Count a shared summary instance once in the failure-probability split (#669 follow-up) #711

Problem

#669 part b sizes every approximate input of a query for an equal share of its failure probability: with n approximate inputs, each gets δ/n (and δ/(n·G) over a whole result of G values). When two inputs read the same summary instance (for example quantile(0.99, …) - quantile(0.5, …) answered from one KLL that Pass 2 shares, or several statistics from one UnivMon), that instance is counted once per input. The union bound over the inputs then counts the same failure event several times, so the shared summary is sized deeper than it needs to be.

#669's derivation (§2, "Shared summary") notes that a summary sized for a simultaneous guarantee over all of its readouts fails once, not once per input.

Goal

Count a shared summary instance once in the failure-probability split, where the summary's guarantee holds for all of its readouts simultaneously:

• Stage 1 composition: n counts distinct summary instances whose readouts are simultaneous, not inputs.
• Stage 3 sizing for δ/(n·G): the same count.
• Only for families whose guarantee is simultaneous over the readouts in question (for example KLL over several quantiles of one stream); per-readout guarantees keep counting each readout.

Out of scope

Unequal splits; the decision for #669b is an equal split only.

Related: #669, #509 (Pass 2 sharing).