The Kronecker spectrum of the inverse of a Kronecker sum on 8 points a side

At its defaults it draws the kronecker spectrum of the inverse of a kronecker sum on 8 points a side. A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 8 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.19, 0.0262, 0.00214 of the first, and the number of Kronecker terms needed runs 3, 5, 6, 7, 8, 8 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.50 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

kron-inverse is one function in lib/figures/tensor.js — an index that is a tuple — a matrix that is d small ones, and the inverse that is nearly one. 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.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 8 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.19, 0.0262, 0.00214 of the first, and the number of Kronecker terms needed runs 3, 5, 6, 7, 8, 8 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.50 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 8 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.19, 0.0262, 0.00214 of the first, and the number of Kronecker terms needed runs 3, 5, 6, 7, 8, 8 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.50 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 4 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.131, 0.00641, 9·10⁻⁵ of the first, and the number of Kronecker terms needed runs 2, 3, 4, 4, 4, 4 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.20 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 11 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.205, 0.045, 0.00694 of the first, and the number of Kronecker terms needed runs 3, 5, 7, 8, 10, 11 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.80 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 6 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.171, 0.0167, 0.000814 of the first, and the number of Kronecker terms needed runs 3, 4, 5, 6, 6, 6 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.30 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

A Kronecker product B ⊗ C, read as a four-index array and cut between its two index pairs, is exactly rank one. The inverse of T ⊕ T is rank 9 at the same cut, so the format is not closed under inversion — which is why a solve in it goes through the eigenbasis rather than through an inverse. What the spectrum says is that it is nearly closed: the singular values are 1, 0.199, 0.034, 0.00373 of the first, and the number of Kronecker terms needed runs 3, 5, 6, 7, 8, 9 at 10⁻², 10⁻⁴, 10⁻⁶, 10⁻⁸, 10⁻¹⁰ and 10⁻¹². That is 0.60 terms a decade — the same shape, and nearly the same number, as the 0.554 columns a decade the hierarchy field measures for a kernel block, arrived at from a different direction entirely.

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.

6 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.