orthogonality-spread

At its defaults it draws an inner product whose sign is a property of the thread count. Two columns of a 48 × 48 orthogonal factor from a Householder QR of a matrix with κ = 10^8. Their inner product is zero in exact arithmetic and its terms cancel completely, so its summation condition number is 1.09·10¹⁷. The dots are its computed value at ten partition counts, in units of 2.78·10⁻¹⁷: 5 distinct values, on both sides of zero, including exact zeros. Across the whole factor, 125 of the 1128 off-diagonal pairs have a sign that is not a property of the matrix. And the aggregate is fine: ‖QᵀQ − I‖ over the same ten partitionings moves by a factor of 1.0243, because a norm of 2304 independently rounded quantities averages their disagreements away. Report the norm; never read one entry.

orthogonality-spread is one function in lib/figures/repro.js — one sum, several answers — what a reduction's division costs, and the two policies that remove it. 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.

Two columns of a 48 × 48 orthogonal factor from a Householder QR of a matrix with κ = 10^8. Their inner product is zero in exact arithmetic and its terms cancel completely, so its summation condition number is 1.09·10¹⁷. The dots are its computed value at ten partition counts, in units of 2.78·10⁻¹⁷: 5 distinct values, on both sides of zero, including exact zeros. Across the whole factor, 125 of the 1128 off-diagonal pairs have a sign that is not a property of the matrix. And the aggregate is fine: ‖QᵀQ − I‖ over the same ten partitionings moves by a factor of 1.0243, because a norm of 2304 independently rounded quantities averages their disagreements away. Report the norm; never read one entry.

Two columns of a 48 × 48 orthogonal factor from a Householder QR of a matrix with κ = 10^8. Their inner product is zero in exact arithmetic and its terms cancel completely, so its summation condition number is 1.09·10¹⁷. The dots are its computed value at ten partition counts, in units of 2.78·10⁻¹⁷: 5 distinct values, on both sides of zero, including exact zeros. Across the whole factor, 125 of the 1128 off-diagonal pairs have a sign that is not a property of the matrix. And the aggregate is fine: ‖QᵀQ − I‖ over the same ten partitionings moves by a factor of 1.0243, because a norm of 2304 independently rounded quantities averages their disagreements away. Report the norm; never read one entry.

Two columns of a 48 × 48 orthogonal factor from a Householder QR of a matrix with κ = 10^2. Their inner product is zero in exact arithmetic and its terms cancel completely, so its summation condition number is 6.79·10¹⁶. The dots are its computed value at ten partition counts, in units of 2.78·10⁻¹⁷: 6 distinct values, on both sides of zero, including exact zeros. Across the whole factor, 112 of the 1128 off-diagonal pairs have a sign that is not a property of the matrix. And the aggregate is fine: ‖QᵀQ − I‖ over the same ten partitionings moves by a factor of 1.0181, because a norm of 2304 independently rounded quantities averages their disagreements away. Report the norm; never read one entry.

Two columns of a 48 × 48 orthogonal factor from a Householder QR of a matrix with κ = 10^12. Their inner product is zero in exact arithmetic and its terms cancel completely, so its summation condition number is 2.16·10¹⁹. The dots are its computed value at ten partition counts, in units of 6.94·10⁻¹⁸: 4 distinct values, on both sides of zero, including exact zeros. Across the whole factor, 94 of the 1128 off-diagonal pairs have a sign that is not a property of the matrix. And the aggregate is fine: ‖QᵀQ − I‖ over the same ten partitionings moves by a factor of 1.0150, because a norm of 2304 independently rounded quantities averages their disagreements away. Report the norm; never read one entry.

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.

7 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 qrHouseholder, 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.