det A = 30030, and its residue at 25 primes — 5 of which divide it
At its defaults it draws det a = 30030, and its residue at 25 primes — 5 of which divide it. Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 25 primes here are unlucky, a rate of 20 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
unlucky-prime 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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 25 primes here are unlucky, a rate of 20 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
n: 6, scale: 1
The arguments are the ones A fraction recovered from one remainder 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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 25 primes here are unlucky, a rate of 20 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
n: 6, scale: 12
The arguments are the ones A fraction recovered from one remainder 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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 36 primes here are unlucky, a rate of 13.9 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
n: 8, scale: 20
The arguments are the ones A fraction recovered from one remainder 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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 44 primes here are unlucky, a rate of 11.4 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
n: 6, scale: 20
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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 44 primes here are unlucky, a rate of 11.4 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
n: 4, scale: 1
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.
Each dot is det A mod p, drawn as a fraction of p so the primes are comparable. A dot on the floor is a prime at which the elimination reports a singular matrix — and the matrix is not singular: its determinant is 30030. 5 of 25 primes here are unlucky, a rate of 20 per cent, which is what it looks like when the primes are small. At the 26-bit primes a modular determinant actually uses, a prime divides a 15-bit determinant with probability of order 2.24·10⁻⁷ — but the consequence is unchanged, and there is no way to see it from one residue. What sees it is the budget: computing more residues than Hadamard's bound requires, and refusing to reconstruct when the extra ones disagree.
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.
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.
a residue is zero exactly when the prime divides the determinant
a size to sweep primes on
every Bareiss division is exact
how far down the prime list to sweep
the construction's determinant is the product of the first six primes
the matrix is nonsingular over the rationals
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 fraction recovered from one remainder
A solution over the rationals can be computed modulo a prime power and then recovered — the residue determines the fraction uniquely, but only once the modulus is twice the square of the fraction's longer part. Below that there is no partial credit: the algorithm returns a different fraction with the same residue, and it is a perfectly good one.
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 answer with no error in it
An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.
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.