The smallest defect a regression test can catch, on a machine that will not repeat itself

At its defaults it draws the smallest defect a regression test can catch, on a machine that will not repeat itself. A 150 × 150 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 3.24·10⁻¹², relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹³ overlaps and is invisible; 10⁻¹² is the smallest that separates, at 4.61·10⁻¹². So the tolerance exists, it is bracketed between 3.24·10⁻¹² and 4.61·10⁻¹² — a factor of 1.42 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

regression-window is one function in lib/figures/race.js — what a reduction decides when the decision is discrete — a step count, a rank, and a test's tolerance. 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.

A 150 × 150 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 3.24·10⁻¹², relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹³ overlaps and is invisible; 10⁻¹² is the smallest that separates, at 4.61·10⁻¹². So the tolerance exists, it is bracketed between 3.24·10⁻¹² and 4.61·10⁻¹² — a factor of 1.42 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

A 150 × 150 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 3.24·10⁻¹², relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹³ overlaps and is invisible; 10⁻¹² is the smallest that separates, at 4.61·10⁻¹². So the tolerance exists, it is bracketed between 3.24·10⁻¹² and 4.61·10⁻¹² — a factor of 1.42 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

A 80 × 80 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 6.66·10⁻¹², relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹⁴ overlaps and is invisible; 10⁻¹³ is the smallest that separates, at 6.93·10⁻¹². So the tolerance exists, it is bracketed between 6.66·10⁻¹² and 6.93·10⁻¹² — a factor of 1.04 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

A 100 × 100 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 9.01·10⁻¹², relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹² overlaps and is invisible; 10⁻¹¹ is the smallest that separates, at 2.92·10⁻¹¹. So the tolerance exists, it is bracketed between 9.01·10⁻¹² and 2.92·10⁻¹¹ — a factor of 3.24 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

A 220 × 220 system with κ = 10⁴, solved by conjugate gradients to 10⁻¹⁰ on six partitionings. The shaded band is how far the answer moves across those six when nothing at all is wrong: 1.88·10⁻¹¹, relative. Each pair of dots is one defect — a single entry of the matrix changed by a relative δ, an assembly slip or a stale coefficient — solved on the same six machines, showing its nearest and furthest run. A defect is catchable only when its nearest run clears the band, because a test whose threshold sits inside the band fails a correct build. Everything at or below 10⁻¹¹ overlaps and is invisible; 10⁻¹⁰ is the smallest that separates, at 1.33·10⁻¹⁰. So the tolerance exists, it is bracketed between 1.88·10⁻¹¹ and 1.33·10⁻¹⁰ — a factor of 7.07 — and it is neither zero nor the 10⁻⁸ that usually gets typed.

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.

6 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 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.