The floor against the mass, under a partition that keeps the table's job fixed and one that does not

The floor against the mass, under a partition that keeps the table's job fixed and one that does not

The smallest counter a Space-Saving summary of 32 settles at, against the arrivals it saw, on log axes where a power law is a straight line. A round-robin shard is a sample of the whole stream and always holds more distinct keys than 32 counters can — the table's share of the shard's key space runs 0.01 to 0.21 — so nothing about the arithmetic changes across the range and the floor is linear: 2.03e-2·n^1.018, worst residual 2.7%. A hashed shard is a scaled copy — 0.077 distinct keys per arrival at every size, to three figures — and what moves instead is the table's share, from 1.33 at the small end, where 32 counters hold the shard's entire key space and the floor is 0.1, to 0.01 at the large end. A range that crosses from an exact structure to a summary is not a line: the local exponent runs from n^5.17 to n^1.19, a least-squares fit would report n^1.73 at a 441% residual, and it is refused. The two meet at the right-hand end because one shard is the whole stream either way.

Drawn at 700 × 396, wide on the page. Everything above is what floor-regime returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

5 essays call floor-regime. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.