What one symmetric permutation does to the storage, on a matrix it does not change

At its defaults it draws what one symmetric permutation does to the storage, on a matrix it does not change. The same 256 × 256 matrix, its rows and columns renumbered by one permutation and its columns by the same one. The condition number is 24.3948 either way, to eight digits; the Frobenius norm is 6139.964 either way, to twelve. Nothing a norm can see has moved. Clustered, the partition stores 27,008 numbers. Shuffled, the admissibility test finds no admissible pair anywhere — every cluster of a shuffled numbering spans the whole interval, so every q is infinite — and the format degenerates to dense storage exactly. The rule with no test to fail does worse than that: it compresses every off-diagonal block regardless, gets ranks up to 119 out of 128, and stores 118,208 numbers — 1.80 times the matrix it was compressing. A rank-119 factorisation of a 128-column block is a more expensive way to write down the block than the block.

ordering-cost 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 same 256 × 256 matrix, its rows and columns renumbered by one permutation and its columns by the same one. The condition number is 24.3948 either way, to eight digits; the Frobenius norm is 6139.964 either way, to twelve. Nothing a norm can see has moved. Clustered, the partition stores 27,008 numbers. Shuffled, the admissibility test finds no admissible pair anywhere — every cluster of a shuffled numbering spans the whole interval, so every q is infinite — and the format degenerates to dense storage exactly. The rule with no test to fail does worse than that: it compresses every off-diagonal block regardless, gets ranks up to 119 out of 128, and stores 118,208 numbers — 1.80 times the matrix it was compressing. A rank-119 factorisation of a 128-column block is a more expensive way to write down the block than the block.

The same 128 × 128 matrix, its rows and columns renumbered by one permutation and its columns by the same one. The condition number is 20.8802 either way, to eight digits; the Frobenius norm is 2155.267 either way, to twelve. Nothing a norm can see has moved. Clustered, the partition stores 10,112 numbers. Shuffled, the admissibility test finds no admissible pair anywhere — every cluster of a shuffled numbering spans the whole interval, so every q is infinite — and the format degenerates to dense storage exactly. The rule with no test to fail does worse than that: it compresses every off-diagonal block regardless, gets ranks up to 57 out of 64, and stores 28,096 numbers — 1.71 times the matrix it was compressing. A rank-119 factorisation of a 128-column block is a more expensive way to write down the block than the block.

The same 256 × 256 matrix, its rows and columns renumbered by one permutation and its columns by the same one. The condition number is 24.3948 either way, to eight digits; the Frobenius norm is 6139.964 either way, to twelve. Nothing a norm can see has moved. Clustered, the partition stores 27,008 numbers. Shuffled, the admissibility test finds no admissible pair anywhere — every cluster of a shuffled numbering spans the whole interval, so every q is infinite — and the format degenerates to dense storage exactly. The rule with no test to fail does worse than that: it compresses every off-diagonal block regardless, gets ranks up to 119 out of 128, and stores 118,208 numbers — 1.80 times the matrix it was compressing. A rank-119 factorisation of a 128-column block is a more expensive way to write down the block than the block.

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.

12 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, orderComparison, 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.