Matrix–vector products before the curvature test fires, against the size of the negative eigenvalue, n = 40

At its defaults it draws matrix–vector products before the curvature test fires, against the size of the negative eigenvalue, n = 40. A 40×40 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 9. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 12 and 42 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

detection-onset 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.

A 40×40 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 9. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 12 and 42 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

A 50×50 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 10. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 14 and 34 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

A 40×40 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 9. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 12 and 42 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

A 20×20 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 11. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 6 and 33 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

A 80×80 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 10. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 24 and 34 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

A 30×30 matrix whose spectrum is positive over one decade apart from a single eigenvalue at −λ. At λ = 3 conjugate gradients meets a non-positive curvature after 3 products; at λ = 0.001 it takes 10. The trend is monotone and it runs the reassuring way: a Krylov space finds large eigenvalues first, so the indefiniteness that takes longest to detect is the indefiniteness that matters least. The direction that fires the test recovers between 4 and 24 per cent of λ, so it is a certificate rather than an estimate of the eigenvalue.

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.

12 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 detectionSweep, 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.