delay-branches

At its defaults it draws a spectrum with no last eigenvalue: 12 of the infinitely many solutions of a delay problem. T(λ) = A − λI + 1e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.6·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 4 of them.

delay-branches is one function in lib/figures/nlevp.js — infinitely many eigenvalues — a contour that counts, and the probe block that is a ceiling. 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.

T(λ) = A − λI + 1e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.6·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 4 of them.

T(λ) = A − λI + 1e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.6·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 4 of them.

T(λ) = A − λI + 0.5e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.2·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 4 of them.

T(λ) = A − λI + 10e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.3·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 4 of them.

T(λ) = A − λI + 2e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.8·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 6 of them.

T(λ) = A − λI + 5e^−λI for the model matrix at n = 4. The problem decouples in A's eigenvector basis, so each of its 4 eigenvalues μ contributes the scalar equation μ − λ + γe^−λ = 0, whose solutions are λ = μ + W_k(γe^−μ) — one for every branch of the Lambert W function, and therefore countably infinitely many, running off to the left along two curves. The marks are the closed form, each checked by substitution to 1.7·10⁻¹⁵. There is no computation that returns this set: the only well-posed question is which eigenvalues lie inside a region, and the circle of radius 4 contains 6 of them.

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 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 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.