ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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.

The sl2 modular data at ell=3

Statement

Assume full AC, inherited from the weighted RT torus-space construction. For the established standard $ℓ = 3$ root-of-unity ribbon quotient, the labels are $0, 1$ , both dimensions are $1$ , and

$S = \frac{1}{\sqrt{2}} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}), (θ_{0}, θ_{1}) = (1, i), 1 \otimes 1 = 0.$

The torus state dimension is $2$ .

Facts & Assumptions

Given: Full AC, $ℓ = 3$ , $q = e^{π i / 3}$ , and the positive-trace convention.

[F1]

The actual quotient has $d_{a} = \sin ((a + 1) π / 3) / \sin (π / 3)$ and $θ_{a} = q^{a (a + 2) / 2}$ by The standard root-of-unity sl2 tilting ribbon quotient is modular. Applying the conditional sine calculation to these established data gives $S_{a b} = \sqrt{2 / 3} \sin ((a + 1) (b + 1) π / 3)$ (Truncated sl2 fusion data give the sine Hopf-link matrix).

[F2]

The actual quotient has the truncated fusion rule of The rank-one tilting quotient is the truncated fusion category; its retained range for $a = b = 1$ , $ℓ = 3$ is just $c = 0$ . The torus dimension formula is RT torus operators and the state-space dimension formula.

[A1]

Full AC (The Axiom of Choice) is assumed for the weighted RT functor and its state-space construction; it supplies the inherited countable choice. The actual tilting quotient and full weighted RT functor are established by their exact suppliers (The standard root-of-unity sl2 tilting ribbon quotient is modular, The weighted Reshetikhin–Turaev TQFT and modular-functor factorization).

Proof

1.1F1F2givenalgebra

The dimension formula gives $d_{0} = d_{1} = 1$ . The twist exponents are $0$ and $3 / 2$ , so $θ_{0} = 1$ and $θ_{1} = e^{π i / 2} = i$ . Substitution in the sine formula gives $S_{00} = S_{01} = S_{10} = 1 / \sqrt{2}$ and $S_{11} = - 1 / \sqrt{2}$ . The truncated fusion range for $1 \otimes 1$ contains only $0$ .

2.1F2A1step 1.1algebra∎

There are exactly two labels; hence the torus dimension formula gives $\dim V (T^{2}) = 2$ . Also $S^{2} = I$ by direct multiplication of the displayed matrix, verifying nondegeneracy in this finite case.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 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

Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.3 (standard reference, not scraped)
Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §3.3 (standard reference, not scraped)