Regrouping a chunk system, preserving trivial initial information

Let $C$ be a chunk system with online escapes on a metric space $X$ with marked points $s,t$: a finitely supported random sequence of $m$ chunks of set requests, equipped with a filtration $\\mathcal F_0\\subseteq\\mathcal F_1\\subseteq\\cdots$, adapted sizes $c_0,\\dots,c_{m-1}\\in[0,c_{\\max}]$ of expected total at least $T$, and the conditional cost bound: for every chunk index $i$, every evader $E$ and every online escape rule with escape price $p$,

Chunks produced by the recursive construction of the randomized $\\Omega(\\log^2 k)$ lower bound come in a wide range of sizes. The regrouping step of Bubeck-Coester-Rabani's Lemma 15 merges consecutive chunks into super-chunks of an almost prescribed size $\\Theta$, which is what lets the induction be applied on the next level.

Theorem. Suppose the sizes of $C$ lie in $[0,c_{\\max}]$, the initial information is trivial, the expected total satisfies $M\\Theta\\le T$ for some $M\\ge 1$ and some $\\Theta\\ge c_{\\max}$, and the conditional expected remaining size $F_h=\\mathbb E[\\sum_{j\\ge h}c_j\\mid\\mathcal F_h]$ satisfies both

pointwise. Then for every $p'\\ge p+\\Theta+3c_{\\max}$ there exists a chunk system with trivial initial information on the same space, serving the same request sequences, with exactly $M$ chunks, all of whose sizes lie in the narrow window $[\\Theta-c_{\\max},\\Theta+c_{\\max}]$, of expected total $M\\Theta$ and escape price $p'$.

The super-chunks are cut at the stopping times $h_i=\\min\\{h: F_h\\le F_0-i\\Theta\\}$, and the size assigned to the $i$-th super-chunk is $\\mathbb E[F_{h_i}-F_{h_{i+1}}\\mid\\mathcal F_{h_i}]$. Slow decrement is what keeps this quantity from undershooting $\\Theta$ by more than $c_{\\max}$. The cost bound for a super-chunk is obtained by charging the original chunks one by one up to the first moment the escape rule fires inside the super-chunk; bounded surprise is what bounds the mass of the chunks after that moment by $\\Theta+3c_{\\max}$, which is exactly the extra escape price that has been reserved.

Formalization note. Both hypotheses are branchwise strengthenings of statements that hold automatically in conditional expectation, and they are stated here explicitly because a branchwise stopping-time argument consumes them branchwise. The conclusion is phrased as non-emptiness of the type of output chunk systems.