kernel-spectra

At its defaults it draws the singular values of one off-diagonal block, for four kernels on the same 96 points. Two intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

kernel-spectra is one function in lib/figures/kernel.js — one block — where a matrix with no zero entries turns out to have five useful columns. 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 intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

Two intervals that do not touch — [0, 1] and [2, 3] — and the 96 × 96 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 96 singular values above 10⁻⁸ out of 96, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

Two intervals that do not touch — [0, 1] and [2, 3] — and the 32 × 32 block between them, for four kernels. 1/r falls a factor of 53 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 32 singular values above 10⁻⁸ out of 32, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

Two intervals that do not touch — [0, 1] and [2, 3] — and the 256 × 256 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 256 singular values above 10⁻⁸ out of 256, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

Two intervals that do not touch — [0, 1] and [2, 3] — and the 128 × 128 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 128 singular values above 10⁻⁸ out of 128, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about the matrix.

Two intervals that do not touch — [0, 1] and [2, 3] — and the 64 × 64 block between them, for four kernels. 1/r falls a factor of 52 a column and is below 10⁻⁸ after 5; log r behaves the same way, for the reason the expansion makes obvious. An independent draw per entry gives 64 singular values above 10⁻⁸ out of 64, on the same size and the same density, which is what makes the other curves a measurement rather than a property of sorted numbers. The block has full algebraic rank in every case; what differs is where the numbers stop mattering, and that is a decision rather than a fact about 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 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 spectraByKernel, 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 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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.