What a factorisation survives across one barrier step, against the step's reduction factor

At its defaults it draws what a factorisation survives across one barrier step, against the step's reduction factor. From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 1.02 and 2.88·10¹¹. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

barrier-reuse is one function in lib/figures/barrier.js — the barrier — a condition number sent to infinity on purpose, and the number that describes the error. 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.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 1.02 and 2.88·10¹¹. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 1.02 and 2.88·10¹¹. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 4 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 0.0527 and 1.49·10¹⁰. The reason is in the entry count: the 4 entries that moved are the 4 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 1.13 and 3.27·10¹¹. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 0.263 and 7.43·10¹⁰. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

From one interior-point iteration to the next, H, C and the sparsity pattern are identical and exactly 6 entries of the matrix change — the diagonal of the (2, 2) block, and nothing off it. That is the sparsity pattern reused for free forever. The factorisation is a different question, and the answer is a threshold: carrying the decomposition of K(μ) to K(σμ) and cleaning up with iterative refinement holds at the rounding level while σ is within a per cent of one, and by σ = 0.9 it buys one step. At the schedules an interior-point method actually uses — σ between 0.5 and 0.1 — the first reuse is already at 0.822 and 2.32·10¹¹. The reason is in the entry count: the 6 entries that moved are the 6 that dominate the matrix, and they moved by a factor of 1/σ, so the relative change in K across one step at σ = 0.1 is 9 rather than the 10⁻³ the collection's reuse essays are about.

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.

11 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 reuseThreshold, 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.