The same 16 answers, backward stable under one reading and not under another

At its defaults it draws the same 16 answers, backward stable under one reading and not under another. Every real eigenpair of an overdamped chain of 8 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.06 and 45.69, and allowing only M costs between 1.99 and 6367. At a tolerance of 10⁻¹⁵ 15 of these pairs are stable against the whole polynomial and 7 against K alone.

restriction-ratio is one function in lib/figures/graphfit.js — three approximations the data chose the shape of — support points, a restricted perturbation, and a model made of measurements. 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.

Every real eigenpair of an overdamped chain of 8 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.06 and 45.69, and allowing only M costs between 1.99 and 6367. At a tolerance of 10⁻¹⁵ 15 of these pairs are stable against the whole polynomial and 7 against K alone.

Every real eigenpair of an overdamped chain of 8 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.06 and 45.69, and allowing only M costs between 1.99 and 6367. At a tolerance of 10⁻¹⁵ 15 of these pairs are stable against the whole polynomial and 7 against K alone.

Every real eigenpair of an overdamped chain of 4 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.19 and 45.73, and allowing only M costs between 2 and 717.5. At a tolerance of 10⁻¹⁵ 8 of these pairs are stable against the whole polynomial and 3 against K alone.

Every real eigenpair of an overdamped chain of 6 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.1 and 45.73, and allowing only M costs between 2 and 2449. At a tolerance of 10⁻¹⁵ 12 of these pairs are stable against the whole polynomial and 5 against K alone.

Every real eigenpair of an overdamped chain of 10 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.04 and 45.64, and allowing only M costs between 1.99 and 1.385·10⁴. At a tolerance of 10⁻¹⁵ 19 of these pairs are stable against the whole polynomial and 8 against K alone.

Every real eigenpair of an overdamped chain of 14 masses, computed once through the first companion linearisation, and its backward error measured four ways — against perturbations of all three coefficients, of K alone, and of M alone. The unrestricted number is the one every published formula computes and it sits at the rounding level across the spectrum. Restricting the perturbation charges the same residual to less data, so it can only raise the quotient, and the factor has a closed form: (|λ|²‖M‖ + |λ|‖C‖ + ‖K‖) divided by the allowed part, checked here against the two computed numbers to 10⁻⁸. Because it carries |λ|², it is not a property of the polynomial: allowing only K to move costs between 1.02 and 45.56, and allowing only M costs between 1.99 and 4.68·10⁴. At a tolerance of 10⁻¹⁵ 28 of these pairs are stable against the whole polynomial and 15 against K alone.

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.

109 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 restrictionSweep, verdictTable, 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.