Endpoint regularization and associator coherence

For a degree-$d$ strand diagram, ${\mathrm{\Delta }}_i$ is the finite sum over the $2^k$ assignments of its $k$ endpoints on strand $i$ to the two new copies. Under this sum, each 4T generator expands into 4T generators on the cabled skeleton: fix the copy choices for endpoints outside the local 4T disk, and the four terms inside that disk are precisely one 4T relation. The same local check shows that deletion either kills all four terms or leaves a 4T relation. Endpoint assignments on stacked diagrams are independent, so cabling preserves stacking; deletion does as well. Thus these operations are algebra maps on the framed 4T quotient and extend degreewise to the completion.

In a collision chart for a cluster $S$, write $z_j=z_S+\rho u_j$ for $j\in S$. For internal pairs, $d\log (z_i-z_j)=d\log \rho +d\log (u_i-u_j)$; pairs with one label outside $S$ have nonzero limiting denominator and hence are regular in $\rho$. Therefore the singular radial part of ${\mathrm{\Omega }}_q$ is $A_{S}d\rho \mathrm{/}\rho$ with $A_S$ from [F2]. Removing ${\rho }^{A_S}$ gives the unique tangential transport from [F2]; its degree-zero term is the unit, hence $\mathrm{\Phi }$ and every endpoint transport are invertible in the product completion.

For a concatenation of paths ${\gamma }_1$ followed by ${\gamma }_2$, split the ordered integration simplex for each degree according to the number of integration times lying in the second segment. The later times occur on the left in the integrand, so summing the pieces gives $U({\gamma }_2\ast {\gamma }_1)=U({\gamma }_2)U({\gamma }_1)$, the declared bottom-to-top convention. This also proves that subdivision does not change the total holonomy.

For four ordered real strand positions, quotient by translation and positive scaling and write the three normalized gaps as $g_1,g_2,g_3>0$ with $g_1+g_2+g_3=1$. Their chamber is an open triangle. The two corners $g_1=g_2=0$ and $g_2=g_3=0$ are triple collisions; real oriented blow-up replaces each corner by the interval of relative gap scales. The compactified chamber is a pentagon. Its five vertices are $(((12)3)4)$, $((12)(34))$, $1(2(34))$, $1((23)4)$, and $((1(23))4)$. One route from $(((12)3)4)$ to $1(2(34))$ follows the two edges through $((12)(34))$; the other follows the three edges through $((1(23))4)$ and $1((23)4)$. The edge transports are respectively ${\mathrm{\Delta }}_1\mathrm{\Phi },{\mathrm{\Delta }}_3\mathrm{\Phi }$ and $\mathrm{\Phi }\otimes \mathrm{i}\mathrm{d},{\mathrm{\Delta }}_2\mathrm{\Phi },\mathrm{i}\mathrm{d}\otimes \mathrm{\Phi }$: a rotation inside a block doubles the strand occupied by that block. Choose the same small interior truncation point in each endpoint collar for both routes. The two truncated paths then have fixed common endpoints in the open triangle of positive gaps; their straight-line homotopy stays in that triangle and avoids every collision. Flatness [F3] and the concatenation rule of step 1.3 equate their transports. Taking the regularized limits supplied by [F2] gives $(\mathrm{i}\mathrm{d}\otimes \mathrm{\Phi })({\mathrm{\Delta }}_2\mathrm{\Phi })(\mathrm{\Phi }\otimes \mathrm{i}\mathrm{d})=({\mathrm{\Delta }}_3\mathrm{\Phi })({\mathrm{\Delta }}_1\mathrm{\Phi })$.

For two points, a positive half-turn changes $\arg (z_1-z_2)$ by $\pi$. Integrating the single commuting KZ term $(2\pi i{)}^{-1}{t}_{12}d\log (z_1-z_2)$ gives $R_{12}=\exp (t_{12}\mathrm{/}2)$. For three points, let strands 2 and 3 approach their prescribed nested collision while strand 1 makes a positive half-turn around their common limiting position. In the limit, $d\log (z_1-z_2)$ and $d\log (z_1-z_3)$ become the same differential, so the connection along this path is $(2\pi i{)}^{-1}(t_{12}+t_{13})d\log (z_1-z_2)$ and its transport is $\exp ((t_{12}+t_{13})\mathrm{/}2)={\mathrm{\Delta }}_{2}R$. The underlying braid has two positive crossings, first between labels 1 and 2, then between labels 1 and 3. Cut it at the nested collision endpoints: from bottom to top the factors are ${\mathrm{\Phi }}^{-1},R_{12},{\mathrm{\Phi }}_{213},R_{13},{\mathrm{\Phi }}_{231}^{-1}$. The associator factors change only the indicated parenthesizations. The factorized path has the same ordered pairwise crossings; moving their heights continuously while preserving their order is a collision-free homotopy, because each half-turn keeps its crossing pair distinct and the third point remains outside its crossing disk. Flatness [F3] and the path-composition rule of step 1.3 equate the transports, and taking the regularized limits at the nested endpoints using [F2] gives ${\mathrm{\Delta }}_{2}R={\mathrm{\Phi }}_{231}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}{\mathrm{\Phi }}^{-1}$.

Reversing the associating path interchanges its two endpoint clusters. Relabeling strands $1,2,3$ as $3,2,1$ identifies that reversed path with the KZ path defining ${\mathrm{\Phi }}_{321}$; path reversal inverts holonomy, so ${\mathrm{\Phi }}^{-1}={\mathrm{\Phi }}_{321}$. Under deletion of strand 1 or 3, one of the two residues $a,b$ in the reduced equation of [F1] is zero; the equation is then scalar in its sole generator and its two unit-leading tangential solutions agree, so their ratio is $1$. Deleting strand 2 sets both residues to zero. Hence ${ϵ}_i(\mathrm{\Phi })=1$ for each $i$.

When strand 1 is doubled, let its two copies form the left cluster and move strand 3 in a positive half-turn around that cluster. The limiting logarithmic forms agree, giving transport $\exp ((t_{13}+t_{23})\mathrm{/}2)={\mathrm{\Delta }}_{1}R$. Its underlying braid crosses labels 2 and 3 first and then labels 1 and 3. Cutting at the nested collision endpoints gives, from bottom to top, $\mathrm{\Phi },R_{23},{\mathrm{\Phi }}_{132}^{-1},R_{13},{\mathrm{\Phi }}_{312}$. The factorization preserves this ordered crossing list; moving the crossing heights while preserving their order gives the collision-free homotopy described in step 2.2. Flatness [F3] and the regularized endpoint limits [F2] therefore give ${\mathrm{\Delta }}_{1}R={\mathrm{\Phi }}_{312}{R}_{13}{\mathrm{\Phi }}_{132}^{-1}{R}_{23}\mathrm{\Phi }$. Inverting each positive equality reverses the factor order and inverts every factor, yielding the two displayed negative hexagons.

For a fixed parenthesized braid, first fix its collision-free path in ${\mathrm{Conf}}_q(\mathbb{C})$. Any two subdivisions of that path into ordinary braid pieces have the same transport by the concatenation rule of step 1.3; homotopic choices have the same transport by flatness [F3]. A parenthesization change is a path in the resolved real chamber of ordered positions. Translation and positive scaling identify its interior with the convex simplex of normalized positive gaps. For each tangential endpoint, use [F4] to choose one interior truncation point by assigning each nested collision radius a fixed positive power of $\epsilon$; use the same point for every decomposition. After joining each truncated edge path to these common collar points, the paths lie in the simplex with fixed endpoints. Their straight-line homotopy stays in the simplex and avoids collisions, so flatness equates their transports. Letting $\epsilon \downarrow 0$ and removing the nested residue factors gives equality of the regularized transports by [F2]. Thus changing the sequence of parenthesization changes does not change the transport assigned to the fixed geometric braid.