sketch-signal

At its defaults it draws what each round of sketch-and-deflate actually sees, for a sketch kept, half redrawn, and redrawn. A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 2.62 at the first round and 9.03·10⁻¹⁵ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 3.3, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.2845 and the redrawn one at 0.1126.

sketch-signal is one function in lib/figures/sketchreuse.js — spent randomness — the sketch whose second use sees nothing at all. 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 randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 2.62 at the first round and 9.03·10⁻¹⁵ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 3.3, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.2845 and the redrawn one at 0.1126.

A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 2.62 at the first round and 9.03·10⁻¹⁵ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 3.3, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.2845 and the redrawn one at 0.1126.

A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 1.42 at the first round and 2.26·10⁻¹⁵ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 2.6, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.5355 and the redrawn one at 0.2255.

A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 4.23 at the first round and 2.26·10⁻¹⁴ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 4.7, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.1548 and the redrawn one at 1.686·10⁻¹⁶.

A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 2.33 at the first round and 9.28·10⁻¹⁵ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 2.3, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.4114 and the redrawn one at 0.1802.

A randomised range finder forms Y = AΩ, takes Q spanning it, and deflates: A ← A − QQᵀA. With the same Ω the next sketch is A₂Ω = AΩ − QQᵀ(AΩ) = Y − QQᵀY, which is zero exactly, because Q spans Y. Measured, the kept sketch sees 3.40 at the first round and 1.42·10⁻¹⁴ at the second — the unit roundoff, which is to say nothing at all, and the subspace it produces after that is whatever the rounding error happened to point at. A fresh sketch stays at about 3.8, and redrawing half the columns sees about half as much. After 5 rounds of identical arithmetic the kept run is at 0.2358 and the redrawn one at 4.372·10⁻¹⁵.

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 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.

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 113 of 238 generators — 98 print a residual and 15 are exempt with a published reason; 125 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.