Multicolored ribbon graphs with arbitrary coupons have a unique evaluation

Work first in [F4]'s strict model. Write $b_Y:\mathbf{1}\to Y\otimes Y^{\ast }$, $e_Y:Y^{\ast }\otimes Y\to \mathbf{1}$, $r_Y=e_{Y^{\ast }}(j_Y\otimes 1)$ and $b_Y^R=(1\otimes j_Y^{-1})b_{Y^{\ast }}$. Assign the listed generator values and each coupon its morphism. Slice any generic diagram between all crossings, extrema, recorded twists and coupon levels, then evaluate by categorical tensor product/composition. The strand-only equations of [F1] hold with arbitrary branch colors: Yang–Baxter follows by the naturality/hexagon expansion on $U,V,W$; each inverse and snake has its own actual color; the bent-crossing equation is the arbitrary-$V,W$ composite in [F2]; its inverse-order proof and opposite braiding give all signs; and the twist-square computation of [F2] applies to each arbitrary single $Y$. Redundant orientation variants are its right-dual composites applied separately to each color. Thus no equality of unequal objects or single-color specialization is used to justify multicolored relations.

For $f:\mathrm{\mid }A\mathrm{\mid }\to \mathrm{\mid }B\mathrm{\mid }$, the two displayed crossing/coupon equations are exactly naturality of $c$; multiplying by its inverses gives both underpass versions. Expanding each tensor-list crossing by the two hexagons writes the actual finite word of crossings of one band with all inputs/outputs of the coupon. Upward bands are their dual tensor objects and use the already checked bent-crossing formulas, so these equations cover every orientation. For full rotations define ${\mathrm{\Theta }}_{\mathrm{\varnothing }}=1$ and ${\mathrm{\Theta }}_{A\otimes Y}=c_{Y,A}{c}_{A,Y}({\mathrm{\Theta }}_A\otimes {\theta }_Y)$. Balancing inductively gives ${\mathrm{\Theta }}_A={\theta }_{\mathrm{\mid }A\mathrm{\mid }}$, including all individual twists and all pairwise full-bunch crossings. Naturality gives ${\theta }_{\mathrm{\mid }B\mathrm{\mid }}f{\theta }_{\mathrm{\mid }A\mathrm{\mid }}^{-1}=f$, exactly the input-bunch inverse/output-bunch full rotation, with no deleted framing twist.

For $f:A\otimes Y\to B$ define its typed left mate $f^{\mathrm{♭}}=(f\otimes 1_{Y^{\ast }})(1_A\otimes b_Y):A\to B\otimes Y^{\ast }$. The inverse is $(1_B\otimes e_Y)(f^{\mathrm{♭}}\otimes 1_Y)$. Substitution and the two snakes show both composites are identities on these Hom spaces. The other bends use $b_Y^R,r_Y$ and their two snakes. Iteration supplies every signed-list orientation and the needed double-dual comparison $j$; two full dualizations are consistent because $j_{B}f=f^{\ast \ast }{j}_A$. Coupon composition/tensor rules are the actual equalities $gf$ and $f\otimes g$. Thus all evaluations of bent or subdivided coupons have explicitly inverse composites, with no unnamed duality scalar.

To prove that these equations suffice geometrically, take a boundary-relative graph isotopy and choose disjoint solid rectangular neighborhoods of its finitely many coupons, carried along their tracks. Compactness and finite time subdivision permit smaller such moving neighborhoods disjoint from all other bands except their incident fixed collars. Any homotopy of their frame motions is realized by an ambient isotopy there: prescribe its velocity on each rectangle, multiply by a smooth cutoff in its disjoint moving neighborhood and integrate the resulting time-dependent vector field. Its flow fixes the external collars and carries the incident band collars. Time subdivision keeps these supports disjoint. This reduces frame control to finitely many paths in $SO(3)$ with fixed endpoint frames.

Every resulting rotation loop can be homotoped relative endpoints to planar rotations. Explicitly, unit quaternions form $S^3$ and $q:v\mapsto qv{q}^{-1}$ gives a locally invertible two-sheeted map onto $SO(3)$ with kernel $\{\pm 1\}$. A smooth loop in $S^3$ misses a point after a small perturbation and contracts by stereographic projection to ${\mathbb{R}}^3$, so $S^3$ is simply connected. Hence an $SO(3)$ loop is classified by whether its lift ends at $+1$ or$-1$. A planar $2\pi$ rotation lifts from $+1$ to$-1$, so planar rotations realize both classes. The cutoff extension of step 2.3 realizes the homotopies on the graph. It follows that the isotopy reduces to motions keeping coupons parallel to the projection plane with horizontal based edges, together with finitely many planar $\pm 2\pi$ coupon rotations. The latter are the complete twist relation of step 2.1.

For the remaining motions protect smaller coupon rectangles and keep their incident band collars fixed. Apply [F1]'s generic movie construction to the exterior core arcs, with its perturbation cutoffs vanishing in those collars. Away from coupons its event list is unchanged. At a protected rectangle there are also generic isolated edge tangencies and corner passages; they are retained, not excluded by a false codimension count. In a small disk a smooth-edge tangency has the form $y=u^2+s$ against the edge $y=0$, possibly with both signs reversed. The spatial strand is strictly above or strictly below the coupon in that disk, since the actual graph is embedded; compactness gives a fixed nonzero height separation there. Attachments lie strictly inside the coupon bases, so the disk may be chosen disjoint from every attachment collar. A corner passage is a strand crossing the two-edge junction transversely; its local resolution is given below. Multiple simultaneous events and higher-order edge contacts add a condition to these already retained normal forms, and only those higher degeneracies are excluded by independent jets.

Resolve an edge tangency as follows. Shrink the coupon a little inside its protected neighborhood, replacing each moved attachment by a short parallel identity strip; this changes neither its color nor its input/output tensor order. The same spatial height separation keeps the external strand disjoint throughout. The small U-shaped strand segment $y=u^2+s$ now gives two successive entries/exits past the shrunken edge. Separate them in slicing height and bend its turn by the explicit mate bijection of step 2.2. Each straight passage is a naturality equation from step 2.1 or its inverse. When only part of the short based strips lies in that small disk, enlarge the local word by inserting the missing crossings paired immediately with their inverses outside the coupon. In algebraic terms the full naturality word is $C_B(f\otimes 1)=(1\otimes f)C_A$, where $C_A=c_{\mathrm{\mid }A\mathrm{\mid },Z}$ and $C_B=c_{\mathrm{\mid }B\mathrm{\mid },Z}$, or its other-side/inverse version. The completed event is this exact equation pre/post-composed with the same exterior words $L,R$; expand $C_A,C_B$ by the hexagon to match the actual based-strip order. Removing the adjacent inserted inverse pairs recovers the two original readings of the event. No cancellation is moved through $f$, and no naturality for a partial input subobject of $f$ is asserted: the missing input and output crossing words can differ. For the two consecutive passages of the U segment, complete both together, so their exterior words on each fixed side are inverse. Those words reduce to $C_A^{-1}{C}_A=1$ on the input side and $C_B{C}_B^{-1}=1$ on the output side, with their respective actual tensor types. Retract the auxiliary bend by its snake. Thus the entire edge-tangency event changes no evaluated morphism. The negative quadratic contact reverses this derivation. A cap instead of a cup uses the inverse mate, an upward strand uses $Z^{\ast }$, and an underpass uses the inverse-crossing naturality, all explicitly established in step 1.1, step 2.1, step 2.2. This covers both orientations, over/under choices, both edge sides and both contact signs. The local shrink only alters attachment-free portions of the protected rectangle, so a base-edge event is treated exactly like a side-edge event; short based strips are included in the sublist completion just described.

For a corner passage choose a disk containing that corner but no attachments. Round or cut off the corner of the protected coupon inside that disk, joining the unchanged two based-edge portions by a smooth arc. This is a planar isotopy of an attachment-free part of the coupon and changes no morphism or tensor list. The external strand stays spatially separated above/below it. Perturb the rounded arc slightly so the corner passage is replaced by two ordinary edge passages and, if the strand turns back in that disk, one quadratic edge tangency. The two possible routes around the old corner differ by the consecutive passage and its inverse; their expanded words cancel as in step 5.1. Thus restoring the corner yields the same value. The four corners differ only in which edge is a base edge; all attachments remain outside the disk, so step 5.1's sublist completion applies to each. This supplies the corner events instead of excluding their generic isolated times.

A coupon level may cross an exterior extremum level or a crossing level at an isolated time; these slicing events are also retained. If their support disks are disjoint from the protected rectangle, their generator words exchange by interchange. If adjacent in projection, shrink the rectangle as in step 5.1 so the extremum/crossing lies outside it. The induced strand passages are the partial/full naturality words already resolved there; a cup or cap adjacent to the coupon is handled by step 2.2's mate bijection and its inverse snake, and a crossing of its short incident strips is moved by [F1]'s cup passage or third move before the naturality word. Moving the coupon back cancels the inserted inverse words. Thus neither a slicing extremum nor a crossing-level event is silently removed from the movie. If two projected coupon rectangles pass each other, shrink one and move its short incident strips past the other successively, then exchange their disjoint levels. The complete identity is $c_{\mathrm{\mid }B\mathrm{\mid },\mathrm{\mid }D\mathrm{\mid }}(f\otimes g)=(g\otimes f)c_{\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }C\mathrm{\mid }}$, obtained by the two naturality equations successively; edge/corner events during this shrink were handled in step 5.1, step 6.1. This covers all valences and the empty tensor list. The retained event list is exhaustive because every remaining regular projected arc/coupon incidence is transverse, every nonregular one has precisely one of the stated edge/corner normal forms after the generic perturbation, and all exterior core singularities and framing events are [F1]'s list.

There are finitely many events by [F1]'s compactness/relative genericity argument applied to this finite graph with its protected rectangle collars. Every event preserves the composite by step 1.1, step 2.1, step 2.2, step 4.1, step 5.1, step 6.1, step 7.1; between them only slicing levels and disjoint-factor interchange change. The whole framed isotopy therefore preserves evaluation. The generator assignment preserves composition/tensor products and the empty diagram by definition; every diagram has the sliced form, so it also proves uniqueness. Transport through [F4]'s strong monoidal equivalence gives the claimed evaluation in the original category, with all chosen dual comparisons inserted as in [F2]. Only [A1]'s inherited countable choice is used; all additional neighborhoods, frame adjustments and word operations are finite.