Two kernels on identical geometry, as the picture is scaled up at a fixed wavelength

At its defaults it draws two kernels on identical geometry, as the picture is scaled up at a fixed wavelength. Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 40 needs 12, 16, 22, 33, 53, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 3.2, 6.4, 12.7, 25.5, 50.9 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

oscillatory-scale 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 parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 40 needs 12, 16, 22, 33, 53, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 3.2, 6.4, 12.7, 25.5, 50.9 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 40 needs 12, 16, 22, 33, 53, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 3.2, 6.4, 12.7, 25.5, 50.9 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 20 needs 9, 12, 16, 22, 33, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 1.6, 3.2, 6.4, 12.7, 25.5 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 80 needs 16, 22, 33, 53, 92, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 6.4, 12.7, 25.5, 50.9, 101.9 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 5 needs 6, 7, 9, 12, 16, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 0.4, 0.8, 1.6, 3.2, 6.4 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

Two parallel segments of length L a distance 2L apart, so q = 0.50 at every point on this sweep and nothing about the arrangement changes. 1/r needs 6, 6, 6, 6, 6 columns — one number. cos(κr)/r at κ = 10 needs 7, 9, 12, 16, 22, growing like a power of the size with no size at which it stops, because what decides it is the number of wavelengths across the pair — 0.8, 1.6, 3.2, 6.4, 12.7 — and no ratio of lengths can see that. The segments are parallel rather than collinear on purpose: in one dimension cos(κ|x − y|) obeys an addition formula and is exactly rank two, which makes the collinear version of this look like a confirmation of the smooth case and is an artefact of the arrangement.

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.

10 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 rankAgainstScale, 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.