The 14 eigenvalues of a saddle-point matrix with 10 unknowns and 4 constraints, inside their closed-form brackets

At its defaults it draws the 14 eigenvalues of a saddle-point matrix with 10 unknowns and 4 constraints, inside their closed-form brackets. K = [[H, Aᵀ], [A, 0]] with κ(H) = 100 and κ(A) = 100. The shaded bands are the Rusten–Winther brackets, computed from four numbers — the extreme eigenvalues of H and the extreme singular values of A — before the matrix was assembled: negative eigenvalues in [-0.995, -10·10⁻⁵] and positive ones in [0.01, 1.62]. The marks are the computed spectrum. There are exactly 10 above zero and 4 below, which is Sylvester's law of inertia and not a property of this matrix: K is congruent to blkdiag(H, −AH⁻¹Aᵀ), both blocks are definite, and congruence preserves signs. The two groups are separated by a gap containing zero, at a ratio of 39.3 between the innermost positive eigenvalue and the innermost negative one.

saddle-inertia is one function in lib/figures/kkt.js — the matrix a constraint makes — an inertia known before assembly, and a failure with an address. 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.

K = [[H, Aᵀ], [A, 0]] with κ(H) = 100 and κ(A) = 100. The shaded bands are the Rusten–Winther brackets, computed from four numbers — the extreme eigenvalues of H and the extreme singular values of A — before the matrix was assembled: negative eigenvalues in [-0.995, -10·10⁻⁵] and positive ones in [0.01, 1.62]. The marks are the computed spectrum. There are exactly 10 above zero and 4 below, which is Sylvester's law of inertia and not a property of this matrix: K is congruent to blkdiag(H, −AH⁻¹Aᵀ), both blocks are definite, and congruence preserves signs. The two groups are separated by a gap containing zero, at a ratio of 39.3 between the innermost positive eigenvalue and the innermost negative one.

For a 10 × 4 saddle-point problem at a constrained minimum whose fourth constraint has singular value σ, the largest σ on a quarter-decade grid at which each route's count is first not (10, 4, 0), median of eight draws, against the curvature h of H along that constraint's weak direction, on logarithmic axes. The Bunch–Kaufman LDLᵀ count first fails at 1.8·10⁻¹¹ for h = -1·10⁻⁴, 1.8·10⁻¹⁰ for h = -0.01, 1.4·10⁻⁹ for h = -1, 1.8·10⁻⁸ for h = -100 — a fitted slope of 0.49 against |h|, with the dashed line of slope one half through the value at h = −1. The unpivoted elimination fails slightly higher. The count through an orthonormal null-space basis is right down to rounding, and a rank test at tolerance 10·u·‖A‖ drops the constraint only at 1.1·10⁻¹⁵.

For σ from 1 down to 10⁻¹⁶ in half decades, eight draws of a 10 × 4 problem at a constrained minimum whose fourth constraint has singular value σ and whose Hessian has curvature -1 along that constraint's weak direction. Each bar stacks the draws whose count is the full problem's verdict, the verdict of the problem with the weak constraint dropped, and neither. The upper panel is a Bunch–Kaufman LDLᵀ of K, the lower one the count through an orthonormal null-space basis. The LDLᵀ count is the full verdict on every draw down to 10⁻⁸, and below that it is split between the full verdict and the dropped one at every σ, without settling.

For σ from 1 down to 10⁻¹⁶ in half decades, eight draws of a 10 × 4 problem at a constrained minimum whose fourth constraint has singular value σ and whose Hessian has curvature 1 along that constraint's weak direction. Each bar stacks the draws whose count is the full problem's verdict, the verdict of the problem with the weak constraint dropped, and neither. The upper panel is a Bunch–Kaufman LDLᵀ of K, the lower one the count through an orthonormal null-space basis. The LDLᵀ count is the full verdict on every draw down to 10⁻⁸, and below that it is split between the full verdict and counts that are neither at every σ, without settling.

The shift the inertia-correction loop settles on — 0, 10⁻⁴, then a hundred times that, then eight times each failure — for eight draws of a 10 × 4 problem that is at a constrained minimum and needs no shift, against the singular value σ of its nearly dependent fourth constraint, with curvature -1 along the constraint's weak direction. Dots are the largest shift over the eight draws and bars at the floor mark draws left unshifted. Down to σ = 10⁻⁷ no draw is shifted. Below 10⁻⁸ 22 of the 88 draws are, by as much as 2620 at σ = 10⁻¹⁰.

For σ from 1 down to 10⁻¹⁶ in half decades, eight draws of a 10 × 4 problem at a constrained minimum whose fourth constraint has singular value σ and whose Hessian has curvature -1 along that constraint's weak direction. Each bar stacks the draws whose count is the full problem's verdict, the verdict of the problem with the weak constraint dropped, and neither. The upper panel is a Bunch–Kaufman LDLᵀ of K with 10⁻⁸ subtracted along the constraint block's diagonal, the lower one the count through an orthonormal null-space basis. Every draw reads the full verdict above σ = 10⁻⁴, the square root of |h| times δc, and the dropped one below it.

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.

76 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 inertia, rustenWinther, 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.