$ be a chunk system with online escapes on a metric space $X$ with marked points $s,t$: sizes in $[c_A, c_B]$ with $0 < c_A \le c_B$, escape price $p_e$, trivial initial knowledge ($\mathcal F_0$ trivial), and total expected mass $T_0 = \sum_\omega P(\omega)\sum_i c_i(\omega)$. Assume the Doob jump bound in the descending regime: the martingale $M_h = \mathbb E[\sum_i c_i \mid \mathcal F_h]$ has pointwise jumps at most $jbS \le c_A$. Then for every target count $M > 0$ the chunks can be regrouped into $M$ windows delimited by the hitting times of the uniformly spaced levels $T_0(1 - k/M)$ of the conditional future mass $g_h = \mathbb E[\sum_{i \ge h} c_i \mid \mathcal F_h]$, yielding a chunk system with the same outcomes, weights and request sequence, whose $M$ conditional window sizes all lie within $c_B + jbS$ of the spacing $\delta = T_0/M$:

for any escape price $p' \ge p_e + \delta + (c_B + jbS)$. The expected total is preserved exactly (the tower property), so the same lower bound $T$ carries over. The proof pins the boundary overshoots by one chunk plus one Doob jump (since $jbS \le c_A$ the conditional future mass descends monotonically, so no up-crossings occur), computes the conditional window sizes by optional stopping, and charges a bailing adversary's untouched window tail — at most one spacing plus slack in conditional expectation — to the price difference $p' - p_e$, applying the input's conditional cost bound chunkwise on the quiet, still-in-window events, which are measurable in the fine filtration. This is the engine that turns the stage-constructed level-$(w+1)$ system into the standard windowed form in the induction of Lemma 6.