two-dimensional-limit
At its defaults it draws κ of the 0.8 kernel on m×m grids, against its two-dimensional limit. Condition number against the grid side on a logarithmic vertical axis, with the asymptotic value ((1+ρ)/(1−ρ))⁴ = 6561 drawn as a horizontal line. The measured values are 581, 1196, 1794, 2337, reaching 35.6% of the limit on the largest grid — where the one-dimensional section of the same length reaches far more.
two-dimensional-limit is one function in lib/figures/bttb.js —
block toeplitz — the second dimension, where the cluster thins. 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.
Condition number against the grid side on a logarithmic vertical axis, with the asymptotic value ((1+ρ)/(1−ρ))⁴ = 6561 drawn as a horizontal line. The measured values are 581, 1196, 1794, 2337, reaching 35.6% of the limit on the largest grid — where the one-dimensional section of the same length reaches far more.
rho: 0.8
The arguments are the ones A limit the matrix never reaches 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.
Condition number against the grid side on a logarithmic vertical axis, with the asymptotic value ((1+ρ)/(1−ρ))⁴ = 6561 drawn as a horizontal line. The measured values are 581, 1196, 1794, 2337, reaching 35.6% of the limit on the largest grid — where the one-dimensional section of the same length reaches far more.
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.
9 distinct claims across 2 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.
the 4×4 grid is below the limit — asserted 4 times
and climbs towards it at m = 6 — asserted 3 times
a correlation whose two-dimensional limit is finite and drawable
and has not arrived at the largest grid drawn
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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 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.
Iterating, instead of factorisingThe coarse problem is a different problem
In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.
Structure, and the solver that cannot see itThe matrix that is one row
A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.
Structure, and the solver that cannot see itTwo dimensions, and the cluster that thins
The same kernel, the same averaging, the same transform — applied along two axes instead of one. In one dimension the preconditioned step count is 7, 10, 10, 10; on square grids with the same unknown counts it is 10, 18, 20, 21, and the share of the spectrum near one falls from 56% to 17%.