Oriented minimal-parabolic normal forms and the cyclotomic cap ideal

A finite planar routing schedule. Forget dots temporarily and count all transverse intersections, including self-intersections, by $C$. An arc with a self-intersection has an innermost monogon or bigon; two simple arcs with more intersections than their endpoint alternation requires have a bigon. To see the latter, cut along one arc: successive entrances and exits of the other arc in the resulting disk either delimit a bigon or delimit a subarc containing a self-intersection, to which the first case applies. A closed component similarly has a monogon, a bigon with another component, or is a simple disjoint circle: cut along a path from the circle to the exterior and use successive entrances and exits. Clear the chosen disk as follows. An interior arc returning to the same side cuts off a smaller disk; choose that disk first. Otherwise its segment joins two sides, and push the disk corner past its nearest intersection by a triangle move. This removes that intersection from the chosen disk. A whole closed component in the disk is handled first by the same procedure. Thus the number of intersection vertices on or inside the chosen disk, apart from its defining corners, strictly decreases on every main triangle move, until its monogon or bigon is empty. This is a finite schedule: the diagram has finitely many vertices, and selection of a smaller disk strictly decreases that number too. Local extrema are carried along by isotopy; a crossing carried past an extremum is its specified adjunction mate. No new crossing pair is inserted. Once every component is simple and every pair crosses at most once, cut the disk along one component at a time and repeat the same nearest-intersection triangle sweep to route that component to its chosen reduced path. The already routed components remain fixed; the number of unrouted components decreases. This gives a finite sequence of triangle moves between reduced drawings of the same matching.

Algebraic use of the schedule. A triangle move is a downward KLR braid equation or its adjunction mate. Its leading term has the same three crossings, and every correction has fewer crossings. An empty downward bigon is reduced by the quadratic KLR equation; an opposite-orientation bigon is reduced by the prescribed cancellation equation, whose correction has no crossings but may insert a cup and cap. A monogon is reduced by the appropriate curl equation or its adjunction mate, removing its one crossing. Distinct-color opposite bigons have their specified unit cancellation. Move dots along components by cyclicity and the KLR dot equations; every correction again has fewer crossings. Induct first on $C$. At a fixed $C$ use the finite disk schedule of step 1.1, and apply the induction hypothesis separately to each correction. All sums are finite: curl and cancellation coefficients have nonnegative open-dot and bubble indices with fixed total. In particular an inserted cup and cap cannot defeat this induction, since its branch has already lowered $C$. After intersections are reduced, extra extrema on a simple component cancel by the triangles; these decrease the finite extremum count without changing $C$. Move each isolated circle to the far left by the coefficient slides, in decreasing order of its distance from the left edge. For a fixed coefficient every slide is a finite polynomial identity; it creates no intersection and decreases that distance. Replace the reciprocal bubble family by its finite inverse-coefficient recursion. Finally route the reduced open matching as in step 1.1, handling its shorter-crossing corrections by induction, and place the dots at the chosen positions. This proves finite spanning by the stated normal forms; it does not presuppose their independence.

Leading matching coefficients in the actual model. Work over the universal framing ring and then its generic splitting fraction field, localizing distinct root differences and the finitely many nonzero spectator $Q$ values used in the diagram. This is permitted solely to find a nonzero witness, not to prove an identity by specialization. Use the flag summands in which every background or visited $i$ root is a distinct root of the free universal framing polynomial. These summands are indexed by root additions and removals and are one-dimensional over the splitting fraction field. They exist because $r$ may grow while the framing degree grows as $2r+h-A_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}$. The spectator factors of $p$ need not be separable: they are placed entirely in the complementary factor, and their values at these free framing roots are nonzero. Chinese remainder against the separable free framing factor gives the indicated summands even when a spectator factor has repeated roots. A dot multiplies by that root. A native equal-color crossing is $(1-s)\mathrm{/}(x-y)$: its root-switching term has coefficient $-1\mathrm{/}(x-y)$, and its other term retains the labels and deletes that crossing. A distinct-color crossing switches the labeled roots with coefficient either a unit or a nonzero $Q$ value. The monic coefficient cap at a simple root $z$ has coefficient $1\mathrm{/}{f}^{\mathrm{\prime }}(z)$ up to its specified unit normalization, and its dual cup has the inverse coefficient; these follow by evaluating the dual-power Casimir $(f(X)-f(Y))\mathrm{/}(X-Y)$ on the root idempotents. All these coefficients are nonzero. Consequently expanding a reduced matching diagram on these summands has a leading term that carries a root label along each of its matched components, with a nonzero product coefficient. All other terms delete at least one crossing, and hence are smaller in the minimal crossing number of their boundary matching after step 2.1. Choose pairwise distinct free framing roots for the different open $i$ components and independent spectator roots for the other components. The endpoint equalities of this leading term specify exactly its matching: a different matching must identify two labels assigned to different components. In a finite proposed relation select a matching of maximal minimal crossing number. No smaller matching can contribute to its root-labeled leading coefficient, and another matching of the same crossing number has different endpoint equalities. Its coefficient is therefore its dot polynomial on the selected component roots times its bubble polynomial, times a nonzero rational factor. This replaces an invalid independent bending of interior upward endpoints; only the explicit root flags and their already proved cup/cap maps are used.

A faithful witness at a fixed physical weight. Let $h=⟨h_i,\mu ⟩$ be fixed, choose spectator background counts $m_j$, and put $A_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}={\sum }_{j\mathrm{\ne }i}(-a_{ij})m_j$. For increasing $r$ put $N=h+2r$ and ${\lambda }_i=N-A_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}$. Choose the universal monic framing $a_i(u)=u^{{\lambda }_i}+z_1{u}^{{\lambda }_i-1}+\cdots +z_{{\lambda }_i}$ with $\mathrm{deg}{z}_s=2d_{i}s$, independent of the root variables. In the flag model, $p=a_i{\prod }_{j\mathrm{\ne }i}{Q}_{ij}\mathrm{/}{t}_{ij}$ and $\mathcal{B}=p^{\mathrm{r}\mathrm{e}\mathrm{v}}\mathrm{/}(c_r^{\mathrm{r}\mathrm{e}\mathrm{v}}{)}^2=a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}U$ with $U_0=1$. Its first $D$ coefficients have the unit triangular form ${\mathcal{B}}_s=z_s+{\sum }_{0<\mathrm{\ell }<s}{z}_{\mathrm{\ell }}{U}_{s-\mathrm{\ell }}+U_s$ for $s\le D$ whenever ${\lambda }_i\ge D$. Thus substituting bubbles in any nonzero finite downward PBW/bubble combination leaves a nonzero polynomial-operator combination before truncation. By the faithful polynomial representation it has a test vector with nonzero output. Take a homogeneous component of that vector and output, and let $B$ bound their degrees and all framing indices used. These bounds can be fixed while $r$ grows: only the finitely many appended strand variables, bubble indices, and fixed-degree symmetric background coefficients occur in the operator and test vector. Let $L$ bound the number of appended $i$ flags. Every flag/factor coefficient relation encountered has degree at least $2d_i(h+r-L+1)$; all variables have nonnegative degrees. Choose $r$ with ${\lambda }_i\ge D$ and this bound greater than $B$. No nonzero output of degree at most $B$ can belong to the ideal of these higher-degree relations, so it survives in that actual flag model. For the general-boundary leading coefficient of step 3.1, choose $r$ also large enough to supply distinct free framing roots for every component and at least $D$ unused free complementary roots. The selected component roots are algebraically independent; the first $D$ elementary coefficients of those unused roots are algebraically independent over them. In the factor identity $\mathcal{B}=b_{N-r}^{\mathrm{r}\mathrm{e}\mathrm{v}}\mathrm{/}{c}_r^{\mathrm{r}\mathrm{e}\mathrm{v}}$, each of the first $D$ coefficients is a unit triangular polynomial in these complementary elementary coefficients. Thus the nonzero dot/bubble coefficient of step 3.1 remains nonzero in this actual localized flag model. Clearing its finitely many nonzero denominators gives an actual nonzero model operator. This proves independence at all boundaries in every characteristic; it uses unit framing or complementary-factor coefficients, not a root coefficient such as $2e_1$.

Basis and confluence. Step 2.1 gives spanning at all boundaries, and steps 3.1–4.1 prove independence in the actual model at all boundaries. Two finite straightening schedules have outputs whose difference represents zero; their basis coefficients therefore agree. This proves the asserted confluence of the output without assuming unchecked critical-pair identities. The same separating witnesses prove joint faithfulness before quotienting.

Forced vacuum coefficients. Write $h_{\mathrm{\ell }}$ for the counterclockwise bubble coefficient with literal ${\lambda }_i-1+\mathrm{\ell }$ dots, with $h_0=1$ and $h_{\mathrm{\ell }<0}=0$. Closing the marked relation $a_i(x)x^{\mathrm{\ell }-1}=0$, for $\mathrm{\ell }\ge 1$, is lawful in $J$ and gives $h_{\mathrm{\ell }}+c_{i,{\lambda }_i-1}{h}_{\mathrm{\ell }-1}+\cdots +c_{i,0}{h}_{\mathrm{\ell }-{\lambda }_i}=0$. Thus ${\mathsf{H}}_{\mathrm{\Lambda }}{a}_i^{\mathrm{r}\mathrm{e}\mathrm{v}}=1$ and $A_{\mathrm{\Lambda }}=a_i^{\mathrm{r}\mathrm{e}\mathrm{v}}$. Let $S$ be this coefficient-specialization ideal; then $S\subset J$. The basis of step 5.1 shows that specializing these far-left central coefficients alone leaves the ordinary downward corner exactly $R(\beta )$. This conclusion concerns $S$ alone and does not assume the conclusion about $J$. For ${\lambda }_i=0$, $a_i=1$ kills the first $i$ word and the same recurrence gives $A=\mathsf{H}=1$.

The correctly typed local closure. Work modulo $S$. A straight vacancy closure with $q\ge 0$ dots has clockwise $A$-index ${\lambda }_i+1+q$, so is zero. The crossed closure on the first physical $G_i$ at input $\mathrm{\Lambda }$ is the first-adjunction mate of $R_{\mathrm{\Lambda }}$, whose $H_i$ input is $\mathrm{\Lambda }-{\alpha }_i$ and whose clockwise bubble region is $\mathrm{\Lambda }$. In its exact native equation the bubble literal count is $b=-{\lambda }_i-1+m$ and $a+b=-1$, hence $a={\lambda }_i-m$. Its value is consequently $-{\sum }_{m=0}^{{\lambda }_i}{A}_m{x}^{{\lambda }_i-m}=-a_i(x)$; its degree is $2d_i{\lambda }_i$. Taking the mate preserves this polynomial because the dot is cyclic. If $P(u,x)$ gives additional dots on the added and physical strands, the native relation ${\tau }_{1}u=x{\tau }_1+1$ gives $C(uP)=xC(P)+B(P)$ with $B$ a straight closure, up to the fixed cap convention's unit sign. Since $B(P)=0$, induction on the nonnegative exponents gives $C(P)=-a_i(x)P(x,x)$. Before these closures, slide any bubble at the bent source $\mathrm{\Lambda }+{\alpha }_i$ across the added root $z$: $A_{\mathrm{\Lambda }+{\alpha }_i}=(1-zt{)}^2{A}_{\mathrm{\Lambda }}$ and the reciprocal equation for $\mathsf{H}$. Each coefficient is finite, so this reduces it to polynomial dots and the already specialized vacuum coefficients. No bubble is assigned the wrong ambient weight.

The marked operator at the dominant vacuum. Modulo the killed-upward ideal, the actual positive rank-one matrix identifies $H_i{G}_i{v}_{\mathrm{\Lambda }}$ with $D_{{\lambda }_i}$. Its columns are $v_r=x^r\eta$, $0\le r<{\lambda }_i$, and its dual rows are the finite bubble-weighted counit rows of the prescribed inverse. A raw row with $s\ge 0$ extra dots sends $x^r{v}_0$ to $h_{r+s-{\lambda }_i+1}$. Its value on $a_i(x)v_0$ is therefore the zero recurrence of step 6.1 at index $s+1$. Every actual dual row is a finite combination of raw rows, so all projections of $a_i(x)v_0$ vanish; the matrix inverse equation implies the column itself is zero. Multiplying by powers of $x$ kills $a_i(x)$ on all columns and hence on $H_i{G}_i{v}_{\mathrm{\Lambda }}$. When ${\lambda }_i=0$ this object is already zero. This uses the actual inverse rows, rather than assuming raw rows are dual bases.

Every killed-upward factor. A term through $W{H}_i{v}_{\mathrm{\Lambda }}$ has factor maps $f:G_u{v}_{\mathrm{\Lambda }}\to W{H}_i{v}_{\mathrm{\Lambda }}$ and $g:W{H}_i{v}_{\mathrm{\Lambda }}\to G_v{v}_{\mathrm{\Lambda }}$. Bend that vacuum $H_i$ in both maps. The triangles recover $gf$ by closing the composite of the bent maps, which has all-downward boundaries with an additional first $i$ strand at source $\mathrm{\Lambda }+{\alpha }_i$. The interior word $W$ need not be downward. Apply step 5.1 to the composite and then the coefficient slide of step 7.1. Put $n=\mathrm{\mid }\beta \mathrm{\mid }$ and let $W_P\subset S_{n+1}$ fix position one. There are two double cosets, $W_P$ and $W_P{s}_1{W}_P$. Removing the distinguished input and output and inserting them in their required positions supplies length-additive representatives $u{s}_{1}v$, $u,v\in W_P$, in the second case; the first case uses only $s_2,\dots ,s_n$. Choose these reduced words for PBW. Shorter braid corrections still straighten in this family. In the first case the first strand closes to the zero straight vacancy loop. In the second the exterior words fix position one, whose color is $i$ at both boundaries; therefore both colors at the unique $s_1$ crossing are $i$. The exterior downward diagrams pull past the closure, and its local value from step 7.1 is $-a_i(x_1)P(x_1,x_1)$. Every such term is in the ordinary first-strand $i$ ideal. For $n=0$ only the zero straight closure occurs.

Every marked monic whisker. In a term vertically sandwiching $W(a_j(x)1_{G_j{v}_{\mathrm{\Lambda }}})$, select the chronologically earliest $H_i$ in its intermediate word and move it toward the vacuum. Distinct-color commutation is natural and commutes with the marked $j$ dot. For $h\ge 0$ the rank-one isomorphism moves $H_i{G}_i$ to $G_i{H}_i$ plus finitely many identity branches; the latter remove an $H_i$. For $h<0$ use the reverse isomorphism $r_h:G_i{H}_i\to H_i{G}_i\oplus D_{-h}$ and its actual retract maps $\iota =r_h^{-1}{j}_H$, $\pi =p_H{r}_h$, with $\pi \iota =1$. Naturality gives $(H_i{G}_i)(b)=\pi (G_i{H}_i)(b)\iota$ for every input endomorphism $b$. Absorb these retract maps into the outer factor maps; the intermediate word has moved its earliest $H_i$ one place toward the vacuum. At the marked first strand, $j\mathrm{\ne }i$ gives a killed-upward factor by mixed dot compatibility; $j=i$ gives zero by step 7.2 and the dominant vacuum matrix. The lexicographic measure (number of $H_i$ letters in the intermediate word, distance of its earliest $H_i$ from the vacuum) strictly decreases on each finite branch. It counts the intermediate word, not cups and caps in adapted factor maps. After that word is downward, step 5.1 straightens its factor maps. Consequently all marked whiskers reduce to ordinary first-strand ideal insertions and killed-upward factors, already handled in step 8.1.

Exact intersection. Express an ordinary downward element $x\in J$ as a finite sum of factored whiskers of the defining generators. Specialize $S\subset J$ first. The killed-upward summands land in $I_a$ by step 8.1, and the marked monic summands do so by step 9.1; specialization summands vanish. Step 6.1 shows that specialization leaves the included ordinary element $x$ unchanged. Thus $J\cap e(v)R(\beta )e(u)\subset e(v)I_{a}e(u)$. The reverse inclusion follows from the defining monic generators and downward compositions. Every downward quotient morphism is represented by an ordinary downward element by step 6.1, giving the corner isomorphism, with composition equal to multiplication.

Generation and transport. Apply the word induction of step 9.1 without a marker. At the vacuum $H_i{v}_{\mathrm{\Lambda }}$ is zero; positive rank-one branches give finite sums and negative branches give actual retracts, so every word is a retract of a finite sum of downward words. Downward words under the corner identification of step 10.1 are the representable word projectives $e(u)R\mathrm{/}{I}_a$. They generate the finite projectives by additive sums and idempotents; the corner isomorphisms extend to a fully faithful and essentially surjective equivalence of their additive idempotent completions. Appending $G_j$ acts on these representables by the ordinary appended-strand bimodule, hence by actual induction. Transporting the first adjunction identifies $H_i$ with its normalized projective restriction by uniqueness of adjoints. Restriction preserves finite projectives because its values are the transported retracts just proved. The second units and counits transport through the same fully faithful equivalence, so their naturality and both triangles hold as actual bimodule natural transformations. This transports the maps; a claim that their second cap equals a separately prescribed trace would require an additional calculation. All constructions used finite coefficient operations and unit identities over the universal coefficient ring; arbitrary monic specialization gives the ungraded claims.