ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An order-three noncritical link-state form with a one-dimensional radical

Example

Take $q = e^{π i / 3}$ , so $δ = 1$ and $κ = ord (q^{2}) = 3$ . At rank $n = 4$ with $p = 2$ arcs and $t = 0$ defects, the link-state Gram matrix has a one-dimensional radical even though this label is noncritical.

Diagram

The sites are labeled from left to right. In each half-diagram, an arc joins its two labeled sites; the left panel is $x = (12) (34)$ and the right panel is $y = (14) (23)$ . These are the two noncrossing perfect matchings on four sites; the picture records their basis labels, while the Gram entries below are computed from the overlay loop count.

\begin{tikzpicture}[scale=0.9]
  \foreach \x in {0,1,2,3} { \fill (\x,0) circle (1.5pt); }
  \node[below] at (0,-0.05) {$1$};
  \node[below] at (1,-0.05) {$2$};
  \node[below] at (2,-0.05) {$3$};
  \node[below] at (3,-0.05) {$4$};
  \draw (0,0) .. controls (0,0.7) and (1,0.7) .. (1,0);
  \draw (2,0) .. controls (2,0.7) and (3,0.7) .. (3,0);
  \node at (1.5,1.0) {$x=(12)(34)$};

  \foreach \x in {5,6,7,8} { \fill (\x,0) circle (1.5pt); }
  \node[below] at (5,-0.05) {$1$};
  \node[below] at (6,-0.05) {$2$};
  \node[below] at (7,-0.05) {$3$};
  \node[below] at (8,-0.05) {$4$};
  \draw (5,0) .. controls (5,1.45) and (8,1.45) .. (8,0);
  \draw (6,0) .. controls (6,0.7) and (7,0.7) .. (7,0);
  \node at (6.5,1.7) {$y=(14)(23)$};
\end{tikzpicture}

Facts & Assumptions

Given: $q = e^{π i / 3}$ , $n = 4$ , $p = 2$ , and the link-state capping form.

[F1]

The standard-module basis is the set of noncrossing partial matchings with the indicated number of arcs and defects (Temperley–Lieb link-state standard modules by defect number).

[F2]

The form value on two basis states is $δ^{c}$ when all defect paths propagate, with $c$ the number of overlay loops (The link-state Gram form is symmetric and invariant).

[F3]

The Gram determinant is the product of quantum-integer ratios with polynomial continuation at removable zero-over-zero factors (The Temperley–Lieb link-state Gram determinant).

[F4]

A label is critical exactly when $κ < \infty$ and $κ ∣ (t + 1)$ (Complex Temperley–Lieb critical labels and reflection).

Verification

Given: $δ = 1$ , $κ = 3$ , and the two perfect matchings $x, y$ shown above.

1.1F1F2givenconstruct

There are exactly two noncrossing perfect matchings on four ordered sites: $x = (12) (34)$ and $y = (14) (23)$ . Overlaying either matching with itself gives two loops and hence Gram value $δ^{2} = 1$ ; overlaying $x$ with $y$ gives one loop and hence Gram value $δ = 1$ . Thus the matrix in the ordered basis $(x, y)$ is $G_{4, 2} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}) .$

2.1step 1.1algebra

A vector $a x + b y$ is orthogonal to both $x$ and $y$ exactly when $a + b = 0$ , by the two equal rows of $G_{4, 2}$ . Therefore $rad ϕ_{4, 0} = C (x - y)$ , a one-dimensional subspace.

3.1F3F4givenstep 1.1algebra∎

Here $t = n - 2 p = 0$ , so $t + 1 = 1$ is not divisible by $κ = 3$ ; the label is noncritical by [F4]. The determinant formula [F3] gives $\det G_{4, 2} = {(\frac{[2]}{[1]})}^{3} (\frac{[3]}{[2]}) = 0,$ because $[1] = [2] = 1$ and $[3] = 0$ at $q = e^{π i / 3}$ . This agrees with the explicitly computed radical.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra (standard reference, not scraped)
Frederick M. Goodman and Hans Wenzl, The Temperley-Lieb Algebra at Roots of Unity (standard reference, not scraped)