reversal

At its defaults it draws the same four routes on a bidiagonal whose entries are all 1 or 2^12. Nothing about this matrix is graded: every entry is within a factor of 4096 of every other. Its singular values are not — 7 of them are clustered near 4096.9 and the last is 5.17·10⁻²⁶, a spread of 29 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 0.0145. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 5.55·10⁻¹⁶ in 16.

reversal is one function in lib/figures/relacc.js — relative accuracy — small compared to what, against an exact rational answer. 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.

Nothing about this matrix is graded: every entry is within a factor of 4096 of every other. Its singular values are not — 7 of them are clustered near 4096.9 and the last is 5.17·10⁻²⁶, a spread of 29 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 0.0145. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 5.55·10⁻¹⁶ in 16.

Nothing about this matrix is graded: every entry is within a factor of 4096 of every other. Its singular values are not — 7 of them are clustered near 4096.9 and the last is 5.17·10⁻²⁶, a spread of 29 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 0.0145. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 5.55·10⁻¹⁶ in 16.

Nothing about this matrix is graded: every entry is within a factor of 16 of every other. Its singular values are not — 7 of them are clustered near 16.929 and the last is 3.711·10⁻⁹, a spread of 10 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 1.6·10⁻¹³. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 8.66·10⁻¹⁶ in 17.

Nothing about this matrix is graded: every entry is within a factor of 256 of every other. Its singular values are not — 7 of them are clustered near 256.92 and the last is 1.388·10⁻¹⁷, a spread of 19 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 1.32·10⁻⁷. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 7.8·10⁻¹⁶ in 17.

Nothing about this matrix is graded: every entry is within a factor of 64 of every other. Its singular values are not — 7 of them are clustered near 64.925 and the last is 2.273·10⁻¹³, a spread of 14 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 9.44·10⁻¹⁰. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 1.67·10⁻¹⁵ in 17.

Nothing about this matrix is graded: every entry is within a factor of 1024 of every other. Its singular values are not — 7 of them are clustered near 1024.9 and the last is 8.47·10⁻²², a spread of 24 decades. Every ordering from the graded family turns over. One-sided Jacobi, which held everything there, returns the 7 large values exactly and σ_min wrong by a relative 3.39·10⁻⁵. The zero-shift sweep, which has the theorem behind it, does not converge at all — 400 sweeps and still running. The shifted sweep, which the theorem is a warning about, gets every value to 9.99·10⁻¹⁶ in 16.

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.

11 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 viaJacobi, viaNormalEquations, bidiagQR, exactSingularValues, 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.