The largest train rank of a 5-index reciprocal tensor on 8 points a side, against the accuracy asked for

At its defaults it draws the largest train rank of a 5-index reciprocal tensor on 8 points a side, against the accuracy asked for. Every point is a decomposition that met its tolerance: the measured errors are 0.0041, 4.2·10⁻⁴, 3.4·10⁻⁵, 3.3·10⁻⁷, 5.7·10⁻⁹, 6.6·10⁻¹², 1.6·10⁻¹³ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 7, 8, 10, 11 and the storage 264, 448, 680, 1,160, 1,664, 2,208, 2,504, against 32,768 entries. That is 0.80 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

train-accuracy is one function in lib/figures/ttrain.js — the train — one rank per cut of the index list, and the iterate that must be cut back. 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 point is a decomposition that met its tolerance: the measured errors are 0.0041, 4.2·10⁻⁴, 3.4·10⁻⁵, 3.3·10⁻⁷, 5.7·10⁻⁹, 6.6·10⁻¹², 1.6·10⁻¹³ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 7, 8, 10, 11 and the storage 264, 448, 680, 1,160, 1,664, 2,208, 2,504, against 32,768 entries. That is 0.80 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

Every point is a decomposition that met its tolerance: the measured errors are 0.0012, 4.2·10⁻⁵, 4.2·10⁻⁵, 8.3·10⁻⁷, 1.2·10⁻¹⁵, 1.2·10⁻¹⁵, 1.2·10⁻¹⁵ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 4, 5, 6, 6, 6 and the storage 90, 144, 144, 210, 288, 288, 288, against 216 entries — which the train exceeds at the tight end, so the format stops being a compression before it stops working. That is 0.30 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

Every point is a decomposition that met its tolerance: the measured errors are 0.0023, 1.3·10⁻⁴, 5.1·10⁻⁶, 1.4·10⁻⁷, 2.2·10⁻⁹, 8.6·10⁻¹⁶, 8.6·10⁻¹⁶ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 6, 7, 8, 8 and the storage 120, 192, 280, 384, 504, 640, 640, against 512 entries — which the train exceeds at the tight end, so the format stops being a compression before it stops working. That is 0.50 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

Every point is a decomposition that met its tolerance: the measured errors are 0.0034, 2.5·10⁻⁴, 1.4·10⁻⁵, 5.7·10⁻⁷, 4.1·10⁻¹⁰, 5.7·10⁻¹², 1.7·10⁻¹⁵ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 6, 8, 9, 10 and the storage 150, 240, 350, 480, 800, 990, 1,200, against 1,000 entries — which the train exceeds at the tight end, so the format stops being a compression before it stops working. That is 0.70 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

Every point is a decomposition that met its tolerance: the measured errors are 0.0045, 3.9·10⁻⁴, 2.7·10⁻⁵, 6.4·10⁻⁸, 2.2·10⁻⁹, 6·10⁻¹¹, 1.5·10⁻¹⁴ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 7, 8, 9, 11 and the storage 180, 288, 420, 756, 960, 1,188, 1,716, against 1,728 entries. That is 0.80 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

Every point is a decomposition that met its tolerance: the measured errors are 0.0036, 2.9·10⁻⁴, 1.8·10⁻⁵, 2.6·10⁻⁷, 5.4·10⁻⁹, 3.6·10⁻¹¹, 1.4·10⁻¹⁴ against tolerances of 10⁻², 10⁻³, 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². The rank runs 3, 4, 5, 7, 8, 9, 11 and the storage 192, 320, 480, 768, 1,008, 1,280, 1,536, against 4,096 entries. That is 0.80 of rank per decade of accuracy over the whole range — a constant, with no cliff and no regime where a digit costs more than the last one, which is the same shape the hierarchy field measured for a kernel matrix and is not something either field's geometry promises.

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.

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.

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.