A 256 × 256 kernel matrix partitioned by the strong rule, with each compressed block's rank

At its defaults it draws a 256 × 256 kernel matrix partitioned by the strong rule, with each compressed block's rank. 112 blocks: 46 kept dense and 66 stored as two thin factors, whose ranks run from 4 to 5. The rule decides one thing — whether a pair of index clusters may be compressed — and it decides it from four numbers about where those clusters sit, before any entry of the matrix is read. The strong rule refuses any pair whose clusters touch, and subdivides instead, so the diagonal is fringed with small dense blocks and no rank on the picture exceeds 5. The whole thing stores 27,008 numbers against 65,536 entries, and reproduces the matrix to 3.41·10⁻¹⁰.

block-partition 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.

112 blocks: 46 kept dense and 66 stored as two thin factors, whose ranks run from 4 to 5. The rule decides one thing — whether a pair of index clusters may be compressed — and it decides it from four numbers about where those clusters sit, before any entry of the matrix is read. The strong rule refuses any pair whose clusters touch, and subdivides instead, so the diagonal is fringed with small dense blocks and no rank on the picture exceeds 5. The whole thing stores 27,008 numbers against 65,536 entries, and reproduces the matrix to 3.41·10⁻¹⁰.

46 blocks: 16 kept dense and 30 stored as two thin factors, whose ranks run from 8 to 12. The rule decides one thing — whether a pair of index clusters may be compressed — and it decides it from four numbers about where those clusters sit, before any entry of the matrix is read. The weak rule compresses every off-diagonal block there is, including the two touching halves at the top level, whose rank is the largest number on the picture and is the one that climbs as the matrix grows. The whole thing stores 24,064 numbers against 65,536 entries, and reproduces the matrix to 1.08·10⁻⁹.

112 blocks: 46 kept dense and 66 stored as two thin factors, whose ranks run from 4 to 5. The rule decides one thing — whether a pair of index clusters may be compressed — and it decides it from four numbers about where those clusters sit, before any entry of the matrix is read. The strong rule refuses any pair whose clusters touch, and subdivides instead, so the diagonal is fringed with small dense blocks and no rank on the picture exceeds 5. The whole thing stores 27,008 numbers against 65,536 entries, and reproduces the matrix to 3.41·10⁻¹⁰.

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.

11 distinct claims across 3 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 represent, 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.