Cyclotomic word traces assemble a perfect symmetric form

Composed actual word adjunctions. If $\nu =({\nu }_1,\dots ,{\nu }_n)$ and ${\gamma }_r={\sum }_{a\le r}{\alpha }_{{\nu }_a}$, compose [F1] in chronological right-append order. This gives $E_{\nu }\{t_{\beta }\}⊣F_{\nu }⊣E_{\nu }$, because ${\sum }_r{t}_{{\nu }_r}({\gamma }_{r-1})={\sum }_r[2d_{{\nu }_r}-2(\mathrm{\Lambda },{\alpha }_{{\nu }_r})+2({\gamma }_{r-1},{\alpha }_{{\nu }_r})]=(\beta ,\beta )-2(\mathrm{\Lambda },\beta )$. The equality uses $({\alpha }_i,{\alpha }_i)=2d_i$ and expansion of the square of the sum, so it is independent of the chosen word. Composing the actual counits gives ${ϵ}_{\nu }$, and composing the actual units gives the reverse word Casimir; each composition obeys both triangles by successive applications of the component triangles. This is a construction of maps, not a inference from a Grothendieck-module relation.

Reduce an arbitrary crossing to the terminal pair. Put $\nu =(\eta ,i,j,\zeta )$, where the singled crossing is between positions $r,r+1$ and $\mathrm{\mid }\eta \mathrm{\mid }=r-1$. First apply the actual caps for the suffix $\zeta$, in reverse right-append order, to each product $p_{\nu }a$ and $a{p}_{\nu }$. Each cap is bilinear over its old algebra by [F1]; the singled crossing belongs to that old algebra, so each suffix cap moves it outside on its own side. The same suffix map therefore gives an opposite-corner element $a_0\in e(\eta ij)R^{\mathrm{\Lambda }}(\gamma +{\alpha }_i+{\alpha }_j)e(\eta ji)$, with products $p_0{a}_0$ and $a_0{p}_0$. Here $p_0={\tau }_{r}e(\eta ij)$ is the terminal crossing in the shorter tower. Dots and crossings in the suffix need not commute with $a$: they are included in $a$, and only bimodule linearity moves the singled old crossing. Thus the same incoming algebra element is used in both terminal calculations.

The actual arbitrary-prefix two-return contraction. Use Actual terminal dot caps have a prefix residue formula at the remaining prefix $\eta$, with the $i,j$ dots denoted $u,v$. Its proof identifies the constructed $T_i$ itself with $[u^{-1}]u^m{R}_i^{\eta }(u)$, including its canonical first-unit transport. For every old algebra coefficient and terminal dot polynomial $g$, it constructs a common residual map ${\mathcal{D}}_{\eta }(g)=[u^{-1}{v}^{-1}]g(u,v)R_i^{\eta }(u)R_j^{\eta }(v)$ and proves $T_i{T}_j(Q_{ij}g)=t_{ji}{\mathcal{D}}_{\eta }(g)$ and $T_j{T}_i(Q_{ji}g)=t_{ij}{\mathcal{D}}_{\eta }(g)$. The inner cap cancels its own spectator factor before extraction; the two remaining finite coefficient extractions commute in the old algebra corner. Thus the two-return quadratic pattern has the required ratio for arbitrary prefixes and all weights, not only at the vacuum. This does not identify the actual residual external maps for the zero- and one-return patterns.

The actual functional and its whole regular pairing. Word induction from the vacuum gives $F_{\nu }k=Be(\nu )$, and word restriction is $E_{\nu }M=e(\nu )M$. The composed second adjunction sends $f\in {\mathrm{HOM}}_B(M,Be(\nu ))$ to the functional $e(\nu )m\mapsto {ϵ}_{\nu }(e(\nu )f(m))$, with shift $\{-t_{\beta }\}$. This is an isomorphism by the actual word unit/counit triangles. There are finitely many words of the fixed root weight, so summing gives ${\mathrm{HOM}}_B(M,B)\cong DM\{-t_{\beta }\}$. At $M=B$, right multiplication by $a$ maps to $b\mapsto {\tau }_{\beta }(ba)$; hence that map is an isomorphism $B\to DB\{-t_{\beta }\}$. In particular the mixed corner $e(\eta )Be(\nu )$ pairs perfectly with its opposite corner; no diagonal-only nondegeneracy argument is used. The counit degrees in step 1.1 give $\mathrm{deg}{\tau }_{\beta }=t_{\beta }$.

All terminal contractions and the actual double mate. Apply [F5] to $a_0$ from step 1.2. Its two cap compositions give $t_{ij}{\mathcal{D}}_{\eta }(a_0)$ for $p_0{a}_0$ and $t_{ji}{\mathcal{D}}_{\eta }(a_0)$ for $a_0{p}_0$ in the same old corner. Closing the unchanged prefix $\eta$ by its actual composed counit proves $\tau (p_{\nu }a)=(t_{ij}\mathrm{/}{t}_{ji})\tau (a{p}_{\nu })$. This includes all zero-, one- and two-return matchings and their lower-crossing corrections; step 1.3 identifies the two-return polynomial special case directly. For dots and equal-color crossings use the actual cyclic normalization of [F1], whiskered through the same suffix/prefix argument. Choose finite homogeneous dual bases $x_{\alpha },y_{\alpha }$ for the actual word form of step 2.1, so $b={\sum }_{\alpha }{x}_{\alpha }\tau (y_{\alpha }b)={\sum }_{\alpha }\tau (b{x}_{\alpha })y_{\alpha }$. Contracting the actual word unit and counit gives the right rotation $p_{\nu }^{\ast \ast }={\sum }_{\alpha }{x}_{\alpha }\tau (p_{\nu }{y}_{\alpha })$; its order is fixed by using the right mate, and the opposite rotation would use the inverse order. Hence $\tau (a{p}_{\nu }^{\ast \ast })={\sum }_{\alpha }\tau (a{x}_{\alpha })\tau (p_{\nu }{y}_{\alpha })=\tau (p_{\nu }a)$ by the second dual-basis identity. The perfect opposite-corner pairing therefore identifies $p_{\nu }^{\ast \ast }=(t_{ij}\mathrm{/}{t}_{ji})p_{\nu }$. This contraction is the unit/counit definition of the actual mate, independent of the chosen dual bases. To obtain equality of natural transformations at every input algebra, apply the calculation to every vacuum word projective of that input: their finite sum is its regular module, so equality there is equality of the actual tensor-bimodule maps. Tensoring those maps gives the same equality on arbitrary modules. Thus the result concerns the actual separately constructed color adjunctions, with all prefix and suffix triangles, rather than a single scalar closure test.

Diagonal weights and symmetry. Fix a total order on the finite color support of $\beta$ and define $c_{\nu }$ by the inversion product in the Statement. An adjacent pair $(i,j)$ creates or removes exactly its own inversion, so $c_{s_r\nu }\mathrm{/}{c}_{\nu }=t_{ji}\mathrm{/}{t}_{ij}$; an equal-color swap changes nothing. The product is intrinsic to the word, hence independent of any sorting path. Equivalently the square move multiplies reciprocal factors, a far move commutes disjoint factors and the three-strand move has the same three pair factors on both sides. All factors are nonzero degree-zero field scalars. With $\mathrm{tr}(a)=\tau (ca)$, the ordered closure identity of step 3.1 gives $\mathrm{tr}(p_{\nu }a)=c_{s_r\nu }(t_{ij}\mathrm{/}{t}_{ji})\tau (a{p}_{\nu })=c_{\nu }\tau (a{p}_{\nu })=\mathrm{tr}(a{p}_{\nu })$. Dots, idempotents and equal-color crossings already have the same cyclicity without changing $c$. Any element is a linear combination of products of these generators, and moving the generators one at a time through the functional gives $\mathrm{tr}(ab)=\mathrm{tr}(ba)$ for every $a,b$. Since multiplication by $c$ is invertible, the perfect pairing of step 2.1 stays perfect; its degree remains $t_{\beta }$.

The symmetric bimodule consequence. The symmetric perfect pairing of step 4.1 identifies $B\{t_{\beta }\}$ with $DB$ as graded left and right bimodules: the map sends $a$ to the functional $b\mapsto \mathrm{tr}(ba)$; symmetry makes left and right multiplication agree with the two dual actions. For a finite graded projective $P$, evaluation on shifted free modules and restriction to a retract gives $D_{\psi }P\cong {\mathrm{HOM}}_B(P,DB{)}^{\psi }\cong {\mathrm{HOM}}_B(P,B{)}^{\psi }\{t_{\beta }\}=P^{\mathrm{\#}}\{t_{\beta }\}$. No assertion that every shifted projective is self-dual, or that every finite module is projective, is made.

Zero, vacuum and deformation cases. If $B=0$, all word projectives and functionals vanish and the pairing is the zero-to-zero isomorphism. At $\beta =0$ the sole empty word has the identity counit on $k$ and degree zero. At $\beta ={\alpha }_i$ the actual cap from [F1] is top-coefficient extraction on $k[z]\mathrm{/}{z}^{{\lambda }_i}$, with degree $-2d_i({\lambda }_i-1)$; if ${\lambda }_i=0$ the positive-root block is zero. For a monic $a_i$ the same remainder coefficient and Casimir work ungraded. The constructions in [F1]–[F3] are actual monic polynomial identities, and only nonzero $t_{ij}$ and units of triangular matrices are inverted, so arbitrary monic specialization preserves the constructed word perfect pairing with shifts forgotten. The actual terminal identities of [F5] are also monic identities descended before quotienting, so step 3.1 and the inversion rescaling retain the ordered mates and symmetry under the same specialization. Finite dimension is supplied separately by [F4]. Fixed root degree involves finitely many colors, words and tensor summands; no additional infinite choice, characteristic-zero or finite-type hypothesis occurs.