How nearly the constraint is satisfied, when a sketch is allowed to see it and when it is not

At its defaults it draws how nearly the constraint is satisfied, when a sketch is allowed to see it and when it is not. Sketching reduces a least-squares problem to 8 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.68·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 1.747·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

sketch-feasibility is one function in lib/figures/lse.js — a constraint as a weight — the limit, the ceiling that belongs to the solver, and the order of the rows. 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.

Sketching reduces a least-squares problem to 8 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.68·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 1.747·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

Sketching reduces a least-squares problem to 8 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.68·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 1.747·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

Sketching reduces a least-squares problem to 24 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.19·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 8.769·10⁻⁹ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

Sketching reduces a least-squares problem to 16 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.68·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 1.259·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

Sketching reduces a least-squares problem to 4 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 4.76·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 5.254·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

Sketching reduces a least-squares problem to 6 rows and keeps its minimiser within (1 + ε) of the original's. That is a statement about a norm, and an equality constraint is a statement that a quantity is zero, which no multiplicative distortion preserves. Kept out of the sketch and imposed exactly, the constraint holds to 1.68·10⁻¹⁶ at every weight. Written as a weight and sketched along with everything else, it holds at 1.841·10⁻⁸ at τ = 10⁸ — and the slope is the finding: the violation falls as 1/τ, one power, where the unsketched weighted problem's falls as 1/τ². The sketch costs exactly half the decades the weight was buying, and buying them back means squaring τ, which the previous figure says the arithmetic stops allowing.

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.

10 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.

It draws a decomposition and prints its residual. It calls weightedFeasibility, sketchedLse, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

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.