Train ranks at each of the 5 cuts of a 6-index tensor on 6 points a side, four families

At its defaults it draws train ranks at each of the 5 cuts of a 6-index tensor on 6 points a side, four families. Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 5 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2, 2, 2. The reciprocal family climbs to 10, the product family is one everywhere, and independent normal entries reach 216, which is the largest rank the cut allows. Storage runs sinsum 120, reciprocal 1,800, product 36, noise 95,976 against 46,656 entries.

train-ranks 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.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 5 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2, 2, 2. The reciprocal family climbs to 10, the product family is one everywhere, and independent normal entries reach 216, which is the largest rank the cut allows. Storage runs sinsum 120, reciprocal 1,800, product 36, noise 95,976 against 46,656 entries.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 4 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2, 2. The reciprocal family climbs to 9, the product family is one everywhere, and independent normal entries reach 36, which is the largest rank the cut allows. Storage runs sinsum 96, reciprocal 1,206, product 30, noise 10,440 against 7,776 entries.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 2 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2. The reciprocal family climbs to 6, the product family is one everywhere, and independent normal entries reach 6, which is the largest rank the cut allows. Storage runs sinsum 48, reciprocal 288, product 18, noise 288 against 216 entries.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 5 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2, 2, 2. The reciprocal family climbs to 10, the product family is one everywhere, and independent normal entries reach 216, which is the largest rank the cut allows. Storage runs sinsum 120, reciprocal 1,800, product 36, noise 95,976 against 46,656 entries.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 6 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2, 2, 2, 2. The reciprocal family climbs to 9, the product family is one everywhere, and independent normal entries reach 64, which is the largest rank the cut allows. Storage runs sinsum 96, reciprocal 1,084, product 28, noise 25,120 against 16,384 entries.

Cut the index list after position k, put the first k indices on the rows and the rest on the columns, and take the rank of the matrix that results. There are 3 such cuts and each rank is an ordinary matrix rank. The sine of a sum of d variables has rank exactly two at every one of them, for every d, because the addition formula separates it into two terms at every cut — a rank written down rather than measured, and the computed values are 2, 2, 2. The reciprocal family climbs to 9, the product family is one everywhere, and independent normal entries reach 36, which is the largest rank the cut allows. Storage runs sinsum 72, reciprocal 720, product 24, noise 2,664 against 1,296 entries.

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.

7 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 ttSvd, 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.