moment-conditioning

At its defaults it draws what the higher moments cost, and the one division that removes it. The condition number of the block Hankel against K, on a contour of radius 6, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.28 to 2.26·10⁶ across six moments. Scaled, it runs 1.28 to 1750, a factor of 1290 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

moment-conditioning is one function in lib/figures/beyn.js — a ceiling with a knob on it — higher contour moments, and the conditioning that rises with them. 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.

The condition number of the block Hankel against K, on a contour of radius 6, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.28 to 2.26·10⁶ across six moments. Scaled, it runs 1.28 to 1750, a factor of 1290 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

The condition number of the block Hankel against K, on a contour of radius 6, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.28 to 2.26·10⁶ across six moments. Scaled, it runs 1.28 to 1750, a factor of 1290 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

The condition number of the block Hankel against K, on a contour of radius 3, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 5.67 to 3071 across six moments. Scaled, it runs 5.67 to 12.2, a factor of 251 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

The condition number of the block Hankel against K, on a contour of radius 5, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.28 to 2.26·10⁶ across six moments. Scaled, it runs 1.28 to 695, a factor of 3250 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

The condition number of the block Hankel against K, on a contour of radius 10, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.28 to 2.26·10⁶ across six moments. Scaled, it runs 1.28 to 3.37·10⁴, a factor of 67.1 better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

The condition number of the block Hankel against K, on a contour of radius 15, integrating in z and in the scaled variable (z − c)/ρ. Unscaled, ‖Aₚ‖ grows like ρᵖ, so the block Hankel of K moments is graded over ρ²ᴷ and its condition number rises at exactly the rate the ceiling does — from 1.22 to 7.37·10⁸ across six moments. Scaled, it runs 1.22 to 3952, a factor of 1.9·10⁵ better at the far end. The scaled variable is on the unit circle, so every moment is the same size; the cost is one division per quadrature point and the eigenvalues come back with c + ρ× applied.

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.

8 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 contourMoments, momentRank, 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 192 of 346 generators — 174 print a residual and 18 are exempt with a published reason; 154 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.