definiteness-verdict

At its defaults it draws two conforming builds, one matrix, and 26 different verdicts in 200 tries. Forty 6 × 6 Gram matrices at each of five conditionings, each formed exactly from a tall matrix whose last column is nearly the sum of its first two. Every one of them is asked *are you positive definite* by a Cholesky, once with the inner products accumulated with a single rounding per term and once with each product rounded before it is subtracted. The bar is how many of the forty came back with different verdicts: 7 at 10⁻⁸, 2 at 10⁻⁹, 4 at 10⁻¹⁰, 10 at 10⁻¹¹, 3 at 10⁻¹². The two shaded parts split those by which build was right, settled by the signs of the leading principal minors computed in BigInt rationals — 9 to the fused form and 17 to the rounded one. So neither is the correct implementation. The verdict is a property of the build, and the answer to which build is right is a property of the matrix.

definiteness-verdict is one function in lib/figures/fma.js — one multiply the compiler removed — a determinant of one computed as zero, and a verdict that is a build's. 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.

Forty 6 × 6 Gram matrices at each of five conditionings, each formed exactly from a tall matrix whose last column is nearly the sum of its first two. Every one of them is asked *are you positive definite* by a Cholesky, once with the inner products accumulated with a single rounding per term and once with each product rounded before it is subtracted. The bar is how many of the forty came back with different verdicts: 7 at 10⁻⁸, 2 at 10⁻⁹, 4 at 10⁻¹⁰, 10 at 10⁻¹¹, 3 at 10⁻¹². The two shaded parts split those by which build was right, settled by the signs of the leading principal minors computed in BigInt rationals — 9 to the fused form and 17 to the rounded one. So neither is the correct implementation. The verdict is a property of the build, and the answer to which build is right is a property of the matrix.

Forty 6 × 6 Gram matrices at each of five conditionings, each formed exactly from a tall matrix whose last column is nearly the sum of its first two. Every one of them is asked *are you positive definite* by a Cholesky, once with the inner products accumulated with a single rounding per term and once with each product rounded before it is subtracted. The bar is how many of the forty came back with different verdicts: 7 at 10⁻⁸, 2 at 10⁻⁹, 4 at 10⁻¹⁰, 10 at 10⁻¹¹, 3 at 10⁻¹². The two shaded parts split those by which build was right, settled by the signs of the leading principal minors computed in BigInt rationals — 9 to the fused form and 17 to the rounded one. So neither is the correct implementation. The verdict is a property of the build, and the answer to which build is right is a property of the matrix.

Forty 4 × 4 Gram matrices at each of five conditionings, each formed exactly from a tall matrix whose last column is nearly the sum of its first two. Every one of them is asked *are you positive definite* by a Cholesky, once with the inner products accumulated with a single rounding per term and once with each product rounded before it is subtracted. The bar is how many of the forty came back with different verdicts: 5 at 10⁻⁸, 3 at 10⁻⁹, 6 at 10⁻¹⁰, 2 at 10⁻¹¹, 8 at 10⁻¹². The two shaded parts split those by which build was right, settled by the signs of the leading principal minors computed in BigInt rationals — 13 to the fused form and 11 to the rounded one. So neither is the correct implementation. The verdict is a property of the build, and the answer to which build is right is a property of the matrix.

Forty 10 × 10 Gram matrices at each of five conditionings, each formed exactly from a tall matrix whose last column is nearly the sum of its first two. Every one of them is asked *are you positive definite* by a Cholesky, once with the inner products accumulated with a single rounding per term and once with each product rounded before it is subtracted. The bar is how many of the forty came back with different verdicts: 3 at 10⁻⁸, 6 at 10⁻⁹, 6 at 10⁻¹⁰, 6 at 10⁻¹¹, 5 at 10⁻¹². The two shaded parts split those by which build was right, settled by the signs of the leading principal minors computed in BigInt rationals — 8 to the fused form and 18 to the rounded one. So neither is the correct implementation. The verdict is a property of the build, and the answer to which build is right is a property of the matrix.

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.

5 distinct claims across 4 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 definitenessCensus, 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 197 of 363 generators — 179 print a residual and 18 are exempt with a published reason; 166 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.