degree-deficit
At its defaults it draws the singular values of b, and the number of infinite eigenvalues the exact polynomial says there are. The pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls in the right place, so the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.
degree-deficit is one function in lib/figures/pencil.js —
pencils — two matrices, an eigenvalue with no value, and a problem with no answer. 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 pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls in the right place, so the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.
k: 2
The arguments are the ones A problem with no 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 pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls in the right place, so the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.
k: 1
The arguments are the ones An eigenvalue with no value 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 pencil has 1 algebraic constraint, so 1 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 5, so n − 5 = 1 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.25, 2.36, 1.04, 0.835, 0.3, 5.4·10⁻¹⁵⁸: the gap is a factor of 5.55·10¹⁵⁶ and it falls in the right place, so the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.
k: 4
The arguments are the ones An eigenvalue with no value 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 pencil has 4 algebraic constraints, so 4 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 2, so n − 2 = 4 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.16, 1, 4.28·10⁻¹⁷, 0, 0, 0: the gap is a factor of −∞ and it falls in the right place, so the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.
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.
7 distinct claims across 4 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 pencil small enough for the exact determinant
a regular pencil
an integer entry, which is what makes the exact route exact
and whose B has exactly that many singular values below the cut
at least one constraint, or there is no deficit to draw
matmul shapes agree
whose polynomial loses one degree per constraint
Against the rule
It draws a decomposition and prints its residual. It calls
infiniteByRank,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 113
of 219 generators —
98 print a residual and
15 are exempt with a published reason;
106 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 problem with no answer
If two matrices share a null vector then det(A − λB) is identically zero and every λ is an eigenvalue, which means none of them is. Perturb such a pencil by a ten-billionth and a solver returns six numbers with residuals below 10⁻⁹. Change the seed and it returns six different numbers, spread over forty-four, with residuals just as small.
Eigenvalues, singular values, rankAn eigenvalue with no value
If the second matrix of a pencil is singular then some of the eigenvalues are infinite, and that is not a degeneracy — it is the algebraic constraints of the model, one per constraint. What survives is a pair of numbers rather than one, and on the line those pairs live on, infinity is an ordinary point with an ordinary residual.
Two errors, and whose fault they areDeciding that a zero has arrived
The previous tolerances were offers — accept this much error, save this much work. A detection threshold is not an offer, because both directions are failures. One matrix here has three genuinely near-invariant subspaces, and the constant somebody typed decides which of them the recurrence stops at; at eight significand bits the same kind of constant produces a proof of something false.