TheoremStatement: AI-adaptedProof: 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.

Correctness and termination of the colored-braid evaluation procedure

Statement

For every valid input of the colored-braid evaluation algorithm, where the braid word has $m$ letters and the total tensor-product dimension is $D$ , the procedure terminates after exactly $m$ crossing-matrix updates and one finite pivotal-trace contraction. It returns the ribbon-category evaluation of that framed colored braid closure in the given matrix realization. The stated bound concerns the number of matrix updates; each update is a finite exact matrix multiplication over the effective field.

Statement references

Colored-braid evaluation algorithm

Turaev, Quantum Invariants of Knots and 3-Manifolds, Chapter I, Theorem 2.5 and §§3–4

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

Facts & Assumptions

Given: A valid effective matrix input, a color-preserving braid word of length $m$ , and the algorithm specified above.

[F1]

The algorithm initializes the identity matrix, multiplies the typed positive or inverse braiding matrix for each word letter, checks final color preservation, and returns the finite contraction ${ev}_{W^{\lor}} ((J_{W} M_{β}) \otimes 1_{W^{\lor}}) {coev}_{W}$ (Ribbon evaluation from a colored braid word).

Proof

technique · direct

Given: The hypotheses in the Statement and [F1].

1.1F1giveninduction

After $j$ updates, induction on $j$ shows that the running matrix is the composite of the first $j$ typed crossing maps in composition order. The base case $j = 0$ is the identity of $W$ ; the update rule in [F1] composes exactly the next generator or its inverse, proving the induction step. There are precisely $m$ letters, so there are precisely $m$ updates.

2.1F1step 1.1givenconstruct∎

At $j = m$ , [F1] identifies the matrix with $M_{β}$ . Its final-color check makes it an endomorphism of $W$ , and the output contraction is the categorical pivotal trace of that endomorphism, hence the defined closure evaluation. Every update is one finite matrix multiplication and the last contraction is finite; the effective-field procedures terminate, so the algorithm terminates and has the asserted value.

Depends on

Ribbon evaluation from a colored braid word

Used by

Finite colored Jones evaluation is exact and representative-independent Theorem

Cited to discharge well-definedness by Ribbon evaluation from a colored braid word.

Dependency tree · two levels

7 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

V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, Chapter I, Theorem 2.5 and §§3–4 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, §8.10 (standard reference, not scraped)