The 216 eigenvalues of the 3-dimensional Laplacian on 6 points a side, computed against their closed form

At its defaults it draws the 216 eigenvalues of the 3-dimensional laplacian on 6 points a side, computed against their closed form. The matrix has 216 rows and 46,656 entries, and is a sum of 3 Kronecker products of one 6 × 6 matrix — 108 numbers. Its eigenvalues are every sum of 3 numbers drawn from 4sin²(kπ/2(n+1)), so the whole spectrum is written down before anything runs. The line is that closed form and the marks are what a Jacobi decomposition of the assembled matrix returns; the largest disagreement anywhere is 7.08·10⁻¹³. The smallest eigenvalue is 0.5942 and the largest 11.41, so the condition number is 19.2 — which is the part the structure does not help with.

kron-spectrum is one function in lib/figures/tensor.js — an index that is a tuple — a matrix that is d small ones, and the inverse that is nearly one. 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.

The matrix has 216 rows and 46,656 entries, and is a sum of 3 Kronecker products of one 6 × 6 matrix — 108 numbers. Its eigenvalues are every sum of 3 numbers drawn from 4sin²(kπ/2(n+1)), so the whole spectrum is written down before anything runs. The line is that closed form and the marks are what a Jacobi decomposition of the assembled matrix returns; the largest disagreement anywhere is 7.08·10⁻¹³. The smallest eigenvalue is 0.5942 and the largest 11.41, so the condition number is 19.2 — which is the part the structure does not help with.

The matrix has 216 rows and 46,656 entries, and is a sum of 3 Kronecker products of one 6 × 6 matrix — 108 numbers. Its eigenvalues are every sum of 3 numbers drawn from 4sin²(kπ/2(n+1)), so the whole spectrum is written down before anything runs. The line is that closed form and the marks are what a Jacobi decomposition of the assembled matrix returns; the largest disagreement anywhere is 7.08·10⁻¹³. The smallest eigenvalue is 0.5942 and the largest 11.41, so the condition number is 19.2 — which is the part the structure does not help with.

The matrix has 216 rows and 46,656 entries, and is a sum of 1 Kronecker products of one 216 × 216 matrix — 46656 numbers. Its eigenvalues are every sum of 1 numbers drawn from 4sin²(kπ/2(n+1)), so the whole spectrum is written down before anything runs. The line is that closed form and the marks are what a Jacobi decomposition of the assembled matrix returns; the largest disagreement anywhere is 2.62·10⁻¹³. The smallest eigenvalue is 0.0002096 and the largest 4, so the condition number is 19080 — which is the part the structure does not help with.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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 3 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 jacobiEigSym, 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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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.