description-ladder

At its defaults it draws one 40×40 matrix described four ways, with its condition number and its backward error at each. The matrix is ρ^|i−j| at ρ = 0.999, where κ = 7.878·10⁴. Described as n² entries its condition number is that; as 79 constant diagonals it is 3.301·10⁴; as the 40 a symmetric Toeplitz matrix has, 3.178·10⁴; and as the one number ρ that the matrix actually holds, 61.88. The backward error of the same computed solution runs the other way — 2.01·10⁻¹⁷, 1.15·10⁻¹⁴, 5.62·10⁻¹² — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 99 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

description-ladder is one function in lib/figures/structbe.js — structured backward error — whether the nearby problem is the same kind of problem. 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 matrix is ρ^|i−j| at ρ = 0.999, where κ = 7.878·10⁴. Described as n² entries its condition number is that; as 79 constant diagonals it is 3.301·10⁴; as the 40 a symmetric Toeplitz matrix has, 3.178·10⁴; and as the one number ρ that the matrix actually holds, 61.88. The backward error of the same computed solution runs the other way — 2.01·10⁻¹⁷, 1.15·10⁻¹⁴, 5.62·10⁻¹² — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 99 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

The matrix is ρ^|i−j| at ρ = 0.999, where κ = 7.878·10⁴. Described as n² entries its condition number is that; as 79 constant diagonals it is 3.301·10⁴; as the 40 a symmetric Toeplitz matrix has, 3.178·10⁴; and as the one number ρ that the matrix actually holds, 61.88. The backward error of the same computed solution runs the other way — 2.01·10⁻¹⁷, 1.15·10⁻¹⁴, 5.62·10⁻¹² — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 99 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

The matrix is ρ^|i−j| at ρ = 0.9, where κ = 284.2. Described as n² entries its condition number is that; as 79 constant diagonals it is 145.9; as the 40 a symmetric Toeplitz matrix has, 140.5; and as the one number ρ that the matrix actually holds, 1.65. The backward error of the same computed solution runs the other way — 1.84·10⁻¹⁷, 7.29·10⁻¹⁵, 8.74·10⁻¹⁴ — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 100 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

The matrix is ρ^|i−j| at ρ = 0.9999, where κ = 7.977·10⁵. Described as n² entries its condition number is that; as 79 constant diagonals it is 3.343·10⁵; as the 40 a symmetric Toeplitz matrix has, 3.218·10⁵; and as the one number ρ that the matrix actually holds, 613.2. The backward error of the same computed solution runs the other way — 1.56·10⁻¹⁷, 7.49·10⁻¹⁵, 1.5·10⁻¹¹ — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 100 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

The matrix is ρ^|i−j| at ρ = 0.99, where κ = 6988. Described as n² entries its condition number is that; as 79 constant diagonals it is 2939; as the 40 a symmetric Toeplitz matrix has, 2829; and as the one number ρ that the matrix actually holds, 6.798. The backward error of the same computed solution runs the other way — 4.53·10⁻¹⁷, 2.56·10⁻¹⁴, 7.51·10⁻¹³ — and at the last rung there is none: no ρ whatever has the computed answer as its exact solution, and the closest one leaves 100 per cent of the residual unexplained. The drop that matters is the last one, and it is not a linear-algebra structure at all.

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.

It draws a decomposition and prints its residual. It calls descriptionLadder, 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 113 of 219 generators — 98 print a residual and 15 are exempt with a published reason; 106 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.