Numbers stored against the number of indices, at n = 20 and rank 4: the tensor, its core, and a train

At its defaults it draws numbers stored against the number of indices, at n = 20 and rank 4: the tensor, its core, and a train. The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

tucker-storage is one function in lib/figures/hosvd.js — the higher-order svd — d matrix decompositions, and a core that cannot be diagonal. 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.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 6 it is 3.66·10¹⁵, smaller than the tensor by 2.87·10¹⁰ and still unstorable. A train is (d − 2)nr² + 2nr, which is 13,200 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 2 it is 1.05·10⁶, smaller than the tensor by 9.99·10¹⁹ and still unstorable. A train is (d − 2)nr² + 2nr, which is 1,520 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 3 it is 3.49·10⁹, smaller than the tensor by 3.01·10¹⁶ and still unstorable. A train is (d − 2)nr² + 2nr, which is 3,360 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 5 it is 9.54·10¹³, smaller than the tensor by 1.1·10¹² and still unstorable. A train is (d − 2)nr² + 2nr, which is 9,200 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.

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.

3 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.