Bounded surprise forces every branch total to be at most the mean plus the slack

Bounded surprise pins the total of every branch.

Let $C$ be a chunk system with online escapes (ChunkSystemB) on a metric space with marked points $s,t$, with sizes $c_0,\dots,c_{m-1}$ adapted to its filtration, and with trivial initial information (the time-$0$ history is the same for every outcome). Write $\Sigma(\omega)=\sum_i c_i(\omega)$ for the total size of the branch $\omega$, $\mathbb{E}[\Sigma]$ for its expectation, and $F_h=\mathbb{E}\bigl[\sum_{j\ge h}c_j\mid \mathcal F_h\bigr]$ for the conditional expected remaining size (condRemaining).

The bounded-surprise property with slack $c_{\max}$ (BoundedSurprise) demands, in every branch and for every window $[a,b)$ of chunk indices,

Statement. If $C$ has trivial initial information and satisfies bounded surprise with slack $c_{\max}$, then for every outcome $\omega$

Why this matters. Since $c_j$ is $\mathcal F_j$-measurable, one has $F_a=\mathbb{E}[\Sigma\mid\mathcal F_a]-\sum_{j<a}c_j$, so the bounded-surprise inequality for the window $[a,b)$ is equivalent to $\mathbb{E}[\Sigma\mid\mathcal F_b]-\mathbb{E}[\Sigma\mid\mathcal F_a]\le c_{\max}$: the Doob martingale of the total never rises by more than $c_{\max}$ between any two times. The statement is the case $a=0$, $b=m$ of this observation, where $F_0=\mathbb{E}[\Sigma]$ (trivial initial information) and $F_m=0$. It shows that bounded surprise with a slack of the order of a single chunk is a near-determinism condition on the total, and therefore cannot be satisfied by a chunk system whose expected total is boosted by a random race imbalance of larger order, such as the stage-2 construction of Claim 13 of Bubeck--Coester--Rabani (STOC 2023), whose total fluctuates by $\Theta(\sqrt{\kappa})$ chunk sizes. Any regrouping lemma meant to be applied to such systems must therefore use a weaker hypothesis than BoundedSurprise with slack equal to the chunk ceiling.