The cyclotomic cap-off quotient has no extra downward relations

Coefficient specialization. The lawful closure of $a_i(x)x^{\mathrm{\ell }-1}$ for $\mathrm{\ell }\ge 1$ gives the finite recurrence $h_{\mathrm{\ell }}+c_{i,{\lambda }_i-1}{h}_{\mathrm{\ell }-1}+\cdots +c_{i,0}{h}_{\mathrm{\ell }-{\lambda }_i}=0$, with $h_0=1$ and negative indices zero, as proved in [F4]. Hence ${\mathsf{H}}_{\mathrm{\Lambda }}=(a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}{)}^{-1}$ and $A_{\mathrm{\Lambda }}=a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}$. The resulting coefficient ideal $S$ belongs to $J$. The tensor-product basis [F2] shows that passing to ${\mathcal{P}}_i\mathrm{/}S$ leaves each ordinary downward element unchanged. Actual clockwise vacancy closures have coefficient index ${\lambda }_i+1+r$ for $r\ge 0$ and vanish; fake coefficients of indices $1,\dots ,{\lambda }_i$ retain the prescribed monic values. The case ${\lambda }_i=0$ gives $A=\mathsf{H}=1$.

Every killed-upward factor. The crossed closure at the first physical $G_i$ input $\mathrm{\Lambda }$ is the first-adjunction mate of the exact $R_{\mathrm{\Lambda }}$ composite, whose $H_i$ input is $\mathrm{\Lambda }-{\alpha }_i$ and bubble region is $\mathrm{\Lambda }$. Its native value is $-{\sum }_{m=0}^{{\lambda }_i}{A}_m{x}^{{\lambda }_i-m}=-a_i(x)$; with extra dots $P(u,x)$ it is $-a_i(x)P(x,x)$ because the dot-crossing correction is a zero straight closure. This is the typed local calculation of [F4], with no ambient weight substituted by analogy. Bend a factor through any $W{H}_i{v}_{\mathrm{\Lambda }}$ at its vacuum $H_i$ endpoint; the triangles give a composite with all-downward boundaries and one added first $i$ strand at source $\mathrm{\Lambda }+{\alpha }_i$. This does not require $W$ to be downward. Apply [F2], slide its bubble coefficients across the added root by $A_{\mathrm{\Lambda }+{\alpha }_i}=(1-zt{)}^2{A}_{\mathrm{\Lambda }}$, and specialize $S$. With the stabilizer $W_P$ of position one in $S_{n+1}$, [F4] uses exactly the two double cosets $W_P$ and $W_P{s}_1{W}_P$, with a length-additive word $u{s}_{1}v$ in the second. The first gives the zero vacancy loop; the second has its sole $s_1$ crossing equal-colored, since the first boundary color is $i$ on both sides. Exterior diagrams pull past the closure, whose value lies in the ordinary first-strand $i$ ideal. Thus every such factor lands in $I_a$, including arbitrary internal words and factor maps.

Every marked monic whisker. At the dominant vacuum the actual dotted columns and bubble-convolution dual rows of the positive matrix, modulo the killed-upward factor, identify $H_i{G}_i{v}_{\mathrm{\Lambda }}$ with $D_{{\lambda }_i}$. A raw cap row with $s$ extra dots evaluates $a_i(x)v_0$ by the zero recurrence at index $s+1$. Every actual dual row is a finite combination of those rows, so the inverse matrix kills $a_i(x)$ on the entire direct sum. To normalize an arbitrary marked whisker, move its earliest $H_i$ toward the initial marked strand. Mixed dot-natural isomorphisms preserve that marker; a nonnegative same-color branch uses the matrix decomposition, while a negative branch uses the actual retract $\iota =r_h^{-1}{j}_H$, $\pi =p_H{r}_h$ and $(H_i{G}_i)(b)=\pi (G_i{H}_i)(b)\iota$. Absorb retract maps into the factor maps. The intermediate word's pair (number of $H_i$ letters, distance of its earliest one from the vacuum) decreases on every finite branch. At the marked first strand, a different color gives a killed-upward factor; color $i$ gives the zero vacuum matrix calculation. Once the intermediate word is downward, [F2] reduces its factor maps. This complete normalization of [F4] leaves only ordinary first-strand ideal insertions and the factors already treated in step 2.1.

Exact intersection and corners. Expand $x\in J\cap e(\nu )R(\beta )e(\mu )$ as a finite sum of vertically factored whiskers of the defining generators and pass first to ${\mathcal{P}}_i\mathrm{/}S$. Steps 2.1–3.1 put every surviving generator summand in $e(\nu )I_{a}e(\mu )$, while coefficient-specialization summands vanish. Step 1.1 leaves the ordinary $x$ unchanged, proving the reverse inclusion. Every ordinary first-strand ideal insertion belongs to $J$ by its definition, proving the forward inclusion. The basis [F2] and the coefficient specialization represent every downward quotient morphism by an ordinary downward element, so the resulting monic cyclotomic corner map is bijective and respects multiplication. All computations hold over the universal graded coefficient ring and then under arbitrary monic specialization with grading forgotten; no finite-type, characteristic-zero, quotient-faithfulness or generic Frobenius assumption is used.