cholesky-stops

At its defaults it draws the pivots cholesky computes on a saddle-point matrix, and the one it stops at — row 11 of 14. Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -328.56, and the pivot the routine actually reports is -328.56 — the same number to 13 digits. The failure has an address and a value, and both were available before the factorisation started.

cholesky-stops 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.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -328.56, and the pivot the routine actually reports is -328.56 — the same number to 13 digits. The failure has an address and a value, and both were available before the factorisation started.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -3.7312, and the pivot the routine actually reports is -3.7312 — the same number to 15 digits. The failure has an address and a value, and both were available before the factorisation started.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -328.56, and the pivot the routine actually reports is -328.56 — the same number to 13 digits. The failure has an address and a value, and both were available before the factorisation started.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -0.082076, and the pivot the routine actually reports is -0.082076 — the same number to 16 digits. The failure has an address and a value, and both were available before the factorisation started.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -3.1551·10⁶, and the pivot the routine actually reports is -3.1551·10⁶ — the same number to 9 digits. The failure has an address and a value, and both were available before the factorisation started.

Every pivot Cholesky forms on K = [[H, Aᵀ], [A, 0]], in order, with the 10 taken inside H drawn as positive and the 11-th, which is not, drawn as the failure. It is not a near miss and it is not a rounding accident: the leading n × n block is positive definite, so the first 10 pivots are positive at any conditioning, and the next one is 0 − a₁ᵀH⁻¹a₁ where a₁ᵀ is the first row of A. That equals −(AH⁻¹Aᵀ)₁₁ = -3.189·10⁴, and the pivot the routine actually reports is -3.189·10⁴ — the same number to 11 digits. The failure has an address and a value, and both were available before the factorisation started.

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 choleskyPivots, schurComplement, 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.