szego-approach

At its defaults it draws κ of the ρ = 0.8 toeplitz family, against the limit it never reaches. The condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.8, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 81 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 98.9% of it at n = 128.

szego-approach is one function in lib/figures/structure.js — structure — the matrix that is one row, and the solver that cannot see it. 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 condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.8, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 81 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 98.9% of it at n = 128.

The condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.8, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 81 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 98.9% of it at n = 128.

The condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.95, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 1521 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 87.9% of it at n = 128.

The condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.5, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 9 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 99.9% of it at n = 128.

The condition number of the n×n section of the Kac–Murdock–Szegő matrix ρ^|i−j| at ρ = 0.9, plotted against the size on logarithmic axes, with Szegő's asymptotic value ((1+ρ)/(1−ρ))² = 361 drawn as a horizontal line. The measured curve climbs towards it from below and reaches 96.0% of it at n = 128.

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.

14 distinct claims across 5 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 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 52 of 99 generators — 37 print a residual and 15 are exempt with a published reason; 47 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.