null-basis-sweep

At its defaults it draws the reduced hessian and the answer, against how badly the basic columns were chosen. γ is the condition number of the first 4 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 83.77 to 4.871·10¹⁸ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 31.72 and 43.09 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.3173 against 8.07·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

null-basis-sweep is one function in lib/figures/nullbasis.js — the basis of a constraint — three spans of one null space, and the square between 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.

γ is the condition number of the first 4 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 83.77 to 4.871·10¹⁸ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 31.72 and 43.09 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.3173 against 8.07·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

γ is the condition number of the first 4 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 83.77 to 4.871·10¹⁸ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 31.72 and 43.09 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.3173 against 8.07·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

γ is the condition number of the first 2 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 148.6 to 2.711·10¹⁹ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 64.71 and 75.33 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.8965 against 2.01·10⁻¹⁵ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

γ is the condition number of the first 6 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 27.85 to 7.309·10¹³ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 6.175 and 20.89 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.05908 against 5.51·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

γ is the condition number of the first 3 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 277.2 to 8.131·10¹⁷ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 43.16 and 55.06 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.5617 against 3.07·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

γ is the condition number of the first 5 columns of A, and the naive rule calls exactly those columns basic. Its reduced Hessian's condition number climbs from 36.88 to 6.626·10¹⁶ across ten decades of γ — the square, because κ(ZᵀHZ) ≤ κ(H)κ(Z)² and the bound is attained. Choosing the basic columns by a column-pivoted QR instead holds it between 24.27 and 27.3 at every stop. The two forward errors, measured against a BigInt answer, follow their own condition numbers: 0.3844 against 2.1·10⁻¹⁶ at γ = 10¹⁰. κ(A) is 10⁶ throughout and never moves — nothing about the problem's difficulty changes across this axis, only which columns a one-line rule happens to pick.

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.

10 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 basisComparison, 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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.