The largest rank in each partition, and what refusing to compress a touching pair costs

At its defaults it draws the largest rank in each partition, and what refusing to compress a touching pair costs. The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

partition-ranks is one function in lib/figures/hodlr.js — the partition — a rule that reads four numbers a pair and no entry of the matrix. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

The strong rule's worst block is rank 5 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 9, 10, 12, 13, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 67,968 numbers against 61,440 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

The strong rule's worst block is rank 7 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 12, 14, 16, 18, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 86,784 numbers against 79,872 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

The strong rule's worst block is rank 4 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 7, 8, 9, 10, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 58,560 numbers against 49,152 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

The strong rule's worst block is rank 3 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 5, 6, 7, 7, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 49,152 numbers against 38,912 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

The strong rule's worst block is rank 2 at every size — one number across a factor of eight — because it never compresses a pair of clusters that touch. The weak rule compresses them and its worst rank climbs 3, 3, 4, 4, at about one per doubling, which is the touching block's logarithm arriving inside a whole partition. That is what the test buys. What it costs is on the badge: at every size measured, the finer partition stores MORE — 39,744 numbers against 25,600 at n = 512 — because it pays in blocks what it saves in rank, and the blocks near the diagonal are dense. The bounded rank is an asymptotic argument and the sizes here are not asymptotic.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

16 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls storageAgainstSize, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.