TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Finite colored Jones evaluation is exact and representative-independent

Statement

Assume the Axiom of Choice and the prescribed generic matrix ribbon data for the $N$ -dimensional color, where $N \geq 1$ . For every finite braid word $β$ on $n \geq 1$ strands whose closure is a knot, the algorithm performs finitely many exact operations, including exactly one crossing-matrix update per letter and one finite pivotal-trace contraction, and returns the generic normalized value $J_{N}^{g e n} (\hat{β}; q)$ . If two such braid words close to the same oriented unframed knot, their returned generic values agree. The unique Laurent representative belongs to $Z [q, q^{- 1}]$ , and its evaluation at any nonzero complex $q$ is defined, including at $q_{N} = e^{2 π i / N}$ .

Statement references

Finite colored Jones procedure

Generic normalization and color data

Colored-braid evaluator correctness and termination

Scalar twist correction and Markov invariance

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.9–8.10

Murakami, An Introduction to the Volume Conjecture, §§2.4–2.5

Facts & Assumptions

Given: The Axiom of Choice, an integer $N \geq 1$ , the prescribed generic matrix ribbon data, a finite braid word $β$ of length $m$ on $n \geq 1$ strands, and a knot closure.

[F1]

The procedure forms the positive and inverse crossing matrices and $J_{W} = G_{N}^{\otimes n}$ on $W = V_{N}^{\otimes n}$ , calls the finite matrix ribbon evaluator with this pivotal matrix, computes the exponent sum, and returns $λ_{N}^{- w (β)} t_{N} (β) / d_{N}$ over the exact field $Q (A)$ (Colored Jones evaluation from ribbon data).

[F2]

For valid finite-dimensional matrix data over an effective field and a color-preserving braid word of length $m$ , the ribbon evaluator makes exactly $m$ crossing-matrix updates and one finite pivotal-trace contraction and returns the framed colored-braid value (Correctness and termination of the colored-braid evaluation procedure).

[F3]

The generic data lie in an $F$ -linear ribbon category with $End (1) = F$ , where $F = Q (A)$ ; $V_{N}$ is absolutely simple with twist $λ_{N} I$ , $d_{N} \neq 0$ , $λ_{N} \neq 0$ , and $J_{N}^{g e n} = λ_{N}^{- w (β)} t_{N} (β) / d_{N}$ (Normalized colored Jones sequence).

[F4]

Under the Axiom of Choice, in a $k$ -linear ribbon category with $k$ -bilinear tensor product and $End (1) = k$ , an absolutely simple color with scalar twist $λ I$ gives a corrected framed trace $λ^{- w (β)} t_{n} (β)$ depending only on the oriented unframed link type when all components have that color (Scalar twist correction removes the framing).

[F5]

The framed ribbon-tangle evaluator is defined under Countable Choice (Colored framed-link evaluation from a ribbon category).

[F6]

The Axiom of Choice implies Countable Choice (AC implies DC implies countable choice).

[F7]

The quotient has a unique integral Laurent representative in $q = A^{4}$ , proved by The normalized generic colored Jones quotient is an integral Laurent polynomial.

[A1]

The Axiom of Choice (The Axiom of Choice).

[A2]

Countable Choice has its usual sequence-of-nonempty-sets meaning (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: The hypotheses in the Statement and [A1].

1.1F1F2F3givenconstruct

The universal- $R$ sum used for $c_{N}$ has only the terms $0 \leq r < N$ , since $E^{N} = F^{N} = 0$ on the displayed $N$ -dimensional module. The invertible crossing matrix has finite size, so its inverse is computable by finite Gaussian elimination over $Q (A)$ ; forming $J_{W} = G_{N}^{\otimes n}$ also takes finitely many field operations. The braid word has finitely many letters, and [F2] performs exactly one matrix update per letter followed by one finite trace contraction. The remaining normalization uses finite arithmetic and integer powers of the nonzero generic scalars $d_{N}$ and $λ_{N}$ in the effective field $Q (A)$ , so every operation terminates.

2.1F1F2F3step 1.1given

By [F2], the evaluator returns $t_{N} (β)$ , the pivotal trace of the framed colored closure. The last step in [F1] multiplies this value by $λ_{N}^{- w (β)}$ and divides by $d_{N}$ , exactly the formula in [F3]. Hence the returned value is $J_{N}^{g e n} (\hat{β}; q)$ .

3.1A1A2F3F4F5F6F7step 2.1given∎

Countable Choice, with the meaning in [A2], follows from [A1] by [F6], so the framed evaluator's hypothesis [F5] is satisfied. By [F3], the prescribed category and color satisfy every hypothesis of [F4], including absolute simplicity and the scalar twist. Hence if $β$ and $β^{'}$ close to the same oriented unframed knot, [F4] directly gives equality of their corrected framed traces $λ_{N}^{- w} t_{N}$ . Dividing by the same nonzero $d_{N}$ and applying step 2.1 proves equality of the algorithm outputs. Finally [F7] gives their unique common representative in $Z [q, q^{- 1}]$ . Substituting any nonzero complex $q$ into its finite Laurent sum is defined, including at $q_{N}$ , without division by a specialized quantum dimension.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, §§8.9–8.10 (standard reference, not scraped)
Hitoshi Murakami, An Introduction to the Volume Conjecture, §§2.4–2.5 (standard reference, not scraped)