Along a sequence of exactly rank-two tensors: the step's condition number, and what a fit from a random start achieves

At its defaults it draws along a sequence of exactly rank-two tensors: the step's condition number, and what a fit from a random start achieves. Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 4.5·10⁻¹³. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0118 at n = 128. Started at the answer instead, the same code stays within 1.4·10⁻¹⁰ of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

decomposition-kappa is one function in lib/figures/cpals.js — alternating least squares — an iteration that walks out of the set it is searching. 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 Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 4.5·10⁻¹³. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0118 at n = 128. Started at the answer instead, the same code stays within 1.4·10⁻¹⁰ of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 4.5·10⁻¹³. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0118 at n = 128. Started at the answer instead, the same code stays within 1.4·10⁻¹⁰ of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 2.2·10⁻¹⁶. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0103 at n = 8. Started at the answer instead, the same code stays within 1.5·10⁻¹² of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 8·10⁻¹⁵. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0117 at n = 32. Started at the answer instead, the same code stays within 8.9·10⁻¹² of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 1.1·10⁻¹¹. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0118 at n = 512. Started at the answer instead, the same code stays within 2.3·10⁻⁹ of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

Every Aₙ on this sequence *is* a rank-two tensor and its two rank-one terms are written down, so nothing here is about existence. The rising curve is the condition number of the r × r system each alternating sweep solves, which on this sequence has a closed form in n whose asymptote is 2n² — the marks are measured and the dashed line is that closed form, agreeing to 1.1·10⁻¹¹. The lower marks are what a three-hundred-sweep fit from a random start returns: 2.1·10⁻⁴ at n = 2 rising to 0.0118 at n = 1024. Started at the answer instead, the same code stays within 9.2·10⁻⁹ of it at every n — not the rounding level, because the drift from an exact start is itself about κ times the unit roundoff, but nine orders below what a random start reaches. That is the control that says the failure is the conditioning and not the implementation. A tensor away from the boundary conditions its step at 6.03.

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.

33 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.