On a constant filtration the bounded-surprise property holds with the optimal parameter $cmax=0$

The tame pair is free on a constant filtration; bounded surprise is then exact.

Let C be a chunk system with online escapes on a metric space with marked points s, t, with size ceiling cHi, and suppose the filtration is constant: every outcome shares its history at every time, i.e. C.hist h ω = C.hist h ω' for all h, ω, ω'. Then the time-h atom C.atom h ω is all of Ω, so the conditional expected remaining size F_h(ω) = condRemaining h ω is, for every level h, the plain tail sum ∑_{j ≥ h} c_j of the sizes, and is therefore independent of ω. The realised window size S_{[a,b)}(ω) = windowSize a b ω is also independent of ω, and

S_{[a,b)} = ∑_{a ≤ j < b} c_j = F_a - F_b ,

so BoundedSurprise holds with the optimal parameter cmax = 0, with equality, for every window.

The hypothesis hist 0 constant that appears throughout the BCR level invariant is the level-0 instance of the constancy assumption; this lemma isolates the observation that the extra branchwise conditions of KServer.BCRInductiveSubchunksTame — namely SlowDecrement and BoundedSurprise at cmax = 3 * 3^w / 2 — carry no content that is not already forced by hsmeas and hsize once the filtration is constant, and that the regrouping of Lemma 15 therefore needs them only for genuinely refining filtrations.

The proof uses only the definitions of atom, condRemaining, windowSize and BoundedSurprise: the atom is Finset.univ by Finset.filter_true_of_mem, so condRemaining collapses to ∑ ω, P ω * (tail sum) / ∑ ω, P ω and the denominator is 1 by hPsum, leaving a ω-independent tail sum.