ExampleConstruction: AI-generatedVerification: Literature-sourcedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Worked root-of-unity fusion and torus-state cases

Example

Use the BHMV convention $A$ primitive of order $2 p$ , $p$ even, $r = p / 2$ , and labels $0, \dots, r - 2$ . At $p = 4$ the label set is ${0}$ and the torus state space has dimension $1$ . At $p = 6$ the label set is ${0, 1}$ , the fusion rule is $1 \otimes 1 ≅ 0$ , and the torus state space has dimension $2$ .

Example references

Root and label conventions

Root fusion rules

Torus state-space definition

Blanchet, Habegger, Masbaum, and Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §4

Chen, The Temperley-Lieb Categories and Skein Modules, §§5.3–5.4

Facts & Assumptions

Given: The fixed BHMV roots $p = 4$ and $p = 6$ and the corresponding specialized Temperley–Lieb categories.

[F1]

The BHMV root-of-unity skein parameter: at $p = 4$ , $r = 2$ ; at $p = 6$ , $r = 3$ ; the label set is ${0, \dots, r - 2}$ .

[F2]

Jones–Wenzl projectors exist up to the first zero quantum integer: at $p = 6$ , $p_{2}$ is defined and has zero trace.

[F3]

Root fusion rules and completeness of the simple objects supplies the resolved full root fusion/completeness interface; no accepted proof is replayed.

[F4]

The solid-torus skein module has a core-multicurve basis: the framed skein module of the solid torus has the core multicurves as a free basis; its authorship record assumes full AC.

[F5]

Handlebody pairing nondegeneracy and gluing supplies the normalized perfect pairing and gluing at this root.

[F6]

External input — exact torus basis. BHMV Corollary4.10 p.909 gives the solid-torus core states as an actual basis of the torus state space; at even $p = 2 r$ its colors are $0, \dots, r - 2$ . Proposition1.9 identifies this space with the declared complementary-handlebody radical quotient. This is an actual basis statement, not a dimension inferred merely from a finite spanning list.

Verification

Given: The two root choices in the Example.

1.1F1givenalgebra

At $p = 4$ , $r = 2$ and $r - 2 = 0$ , so the displayed label set is exactly ${0}$ . At $p = 6$ , $r = 3$ and $r - 2 = 1$ , so it is exactly ${0, 1}$ .

2.1F2F3step 1.1algebra

At $p = 6$ , $δ = - 1$ , $p_{2} = 1_{2} + U_{1}$ , and $e_{0} = - U_{1}$ is an idempotent isomorphic to the tensor unit. Since $p_{2}$ is negligible by [F2], the decomposition $1_{2} = p_{2} + e_{0}$ gives $1 \otimes 1 ≅ 0$ in the quotient. This verifies the stated fusion calculation.

3.1F1F3F4F5F6step 1.1step 2.1algebra∎

By [F6] the torus dimension is the number of these basis labels, namely $r - 1$ . Step1.1 gives one label for $p = 4$ and two labels for $p = 6$ , so the dimensions are $1$ and $2$ . Together with the retained local fusion calculation in step2.1 this proves every Example clause. No polynomial skein basis before radical quotient is confused with the finite TQFT basis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

33 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

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, §4 (standard reference, not scraped)
Joshua Chen, The Temperley-Lieb Categories and Skein Modules, §§5.3–5.4 (standard reference, not scraped)
BHMV, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), exact external inputs (standard reference, not scraped)
Gilmer–Masbaum, Maslov index, Lagrangians, Mapping Class Groups and TQFT, arXiv:0912.4706v3 (standard reference, not scraped)