The rank of an admissible block and of a touching one, against how finely they are sampled

At its defaults it draws the rank of an admissible block and of a touching one, against how finely they are sampled. Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻⁸. The admissible pair — [0, 1] against [2, 3] — needs 5, 5, 5, 5 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 9, 11, 12, 13, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

rank-size is one function in lib/figures/kernel.js — one block — where a matrix with no zero entries turns out to have five useful columns. 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.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻⁸. The admissible pair — [0, 1] against [2, 3] — needs 5, 5, 5, 5 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 9, 11, 12, 13, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻⁸. The admissible pair — [0, 1] against [2, 3] — needs 5, 5, 5, 5 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 9, 11, 12, 13, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻¹⁴. The admissible pair — [0, 1] against [2, 3] — needs 9, 9, 9, 9 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 14, 16, 19, 21, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻¹⁰. The admissible pair — [0, 1] against [2, 3] — needs 6, 6, 6, 6 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 11, 13, 14, 16, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻⁶. The admissible pair — [0, 1] against [2, 3] — needs 4, 4, 4, 4 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 8, 9, 10, 11, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 0.01. The admissible pair — [0, 1] against [2, 3] — needs 2, 2, 2, 2 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 3, 4, 4, 5, climbing by about one per doubling, which is a logarithm. Neither of them grows like the block, and only one of them stops. That difference is what the admissibility test in a partition is buying, and it is why the touching pair is kept dense rather than compressed at all.

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.

5 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 rankAgainstSize, 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.