One 6 × 6 integer matrix, 3 different ranks
At its defaults it draws one 6 × 6 integer matrix, 3 different ranks. The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
rank-by-field 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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 6
The arguments are the ones A basis that describes its lattice badly passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 5
The arguments are the ones A basis that describes its lattice badly passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 5; over 𝔽2 and 𝔽3 it is 3 and 4. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 8
The arguments are the ones A prime that divides the answer passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 8; over 𝔽2 and 𝔽3 it is 6 and 7. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 4
The arguments are the ones A prime that divides the answer passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 4; over 𝔽2 and 𝔽3 it is 2 and 3. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 7
The arguments are the ones A prime that divides the answer passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 7; over 𝔽2 and 𝔽3 it is 5 and 6. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions 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.
a size to compute the rank of
and the rank modulo 3 loses the one divisible by three
every Bareiss division is exact
the assertion refuses a counterexample
the left transform is unimodular
the matrix is nonsingular over the rationals
the rank modulo 2 loses both invariants divisible by two
the rank over a finite field is never larger than the rank over ℚ
the right transform is unimodular
while every prime coprime to the invariants returns the rank over ℚ
Against the rule
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 214
of 397 generators —
194 print a residual and
20 are exempt with a published reason;
183 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A basis that describes its lattice badly
The same set of points has infinitely many bases, they are all correct, and they are not equally useful. One measurement separates them — the product of the vectors' lengths over the lattice determinant — and the determinant is the invariant the reduction may not change, which is what makes the reduction checkable.
Exact arithmetic, and what it costs insteadA prime that divides the answer
A modular elimination reports a singular matrix and is telling the truth — over the field with p elements the matrix is singular. Over the rationals it is not. Nothing in the residue distinguishes the two cases, no quantity is small enough to be suspicious, and the wrong answer is a correct computation of a different question.
Exact arithmetic, and what it costs insteadAn exact answer to a measured problem
The residual is the zero vector, nothing was rounded at any step, and the answer is wrong in its first digit. Data accurate to fourteen places, an exact solve of the system it defines, and an error of 10⁻⁵ — because conditioning was never a statement about arithmetic and removing the arithmetic error removes none of it.
Exact arithmetic, and what it costs insteadEvery intermediate is a minor
Fraction-free elimination divides by the previous pivot at every step and the division is always exact. Not usually, not for these entries — always, because the number being divided is a determinant with that pivot as a factor, which is a theorem and is checked here against the minors themselves.
Exact arithmetic, and what it costs insteadHow many primes the answer needs
Work modulo a word-sized prime and no intermediate can exceed twenty-six bits, whatever the matrix does. The catch is that the answer must be reassembled from several such computations, and the number of them has to be fixed before the first one runs — by a theorem about how large a determinant can be, not by trying more until it settles.
Exact arithmetic, and what it costs insteadThe answer is longer than the question
An exact solution of an integer system is a vector of fractions, each of them a ratio of two determinants. So the output carries 2n long integers where the input carried n² short ones, and no algorithm can write it down more cheaply — the length of the answer is a floor under every exact solver rather than a property of one.
Exact arithmetic, and what it costs insteadThe rank depends on the ring
A floating-point rank is a decision about a threshold. Remove the arithmetic error entirely and the threshold goes away — and the answer still is not a property of the array of numbers, because one integer matrix has rank six over the rationals, five modulo three and four modulo two, with nothing rounded and nothing decided.
Exact arithmetic, and what it costs insteadWhat a determinant does not determine
Two integer matrices can have the same determinant, the same rank and the same size, and define genuinely different maps. What separates them is a list of integers each dividing the next — computed here twice, once by unimodular elimination and once from the gcds of every minor, which share no algorithm at all.