κ of the 0.8 kernel on m×m grids, against its 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: 0.5
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Condition number against the grid side on a logarithmic vertical axis, with the asymptotic value ((1+ρ)/(1−ρ))⁴ = 81 drawn as a horizontal line. The measured values are 31, 46, 56, 62, reaching 76.7% of the limit on the largest grid — where the one-dimensional section of the same length reaches far more.
rho: 0.9
The arguments are the ones Four orders of conditioning, and four steps passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Condition number against the grid side on a logarithmic vertical axis, with the asymptotic value ((1+ρ)/(1−ρ))⁴ = 130321 drawn as a horizontal line. The measured values are 3303, 7725, 12869, 18357, reaching 14.1% 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 The matrix that is one row passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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 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.
9 distinct claims across 4 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 — checked 4 times
and climbs towards it at m = 6 — checked 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 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.
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.
Four orders of conditioning, and four steps
On a 10×10 grid the two-dimensional kernel's condition number runs from 62 at ρ = 0.5 to 818,561 at ρ = 0.98. The preconditioned step count over the same range runs 18, 21, 21, 22, 21, 19, 18, and the count of eigenvalues the preconditioner actually brings within half a unit of one does not move at all — it is 9, 11, 13, 17 at every correlation the figure will draw.
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.