Where the missing eigenvalues went: an 6 × 6 quadratic with 1 of its masses removed

At its defaults it draws where the missing eigenvalues went: an 6 × 6 quadratic with 1 of its masses removed. A chain of 6 masses with the first 1 of them set to zero, so M is singular of rank 5. det Q(λ) is a polynomial of degree at most 12; interpolated exactly in BigInt rationals at 13 nodes it has degree 11, so 1 of the 12 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

qep-count is one function in lib/figures/quadratic.js — a matrix that depends on its own eigenvalue — 2n answers for n rows, and a count that is a degree. 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 chain of 6 masses with the first 1 of them set to zero, so M is singular of rank 5. det Q(λ) is a polynomial of degree at most 12; interpolated exactly in BigInt rationals at 13 nodes it has degree 11, so 1 of the 12 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

A chain of 6 masses with the first 1 of them set to zero, so M is singular of rank 5. det Q(λ) is a polynomial of degree at most 12; interpolated exactly in BigInt rationals at 13 nodes it has degree 11, so 1 of the 12 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

A chain of 7 masses with the first 1 of them set to zero, so M is singular of rank 6. det Q(λ) is a polynomial of degree at most 14; interpolated exactly in BigInt rationals at 15 nodes it has degree 13, so 1 of the 14 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

A chain of 8 masses with the first 1 of them set to zero, so M is singular of rank 7. det Q(λ) is a polynomial of degree at most 16; interpolated exactly in BigInt rationals at 17 nodes it has degree 15, so 1 of the 16 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

A chain of 3 masses with the first 1 of them set to zero, so M is singular of rank 2. det Q(λ) is a polynomial of degree at most 6; interpolated exactly in BigInt rationals at 7 nodes it has degree 5, so 1 of the 6 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

A chain of 4 masses with the first 1 of them set to zero, so M is singular of rank 3. det Q(λ) is a polynomial of degree at most 8; interpolated exactly in BigInt rationals at 9 nodes it has degree 7, so 1 of the 8 eigenvalues are at infinity — the same object a descriptor pencil has, arriving here because a degree of freedom with no inertia is an algebraic constraint. The float route counts the singular values of M judged to be zero and reaches 1, backed by a gap of ∞ between consecutive singular values. One integer, two routes, and only the second of them is a decision.

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.

8 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.

It draws a decomposition and prints its residual. It calls charPolyQuadExact, 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 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.