All generic colored Jones matrices satisfy the ribbon identities

Put $K=K_L^{-1}$. Then the generator relations and coproduct in [F1] become exactly $KE{K}^{-1}=q_K^{2}E$, $KF{K}^{-1}=q_K^{-2}F$, $[E,F]=(K-K^{-1})\mathrm{/}(q_K-q_K^{-1})$, $\mathrm{\Delta }(E)=E\otimes K+1\otimes E$, $\mathrm{\Delta }(F)=F\otimes 1+K^{-1}\otimes F$. The antipodes also agree: $S(E)=-E{K}^{-1}$, $S(F)=-KF$. Thus this is a change of parameter and Cartan generator, not an assumption equating the two parameters. Every weight is an integer, and $E,F$ are nilpotent on every fixed generated object by its finite weight range. The finite $R$ series and all dual matrices are therefore rational matrices in $A$.

Fix finitely many objects, module maps, and a single proposed ribbon identity involving them. Realize every image of an idempotent inside its ambient finite tensor sum. There are finitely many rational entries, denominators, weights and matrix sizes in this identity. Specialize $A=s_{\mathrm{\ell }}=e^{\pi i\mathrm{/}(2\mathrm{\ell })}$ for integers $\mathrm{\ell }$ greater than every color, every weight-range width and every divided-power index which can act nontrivially. Exclude the finitely many zeros of the finitely many nonzero denominator polynomials. Every color specializes to $\mathrm{\Delta }(N-1)$ in [F3]; ordinary powers divided by $[r]!$ specialize to its divided powers for all possibly nonzero indices, since those indices are below $\mathrm{\ell }$. Larger divided powers are zero by weight range. A generic module map preserves weights, because the generic $K$ eigenvalues distinguish them, and commutes with these divided powers and the scalar toral characters. It is consequently a full divided-power module map after specialization, not merely a small-quantum-group map. An idempotent specializes to an idempotent module map, whose image is a tilting summand.

The contragredient action and ordinary evaluation and coevaluation in the color definition are module maps by the antipode identities: the action on their composites uses $m(S\otimes 1)\mathrm{\Delta }=m(1\otimes S)\mathrm{\Delta }=\eta \epsilon$. Their vector-space contractions send each basis vector or dual basis vector to itself, proving both snakes. Tensor, sum and idempotent duals use the corresponding transpose maps. Therefore the whole generated category is rigid. The unsigned pivotal comparison is $j_X^{+}(x)(f)=f(Kx)$: $S^2=\mathrm{Ad}K$, and $K$ is group-like, so it is a natural monoidal module isomorphism to the double dual.

At each such specialization the finite generic series becomes exactly the one in [F2]: replacing ordinary powers by divided powers changes its coefficient to $q_K^{r(r-1)\mathrm{/}2}(q_K-q_K^{-1}{)}^r[r]!=q_K^{r(r+1)\mathrm{/}2}(1-q_K^{-2}{)}^r[r]!$. The weight factor is $A^{hk}=s_{\mathrm{\ell }}^{hk}$. Thus the specialized series is a full natural braiding on all the chosen objects and maps, with both hexagons. Each entry of its generic intertwining, naturality, or hexagon equation is a rational function vanishing at infinitely many distinct $s_{\mathrm{\ell }}$. Its numerator polynomial has infinitely many roots and hence is zero. These identities therefore hold generically. Invertibility follows as well: the nilpotent series has constant term $1$, hence a finite geometric-series inverse on each pair, and $D$ is an invertible diagonal matrix. Idempotent restrictions and their inverses are compatible by naturality. This proves braiding on the full generated category.

Let $u$ be [F4]'s Drinfeld isomorphism, formed with the just-proved generic braiding and duality, and put ${\theta }^{+}=u^{-1}{j}^{+}$. Its tensor defect gives the balancing identity. Every matrix entry of its dual-compatibility equation is rational in $A$. Upon each sufficiently large specialization in step 2.1 it is precisely [F2]'s ribbon twist, so the equation holds by the same infinite-root argument as step 3.1. On $V_N$, its specialized matrix is $s_{\mathrm{\ell }}^{(N-1)(N+1)}I$ by [F2]; infinite-root interpolation therefore gives ${\theta }_{V_N}^{+}=A^{N^2-1}I$ generically. This establishes the unsigned ribbon structure without importing an unproved generic universal-$R$ assertion.

On a weight-$h$ vector define $p_X=(-1{)}^h$. This is a module map because $E,F$ change weight by $2$; it is natural because module maps preserve weights, monoidal because weights add, and involutive. On dual weights $-h$ it satisfies $p_{X^{\ast }}=(p_X{)}^{\ast }$, and $p$ commutes with the braiding. Set $j=p{j}^{+}$ and $\theta =p{\theta }^{+}$. Substitution into the unsigned balancing and dual-compatibility equations proves the corresponding equations for $\theta$; monoidality of $p$ proves that $j$ is a pivotal comparison. Since all weights of $V_N$ have parity $N-1$, its pivotal matrix is $G_N=(-1{)}^{N-1}\mathrm{diag}(A^{2(N-1-2j)})$ and its twist is exactly ${\lambda }_N=(-1{)}^{N-1}{A}^{N^2-1}$. The trace of the identity is $(-1{)}^{N-1}{\sum }_{j=0}^{N-1}{A}^{2(N-1-2j)}=d_N$. Every generic ribbon assertion required in the Statement has now been proved, and $N$ was arbitrary.