The exact solution of a random system is longer than the system

At its defaults it draws the exact solution of a random system is longer than the system. Ax = b with A a random integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 11 the question occupies 484 bits — 11² entries of 4 bits each — and the answer occupies 729, a ratio of 1.51. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 33 bits, while a numerator can be, at 37 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

solution-length is one function in lib/figures/integer.js — exact arithmetic — no residual to print, and a cost measured in the length of the numbers. 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.

Ax = b with A a random integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 11 the question occupies 484 bits — 11² entries of 4 bits each — and the answer occupies 729, a ratio of 1.51. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 33 bits, while a numerator can be, at 37 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

Ax = b with A a random integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 11 the question occupies 484 bits — 11² entries of 4 bits each — and the answer occupies 729, a ratio of 1.51. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 33 bits, while a numerator can be, at 37 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

Ax = b with A a hilbert integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 8 the question occupies 1216 bits — 8² entries of 19 bits each — and the answer occupies 117, a ratio of 0.0962. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 40 bits, while a numerator can be, at 16 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

Ax = b with A a wide integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 10 the question occupies 1000 bits — 10² entries of 10 bits each — and the answer occupies 2058, a ratio of 2.06. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 103 bits, while a numerator can be, at 105 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

Ax = b with A a hilbert integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 9 the question occupies 1944 bits — 9² entries of 24 bits each — and the answer occupies 193, a ratio of 0.0993. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 73 bits, while a numerator can be, at 21 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

Ax = b with A a pascal integer matrix and b chosen so the answer is a genuine vector of fractions. At n = 12 the question occupies 2880 bits — 12² entries of 20 bits each — and the answer occupies 100, a ratio of 0.0347. Every entry is a ratio of two determinants by Cramer's rule, checked here by multiplying each one back by det A and comparing it with the determinant of the column-replaced matrix as an integer — so every denominator divides det A and none of them is longer than its 1 bits, while a numerator can be, at 10 here. The answer IS that long, whatever route produced it. That is a floor no exact solver can get under, and it is why the modular route reconstructs the answer rather than eliminating its way to it.

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.