The curvature pᵀAp/pᵀp along conjugate gradients on a matrix with one eigenvalue at −10^-1

At its defaults it draws the curvature pᵀap/pᵀp along conjugate gradients on a matrix with one eigenvalue at −10^-1. The matrix is 40×40, positive definite apart from a single eigenvalue at -0.1. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.43, 2.72, 1.71, 0.687 before turning negative at step 6, where it is -0.02656. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 26.6 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 37 steps.

curvature-trail is one function in lib/figures/curvature.js — negative curvature — the division that cannot be done, as an output. 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 40×40, positive definite apart from a single eigenvalue at -0.1. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.43, 2.72, 1.71, 0.687 before turning negative at step 6, where it is -0.02656. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 26.6 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 37 steps.

The matrix is 60×60, positive definite apart from a single eigenvalue at -1. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 6.27, 3.16, 2.21, 0.698 before turning negative at step 5, where it is -0.2437. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 24.4 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 42 steps.

The matrix is 40×40, positive definite apart from a single eigenvalue at -0.1. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.43, 2.72, 1.71, 0.687 before turning negative at step 6, where it is -0.02656. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 26.6 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 37 steps.

The matrix is 40×40, positive definite apart from a single eigenvalue at -0.001. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.44, 2.76, 1.83, 0.884 before turning negative at step 9, where it is -2.811·10⁻⁴. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 28.1 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 39 steps.

The matrix is 40×40, positive definite apart from a single eigenvalue at -0.01. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.44, 2.75, 1.81, 0.865 before turning negative at step 8, where it is -0.004188. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 41.9 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 38 steps.

The matrix is 40×40, positive definite apart from a single eigenvalue at -1. Conjugate gradients divides by pᵀAp at every step, and the quantity runs 5.4, 2.36, 0.566, -0.287 before turning negative at step 4, where it is -0.2868. That direction is a proof: one matrix–vector product from outside the iteration confirms it, and it recovers 28.7 per cent of the eigenvalue. MINRES on the same system never forms this quantity, meets nothing, and returns the solution in 36 steps.

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 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 cgCurvature, minres, 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.