false-certificate

At its defaults it draws where conjugate gradients certifies that a positive definite matrix is indefinite. Every matrix in this grid is 30×30 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 14 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁵. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

false-certificate is one function in lib/figures/tolerance.js — tolerances — the zeros nobody meets and everybody writes. 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.

Every matrix in this grid is 30×30 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 14 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁵. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

Every matrix in this grid is 30×30 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 14 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁵. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

Every matrix in this grid is 16×16 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 18 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁶. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

Every matrix in this grid is 48×48 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 18 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁶. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

Every matrix in this grid is 40×40 and positive definite by construction — its spectrum is κ^(−i/(n−1)), so the smallest eigenvalue is 1/κ and none of them is negative. A filled cell is a run in which pᵀAp came out non-positive and the iteration produced a direction it would report as a proof of indefiniteness. 11 of the 72 runs did. The region is a staircase whose top edge is at twelve significand bits and whose left edge, at eight, is at κ = 10⁵. Nothing rounded incorrectly anywhere: every comparison was performed exactly as written, on a number that was computed as accurately as the format allows.

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.

15 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 falseCertificateGrid, 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.