The 12 eigenvalues of a palindromic quadratic, in 6 pairs whose products are one

At its defaults it draws the 12 eigenvalues of a palindromic quadratic, in 6 pairs whose products are one. λ²A + λB + Aᵀ with A and B symmetric reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. The axis is log₁₀|λ|, the marks are the computed eigenvalues and each arc joins a pair; the vertical line is |λ| = 1, about which the whole picture is symmetric. The spectrum spans 9.56 decades at b = 10⁴. Measured, the general solver's spectrum departs from being closed under λ ↦ 1/λ by 3.27·10⁻⁹, and the structured route's — one symmetric generalised eigenvalue problem of size 6 rather than an unsymmetric one of size 12 — by 2.21·10⁻¹⁶, which is the rounding level and is by construction.

reciprocal-pairs is one function in lib/figures/palindromic.js — reciprocal pairs — a symmetry the solver discards, and the division that is worth seven orders. 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.

λ²A + λB + Aᵀ with A and B symmetric reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. The axis is log₁₀|λ|, the marks are the computed eigenvalues and each arc joins a pair; the vertical line is |λ| = 1, about which the whole picture is symmetric. The spectrum spans 9.56 decades at b = 10⁴. Measured, the general solver's spectrum departs from being closed under λ ↦ 1/λ by 3.27·10⁻⁹, and the structured route's — one symmetric generalised eigenvalue problem of size 6 rather than an unsymmetric one of size 12 — by 2.21·10⁻¹⁶, which is the rounding level and is by construction.

λ²A + λB + Aᵀ with A and B symmetric reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. The axis is log₁₀|λ|, the marks are the computed eigenvalues and each arc joins a pair; the vertical line is |λ| = 1, about which the whole picture is symmetric. The spectrum spans 9.56 decades at b = 10⁴. Measured, the general solver's spectrum departs from being closed under λ ↦ 1/λ by 3.27·10⁻⁹, and the structured route's — one symmetric generalised eigenvalue problem of size 6 rather than an unsymmetric one of size 12 — by 2.21·10⁻¹⁶, which is the rounding level and is by construction.

λ²A + λB + Aᵀ with A and B symmetric reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. The axis is log₁₀|λ|, the marks are the computed eigenvalues and each arc joins a pair; the vertical line is |λ| = 1, about which the whole picture is symmetric. The spectrum spans 3.56 decades at b = 10. Measured, the general solver's spectrum departs from being closed under λ ↦ 1/λ by 1.02·10⁻¹⁴, and the structured route's — one symmetric generalised eigenvalue problem of size 6 rather than an unsymmetric one of size 12 — by 1.74·10⁻¹⁶, which is the rounding level and is by construction.

λ²A + λB + Aᵀ with A and B symmetric reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. The axis is log₁₀|λ|, the marks are the computed eigenvalues and each arc joins a pair; the vertical line is |λ| = 1, about which the whole picture is symmetric. The spectrum spans 17.6 decades at b = 10⁸. Measured, the general solver's spectrum departs from being closed under λ ↦ 1/λ by 9.08·10⁻⁸, and the structured route's — one symmetric generalised eigenvalue problem of size 6 rather than an unsymmetric one of size 12 — by 2.16·10⁻¹⁶, which is the rounding level and is by construction.

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.

15 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 qepEigen, cayleyRoute, 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.