The minimal-parabolic polynomial flag model realizes the graded diagram relations

Monic splitting and the vertex ring. Over universal graded $S=k[p_1,\dots ,p_N]$, successive monic division gives the full splitting Artin basis $X_1^{a_1}\cdots X_N^{a_N}$ with $0\le a_k\le N-k$ and perfect coefficient pairing. The two-block staircase and longest-divided-difference retraction of [F4] shows the block-invariant factor ring $L_r$ is finite graded projective over $S$. Lift a homogeneous basis modulo $S_{+}$; graded Nakayama gives a finite graded free-module surjection onto $L_r$. A projective splitting and the least-degree argument kill its kernel, whose $S_{+}$ fiber is zero, so $L_r$ is graded free. The full splitting algebra has Artin ranks $N!$ over $S$ and $r!(N-r)!$ over $L_r$, hence the free rank is $\left(\genfrac{}{}{0px}{}{N}{r}\right)$. Base change to the coefficients of $p_m$ proves the claimed finite free $C$-module in every characteristic. For $r>N$ the excessive coefficient ideal contains $1$, since a monic degree-$r$ factor cannot divide degree-$N$ $p_m$, so $L_m=0$.

The vertex ring as a matrix quotient. Representing ${\mathrm{N}\mathrm{H}}_r$ as the matrix algebra on its Artin module (step 1.1, [F2]) shows that the two-sided ideal generated by a monic $p(X_1)$ is the matrix ideal of the coefficients of the remainder of $p$ modulo $c_r$, and monic division makes these coefficients a unit triangular change from the consecutive relations; hence ${\mathrm{N}\mathrm{H}}_r\mathrm{/}(p(X_1))$ is the matrix algebra over the monic factor ring $L_m$ of the Statement. The same computation shows the $i$-flag $B_m$ is free of rank $N-r$ over $L_m$ and rank $r+1$ over $L_{m+i}$, giving the two root extensions of the model.

Coefficient maps, triangles and normalized transport. The kernel $K_f(X,Y)=(f(X)-f(Y))\mathrm{/}(X-Y)$ is the trace-dual power-basis Casimir: coefficient extraction gives ${\tau }_X(X^a{K}_f)=Y^a$ in the root quotient. Its contractions prove both adjunction triangles, and symmetry gives cyclic dots. The actual nil-Hecke matrix corners identify induction with $B_m$ and its dual with $g\mapsto [v\mapsto {\tau }_{\mathrm{n}\mathrm{e}\mathrm{w}}(gv)]$ [F4], so evaluation and coevaluation are the stated first cap and cup. The same supplier proves the ordered-residue relative cap normalized by $(-1{)}^{N+1}=(-1{)}^{h_m+1}$ transports exactly to ${\tau }_{\mathrm{o}\mathrm{l}\mathrm{d}}$, and its dual unit to that Casimir. This fixes actual second maps, not merely dimensions or trace values. The coefficient forms have degrees $-2d_{i}r$ and $-2d_i(N-r-1)$; after the basic induction shift their adjunction degrees are $0$ and $-2d_i(h_m-1)$. Normalizing $H_i$ at its input gives first unit/second cap degree $d_i(1-h_m)$ and first cap/second unit degree $d_i(1+h_m)$.

The actual canonical comparison and prescribed inverse. Take the typed native mates $T:G_i{H}_i\to H_i{G}_i$ and $S:H_i{G}_i\to G_i{H}_i$ built from the actual maps of step 3.1 and the downward crossing and its twofold mate. On equal-endpoint generic root paths their matrices are $(-T{)}_{z,w}=c(z)\mathrm{/}[c^{\mathrm{\prime }}(w)(z-w)]$ and $S_{w,z}=b(w)\mathrm{/}[b^{\mathrm{\prime }}(z)(w-z)]$, with $w$ selected and $z$ complementary; swap paths have $T=-1,S=1$ [F4]. Thus they are the two polynomial interpolation maps. At $h_m\ge 0$, divide the degree-$<N-r$ interpolant by $c$: its remainder has degree $<r$, and its quotient degree $<h_m$. Expanding $b\mathrm{/}c=u^{h_m}A(u^{-1})$ identifies quotient coefficients with the prescribed $R_k={\sum }_{a+j=h_m-1-k}{A}_j{\epsilon }^{\mathrm{\prime }}{x}^a$, with columns $C_k=x^k\eta$. The supplier proves all five equations $-ST=1$, $RT=0$, $SC=0$, $RC=1$, $-TS+CR=1$. At $h_m<0$, division of the degree-$<r$ interpolant by $b$ gives $R_k^{-}={\sum }_{a+j=-h_m-1-k}{\mathsf{H}}_j\epsilon x^a$, $C_k^{-}=x^k{\eta }^{\mathrm{\prime }}$ and all five equations $-TS=1$, $T{C}^{-}=0$, $R^{-}S=0$, $R^{-}{C}^{-}=1$, $-ST+C^{-}{R}^{-}=1$. At zero weight both identity blocks are empty. These are identities on actual finite projective tensor bimodules, descended integrally before specialization, with degree-zero components after the prescribed shifts.

Typed curls, cancellation and bubble coefficients. The downward crossing $\kappa =(s-1)\mathrm{/}(X-Y)$ has square zero and obeys the braid and mixed dot equations by expansion; the symmetric Casimir gives dot mates. The actual curls of [F4] are $L_{\mu }={\sum }_{a+j=-h_m}{x}^a{\mathsf{H}}_j$ on $H_i$ at input $\mu$, and $R_{\mu }=-{\sum }_{a+j=h_m}{x}^a{A}_j$ at input $\mu -{\alpha }_i$, of degrees $-2d_i{h}_m$ and $2d_i{h}_m$. Mating the completeness equations of step 4.1 gives the cancellation corrections. Positively, $a+j+c=h_m-1$ and the clockwise bubble has literal dots $b=-h_m-1+j$; negatively, $a+j+c=-h_m-1$ and counterclockwise dots are $b=h_m-1+j$. Both give $a+b+c=-2$, with the prescribed native signs; the single-curl sums give $a+b=-1$. Actual pairings determine the bubble coefficients: ${\epsilon }^{\mathrm{\prime }}{x}^n\eta ={\sum }_{z}c(z)z^n\mathrm{/}{b}^{\mathrm{\prime }}(z)$ is the $u^{-1}$ coefficient of $u^{n}c\mathrm{/}b$, namely ${\mathsf{H}}_{n-h_m+1}$, whereas $\epsilon x^n{\eta }^{\mathrm{\prime }}={\sum }_{w}b(w)w^n\mathrm{/}{c}^{\mathrm{\prime }}(w)=A_{n+h_m+1}$. Here $A(t)=t^{h_m}{p}_m(t^{-1})\mathrm{/}{c}_r(t^{-1}{)}^2$ and $\mathsf{H}=A^{-1}$; negative indices vanish and coefficient zero is $1$. Integral descent proves these identities without distinct-root hypotheses after specialization. Thus ${\mathrm{\Phi }}_i=p_m\mathrm{/}{c}_r^2$ has the prescribed normalized series and reciprocal. Appending $i$ root $z$ divides $A$ by $(1-zt{)}^2$, and appending spectator root $y$ multiplies it by $q_{ij}^{\mathrm{r}\mathrm{e}\mathrm{v}}(y,t)$; reciprocal coefficients give the opposite slides. All recursions are finite at each coefficient. At a monic vacuum, closing $a_i(x)x^{\mathrm{\ell }-1}=0$ gives $\mathsf{H}(t)a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}(t)=1$.

Downward $Q$-relations and exact word grading. Give every flag the ordered shift $s_j(m)$ of the Statement. For $j<k$ the increasing word $(j,k)$ maps to the word $(k,j)$ by a quotient/switch of underlying degree $0$, with target-minus-source unit shift $-b_{jk}$; the reverse operator has underlying degree $-2b_{jk}$ and unit-shift $+b_{jk}$, so both have actual degree $-b_{jk}$, and same-color divided differences have degree $-2d_j$. Native operators are the negatives of the source operators, so their degrees and two-crossing composites are unchanged. The ordinary polynomial operators satisfy the idempotent, dot, far-commutation and quadratic $Q$-relations directly; expanding a three-strand composition gives zero difference for distinct outer colors and the divided difference of $Q$ for equal outer colors, and the global substitution $\kappa =-\tau$ changes all cubic signs to the native numerator. The quotient relations descend: on an $i,j$ pair $D_1$ has $bq=0$ and $D_0$ has $b=0$, and $Qb=t_{ij}bq$ makes multiplication by $Q$ well defined while the other crossing is the literal quotient.

The typed partial-color mixed pair. At input $\mu$ put $\xi =\mu +{\alpha }_i-{\alpha }_j$, $X=G_j{H}_i$, $Y=H_i{G}_j$. The native maps are $a=(H_i{G}_j{\epsilon }_{\mu })(H_i{\tau }^{G_i{G}_j}{H}_i)({\eta }_{\xi }{G}_j{H}_i):X\to Y$ and $b=({\epsilon }_{\xi }^{\mathrm{\prime }}{G}_j{H}_i)(H_i{\tau }^{G_j{G}_i}{H}_i)(H_i{G}_j{\eta }_{\mu }^{\mathrm{\prime }}):Y\to X$. Their middle words are $H_i{G}_i{G}_j{H}_i$ and $H_i{G}_j{G}_i{H}_i$; final counits consume respectively the rightmost $G_i{H}_i$ and leftmost $H_i{G}_i$. For $p^{\mathrm{\prime }}=p{q}_{ij}$, [F4] proves the actual normalized cap equation $\pi {\widehat{\epsilon }}^{p^{\mathrm{\prime }}}(Q_{ij}\mathrm{lift}(z))=t_{ij}{\widehat{\epsilon }}^p(z)$ with lift independence on every matrix-corner element, and its dual cup equation. The ordered-residue sign $(-1{)}^{N+1}$ transports precisely to ${\tau }_{\mathrm{o}\mathrm{l}\mathrm{d}}$ of step 3.1. Substituting the quotient/switch and multiplication-by-$Q$ in the two mate contractions therefore proves $ba=t_{ij}{1}_X$ and $ab=t_{ij}{1}_Y$ for this same pair. Source crossings negate both maps and preserve the products. They commute with the $j$ dot and have degree $d_i(1+h_m)+d_i(-1-h_m+a_{ij})-b_{ij}=0$, giving the unshifted mixed isomorphism. The separately distinguished-$j$ model has its own scalar $t_{ji}$; a full two-color adjunction comparison is not a relation of the category containing only $H_i$.

Conclusion. Steps 3.1-7.1 verify every defining relation of The minimal-parabolic KLR diagram proof category on the explicit flag modules: the generators and their degrees, both adjunction triangles, the rank-one matrices in both branches with their inverses, the bubble series and slides, the cross-and-cap equations, the mixed isomorphism with its scalars, and the downward $Q$-relations. Monic extensions, their Casimirs, the finite rank-one inverses and the coefficient identities survive arbitrary monic specialization as ungraded maps, and the construction is choice-free. This proves only the formal MODEL relations; faithfulness and the cyclotomic word-corner transfer belong to separate suppliers.