What a reciprocal pairing is worth: three ways to the small half of a palindromic spectrum

At its defaults it draws what a reciprocal pairing is worth: three ways to the small half of a palindromic spectrum. A palindromic quadratic of size 6 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 5.52·10⁻¹⁵ at every stop and the small ones to 1.041·10⁻⁷ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.89·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 6.538·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

pairing-use 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.

A palindromic quadratic of size 6 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 5.52·10⁻¹⁵ at every stop and the small ones to 1.041·10⁻⁷ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.89·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 6.538·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

A palindromic quadratic of size 6 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 5.52·10⁻¹⁵ at every stop and the small ones to 1.041·10⁻⁷ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.89·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 6.538·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

A palindromic quadratic of size 3 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 2.54·10⁻¹⁵ at every stop and the small ones to 3.242·10⁻¹³ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 2.62·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 5.871·10⁻⁷, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

A palindromic quadratic of size 4 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 3.46·10⁻¹⁵ at every stop and the small ones to 1.626·10⁻¹¹ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 1.56·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 3.473·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

A palindromic quadratic of size 8 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 4.94·10⁻¹⁵ at every stop and the small ones to 4.533·10⁻⁴ at the far end. Taking the reciprocals of the large ones instead returns the small ones to 2.82·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 9.185·10⁻⁶, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

A palindromic quadratic of size 10 whose spectrum spreads by two decades for every decade of b. The general solver returns the large eigenvalues to 7.31·10⁻¹⁵ at every stop and the small ones to 7.925 at the far end. Taking the reciprocals of the large ones instead returns the small ones to 4.56·10⁻¹⁵ — seven orders, for a division, out of a symmetry the solver discarded. The structure-preserving route, which enforces the pairing exactly, is the worst of the three on the eigenvalues themselves: 2.468·10⁻⁵, because each pair comes from one computed number and both members carry its error. Preserving a symmetry and using one are different acts with different prices.

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.

14 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 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.