rank-size
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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
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.
logEps: -8
The arguments are the ones A block nobody can call sparse passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
logEps: -12
The arguments are the ones A block nobody can call sparse passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Both blocks are of the kernel 1/r; both are 32, 64, 128, 256 points a side; both are truncated at 10⁻¹². The admissible pair — [0, 1] against [2, 3] — needs 7, 7, 7, 7 columns, which is one number. The touching pair — [0, 1] against [1, 2] — needs 13, 15, 17, 19, 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.
logEps: -6
The arguments are the ones The kernel with nothing to compress passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
logEps: -14
The arguments are the ones The size the rank does not notice passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
logEps: -2
The arguments are the ones The size the rank does not notice passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions 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.
a block size the dense SVD below is affordable at
a kernel this file defines
an accuracy inside the range the block has singular values in
and the touching block's does
the admissible block's rank does not move across a factor of eight in size
Against the rule
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 141
of 264 generators —
126 print a residual and
15 are exempt with a published reason;
123 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A block nobody can call sparse
A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.
Structure, and the solver that cannot see itA limit the matrix never reaches
Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.
Neither sparse nor denseA rank that is a number of digits
Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.
Eigenvalues, singular values, rankThe cheap rank and what it cannot see
Almost nobody computes singular values to decide a rank. The standard substitute is QR with column pivoting, read off the diagonal of R — and there is a triangular matrix on which the greedy rule makes no interchange at all, has no better column available at any step, and reports a matrix eight orders of magnitude further from singular than it is.
Sparsity, and what elimination costsThe fill that is not independent
Eliminate both halves of a grid and what is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged. Its off-diagonal block is 11 by 12 and six columns describe it to eight digits. Renumber the separator and the same block needs all eleven.
Neither sparse nor denseThe kernel with nothing to compress
Hold the geometry fixed at q = ½, fix the wavelength, and scale the picture up by sixteen. A smooth kernel needs six columns at every scale. An oscillatory one needs twelve, sixteen, twenty-two, thirty-three, fifty-three, and there is no scale at which it stops.
Neither sparse nor denseThe same matrix, numbered twice
One symmetric permutation. The condition number is 24.3948 either way to eight digits and the Frobenius norm is 6.13996414·10³ either way to twelve. The partition that stored 27,008 numbers now finds no admissible pair anywhere and stores all 65,536, and the format that compresses regardless stores 118,208.
Neither sparse nor denseThe size the rank does not notice
Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.
Neither sparse nor denseThe test that costs what it saves
The partition that refuses to compress a touching pair keeps every rank at five while the other lets them climb from nine to thirteen. It also stores more numbers at every size measured — 67,968 against 61,440 at n = 512 — and which of those two facts matters is a question about how large the problem is going to get.
Where the flop count stopped predicting the timeWhere the format starts paying
A hierarchical solve costs 1.48 times a dense factorisation at 64 unknowns and 0.16 times it at 512. The crossover is between 64 and 128, it walks right when the accuracy is tightened, and the exponent between consecutive sizes is 2.13, 1.93, 1.74 — falling towards one and never arriving.
Neither sparse nor denseWhich pairs are allowed to be small
A hierarchical representation is a partition of the matrix into blocks, and the rule that produces it reads four numbers per pair of index clusters and not one entry of the matrix. On a 256-square it yields 112 blocks, 66 of them stored as two thin factors, none of rank above five.