Actual terminal dot caps have a prefix residue formula

Actual cap and index. Apply the ordinary first unit to $e(\eta )$ and decorate its appended strand with $m$ dots; the resulting element is $e(\eta i)x_{n+1}^m$. The canonical first-adjoint transport preserves that unit by [F1]. Its actual second counit consequently evaluates this element as the transported diagram ${\epsilon }_{\mu }^{\mathrm{\prime }}{x}^m{\eta }_{\mu }$, with $\mu =\mathrm{\Lambda }-\gamma$. This is the counterclockwise bubble of [F2], hence $e(\eta ){\mathsf{H}}_{\eta ,m-h_i+1}$. This calculation identifies the constructed $T_i^{\gamma }$ itself; it does not replace it by a chosen coefficient functional. Negative indices give zero, and no sign or weight shift is omitted: the literal-dot index is $m-h_i+1$, and its degree is $2d_i(m-h_i+1)=2d_{i}m+\mathrm{deg}{T}_i^{\gamma }$.

Slide through the prefix. Starting with the vacuum value in [F2], slide the bubble through the chronological letters of $\eta$. Each coefficient computation is finite because every reversed monic factor has constant coefficient one. The resulting series is ${\mathsf{H}}_{\eta }(t)={\prod }_{a:{\eta }_a=i}(1-x_{a}t{)}^2\mathrm{/}(a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}(t){\prod }_{a:{\eta }_a\mathrm{\ne }i}{q}_{i{\eta }_a}^{\mathrm{r}\mathrm{e}\mathrm{v}}(x_a,t))$. The numerator has degree $2\mathrm{\#}\{a:{\eta }_a=i\}$ in $z$, the denominator has degree ${\lambda }_i+{\sum }_{a:{\eta }_a\mathrm{\ne }i}(-a_{i{\eta }_a})$, and their difference is $-h_i$. Therefore $R_i^{\eta }(z)=z^{-h_i}{\mathsf{H}}_{\eta }(z^{-1})$. Extracting coefficient $z^{-1}$ from $z^m{R}_i^{\eta }(z)$ extracts precisely index $m-h_i+1$ from ${\mathsf{H}}_{\eta }$. Combined with step 1.1 this proves the actual terminal-dot formula, without restriction on the sign of $h_i$.

Centrality and old algebra factors. For an equal-color adjacent pair in $\eta$, the two factors of ${\mathsf{H}}_{\eta }$ are symmetric in their dot variables, so each coefficient commutes with its crossing by [F3]. For a distinct-color pair, swapping the color-variable pairs sends the coefficient in the $\eta$ summand to the coefficient in the swapped-word summand; its crossing has precisely that dot-permutation relation with no correction. Thus the finite sum of these coefficients over the words commutes with every crossing, every dot and every idempotent, which generate $A$. They are central. Appended terminal dots commute with all old generators. For any $b\in e(\eta )Ae(\eta )$, [F1] consequently evaluates ${\iota }_i(b)e(\eta i)x_{n+1}^m$ as $be(\eta )[z^{-1}]z^m{R}_i^{\eta }(z)$. By linearity this identifies the actual cap on every polynomial terminal with arbitrary old algebra coefficients.

First actual quadratic contraction. Put the $i$ dot $u$ on the appended prefix $\eta i$. Its actual terminal-$j$ formula of step 2.1 uses $R_j^{\eta i}(v)=t_{ji}{R}_j^{\eta }(v)\mathrm{/}{Q}_{ji}(v,u)$. In this identity the inverse is expanded at infinity in $v$ and has polynomial coefficients in $u$ and the old dots. Apply step 3.1 termwise to the polynomial $Q_{ij}(u,v)g(u,v)$: its old coefficients lie in the append image of $A[u]$. Since $Q_{ij}(u,v)=Q_{ji}(v,u)$, the denominator cancels in the Laurent series before taking the coefficient. The actual inner cap therefore gives $t_{ji}[v^{-1}]g(u,v)R_j^{\eta }(v)$. This is a polynomial in $u$ with old algebra coefficients: $g$ has finitely many powers of $u,v$, and only finitely many series coefficients contribute to $v^{-1}$. Apply the actual outer $i$ cap by step 3.1. The result is $t_{ji}[u^{-1}][v^{-1}]g(u,v)R_j^{\eta }(v)R_i^{\eta }(u)$.

Second contraction and common map. Interchange $i,j$ and keep the variables labeled by color. Now the inner $i$ cap uses $R_i^{\eta j}(u)=t_{ij}{R}_i^{\eta }(u)\mathrm{/}{Q}_{ij}(u,v)$, giving $t_{ij}[v^{-1}][u^{-1}]g(u,v)R_i^{\eta }(u)R_j^{\eta }(v)$ after the outer $j$ cap. Following the quadratic cancellation, each series depends on only its own terminal variable and the old commuting dots. Their coefficients commute with one another. On the finite polynomial $g$, both extractions use finitely many coefficients and therefore commute. The result of step 4.1 and this result are respectively $t_{ji}{\mathcal{D}}_{\eta }(g)$ and $t_{ij}{\mathcal{D}}_{\eta }(g)$ in the same old algebra corner. This is an equality of actual cap compositions, with an explicit common residual map before any prefix closure; no common finite flag coefficient ring or symmetric trace was assumed.

Quotients and boundaries. Every input is evaluated by the actual append homomorphisms and actual quotient caps; two polynomial inputs with the same terminal quadratic image give the same cap value. Because $t_{ij},t_{ji}$ are nonzero, the identities also show that ${\mathcal{D}}_{\eta }$ vanishes on each kernel of the corresponding quadratic evaluation. No injectivity of the append homomorphisms is required. If an old or terminal corner is zero, the same equations are identities of zero actual maps. For the empty prefix $R_i(z)=1\mathrm{/}{a}_i(z)$, giving ordinary monic top-coefficient extraction; $a_i=1$ and negative bubble indices are included by step 1.1. Every inverse-series coefficient is obtained by finite recursion from a unit constant term. Thus arbitrary monic field specialization preserves the argument, with grading forgotten, and there is no separability, characteristic-zero, finite-type or choice hypothesis. In the standard homogeneous case the degree asserted in step 1.1 agrees with [F1], and the two composite cap degrees agree in the two orders by $d_i{a}_{ij}=d_j{a}_{ji}$. The two-return contraction is proved; no conclusion about other terminal matching patterns is used or needed.