Root Kirby-color handle-slide and blow-up invariance

Given: The root parameter, the semisimplified category, its simple projectors, quantum dimensions, twist, and a finite integral framed surgery link.

The BHMV root-of-unity skein parameter fixes $p$, $A$, $r$, $\delta$, and $\mathbb{C}$.

Semisimplified Temperley–Lieb quotient at the BHMV root defines the quotient and the root-specific projectors.

Root fusion rules and completeness of the simple objects supplies the resolved finite simple list, dimensions, twists and invertible Hopf-link matrix in the registered root/pivotal convention. Its accepted interface is reused, not its body.

Weighted Lagrangian extension and root anomaly data fixes the weighted Lagrangian composition and the selected sixth root $\kappa$.

External input — actual BHMV Kirby color and Gauss units. BHMV, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology34(1995), §2 p.897, the displayed definition of ${\theta }_p$ and equations(*)–(**), together with §5.8 p.913: for the primitive $2p$-th root and even $p=2r$, the Kirby element is ${\mathrm{\Omega }}_p={\sum }_{j=0}^{r-2}{d}_j{p}_j$ and ${\mathrm{\Delta }}_{\epsilon }={\eta }^{-1}{\kappa }^{3\epsilon }(\epsilon =\pm 1),\eta \in {\mathbb{C}}^{\times }\mathrm{.}$ Here $\kappa$ is the selected BHMV SIXTH root from [F4], not the square-root weight parameter used by some later sources. The source's usual signed quantum dimensions are exactly $d_j=[j+1]$ for the registered negative-loop recurrence. Its recalled invariant ${\theta }_p(S^3(L))=⟨L({\mathrm{\Omega }}_p)⟩\mathrm{/}({\mathrm{\Delta }}_{+}^{b_{+}}{\mathrm{\Delta }}_{-}^{b_{-}})$ includes the actual Kirby handle-slide identity. This established identity is imported, not claimed as a new local recoupling proof.

External input — weighted normalization and exact sign. BHMV §2 pp.897–898 and Gilmer–Masbaum, arXiv:0912.4706v3, §2 pp.4–6 and §11 pp.23–25, give the weighted version. Put $\nu ={\kappa }^3$; one weight step multiplies the map by $\nu$. The closed TQFT normalization is $Z_p(M,w)={\eta }^{b_0(M)+b_1(M)}{\nu }^w{\theta }_p(M)$, componentwise with the banded data. It is this additional normalization, not raw ${\theta }_p$ alone, that supplies the finite TQFT. Gilmer–Masbaum's Maslov form on $(L_1+L_2)\cap L_3$ is $(a_1+a_2)\odot (b_1+b_2)=\omega (a_2,b_1)$. Modulo its radical it is the negative of the registered Wall form $\omega (a,b^{\mathrm{\prime }})$ with $a+b+c=0$. Thus their correction $-{\mu }_{GM}$ is precisely the registered $+{\mu }_{Wall}$. Their square-root parameter is $\nu$, explicitly the cube of BHMV's parameter. No choice of an unrelated square root ${\mathcal{D}}_p$ is silently identified with ${\eta }^{-1}$; only ${\mathcal{D}}_p^2={\eta }^{-2}$ follows.