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\title[The quantum Kazhdan--Lusztig conjecture]{The quantum Kazhdan--Lusztig conjecture\ for quantum loop algebras}
\author{Qiyue Tang}
\address{Zhili College, Tsinghua University, Beijing 100084, China}
\email{tangqy24@mails.tsinghua.edu.cn}
\date{November 11, 2026}
\subjclass[2020]{Primary 17B37; Secondary 16G20, 14F08, 14D21}
\keywords{Quantum loop algebra, quantum Grothendieck ring, canonical basis, critical cohomology, Coulomb branch}
\hypersetup{
  pdftitle={The quantum Kazhdan--Lusztig conjecture for quantum loop algebras},
  pdfauthor={Qiyue Tang},
  pdfsubject={Quantum characters and canonical bases},
  pdfkeywords={quantum loop algebra, quantum Grothendieck ring, canonical basis}
}
\begin{document}
\begin{abstract}
We prove the quantum Kazhdan--Lusztig conjecture for finite-dimensional
type~$1$ representations of untwisted quantum loop algebras at every
non-root-of-unity complex parameter.
\end{abstract}
\maketitle
\tableofcontents

\section{Introduction}

Let $\g$ be a finite-dimensional complex simple Lie algebra and let
$q\in\C^\times$ be a parameter which is not a root of unity.
Let $I$ be the Dynkin vertex set and let $Y_{i,a}$, for $i\in I$ and
$a\in\C^\times$, be formal variables. A dominant monomial is a product
$m=\prod_{i,a}Y_{i,a}^{u_{i,a}}$ with nonnegative integer exponents,
only finitely many of which are nonzero. The category
$\cC$ of finite-dimensional type~$1$ representations of the untwisted quantum
loop algebra $\U$ has a commutative Grothendieck ring. The $q$-character
homomorphism of Frenkel--Reshetikhin embeds this ring into a Laurent
polynomial ring and records the generalized eigenvalues of the Drinfeld
Cartan currents \cite{FR99}. Its image admits a noncommutative deformation
$\K$, called the quantum Grothendieck ring. This deformation has a standard
basis $\{E_t(m)\}$ and a bar-invariant canonical basis $\{L_t(m)\}$, both
indexed by dominant monomials \cite{Her04,FHOO22}.

Let $L(m)$ be the finite-dimensional simple type~$1$ module with highest
$\ell$-weight $m$. Hernandez's quantum Kazhdan--Lusztig conjecture identifies the specialization
of $L_t(m)$ at $t=1$ with the $q$-character of the simple module with highest
monomial $m$ \cite[Conjecture~7.3]{Her04}. We prove this identification.

\begin{theorem}\label{thm:main}
Let $\g$ be a finite-dimensional complex simple Lie algebra, let
$q\in\C^\times$ be not a root of unity, and let
$m=\prod_{i\in I,a\in\C^\times}Y_{i,a}^{u_{i,a}}$ be any dominant monomial
with finite support. Then
\begin{equation}\label{eq:main}
                 L_t(m)\big|_{t=1}=\chq L(m).
\end{equation}
\end{theorem}

The convention for the bar and for the ordered standard
character is given in Section~\ref{sec:prelim}. In the notation of the
original formulation \cite[Conjecture~7.3]{Her04}, equation
\eqref{eq:main} is $\pi_+(L_t(m))=L(m)$, after identifying the
Grothendieck ring with its $q$-character image.

\subsection{The character problem}

Chari--Pressley classified the finite-dimensional simple type~$1$
representations by their Drinfeld polynomials \cite{CP95}. Factoring these
polynomials gives the dominant monomial $m$ which labels $L(m)$.
This classification separates the parametrization of simple modules from
the calculation of their characters and of tensor-product multiplicities.
The latter questions depend on the spectral parameters: tensor products
can become reducible when ratios of parameters take particular powers
of $q$.

In 1999, Frenkel--Reshetikhin introduced the $q$-character to retain the spectral
information lost by the ordinary weight character \cite{FR99}. Its
monomials encode the joint generalized eigenvalues of the commuting
Drinfeld Cartan currents. Injectivity of the $q$-character map makes
identities of these Laurent polynomials identities in the Grothendieck
ring. Frenkel--Mukhin described its image by rank-one conditions and
established the downward root-monomial structure of simple characters
\cite[Theorems~4.1 and~5.1]{FM01}.

The analogy with Kazhdan--Lusztig theory concerns the transition from
standard objects to simple objects. In their 1979 paper, Kazhdan and
Lusztig constructed the bar-invariant basis of a Hecke algebra and its
transition polynomials and proposed the corresponding character
formula \cite{KL79}. For quantum loop
algebras, ordered tensor products of fundamental representations play
the role of standard objects. Their ordinary characters are products
of fundamental $q$-characters. The quantum Grothendieck ring refines
this product by a parameter $t$, and the transition from its standard
basis to its canonical basis supplies the corresponding
Kazhdan--Lusztig polynomials. More precisely, write
\[
 E_t(m)=\sum_{n\leq m}P_{n,m}(t)L_t(n),\qquad P_{m,m}(t)=1.
\]
If $M(m)$ denotes the ordered tensor product of fundamental modules
with highest monomial $m$, the conjecture gives
\[
                         [M(m):L(n)]=P_{n,m}(1).
\]
Thus the canonical transition determines ordinary tensor-product
composition multiplicities, as in Hernandez's original formulation
\cite[\S7.2]{Her04}.

\subsection{Geometric and algebraic \texorpdfstring{$t$}{t}-characters}

For simply-laced $\g$, Nakajima realized finite-dimensional
representations through quiver varieties and related standard-to-simple
multiplicities to intersection cohomology \cite{Nak01}.
Varagnolo--Vasserot constructed quantum Grothendieck rings using
perverse sheaves on quiver varieties \cite{VV03}.
Nakajima's subsequent theory of $t$-analogues of $q$-characters
established the canonical specialization formula in types
$A$, $D$, and $E$ \cite[Theorem~8.1(2)]{Nak04}. The grading in this
construction records the cohomological information governing the
transition between the two bases.

Hernandez's algebraic construction extended $q,t$-characters and
canonical bases to arbitrary finite type \cite{Her04}. His
Conjecture~7.3, formulated in the 2002 preprint and published in 2004,
asks that these canonical elements specialize to the characters
of the corresponding simple representations. The construction already
establishes the specialization of standard characters
\cite[Theorem~6.2]{Her04}; the conjecture identifies the resulting
canonical labels with the ordinary simple labels.
Hernandez further studied monomials in the non-simply-laced case
\cite{Her05}. The fundamental character calculations in that work
and the Hernandez--Schedler tables \cite{HS} provide the exceptional
fundamental input used in Section~\ref{sec:fundamental}.
His proof of the Kirillov--Reshetikhin conjecture and the associated
$T$-systems supplies another family of ordinary character formulas
\cite{Her06}.

\subsection{Progress on the conjecture}

Hernandez--Leclerc introduced, in simply-laced type, monoidal subcategories whose
Grothendieck rings can be studied by cluster algebras \cite{HL10}.
Hernandez--Oya then constructed quantum Grothendieck-ring
isomorphisms between distinguished categories of types $B_n$ and
$A_{2n-1}$ and proved the specialization formula in the corresponding
type-$B$ categories \cite{HO19}. Fujita--Hernandez--Oh--Oya
extended these comparisons to all non-simply-laced types, proved
positivity of the transition polynomials and structure constants,
and established the specialization conjecture for all simple
modules of type $B$ \cite[Corollary~11.4(2)]{FHOO22}.

One consequence of the same work is a uniform statement in every finite
type. For integral spectral parameters $Y_{i,q^k}$ with
$k\in[a,b]\cap\Z$, the specialization formula holds when
\[
                         b-a<r^\vee h^\vee,
\]
where $r^\vee$ is the lacing number and $h^\vee$ is the dual Coxeter
number \cite[Corollary~9.8(2)]{FHOO22}; see
\cite[Theorem~2.11 and Remark~2.12]{FQ26} for this interval formulation.
Further quantum cluster
comparisons prove the formula for reachable simple modules and
positivity of the monomial coefficients of simple $(q,t)$-characters
\cite[Corollaries~6.10 and~6.12]{FHOO23}. Here reachability refers to the simple modules
represented by cluster monomials in the corresponding monoidal
categorification.

Fujita--Qin introduced freezing operators compatible with ordinary
and deformed characters \cite{FQ26}. For classical types, they
embed a fixed Dynkin diagram into diagrams of larger rank, apply the
spectral-width result there, and freeze back to the original
diagram. This proves the conjecture in all classical types and,
in particular, settles type $C$
\cite[Theorem~2.13]{FQ26}. Together with Nakajima's theorem, these
results cover every finite Dynkin type except $G_2$ and $F_4$.
The argument below proves the specialization formula for every
dominant monomial in these two types.

There are also geometric and filtration constructions relevant to
this problem. Varagnolo--Vasserot constructed actions of
non-symmetric quantum loop groups in critical $K$-theory
\cite{VV24} and related shifted quantum affine representations to
Coulomb branches and weighted-flag algebras \cite{VV25}.
Negu\c t described quantum loop representations by shuffle quotients
and quiver moduli \cite{Neg25a,Neg25b}.
Fujita--Hernandez constructed monoidal Jantzen filtrations from
$R$-matrices and identified the resulting deformation with the
quantum Grothendieck ring in simply-laced type \cite{FH26}.
In the shifted setting, Geiss--Hernandez--Leclerc extended cluster
structures to non-simply-laced types and expressed cluster monomials
as limits of cluster expansions in classes of positive
prefundamental representations \cite{GHL26}.
For a broader account of quantum affine representation theory and its
connections with cluster algebras, see Hernandez's survey \cite{Her26}.

\subsection{The geometric filtration}

The central construction is attached to an actual simple module. Write its
highest Cartan rational function as a quotient whose pole and zero
multiplicities are $\bm$ and $\bl$, and put $\bp=\bm-\bl$.
At a fixed spectral dimension vector $\bnu$, a shuffle realization identifies
the module with a quotient of critical cohomology by the kernel of a
negative-framing Euler operator. The vectors $\bm,\bl,\bnu$ are indexed
by spectral vertices. For finite dimension vectors put
$\mu\cdot\nu=\sum_v\mu_v\nu_v$ and
$E(\mu,\nu)=\sum_{v\to w}\mu_v\nu_w$, with the arrows of
Section~\ref{geo:setup}. The center is
\begin{equation}\label{eq:intro-center}
 d_{\bp}(\bnu)=\bp\cdot\bnu+E(\bnu,\bnu)-\bnu\cdot\bnu.
\end{equation}
A polynomial of degree $r$ has centered degree $2r-d_{\bp}(\bnu)$.
This gives a finite filtration on the actual module and a Laurent polynomial
$S_t(m)$ which specializes, by construction, to $\chq L(m)$.

Several comparisons are needed to extract representation-theoretic
information from this filtration. First, we prove the shuffle-kernel
statement directly at the numerical parameter $q$. A normal-order
calculation for high current modes and finite-jet interpolation identify the
annihilator of the highest vector with a common residue kernel. The
generic shuffle theorem is transferred to numerical parameters by a local
ideal argument, retaining the factors at orthogonal colors.

Second, ordinary flag towers determine the cohomological residue kernel.
A supported nearby-cycle comparison, a resolution of the graph of stable
restriction, and dimensional reduction identify the critical image.
Keeping the framing parameters equivariant gives a finite free
source quotient of the Euler map. Its deformed residue tests identify
the generic highest-vector kernel, and the integral quotient supplies
the standard module at coincident parameters.

Third, we use the Coulomb-branch realization of Varagnolo--Vasserot
\cite{VV25}. Explicit fundamental modules give the ordinary quantum loop
action, including its constants and determinant branches. An integral
generation argument identifies the geometric standard lattice with the
ordered tensor-product lattice. This compares the standard and intrinsic
filtrations in one ordinary flag algebra. Each current is a minuscule
correspondence multiplied by a Riemann--Roch unit. An exact inverse using
source Chern classes recovers the homogeneous correspondence and its
Ext degree. Finite spectral projectors give the ordinary current map
onto a common finite graded quotient.

The resulting finite quotient of the ordinary flag algebra has a
nonnegative basic grading with semisimple degree zero. Its normalized
simple characters are bar-invariant. Hyperbolic restriction computes the
graded standard character as $\overline{E_t(m)}$, whereas homogeneous
generation places every non-head composition factor in strictly positive
external degree. We obtain
\begin{equation}\label{eq:intro-triangle}
 \overline{E_t(m)}=S_t(m)+\sum_{n<m}p_{m,n}(t)S_t(n),
 \qquad p_{m,n}(t)\in t\N[t].
\end{equation}
A finite-height induction first proves $S_t(m)\in\K$. Applying bar to
\eqref{eq:intro-triangle} then gives the triangularity required for
canonical basis uniqueness. This proves $S_t(m)=L_t(m)$ before any
specialization is taken.

\subsection{Organization and conventions}

Section~\ref{sec:prelim} fixes the algebraic conventions.
Section~\ref{sec:kernel} proves the numerical shuffle and residue results.
The subsequent geometric and Coulomb constructions are followed by the
comparison of intrinsic and standard filtrations. We then normalize the
simple characters, establish the exceptional fundamental input, calculate
the standard characters, and prove Theorem~\ref{thm:main}.
An appendix records a finite certificate for the only nonthin fundamental
coefficients used in the proof.

We use $\N=\{0,1,2,\ldots\}$. All vector spaces and cohomology groups are over $\C$, unless a rational
model is explicitly specified. Polynomial degree is half cohomological
degree. Critical cohomology uses the monodromic vanishing-cycle convention
with the suspension normalization stated below. Completions are taken at
specified finite spectral supports; statements about completed actions are
used only on finite continuous modules. All dimension vectors have finite
support, including dimension vectors on infinite spectral quivers.

\section{Quantum characters and canonical bases}\label{sec:prelim}

\subsection{Highest monomials}

Let $I$ be the vertex set of the finite Dynkin diagram, let $C=(c_{ij})$ be
its Cartan matrix, and choose the minimal positive symmetrizer $(d_i)$.
Thus $b_{ij}=d_ic_{ij}=d_jc_{ji}$ and $q_i=q^{d_i}$. Set
\[
 \Y=\Z[Y_{i,a}^{\pm1}:i\in I,\ a\in\C^\times],
 \qquad \mathcal M_+=\langle Y_{i,a}:i\in I,\ a\in\C^\times\rangle.
\]
Elements of $\mathcal M_+$ are called dominant monomials. Products here and
below have finite support. The monomial
$m=\prod Y_{i,a}^{u_{i,a}}$ represents the rational highest $\ell$-weight
\begin{equation}\label{eq:highest-rational}
 \psi_i^m(z)=q_i^{\sum_a u_{i,a}}
       \prod_a\left(\frac{1-aq_i^{-1}/z}{1-aq_i/z}\right)^{u_{i,a}}.
\end{equation}
The finite-dimensional simple objects of $\cC$ are the $L(m)$ with
$m\in\mathcal M_+$. A module is of type~$1$ if each Cartan generator
$k_i$ acts diagonally with eigenvalues in $q_i^\Z$.
The coefficient of $m$ in $\chq L(m)$ is one.

The root monomials are
\begin{equation}\label{eq:root-monomial}
 A_{i,a}=Y_{i,aq_i^{-1}}Y_{i,aq_i}
 \prod_{j\ne i}\prod_{s=1}^{-c_{ji}}
       Y_{j,aq^{-c_{ji}+1-2s}}^{-1}.
\end{equation}
For monomials $n,m$, write $n\leq m$ if
$n=m\prod A_{i,a}^{-v_{i,a}}$ with $v_{i,a}\in\N$. These root
factorizations are unique for finite support. For example, uniqueness
follows by the invertibility of the quantum Cartan matrix as a formal
Laurent series, or by successively removing the smallest spectral degree
on the finite support in each $q^\Z$-orbit. All monomials in $\chq L(m)$ satisfy $n\leq m$.
Let $\omega_i$ and $\alpha_i$ denote the fundamental weights and simple
roots. We put $\wt(Y_{i,a})=\omega_i$, so that
$\wt(A_{i,a})=\alpha_i$.

\subsection{The quantum torus}

Choose one representative for each $q^\Z$-orbit of parameters. In one such
orbit write $Y_{i,k}=Y_{i,bq^k}$. Define the quantum Cartan matrix by
\[
 C_{ii}(z)=z^{d_i}+z^{-d_i},\qquad
 C_{ij}(z)=\frac{z^{c_{ij}}-z^{-c_{ij}}}{z-z^{-1}}\quad(i\ne j).
\]
Write $C(z)^{-1}_{ij}=\sum_{r\in\Z}\widetilde c_{ij}(r)z^r$, using its
expansion at zero. The skew form on the monomial lattice is determined by
\begin{align}\label{eq:gamma-def}
 \gamma(Y_{i,k},Y_{j,l})={}&
 \widetilde c_{ij}(k-l-d_i)-\widetilde c_{ij}(k-l+d_i)\notag\\
 &-\widetilde c_{ij}(l-k-d_i)+\widetilde c_{ij}(l-k+d_i).
\end{align}
The form between distinct parameter orbits is zero. This convention is
independent of the chosen representatives. The based quantum torus $\Yt$
is the free $\Z[t^{\pm1/2}]$-module on the ordinary monomials, with product
\begin{equation}\label{eq:torus-product}
                    m*n=t^{\gamma(m,n)/2}mn.
\end{equation}
The bar is the antilinear anti-involution fixing every based monomial and
sending $t^{1/2}$ to $t^{-1/2}$. In particular
$\overline{x*y}=\overline y*\overline x$.
The quantum Cartan identity gives
\begin{align}
 \gamma(A_{i,k},Y_{j,l})
   &=2\delta_{ij}(\delta_{k-l,d_i}-\delta_{k-l,-d_i}),
       \label{eq:gamma-AY}\\
 \gamma(A_{i,k},A_{j,l})
   &=2(\delta_{k-l,b_{ij}}-\delta_{k-l,-b_{ij}}).
       \label{eq:gamma-AA}
\end{align}
For completeness, put $D(z)=\operatorname{diag}(z^{d_i}-z^{-d_i})$.
The identities
\[
 C(z)^{\mathsf T}D(z)=D(z)C(z),\qquad
 (D(z)C(z))_{ij}=z^{b_{ij}}-z^{-b_{ij}}
\]
show that $D(z)C(z)^{-1}$ is symmetric. Its coefficient of $z^r$
in position $(i,j)$ is
$\widetilde c_{ij}(r-d_i)-\widetilde c_{ij}(r+d_i)$.
The exponent array of $A_{i,k}$ is the $k$th translate of the
$i$th column of $C(z)$. Since $C(z)=C(z^{-1})$, pairing this
array with \eqref{eq:gamma-def} amounts to multiplying on the left
by $C(z)^{\mathsf T}$. The resulting matrix is $D(z)$.
Subtracting its coefficients at $k-l$ and $l-k$ gives
\eqref{eq:gamma-AY}. Pairing with the second root array gives
$D(z)C(z)$ and hence \eqref{eq:gamma-AA}.
These identities fix the grading normalization below.

\subsection{The quantum Grothendieck ring}

For each $i$, let $\mathcal K_{i,t}$ be the subalgebra generated by the elements
$Y_{j,k}^{\pm1}$ for $j\ne i$ and the rank-one characters
$Y_{i,k}(1+A_{i,k+d_i}^{-1})$, in the based-monomial convention, and take
the corresponding generators in every parameter orbit. Then
\[
                         \K=\bigcap_{i\in I}\mathcal K_{i,t}.
\]
This is a bar-stable subalgebra of $\Yt$. The algebraic character theory
\cite{Her04,Her05,FQ26} gives a distinguished element $F_t(m)\in\K$ for
each dominant $m$, with unique dominant monomial $m$ and coefficient one.
All its remaining monomials are below $m$. These properties determine it.

Write $m=Y_1\cdots Y_\ell$, ordering each orbit by decreasing spectral
index. The corresponding fundamental characters at equal spectral index
commute \cite[\S2.7]{FQ26}, and factors from different orbits can be placed in
any order. Define
\begin{equation}\label{eq:standard-definition}
 E_t(m)=t^{-\frac12\sum_{r<s}\gamma(Y_r,Y_s)}
                  F_t(Y_1)*\cdots*F_t(Y_\ell).
\end{equation}
The $E_t(m)$ form a basis of $\K$. The canonical element $L_t(m)$ is
characterized by
\begin{equation}\label{eq:canonical-definition}
 \overline{L_t(m)}=L_t(m),\qquad
 L_t(m)-E_t(m)\in
       \sum_{n<m}t^{-1}\Z[t^{-1}]E_t(n).
\end{equation}
Each expansion has finite support. Existence and uniqueness are the
canonical-basis theorem of \cite{Her04}; the precise convention above is
\cite[Proposition~2.6]{FQ26}. For a single fundamental variable,
\begin{equation}\label{eq:fund-formal}
                F_t(Y_{i,a})=E_t(Y_{i,a})=L_t(Y_{i,a}).
\end{equation}
Their specialization to the fundamental $q$-characters, and the
specialization of every $E_t(m)$ to the corresponding product, are known
independently of Theorem~\ref{thm:main} \cite[Theorem~6.2]{Her04}.

\subsection{Finite support and height}

Let $\rho^\vee$ be half the sum of the positive coroots. For a dominant
monomial set
\begin{equation}\label{eq:height}
                      h(m)=\langle\wt(m),2\rho^\vee\rangle.
\end{equation}
It is a nonnegative integer, and if
$n=m\prod A_{i,a}^{-v_{i,a}}<m$ is dominant, then
\begin{equation}\label{eq:height-drop}
                       h(m)-h(n)=2\sum_{i,a}v_{i,a}>0.
\end{equation}
Thus induction over dominant monomials will always mean induction on
$h(m)$. Every individual module, filtration, or character used below has
finite support.

If $m$ involves finitely many parameter orbits, a dominant $n\leq m$
involves no new orbit. Indeed, on an orbit absent from $m$, its weight
would have the form $-\sum_i v_i\alpha_i$ with all $v_i\geq0$.
Pairing with $2\rho^\vee$ gives a nonpositive number, whereas dominance
gives a nonnegative number, with equality only for the trivial monomial.
Hence all $v_i$ on that orbit vanish. This observation will permit the
final induction to handle arbitrary complex spectral parameters.



\section{Numerical shuffle ideals and highest-vector kernels}
\label{sec:kernel}

Throughout this section $\g$ is of type $G_2$ or $F_4$ and $q$ is a fixed
non-root-of-unity complex number. We work directly with ordinary Drinfeld
generators $e_{i,r},f_{i,r}$, $r\in\Z$, and currents
\[
 e_i(z)=\sum_{r\in\Z}e_{i,r}z^{-r},\qquad
 f_i(z)=\sum_{r\in\Z}f_{i,r}z^{-r},\qquad
 \varphi_i^\pm(z)=\sum_{r\geq0}\varphi_{i,r}^\pm z^{\mp r},
 \quad \varphi_{i,0}^\pm=k_i^{\pm1}.
\]
Set $c_i=q_i-q_i^{-1}$. Define the logarithmic
Cartan modes $p_{j,r}$, $r\ne0$, by
\begin{equation}\label{eq:log-Cartan-modes}
 \varphi_j^\pm(z)=k_j^{\pm1}
   \exp\!\left(\sum_{r\geq1}\frac{p_{j,\pm r}}{r z^{\pm r}}\right).
\end{equation}
The Drinfeld relations then give
\begin{equation}\label{eq:log-Cartan-commutator}
 [p_{j,r},f_{i,s}]
   =-(q^{rb_{ij}}-q^{-rb_{ij}})f_{i,s+r},
 \qquad r\ne0,\quad s\in\Z;
\end{equation}
see \cite[(2.12), (3.12)]{Neg25a}.

\subsection{Two shuffle conventions}

Put
\begin{equation}\label{eq:rat-zeta}
               \zeta^{\rat}_{ij}(z)=\frac{z-q^{-b_{ij}}}{z-1}.
\end{equation}
For a negative word $W=f_{i_1,s_1}\cdots f_{i_N,s_N}$ its rational shuffle
representative is
\begin{equation}\label{eq:negative-word}
 F_W^{\rat}=\Sym_{\mathrm{colors}}
    \left[\prod_{a=1}^Nz_a^{s_a}
              \prod_{a<b}\zeta^{\rat}_{i_bi_a}(z_b/z_a)\right].
\end{equation}
Here $\Sym$ is a sum over color-preserving permutations. The binary shuffle
product includes the factors $(\prod n_i!\prod n_i'!)^{-1}$, so iterating
degree-one factors gives exactly \eqref{eq:negative-word}. Its direction
is that of the negative current algebra.

Fix an order on $I$, and for a dimension vector $\boldsymbol n$ put
\[
 D_{\boldsymbol n}=\prod_{i<j}\prod_{a=1}^{n_i}\prod_{b=1}^{n_j}
                         (1-z_{ia}/z_{jb}).
\]
The polynomial representative is $F^{\pol}=D_{\boldsymbol n}F^{\rat}$.
Its kernel is
\begin{equation}\label{eq:pol-zeta}
 \zeta^{\pol}_{ij}(z)=
 \begin{cases}
 q^{-b_{ij}}-z,&i<j,\\
 (q^{-b_{ii}}-z)/(1-z),&i=j,\\
 1-q^{-b_{ij}}z^{-1},&i>j.
 \end{cases}
\end{equation}
For every color order one has the exact identity
\begin{equation}\label{eq:aug-cancel}
 \frac{F^{\pol}}{\prod_{a<b}\zeta^{\pol}_{i_bi_a}(z_b/z_a)}
   =\frac{F^{\rat}}{\prod_{a<b}\zeta^{\rat}_{i_bi_a}(z_b/z_a)}.
\end{equation}
Thus the change of convention is valid even at a zero of
$D_{\boldsymbol n}$. The image of the word map in the polynomial convention
is denoted $\cS_{\boldsymbol n}(q)$.

The quadratic and Serre relations give a homomorphism from the ordinary
negative current algebra to this image. These checks are rational
identities in the variables and $q$, hence remain valid at our fixed
parameter.

\subsection{High-index highest rows}

Let $v$ be an actual highest vector, with highest rational function
$\bpsi$. Write $\psi_j(z)=\sum_{r\geq0}\psi_{j,r}z^{-r}$ for its
expansion at infinity. For a word of total weight zero, let $\operatorname{HC}_{\bpsi}$
denote its coefficient on $v$. The operator $\CT$ extracts the coefficient
of $z_1^0\cdots z_N^0$ in the nested expansion
$1\ll|z_1|\ll\cdots\ll|z_N|$.

\begin{lemma}\label{alg:high-row}
Suppose $W=f_{i_1,s_1}\cdots f_{i_N,s_N}$ with every $s_a>0$, and
$E=e_{j_1,d_1}\cdots e_{j_N,d_N}$ with every $d_a\geq0$ and matching root
content. Label the variables in the order $j_1,\ldots,j_N$. Then
\begin{equation}\label{eq:high-row}
 \operatorname{HC}_{\bpsi}(EW)=
 \left(\prod_{a=1}^Nc_{j_a}^{-1}\right)
 \CT\left[
 \frac{F_W^{\rat}(\boldsymbol z)
             \prod_{a=1}^Nz_a^{d_a}\psi_{j_a}(z_a)}
      {\prod_{a<b}\zeta^{\rat}_{j_bj_a}(z_b/z_a)}\right].
\end{equation}
The same formula holds in the polynomial convention.
\end{lemma}

\begin{proof}
For $a=q^{-b_{ij}}$, the Cartan-current relation reads
\[
 \varphi_j^+(y)f_i(x)=f_i(x)\varphi_j^+(y)R_{ij}(x/y),\qquad
 R_{ij}(u)=\frac{a-u}{1-au}.
\]
Write $R_{ij}(u)=\sum_{r\geq0}\rho_{ij}(r)u^r$. Explicitly,
\[
 \rho_{ij}(0)=a,\qquad
 \rho_{ij}(r)=(a^2-1)a^{r-1}\quad(r>0).
\]
Comparison of coefficients gives the finite identity
\begin{equation}\label{eq:cartan-normal-order}
 \varphi^+_{j,k}f_{i,s}=
       \sum_{r=0}^k\rho_{ij}(r)f_{i,s+r}\varphi^+_{j,k-r}.
\end{equation}
In particular, this operation only raises the indices of the negative
letters.

Commute the rightmost positive letter $e_{j_N,d_N}$ through $W$.
The terminal term kills $v$. A contribution occurs at every position $k$
with $i_k=j_N$. Since $d_N+s_k>0$, its commutator is
$c_{j_N}^{-1}\varphi^+_{j_N,d_N+s_k}$; there is no negative Cartan
coefficient. Moving it through the suffix by
\eqref{eq:cartan-normal-order} gives
\begin{equation}\label{eq:word-recurrence}
 c_{j_N}^{-1}
 \sum_{\substack{r_l\geq0\ (l>k)\\\sum_{l>k}r_l\leq d_N+s_k}}
 \left(\prod_{l>k}\rho_{i_lj_N}(r_l)\right)
 \psi_{j_N,d_N+s_k-\sum_{l>k}r_l}\,W_{k,\boldsymbol r}v.
\end{equation}
Here $W_{k,\boldsymbol r}$ deletes the $k$th letter and replaces each suffix
index $s_l$ by $s_l+r_l$. Its remaining negative indices are still positive.

Expand the symmetrization in \eqref{eq:negative-word} as the sum over
bijections from word positions to variables of the same color. Partition
these bijections according to the position $k$ sent to $z_N$. The pair
factor with a position before $k$ cancels the corresponding denominator
in \eqref{eq:high-row}. For a position after $k$, the remaining factor is
\[
 \frac{\zeta^{\rat}_{i_lj_N}(z_{\sigma(l)}/z_N)}
      {\zeta^{\rat}_{j_Ni_l}(z_N/z_{\sigma(l)})}
                    =R_{i_lj_N}(z_{\sigma(l)}/z_N).
\]
Taking the $z_N$ constant term therefore gives exactly
\eqref{eq:word-recurrence}, apart from the displayed $c_{j_N}^{-1}$.
The remaining assignments give the shuffle representative of
$W_{k,\boldsymbol r}$. This proves the formula by induction on $N$.
Repeated colors cause no extra factor: the $n_{j_N}!$ bijections split
into $n_{j_N}$ disjoint sets of size $(n_{j_N}-1)!$.

At every step the inequality in \eqref{eq:word-recurrence} bounds the
summation. The formula uses finitely many coefficients of $\bpsi$ and
only the scalar denominators $c_i$. It is consequently an identity at
the given numerical parameter. Finally \eqref{eq:aug-cancel} proves
the polynomial version.
\end{proof}

\subsection{Invertible simultaneous mode shifts}

Let $L=L(m)$ be a finite-dimensional simple ordinary module with
highest vector $v_m$. The span
$W_i=\operatorname{span}\{f_{i,r}|_L:r\in\Z\}$ is a finite-dimensional
subspace of $\End L$. By \eqref{eq:log-Cartan-commutator}, the operators
\[
 T_i=-\frac{\operatorname{ad}(p_{i,1})}{q_i^2-q_i^{-2}},
 \qquad
 T_i^{-1}=\frac{\operatorname{ad}(p_{i,-1})}{q_i^2-q_i^{-2}}
\]
send $f_{i,r}$ to $f_{i,r+1}$ and $f_{i,r-1}$, respectively.
Their denominator is nonzero, and their compositions fix every
spanning generator of $W_i$. Thus they are well-defined inverse
linear maps on $W_i$.

Fix $\beta=\sum_i n_i\alpha_i$. Form the finite direct sum $V_\beta$ of
ordered tensor products $W_{i_1}\otimes\cdots\otimes W_{i_N}$ with this
root content. Evaluation on a highest vector gives a surjection
\[
                 \ev:V_\beta\longrightarrow L_{\wt(m)-\beta}.
\]
Let $A_{i,r}$ be the sum of the $r$th powers of the slot shifts of color
$i$. Let $p^{\mathrm{top}}_{j,r}$ be the scalar by which $p_{j,r}$
acts on $v_m$. The logarithmic Cartan relations give
\begin{equation}\label{eq:slot-Cartan}
 (p^{\mathrm{top}}_{j,r}-p_{j,r})\ev
   =\ev\sum_i(q^{rb_{ij}}-q^{-rb_{ij}})A_{i,r}.
\end{equation}

\begin{lemma}\label{alg:shift}
For $\g$ of type $G_2$ or $F_4$, simultaneous increase of all indices in
a negative word of root content $\beta$ is realized after evaluation by
an invertible operator $P_\beta$ on $L_{\wt(m)-\beta}$.
\end{lemma}

\begin{proof}
Put $x=q^r$ and $B_r=(x^{b_{ij}}-x^{-b_{ij}})$. Up to reversing the
numbering of the Dynkin diagram, direct determinants are
\begin{align}\label{eq:cartan-determinants}
 \det B_r^{G_2}
 &=x^{-8}(x-1)^2(x+1)^2\Phi_3(x)\Phi_6(x)\Phi_{24}(x),\notag\\
 \det B_r^{F_4}
 &=x^{-12}(x-1)^4(x+1)^4\Phi_4(x)^2\Phi_{36}(x).
\end{align}
Here $\Phi_s$ is the $s$th cyclotomic polynomial. The symmetric matrices
used are respectively
\[
 \begin{pmatrix}6&-3\\-3&2\end{pmatrix},\qquad
 \begin{pmatrix}4&-2&0&0\\-2&4&-2&0\\0&-2&2&-1\\0&0&-1&2\end{pmatrix}.
\]
To compute the determinants, write $s_j=x^j-x^{-j}$.
In type $G_2$ the determinant is $s_6s_2-s_3^2$.
In type $F_4$, expansion of successive tridiagonal minors gives
\[
 \begin{aligned}
 \Delta_1&=s_4,&
 \Delta_2&=s_4^2-s_2^2,\\
 \Delta_3&=s_2\Delta_2-s_2^2\Delta_1,&
 \Delta_4&=s_2\Delta_3-s_1^2\Delta_2.
 \end{aligned}
\]
Factoring these Laurent polynomials, using
$\Phi_{24}(x)=x^8-x^4+1$ and
$\Phi_{36}(x)=x^{12}-x^6+1$, gives
\eqref{eq:cartan-determinants}.
Thus $B_r$ is invertible for every $r>0$. Solving
\eqref{eq:slot-Cartan} realizes the slot power sums $A_{i,r}$ as
actual Cartan operators after evaluation. Newton identities, using
$1\leq r\leq n_i$, realize the product of the slot shifts of each color,
and hence their total product $T$, by an operator $P_\beta$ with
$P_\beta\ev=\ev T$. The kernel of $\ev$ is $T$-stable. Since $T$ is
invertible and the space is finite-dimensional, this kernel is actually
$T$-invariant. Its quotient operator $P_\beta$ is invertible, as claimed.
\end{proof}

\subsection{The common residue kernel}

For a fixed color order $\boldsymbol i=(i_1,\ldots,i_N)$ set
\begin{equation}\label{eq:row-rational}
 H_{\boldsymbol i}(F)=
    \frac{F^{\rat}(\boldsymbol z)}
         {\prod_{a<b}\zeta^{\rat}_{i_bi_a}(z_b/z_a)}
                     \prod_{a=1}^N\psi_{i_a}(z_a).
\end{equation}
Starting with the smallest nested contour, shrink successive contours
and retain every residue at a nonzero constant point. Such points include
the poles of $\psi_{i_a}$ and the points obtained from earlier residues
by the shifts prescribed by a remaining denominator. An ordered
constant-point test is one of the resulting iterated residue functionals
applied to $G H_{\boldsymbol i}(F)\prod_a\dd z_a/z_a$, where $G$ is
an arbitrary polynomial in the labeled variables.
At a repeated pole we take the residue of the complete rational
function at that value, including all coincident denominator factors.
Let
$\Theta(\bpsi)_{\boldsymbol n}$ be their common kernel, taking all color
orders and all such $G$.

\begin{theorem}\label{alg:kernel}
Let $L(m)$ be an actual finite-dimensional simple module in type $G_2$ or
$F_4$, with highest vector $v_m$ and highest rational function $\bpsi$.
Evaluation on $v_m$ factors through the numerical shuffle word map.
The induced map on its image has kernel
$\cS_{\boldsymbol n}(q)\cap\Theta(\bpsi)_{\boldsymbol n}$, and gives
an isomorphism
\begin{equation}\label{eq:actual-shuffle-quotient}
 L(m)\cong\bigoplus_{\boldsymbol n}
 \cS_{\boldsymbol n}(q)\Big/
       \bigl(\cS_{\boldsymbol n}(q)\cap
                    \Theta(\bpsi)_{\boldsymbol n}\bigr).
\end{equation}
The statement is an equality of annihilator kernels and is valid at
coincident spectral parameters and repeated poles.
\end{theorem}

\begin{proof}
We first show that the numerical shuffle kernel acts trivially. Let $u$
be in this kernel with root content $\beta$. Increase all indices in
all its words by the same sufficiently large integer $M$, so the new
indices are positive. Denote the resulting element by $u[M]$. Then
\[
 F_{u[M]}=(z_1\cdots z_N)^MF_u=0,\qquad
                        u[M]v_m=P_\beta^Muv_m.
\]
Lemma~\ref{alg:high-row} shows that every positive word of matching
root content with nonnegative indices kills the highest coefficient of
$u[M]v_m$.

These rows detect every vector in the relevant weight space. Indeed,
if $w\ne0$, simplicity gives an element taking $w$ to the highest line.
In its triangular decomposition, a term with a further negative part of
positive root content would first have to raise $w$ to a weight strictly
above the highest weight. Such a term is zero. Thus a positive-current
word suffices. On the finite-dimensional span of positive-current
operators, the same normalized adjoint argument gives an invertible
mode shift. Every positive mode is therefore a linear combination of
arbitrarily high modes of its color. Replacing every factor in the
detecting word proves the assertion for nonnegative indices. We obtain
$u[M]v_m=0$ and then $uv_m=0$ by Lemma~\ref{alg:shift}. Applying the
same argument to $uw$ for every negative word $w$ proves that the
shuffle kernel kills the whole module.

It remains to compare the kernel on the small shuffle image with the
residue tests. For fixed $\boldsymbol n$ the denominator in
\eqref{eq:row-rational} has bounded order, independent of the numerator.
The unshifted cross-color denominators cancel against the inverse
shuffle denominator. The remaining intervariable poles are ordered:
\[
                      z_b=q^{-b_{i_ai_b}}z_a\quad(a<b).
\]
Consequently the first, smallest contour can meet only fixed poles of
$\psi_{i_1}$ and possibly zero. Poles expressed in terms of later
independent variables lie outside that contour. After substitution at
a nonzero constant, only finitely many new shifted constants occur at
the next step. Induction gives finitely many ordered tuples. Each
residue is supported on a finite jet at its tuple. Resonant diagonals
increase the finite jet order by at most the order of their fixed
denominator and do not produce an infinite-order functional.

Zero coordinates can be eliminated uniformly for a given numerator.
For every nonempty set $S$ of coordinates tending to zero, the rational
integrand has a finite lower bound on its total degree in those
coordinates. Factors internal to $S$ have fixed homogeneous degree and
the factors joining $S$ to a nonzero coordinate have nonnegative power
series expansions. Multiplication by $P^D$, where $P=\prod_a z_a$,
increases the bound by $D|S|$. For large $D$ all residues with a zero
coordinate vanish.

On the finite jet algebra at any nonzero tuple, $P$ is a unit. The
Chinese remainder theorem on the labeled polynomial ring separates
all distinct ordered tuples and prescribes arbitrary jets there.
It follows that all constant-point tests vanish if and only if all
nested constant terms with sufficiently high nonnegative powers
$z_1^{d_1}\cdots z_N^{d_N}$ vanish. The finite-jet assertion also
handles repeated coordinates in their single local jet algebra.

Choose a negative preimage $u$ of $F$ and shift its modes as above.
Multiplication by $P^M$ is invertible on the nonzero jet algebras, so
the residue tests vanish for $F_u$ if and only if they vanish for
$F_{u[M]}$. By Lemma~\ref{alg:high-row} and positive-tail detection,
the latter condition is $u[M]v_m=0$. By Lemma~\ref{alg:shift} it is
equivalent to $uv_m=0$. This proves the kernel equality and the
isomorphism.
\end{proof}

The kernel argument also applies over a characteristic-zero coefficient
field when the spectral poles split and the highest-weight module is
absolutely simple. Indeed, all normal-order identities and invertible
mode shifts are finite linear-algebra calculations. The contour
calculation is equivalently the formal iterated-residue expansion in
the specified order, and Chinese remainders apply to its finite jet
algebras. One may extend to an algebraic closure to detect the highest
rows and then descend the equality of kernels by faithful scalar
extension. We will use this version over $\C((a))$.

\subsection{Wheel ideals at every numerical parameter}

In the polynomial convention, the multiplicative Serre-wheel condition
for $i\ne j$ is
\begin{equation}\label{eq:mult-wheel}
 F\big|_{z_{i,r}=z_{j,1}q^{b_{ij}+2(r-1)d_i},\
                    1\leq r\leq1-c_{ij}}=0.
\end{equation}
All choices of distinct labeled variables of these colors are included.
For $c_{ij}=0$ this is the condition at $z_{i,1}=z_{j,1}$, which is
essential in the augmented polynomial convention.

Fix a spectral tuple $\boldsymbol x$. Write its entries as
$x_a=b_{c(a)}q^{k_a}$, where the $b_c$ represent distinct $q^\Z$-orbits.
The associated spectral vertices are $v=(i,b_cq^k)$. There is an arrow
\begin{equation}\label{eq:spectral-arrow}
        (i,b_cq^s)\longrightarrow(j,b_cq^t)
                         \quad\Longleftrightarrow\quad s=t+b_{ij}.
\end{equation}
Let $E(v,w)$ be the number of such arrows and let $\nu_v$ be the
multiplicity of $v$ in the tuple. Set
$R_{\bnu}=\C[y_{v,a}]^{\prod_v\mathfrak S_{\nu_v}}$.
Every multiplicative wheel through $\boldsymbol x$ becomes a diagonal
where the corresponding $y$-variables are equal after writing
$z_a=x_a(1+y_a)$. Let $I_{\Gamma,\bnu}\subset R_{\bnu}$ be the
intersection of the ideals of these diagonals. Equivalently its
elements vanish on all such lifted Serre wheels. This is a homogeneous
ideal, including the diagonals of orthogonal-color pairs.

\begin{theorem}\label{alg:wheel}
At every numerical non-root-of-unity parameter, the completion of
$\cS_{\boldsymbol n}(q)$ at the colored unordered multiset represented
by $\boldsymbol x$ identifies, in the coordinates $z_a=x_a(1+y_a)$,
with the completion of $I_{\Gamma,\bnu}$ at the origin.
Moreover, $I_{\Gamma,\bnu}$ is spanned by the
homogeneous additive words
\begin{equation}\label{eq:additive-word}
 \Sym_{\mathrm{vertices}}
 \left[G(\boldsymbol y)
       \prod_{a<b}(y_b-y_a)^{E(v_a,v_b)-\delta_{v_a,v_b}}\right],
 \qquad G\in\C[y_1,\ldots,y_N],
\end{equation}
over all vertex orders with multiplicity vector $\bnu$. The same
assertion holds over $\Q$ for the combinatorial spectral quiver.
\end{theorem}

\begin{proof}
Let $A=\C[z_{ia}^{\pm1}]$ be the labeled Laurent ring and
$R=A^{\prod_i\mathfrak S_{n_i}}$. For a fixed color order, the word
map $G\mapsto\Sym(GK)$ is $R$-linear. The apparent equal-color
denominators cancel by antisymmetrizing over the corresponding
Vandermonde. The image therefore lies in $R$. Since $A$ is finite
over $R$, each word image is a finite $R$-module. There are finitely
many color orders, so their sum $\cS_{\boldsymbol n}(q)$ is a finite
ideal of $R$. In particular images commute with flat completion and
are closed in the adic topology.

Complete the labeled finite extension over the orbit of
$\boldsymbol x$. It splits as the product of the completions at its
labeled points. In a chosen component full symmetrization becomes
symmetrization by the stabilizer $\prod_v\mathfrak S_{\nu_v}$.
In the coordinates $z_a=x_a(1+y_a)$ every resonant numerator is
$y_b-y_a$ times a unit; every nonresonant factor is a unit; and the
same-vertex denominator is the Vandermonde factor. Thus each completed
word map is
\begin{equation}\label{eq:local-word-unit}
                 G\longmapsto\Sym(G\kappa_\sigma U_\sigma),
\end{equation}
where $\kappa_\sigma$ is the product in
\eqref{eq:additive-word} and $U_\sigma$ is a formal unit. Multiplication
by $U_\sigma$ is an automorphism of the arbitrary labeled input ring.
Removing it from the input proves that the completed word ideal
depends only on the resonance pattern. If one realizes this argument by
polynomial interpolation, one first prescribes the labeled jets to
an order larger by the finite Vandermonde division order.

The local wheel ideal has the same property: a wheel component not
passing through $\boldsymbol x$ disappears on completion, and a
component passing through it becomes exactly its linear diagonal.
The completed wheel ideal is therefore the completion of the
homogeneous ideal $I_{\Gamma,\bnu}$.

Now replace the fixed $q$ by an indeterminate $q'$ and replace the
orbit representatives by algebraically independent $b_c'$. The
tuple $b_{c(a)}'(q')^{k_a}$ has exactly the same resonance pattern.
Indeed, within one orbit all relevant equations are equalities of
integer exponents, since the original $q$ is not a root of unity;
between different orbits no relevant equality holds. Over
$\Q(q',b_c')$, the generic shuffle theorem
\cite[Theorem~1.7 and Definition~5.2]{NT24} identifies the small
shuffle ideal with the wheel ideal. Its completion, followed by
\eqref{eq:local-word-unit}, identifies the completed additive word
ideal with the completed diagonal ideal.

Both additive ideals are homogeneous and finitely generated over the
invariant polynomial ring defined over $\Q$. Extend this polynomial
ring to $\Q(q',b_c')$, localize at its homogeneous maximal ideal,
and complete. Completion is faithfully flat over this local ring.
The two graded quotients of the sum of these ideals by either one
therefore have zero localization there. A nonzero
finitely generated graded module over a nonnegatively graded
polynomial ring cannot have this property: graded Nakayama, or its
lowest nonzero degree, detects it at the homogeneous maximal ideal.
Thus both quotients vanish globally after field extension, and
faithful flatness of the field extension proves equality over $\Q$,
hence over $\C$. Applying the first
two paragraphs again at the original numerical tuple proves the
completion statement. All scalar extensions in this argument precede
local completion.
\end{proof}

Theorem~\ref{alg:wheel} and the finite-jet description of
Theorem~\ref{alg:kernel} reduce every intrinsic local sector to a
homogeneous additive quotient. We next identify this quotient and its
grading geometrically.



\section{The spectral quiver and cyclic moduli}
\label{geo:setup}

We give the geometric realization of the local residue quotient.  All
cohomology groups in this section have rational coefficients; extending
scalars to $\C$ will be understood when we compare them with quantum-loop
modules.  Polynomial degree is half the ordinary cohomological degree.
The constructions are made in each finite dimension vector, and therefore
do not require a bound on the spectral support.

Let $C=(c_{ij})_{i,j\in I}$ be the Cartan matrix, let $(d_i)$ be its
minimal positive symmetrizer, and put $b_{ij}=d_ic_{ij}=d_jc_{ji}$.
For one $q^{\Z}$-orbit of spectral parameters define the quiver $\Gamma$
by
\begin{equation}\label{geo:arrows}
 \Gamma_0=I\times\Z,
 \qquad (i,k)\longrightarrow(j,k-b_{ij})\quad(i,j\in I).
\end{equation}
Different spectral orbits give disjoint copies of this quiver.  An arrow
with $i=j$ will be denoted by $\epsilon_{i,k}$; it goes from $(i,k)$
to $(i,k-2d_i)$.  In particular there is no loop at a spectral vertex.
If $c_{ij}=0$, we retain both arrows between $(i,k)$ and $(j,k)$.
These arrows are part of the polynomial shuffle convention of
\cite[\S3.2]{Neg25b}, and their quadratic contribution to the potential
will be retained throughout.

Here is the potential, including its grading.  In the ungraded quiver
write $a_{ji}:i\to j$ and $\epsilon_i=a_{ii}$, and use right-to-left
composition.  The potential is the cyclic expression
\begin{equation}\label{geo:ungraded-potential}
 W=\sum_{i\ne j}a_{ji}\epsilon_i^{-c_{ij}}a_{ij}.
\end{equation}
Its graded lift $W_\Gamma$ is the sum, over $i\ne j$ and $k\in\Z$, of
the following closed paths:
\begin{equation}\label{geo:graded-potential}
 (j,k)\xrightarrow{a_{ij,k}}(i,k-b_{ij})
 \xrightarrow{\epsilon_i^{-c_{ij}}}(i,k+b_{ij})
 \xrightarrow{a_{ji,k+b_{ij}}}(j,k).
\end{equation}
The middle path consists of $-c_{ij}$ consecutive $\epsilon_i$-arrows.
It has the stated endpoint because $-2d_i(-c_{ij})=2b_{ij}$.
An exponent zero means the constant path.  Thus the two ordered terms
for an orthogonal pair are the same cyclic quadratic monomial and have
sum $2a_{ji}a_{ij}$.  Formula \eqref{geo:graded-potential} is precisely
the graded lift of \cite[(91)]{Neg25b}.  On a finitely supported
representation only finitely many of its terms are nonzero.

Fix finitely supported vectors $\bnu=(\nu_v)$ and
$\bm=(m_v)$ with nonnegative integer entries, and vector spaces
$V_v=\C^{\nu_v}$ and $W_v=\C^{m_v}$.  Put
\begin{align}
 \operatorname{Rep}_{\bnu}(\Gamma)
   &=\bigoplus_{h:v\to w}\Hom(V_v,V_w),\notag\\
 \operatorname{Rep}_{\bnu,\bm}(\Gamma)
   &=\operatorname{Rep}_{\bnu}(\Gamma)
          \oplus\bigoplus_v\Hom(W_v,V_v),\label{geo:representation-space}\\
 G_{\bnu}&=\prod_v\GL(V_v).\notag
\end{align}
A framed representation is \emph{cyclic} if the images of the framing
maps generate it under all paths in $\Gamma$.  Its gauge stabilizer is
trivial: an automorphism fixing the framing fixes every path-generated
vector.  The cyclic locus is open, and its geometric quotient
\[
 X_{\bnu,\bm}
  =\operatorname{Rep}_{\bnu,\bm}(\Gamma)^{\mathrm{cyc}}/G_{\bnu}
\]
is smooth.  The trace $f_{\bnu}=\Tr W_\Gamma$ is gauge invariant and
pulls back to the framed space and descends to $X_{\bnu,\bm}$.  We write
$f$ when the dimension vector is understood.

For finitely supported vectors set
\begin{equation}\label{geo:dimension-form}
 E(\bnu,\boldsymbol\mu)=\sum_{h:v\to w}\nu_v\mu_w,
 \qquad
 d_{\bp}(\bnu)=\bp\cdot\bnu+E(\bnu,\bnu)-\bnu\cdot\bnu.
\end{equation}
The complex dimension of $X_{\bnu,\bm}$ is $d_{\bm}(\bnu)$.
Let $\cV_v$ be its tautological bundle.  For another nonnegative vector
$\bl=(l_v)$, define
\begin{equation}\label{geo:euler-operator}
 \bp=\bm-\bl,\qquad
 \cE_{\bl}=\bigoplus_v\cV_v^{\oplus l_v},\qquad
 L=\bl\cdot\bnu,\qquad D=e(\cE_{\bl})\cup-.
\end{equation}
There is no positivity requirement on $\bp$.  In Chern roots,
$e(\cE_{\bl})=\Delta_{\bl}:=\prod_{v,a}y_{v,a}^{l_v}$,
so $D$ has cohomological degree $2L$. Then
\begin{equation}\label{geo:net-center}
 d_{\bp}(\bnu)=d_{\bm}(\bnu)-L.
\end{equation}
We use $d_{\bp}(\bnu)$ to center the local graded character.

\section{Ordinary cohomology and ordered residue tests}
\label{geo:ordinary}

We first prove the ordinary kernel theorem for a finite quiver $Q$.
This statement permits oriented cycles and loops.  It consequently
applies to every finite part of \eqref{geo:arrows}.  Write
$e_{vw}=\#\{v\to w\}$ and
\[
 R_{\bnu}=\Q[y_{v,a}:1\le a\le\nu_v]^{\prod_v S_{\nu_v}},
 \qquad \deg y_{v,a}=1.
\]
Let $N=\sum_v\nu_v$.  For an order
$\alpha=(v_1,\ldots,v_N)$ with content $\bnu$, relabel the roots as
$y_1,\ldots,y_N$, and define
\begin{equation}\label{geo:ordinary-kernel}
 K_{\alpha,\bm}(y)=
 \prod_{b=1}^N y_b^{-m_{v_b}}
 \prod_{a<b}(y_b-y_a)^{\delta_{v_av_b}-e_{v_av_b}},
 \qquad
 \lambda_{\alpha,\bm}(P)
 =[y_1^{-1}\cdots y_N^{-1}]\,P K_{\alpha,\bm}.
\end{equation}
Every inverse factor is expanded in $y_a/y_b$ for $a<b$.
Equivalently these are constant-point residues, with $y_1$ taken
first.  All multipliers below are polynomials in the \emph{labeled}
variables; no symmetry is imposed on a multiplier.  Set
\begin{equation}\label{geo:residue-ideal}
 J_{\bm}=\{F\in R_{\bnu}:\lambda_{\alpha,\bm}(FG)=0
  \text{ for every $\alpha$ and every }G\in\Q[y_1,\ldots,y_N]\}.
\end{equation}
The same formula defines $J_{\bp}$ when some entries of $\bp$ are
negative.

\begin{theorem}\label{geo:ordinary-kernel-theorem}
The tautological evaluation map induces a graded ring isomorphism
\[
 R_{\bnu}/J_{\bm}\xrightarrow{\sim}
 H^{2*}(X_{\bnu,\bm},\Q).
\]
The cohomology is finite dimensional and has no odd part.
\end{theorem}

\begin{proof}
We construct smooth projective varieties on which the functionals in
\eqref{geo:ordinary-kernel} are actual integrations.  Fix $\alpha$.
At the $b$th stage add a last kernel line $L_b$ of color $j=v_b$;
the old object $V'$ is the cyclic framed quotient by that line.
All arrows strictly raise the flag index.  On the preceding flag
space consider
\begin{equation}\label{geo:flag-exact-sequence}
 0\longrightarrow\cE_j\longrightarrow
 W_j\otimes\cO\oplus\bigoplus_{h:i\to j}\cV'_i
 \xrightarrow{d_j}\cV'_j\longrightarrow0,
 \qquad
 d_j(w,(x_h))=I_j(w)+\sum_h A_h(x_h).
\end{equation}
The last map is surjective.  Indeed each path-generated vector at $j$
is either a framing vector or is obtained by applying the last incoming
arrow of its path.  Cyclicity says that these vectors span $V'_j$.
Consequently $\cE_j$ is a vector bundle of rank
\[
 r_b=m_j+\sum_i e_{ij}\nu'_i-\nu'_j.
\]
To identify the extensions, choose a splitting $V_j=V'_j\oplus L_b$.
The additional framing row and the incoming-arrow rows give a map
$W_j\oplus\bigoplus_{h:i\to j}V'_i\to L_b$.
Changing the splitting adds $u d_j$ for some $u:V'_j\to L_b$.
The quotient of the row data is therefore $\Hom(E_j,L_b)$.
The extension is cyclic precisely when this class is nonzero: otherwise
the old framed representation splits off, whereas a noncyclic extension
has a generated subrepresentation mapping isomorphically to $V'$ and
hence provides such a splitting.  Dividing by scalar changes of $L_b$
gives
\begin{equation}\label{geo:flag-tower}
 F_b=\mathbb P_{F_{b-1}}(\cE_j),\qquad F_0=\mathrm{pt},
\end{equation}
where projectivization parametrizes one-dimensional quotients.
The universal quotient is the actual kernel line $L_b$.
If $r_b=0$, this stage is empty.  Every nonempty final space $F_\alpha$
is a smooth projective iterated projective bundle, and its cohomology
is generated by the labeled classes $y_b=c_1(L_b)$.

For a quotient projective bundle $\pi:\mathbb P(\cE)\to B$ and
$t=c_1(\cO(1))$, the projective-bundle formula reads
\[
 \pi_*P(t)=[t^{-1}]\frac{P(t)}{c_t(\cE)},\qquad
 c_t(\cE)=t^{r}-c_1(\cE)t^{r-1}+\cdots+(-1)^rc_r(\cE),
\]
with expansion at infinity.  The sign is fixed by
$\pi_*(t^{r-1})=1$.  Applying \eqref{geo:flag-exact-sequence} gives
\[
 \frac1{c_t(\cE_j)}=
 t^{-m_j}\prod_{a<b}(t-y_a)^{\delta_{v_aj}-e_{v_aj}}.
\]
Successive integrations consequently yield
\begin{equation}\label{geo:flag-residue}
 \int_{F_\alpha}P(y)=\lambda_{\alpha,\bm}(P).
\end{equation}
For precision, geometric integration removes $y_N$ first, expanding
at infinity relative to the preceding variables.  All stages use the
single chamber $|y_1|\ll\cdots\ll|y_N|$.  Constant-point residues in
the reverse order extract the same Laurent coefficient in that chamber.
For a polynomial numerator only finitely many expansion terms can
contribute: starting with $y_1$, the desired exponent bounds all
powers introduced by factors involving $y_1$, and induction does the
same for the remaining variables.  Thus no exchange of an uncontrolled
infinite sum is involved.  At a rank-zero stage the projective pushforward
is the coefficient of $t^{-1}$ in a polynomial and vanishes, which also
settles the empty-tower case.  If the tower is nonempty, its dimension is
\begin{equation}\label{geo:flag-dimension}
 d_\alpha=\bm\cdot\bnu+
 \sum_{a<b}e_{v_av_b}-\sum_{a<b}\delta_{v_av_b}-N.
\end{equation}
The kernel in \eqref{geo:ordinary-kernel} has degree $-d_\alpha-N$,
so the functional sees only the degree-$d_\alpha$ part of its numerator.

We next show that the flag pullbacks jointly detect cohomology.
The cyclic GIT quotient is projective over
$Y=\operatorname{Rep}_{\bnu}(Q)//G_{\bnu}$; incoming framing introduces
no additional affine invariants.  Scaling every arrow with weight one
and leaving the framing unchanged contracts $Y$ to its origin.
Properness gives a limit as the scaling parameter tends to zero for
every point of $X=X_{\bnu,\bm}$.  The fixed locus is contained in the
projective central fiber, so $X$ is smooth semiprojective.  Its core is
exactly $C=X\times_Y\{0\}$: a limit at infinity can exist only over the
origin, and projectivity of the central fiber gives that limit for
every point over the origin.  The core theorem
\cite[Theorem 1.3.1]{HRV15} gives the actual restriction isomorphism
\begin{equation}\label{geo:core-restriction}
 H^k(X,\Q)\xrightarrow{\sim}H^k(C,\Q).
\end{equation}
Every nullcone representation admits a complete flag with zero-arrow
vertex-simple factors.  One can obtain it from a Hilbert--Mumford
one-parameter subgroup taking the arrow representation to zero and
then refine its weight filtration by vertex lines.  Thus forgetting
flags gives a proper surjection
$\coprod_\alpha F_\alpha\to C$.

Deligne's proper-surjection theorem
\cite[Propositions 8.2.5 and 8.2.7]{Deligne74} says that the kernel of
pullback from a proper variety to a smooth proper variety surjecting
onto it has weights at most $k-1$ in degree $k$.  This statement also
applies to a finite disjoint union of smooth proper varieties.
Here $H^k(C)$ has weights at most $k$ by properness, and at least $k$
by \eqref{geo:core-restriction} and smoothness of $X$.  The kernel is
therefore zero.  Equivalently, proper cohomological descent shows that
all positive simplicial degrees in a smooth proper hypercover have
weights strictly smaller than the total degree, which is precisely
the assertion used here.  We have proved
\begin{equation}\label{geo:joint-flag-injection}
 H^*(X,\Q)\hookrightarrow\bigoplus_\alpha H^*(F_\alpha,\Q).
\end{equation}

If $\ev(F)=0$, equation \eqref{geo:flag-residue} gives every vanishing
in \eqref{geo:residue-ideal}.  Conversely, if $F\in J_{\bm}$,
all its pairings on every $F_\alpha$ vanish.  Labeled line classes
generate the cohomology of a tower, so the allowed multipliers exhaust
all classes against which Poincare duality tests the pullback of $F$.
The pullback is zero on every tower, and
\eqref{geo:joint-flag-injection} gives $\ev(F)=0$.
This proves equality of kernels.  Tautological generation, finite
dimensionality, and evenness for cyclic moduli of arbitrary finite
quivers follow from \cite[Theorems 2.21 and 5.5, (59)--(63)]{FM24}.
They complete the proof.
\end{proof}

\begin{lemma}\label{geo:coordinate-kernel}
The completed kernel of all multiplicative local ordered residue tests
is carried to the completed extension of $J_{\bp}$ by
$z_a=x_a e^{y_a}$, where $x_a$ is the
prescribed nonzero spectral center.  The assertion holds with all
framing weights in a common formal parameter as well.
\end{lemma}

\begin{proof}
A resonant difference becomes a linear difference times a unit:
$e^{y_b}-e^{y_a}=(y_b-y_a)U_{ab}$ and
$e^{y_a}-1=y_aU_a$, with $U_{ab}(0)=U_a(0)=1$.
The residue Jacobian is another unit.  For a fixed labeled order their
product can be nonsymmetric.  Multiplication by it is nevertheless an
automorphism of the space of labeled Taylor jets before taking a
residue.  The finite-jet bound in \eqref{geo:flag-dimension}, or the
finite principal-part recursion for a prescribed multiplicative tuple,
allows its inverse to be replaced by a polynomial to the necessary
order. For signed $\bp$, choose $\bm\geq\bp$ with $\bm\geq0$;
the additional numerator factor is the polynomial
$\Delta_{\bm-\bp}$, so the same finite principal-part bound applies.
Hence the intersections of kernels coincide. Over $\Q[[a]]$
the same argument is applied modulo successive powers of $(a,y)$ and
then passed to the inverse limit.  All nonunit cross-color factors of
the polynomial shuffle convention remain in place; the argument removes
only the displayed units.
\end{proof}

\begin{proposition}\label{geo:ordinary-flavor-kernel}
Let $A=\Gm$ act on each framing space with integral weights $c_{v,r}$,
and put $a=c_1(\C_1)$.  In completed equivariant cohomology the evaluation
kernel is the deformed version of \eqref{geo:residue-ideal}, obtained
by replacing $y_b^{-m_{v_b}}$ with
$\prod_{r=1}^{m_{v_b}}(y_b-c_{v_b,r}a)^{-1}$.
No difference of framing weights is inverted.
\end{proposition}

\begin{proof}
The exact sequence \eqref{geo:flag-exact-sequence} and every tower are
$A$-equivariant.  Their equivariant integrations give the stated
functionals.  Evenness of ordinary cohomology implies that
$H_A^*(X)\otimes_{\Q[a]}\Q[[a]]$ is finite free over $\Q[[a]]$.
Choose finitely many ordinary labeled flag tests separating $H^*(X)$
using Theorem~\ref{geo:ordinary-kernel-theorem}, and lift them
equivariantly.  Their pairing matrix has a square minor nonzero modulo
$a$, hence invertible over $\Q[[a]]$.  The lifted tests therefore detect
every equivariant class.  Tautological generators lift and remain
surjective by graded Nakayama.  This proves both inclusions of kernels,
including when several framing factors have the same reduction.
\end{proof}

\section{Specialization of a potential with several positive weights}
\label{geo:specialization}

We specify the specialization maps before using them.  For a regular
function $f$ on a smooth complex variety $M$, use normalized vanishing
cycles
\[
 \Phi_f(F)=\left.R\Gamma_{\{\operatorname{Re}f\ge0\}}F
                  \right|_{f^{-1}(0)},\qquad \Phi_0=\id.
\]
A complex on the zero fiber is extended by zero when regarded as a
complex on $M$.  If $m=\dim_\C M$ and $\omega_M=\Q_M[2m]$, put
\begin{equation}\label{geo:critical-conventions}
 H^k_{\crit}(M,f)=H^kR\Gamma(M,\Phi_f\Q_M),\qquad
 H_i^{\crit,\BM}(M,f)=H^{-i}R\Gamma(M,\Phi_f\omega_M)
 =H^{2m-i}_{\crit}(M,f).
\end{equation}
All constructions below admit commuting reductive coefficient groups;
they are made at finite free Borel approximations and then stabilized
in each degree.

Assume that $M$ is smooth and projective over an affine variety $B$,
with an equivariant $\Gm$-action $\lambda$ for which every point of $B$
has a limit at zero.  We also allow $M=B$ to be a representation with
nonnegative point weights.  Suppose
\begin{equation}\label{geo:mixed-weights}
 f=\sum_{d>0}f_d,\qquad f_d(\lambda(u)x)=u^d f_d(x).
\end{equation}
The sum is finite.  Properness implies that every point of $M$ has a
limit at zero: apply the valuative criterion to its orbit and its limit
in $B$.  Define
\[
 N=M\times\mathbb A^1_t,\qquad
 \mathcal W(x,t)=\sum_{d>0}t^df_d(x),\qquad
 u\cdot(x,t)=(\lambda(u)x,u^{-1}t).
\]
This action preserves $\mathcal W$.  Its full attracting set is
$C=M\times\{0\}$, and its fixed locus is
$M^\lambda\times\{0\}\subset\mathcal W^{-1}(0)$.
Indeed a point with $t\ne0$ has no limit under $u\to0$, whereas a point
with $t=0$ has the required limit by properness.  The parameter $t$ is
the deformation parameter.  Nearby cycles in the following proof are
taken for the \emph{potential} map
$\mathcal W:N\to\mathbb A^1_s$; the two parameters have different roles.

\begin{proposition}\label{geo:supported-comparison}
Let $V=\mathcal W^{-1}(0)$, let $s_0:V\hookrightarrow N$, and let
$c:C\hookrightarrow V$.  The canonical Milnor map induces a
quasi-isomorphism of global complexes
\begin{equation}\label{geo:eta}
 \eta_M:R\Gamma(C,\omega_C)
       \xrightarrow{\sim}R\Gamma(C,c^!\Phi_{\mathcal W}\omega_N).
\end{equation}
This map is natural for pullback between smooth ambient varieties
respecting the data and for proper trace.  It is compatible with the
coefficient-group equivariance.
\end{proposition}

\begin{proof}
Use the dual normalized Milnor triangle
\begin{equation}\label{geo:milnor-triangle}
 \omega_V=s_0^!\omega_N
 \xrightarrow{\can_{\mathcal W}}\Phi_{\mathcal W}\omega_N
 \longrightarrow\Psi_{\mathcal W}\omega_N\xrightarrow{+1}.
\end{equation}
Here $\Psi$ includes the nearby-cycle shift determined by this triangle.
The map in \eqref{geo:eta} is its first arrow after applying
$R\Gamma(C,c^!-)$; its cone is
$R\Gamma(C,c^!\Psi_{\mathcal W}\omega_N)$.
Thus the comparison is specified independently of any chosen
stratification.

The attracting decomposition of $M$ is filtrable in the present
projective-over-affine situation.  To obtain the needed finite closed
filtration, choose an equivariant relatively ample bundle and an
embedding into $\mathbb P(V_0)\times B$ by finitely many homogeneous
sections.  Ordering the projective-coordinate weights gives closed
unions of attracting strata; components with the same weight may be
grouped.  The same construction works if the affine base has weight-zero
directions.  Write $A_j\subset C$ for a stratum, $F_j$ for its limit
locus, $r_j:A_j\to F_j$ for the limit map, and $a_j:A_j\hookrightarrow V$
for its inclusion.  Hyperbolic localization for the inverse action has
component $(r_j)_*a_j^!$.

Apply the nearby-cycle compatibility of hyperbolic localization
\cite[Theorem 3.3]{Richarz19} to $\mathcal W:N\to\mathbb A^1_s$.
More precisely, base change to the strict henselization of the
$s$-line at zero. The torus action is locally linearizable: the
smooth quasi-projective ambient variety admits invariant affine
neighborhoods, and this property is retained after that base change.
The action on the $s$-line is trivial; the generic-fiber dualizing
complex is equivariant and hence monodromic.  Formation of the fixed
locus and attracting diagram commutes with base change in $s$.
Every nonzero $s$-fiber has empty fixed locus, since the full fixed
locus of $N$ lies in $V$.  The theorem therefore gives
\[
 (r_j)_*a_j^!\Psi_{\mathcal W}\omega_N=0,
 \qquad
 R\Gamma(A_j,a_j^!\Psi_{\mathcal W}\omega_N)=0.
\]
The localization triangles for cohomology with support in the finite
closed filtration of $C$ show that the entire global cone of
$\eta_M$ vanishes.

For the coefficient passage, the cited
hyperbolic-localization theorem is formulated for etale torsion
coefficients.  First use $\Z/\ell^n$ at a finite Borel approximation.
The comparison theorem for nearby cycles over a smooth complex curve
\cite[Expose XIV, Theorem 2.8]{SGA7} applies to the separated finite-type
map $\mathcal W$ and constructible coefficients.  Together with
constructible comparison for support and direct image, it identifies
the resulting vanishing with its Betti counterpart.  The canonical
maps above are defined integrally and commute with derived reduction
of coefficients.  Their cohomology is finitely generated at each
finite approximation.  A nonzero rational cohomology group of the
cone would contain a free integral summand, detected modulo $\ell$,
contradicting the torsion-coefficient vanishing.  The rational statement
and then its complex scalar extension follow.

We give the naturality argument since the map needed later is an
arbitrary map between smooth ambient spaces.  Let $b:M'\to M$ respect
the scaling and potentials, put $B'=b\times\id$, and write
$d=\dim M'-\dim M$.  The central support square is Cartesian, and
both central divisors have the same trivial normal parameter line,
with the identity map on that line.  Local-cohomology exchange gives
\[
 b^*\Phi_f\longrightarrow\Phi_{f\circ b}b^*.
\]
This transformation is defined for every $b$; its being an isomorphism
is needed only for smooth pullback.  With dualizing normalization it
has BM degree $2d$.  Apply the same support exchange to the inclusions
$\{\mathcal W=0\}\subset\{\operatorname{Re}\mathcal W\ge0\}$.
The Milnor canonical map is compatible with these inverse-image support
maps, and the purity identification for the central parameter divisor
is preserved. Writing $N',C',c',\mathcal W'$ for the corresponding
data over $M'$, the induced square is
\[
\begin{CD}
 H_i^{\BM}(M) @>{\eta_M}>>
   \HH^{-i}(C,c^!\Phi_{\mathcal W}\omega_N)\\
 @V{b^!}VV @VV{(b\times\id)^!}V\\
 H_{i+2d}^{\BM}(M') @>{\eta_{M'}}>>
   \HH^{-i-2d}(C',c'^!\Phi_{\mathcal W'}\omega_{N'}).
\end{CD}
\]
The left arrow is the refined l.c.i.\ pullback between smooth
ambient varieties; the right arrow is its supported critical
counterpart, with the normalization of
\cite[\S2.2.3(i)]{COZZ25}. Equivalently one can factor $b$ into its graph embedding
followed by a smooth projection; the graph of $b\times\id$ meets the
central parameter divisor transversely, so the refined pullback has
an excess bundle of rank zero.

If $b$ is proper, use instead proper compatibility of vanishing cycles
and Verdier trace:
\[
 Rb_*\Phi_{f\circ b}\omega_{M'}
 \simeq\Phi_f Rb_*\omega_{M'}\longrightarrow\Phi_f\omega_M.
\]
Proper base change for restriction with support on the central fiber
identifies the ambient trace with the trace on that fiber.  Naturality
of \eqref{geo:milnor-triangle} in its coefficient complex gives the
commuting square with $\eta$ for proper pushforward.  These are squares
of the specified canonical maps, so the inverse of $\eta$ after global
sections is natural too. The attracting filtration establishes
invertibility of the canonically defined map $\eta$.

For equivariance, let the auxiliary scaling act trivially on a finite
free approximation to the coefficient-group classifying space.
The fixed loci, attracting diagrams, and closed filtrations descend
locally to the associated bundles, and their full attracting set is
still the central associated bundle.  The preceding argument applies
there and is compatible with approximation transitions.
\end{proof}

Over $t\ne0$ the isomorphism
\begin{equation}\label{geo:family-trivialization}
 a_M(x,t)=(\lambda(t^{-1})x,t)
\end{equation}
satisfies $a_M^*\mathcal W=f$.  Let
$\alpha\in H_1^{\BM}(\C^*)$ have boundary $1$ at zero, and let
$\epsilon_\alpha$ be exterior product with its
\emph{suspension normalization}: the constant-family boundary satisfies
$\partial\epsilon_\alpha(\gamma)=\gamma$.
In a raw chain convention in which the boundary is
$(-1)^i\gamma$ on a class of BM degree $i$, this means
$\epsilon_\alpha(\gamma)=(-1)^i\gamma\times\alpha$.
Define
\begin{equation}\label{geo:specialization-map}
 \spc_M=\eta_M^{-1}\partial(a_M)_*\epsilon_\alpha:
 H_i^{\crit,\BM}(M,f)\longrightarrow H_i^{\BM}(M).
\end{equation}
In cohomological indexing this is a degree-zero map.
The boundary is that of the localization triangle for the central
support of $\Phi_{\mathcal W}\omega_N$.  The normalization fixes its
sign in every degree.

\begin{corollary}\label{geo:specialization-naturality}
The map $\spc$ commutes with smooth-ambient pullback and proper trace
for scaling-equivariant maps.  It is linear over the cohomology of a
commuting coefficient group.  For the zero potential, $\spc_0=\id$.
\end{corollary}

\begin{proof}
The first assertions follow from Proposition~\ref{geo:supported-comparison}
and the corresponding morphisms of localization triangles.
The BM pullback shifts have even degree, so the suspension convention
introduces no additional sign.  To check the final assertion it is
enough to prove
$(a_M)_*(\gamma\times\alpha)=\gamma\times\alpha$ for ordinary BM
homology.  Represent $\alpha$ by the properly embedded positive real
ray in $\C^*$.  On its product with $M$, the maps
\[
 H_s(x,t)=(\lambda(t^{-s})x,t),\qquad 0\le s\le1,
\]
form a proper homotopy from the identity to $a_M$.
Indeed the joint map $(x,t,s)\mapsto(H_s(x,t),s)$ is a homeomorphism,
and the parameter interval is compact.  Proper homotopy invariance
and pushforward from the ray prove the required identity.
It is an identity on the product with $\alpha$, which is exactly what
the boundary construction uses.  The suspension normalization then
gives $\spc_0=\id$.
\end{proof}

\section{A section of critical stable restriction}
\label{geo:graph-section}

Let $R$ be a linear representation of a reductive group $G$, let
$U\subset R$ be a nonempty invariant open on which $G$ acts freely,
and suppose $X=U/G$ is smooth and projective over
$Y=\Spec\C[R]^G$.  A commuting reductive flavor group $T$ is allowed.
Let $f\in\C[R]^G$ be $T$-invariant.  For specialization assume in
addition that $R$ has nonnegative scaling weights, that the scaling
preserves $U$, and that $f$ satisfies \eqref{geo:mixed-weights}.
These hypotheses hold for \eqref{geo:representation-space}, with
weight one on all arrows and zero on the framing.

\begin{proposition}\label{geo:kirwan-section}
Stable restriction on critical cohomology has a degree-zero,
$H_T^*(\mathrm{pt})$-linear section
\[
 \Psi_f:H^*_{T,\crit}(X,f)\longrightarrow
                   H^*_{G\times T,\crit}(R,f).
\]
With the above scaling,
\begin{equation}\label{geo:section-specialization}
 \spc_{[R/G]}\Psi_f=\Psi_0\spc_X.
\end{equation}
In particular, injectivity of specialization on $[R/G]$ implies
injectivity on $X$.
\end{proposition}

\begin{proof}
Take the closure of the graph of $U\to X$ in $X\times_Y R$.
It is projective over $R$, and above $U$ it is precisely the smooth
open graph.  A projective equivariant resolution, isomorphic above this
open graph, gives a smooth variety $Z$ and maps
\[
 X\xleftarrow{p}Z\xrightarrow{q}R,
 \qquad \dim Z=\dim R,
 \qquad q^{-1}(U)\simeq U.
\]
Characteristic-zero functorial resolution supplies equivariance for
the gauge, flavor, and scaling actions simultaneously.  The map $q$
is proper birational.  Each invariant polynomial, and in particular
each $f_d$, has equal pullbacks by $p$ and $q$, since they agree on the
dense open graph.  Thus the total mixed families also agree on $Z$.
Every point of $Z$ has a scaling limit by properness over $R$; every
point of $X$ has one by properness over $Y$.

Give $X$ the trivial $G$-action. Define
\begin{equation}\label{geo:actual-section}
 \Psi_f=q_*p^*(1\otimes-):
 H^*_{T,\crit}(X,f)\longrightarrow
 H^*_{G\times T,\crit}(X,f)\longrightarrow
 H^*_{G\times T,\crit}(Z,q^*f)\longrightarrow
 H^*_{G\times T,\crit}(R,f).
\end{equation}
The pullback is the exchange transformation of the preceding section,
and the pushforward is proper trace.  Restriction to $U$, proper
base change, and $q^{-1}(U)\simeq U$ identify the composition with
pullback along the principal bundle $U\to X$.  Smooth descent for
vanishing cycles then identifies this restriction with the identity
on $H^*_{T,\crit}(X,f)$.

Here is the equivariant degree check.  If $V_N$ is a free open
approximation to $EG$ of dimension $v_N$, then
$(Z\times V_N)/G$ and $(R\times V_N)/G$ have equal dimension
$r+v_N-g$, where $r=\dim R$ and $g=\dim G$.
The BM pullback to the first space adds $2v_N$ to a class of BM degree
$i$ on $X$.  Converting back to cohomological degree gives
$2(r+v_N-g)-(i+2v_N)=2\dim X-i$.
Thus the section has cohomological degree zero.  Flavor approximations
add the same dimensions on both sides.

Corollary~\ref{geo:specialization-naturality} applies to $p$ and $q$.
The gauge-unit arrow is also compatible: it is pullback along the
product with a gauge classifying-space approximation, on which scaling
is trivial.  This proves \eqref{geo:section-specialization}.
If $\spc_X(x)=0$ and ambient specialization is injective, the equation
gives $\Psi_f(x)=0$, and stable restriction gives $x=0$.
\end{proof}

\section{Dimensional reduction and injective specialization}
\label{geo:cut}

The next identity fixes which dimensional-reduction map is used.
Let $\pi:\cE\to B$ be a vector bundle on a smooth variety, let
$s\in\Gamma(B,\cE^\vee)$, and put
\[
 f(b,v)=\langle v,s(b)\rangle,\qquad
 Z=Z(s),\qquad S=\cE|_Z.
\]
Write $k:S\hookrightarrow\cE_0=f^{-1}(0)$ and
$i:\cE_0\hookrightarrow\cE$.  The coefficient group is assumed to
preserve the bundle and section.  No smoothness or expected-dimension
condition is imposed on $Z$.

\begin{proposition}\label{geo:cut-identity}
Dimensional reduction gives the specified isomorphism
\begin{equation}\label{geo:delta-map}
 \delta_f=\can_f k_*:
 H_i^{\BM}(S)\xrightarrow{\sim}H_i^{\crit,\BM}(\cE,f).
\end{equation}
If the ambient space $\cE$ has a scaling satisfying the hypotheses of
Section~\ref{geo:specialization}, then
\begin{equation}\label{geo:cut-specialization}
 \spc_{\cE}\can_f=i_*,\qquad
 \spc_{\cE}\delta_f=(ik)_*.
\end{equation}
The zero scheme $Z$, and hence $S$, need not be invariant under this
auxiliary scaling.
\end{proposition}

\begin{proof}
The compactly supported dimensional-reduction theorem
\cite[Theorem A.1 and Appendix A.2]{Davison17} concerns the natural
transformation
\[
 \Phi_f\Q_{\cE}\longrightarrow\Q_{\cE_0}
                         \longrightarrow k_*\Q_S.
\]
It becomes an isomorphism after $\pi_!$.  Its naturality under vector
bundle coordinate changes gives the statement for bundles, and its
equivariant form follows at finite free approximations
\cite[Corollary A.9]{Davison17}.  Taking compactly supported cohomology
and Verdier duality identifies the dual of the second arrow with
$k_*$ on BM homology and the dual of the first with the Milnor map
$\can_f$.  This proves \eqref{geo:delta-map} as an equality of maps.
There is no BM degree shift when the source is $S$; replacing it by
$Z$ would introduce the vector-bundle shift $2\rk\cE$.

For the specialization identity use the entire family
\[
 V=\Big\{\sum_{d>0}t^df_d=0\Big\}
           \subset\cE\times\mathbb A^1_t.
\]
Its central fiber is all of $\cE$ and its open part is identified with
$\cE_0\times\C^*$ by \eqref{geo:family-trivialization}.
The inclusion of this family in the constant ambient family is proper.
Its ordinary BM localization boundary therefore commutes with that of
$\cE\times\mathbb A^1_t$.  On the open parts the map is the original
inclusion $i$ under the same family trivialization, and on the central
fiber it is the identity of $\cE$.  The ambient specialization is the
identity by Corollary~\ref{geo:specialization-naturality}.  Thus the
ordinary zero-fiber specialization is the actual pushforward $i_*$.

The Milnor canonical morphism gives a map from this ordinary
localization triangle to the critical localization triangle for the
total potential.  On the central support it is exactly $\eta_{\cE}$
of \eqref{geo:eta}, and on the open part it is $\can_f$ after the
trivialization.  Applying $\eta_{\cE}^{-1}$ therefore gives
$\spc_{\cE}\can_f=i_*$.  Composition with $k_*$ proves the second
equality.  This argument used the whole potential-zero family, so it
requires neither flatness of that family nor scaling invariance of the
particular cut-zero scheme.
\end{proof}

We apply the proposition to the spectral potential.  For each unordered
pair of distinct original colors choose one of the two directions and
put all its graded arrows in a cut.  Each term of
\eqref{geo:graded-potential} contains precisely one cut arrow.  Thus
$f_{\bnu}$ is fiberwise linear on the bundle of cut-arrow variables
over the space of all remaining variables.  The latter space consists
of the $\epsilon$-variables and the opposite cross-color arrows.
The cut equations are linear in those opposite arrows once the
$\epsilon$-variables are fixed.  This remains true for an orthogonal
pair: its equation is $2a=0$ on the non-cut arrow.

\begin{proposition}\label{geo:epsilon-injectivity}
The specialization
\[
 \spc_{\bnu}:H^*_{G_{\bnu},\crit}
     (\operatorname{Rep}_{\bnu}(\Gamma),f_{\bnu})
       \longrightarrow R_{\bnu}
\]
is injective.  Its source has no odd cohomology.
\end{proposition}

\begin{proof}
Only finitely many $\epsilon$-arrows occur for a fixed $\bnu$.
For each original color their components are finite equioriented
quivers of type $A$.  Their representations are direct sums of interval
representations, so there are finitely many orbits.  The stabilizer $H$
of one such representation is connected, with unipotent radical and
reductive quotient a product of general linear groups on its interval
multiplicity spaces.  One sees this directly by writing its automorphism
group as the units in the finite-dimensional endomorphism algebra:
modulo its Jacobson radical that algebra is a product of matrix
algebras, and the kernel of the map on units is a unipotent group.
Consequently $H^*(BH,\Q)$ is an even polynomial ring and restriction to
a maximal torus is injective.

Let $O=G_{\bnu}/H$ be an $\epsilon$-orbit.  Over $O$, write
$\cA_O$ for the vector bundle of non-cut cross-color arrows.
The cut equations form an equivariant linear map on this bundle;
its rank is constant on the orbit, so its kernel $\cK_O$ is a vector
subbundle.  The source $S$ of dimensional reduction over this orbit
is $\cK_O$ times the unrestricted cut-arrow bundle, whereas the full
representation space over the orbit is $\cA_O$ times that same bundle.
The normal bundle is therefore $\cA_O/\cK_O$.

Every weight of this normal bundle is a difference of characters from
distinct original colors.  Indeed all its variables are cross-color
arrows, and the scalar centers for different original colors act
independently while preserving the $\epsilon$-representation.  Hence
no normal weight is zero.  Restriction to a maximal torus expresses
the Euler class as a product of nonzero linear forms.  It is nonzero
in the domain $H^*(BH,\Q)$.

Include the cut-arrow rank in both ranks and write $k$ for
the source rank and $a$ for the target rank over $O$, with $o=\dim O$.
Equivariant Thom isomorphisms identify the stratum map with
\[
 H^{2(o+k)-i}(BH)\xrightarrow{\ e(\cA_O/\cK_O)\ }
 H^{2(o+a)-i}(BH).
\]
It is injective and both stratum BM groups vanish in odd degrees.
Order the finitely many $\epsilon$-orbits so that initial unions are
closed, and pull the resulting filtration back to source and target.
Their BM localization sequences are short exact in even degree,
by induction from the even stratum groups.  In a commuting diagram of
these sequences, a class in the middle source that maps to zero must
restrict to zero on the last stratum, hence come from the preceding
filtered source, where induction kills it.  Thus the entire map
$H_*^{G_{\bnu},\BM}(S)\to
 H_*^{G_{\bnu},\BM}(\operatorname{Rep}_{\bnu}(\Gamma))$
is injective, with even source.  The same argument at finite Borel
approximations applies in every stabilized degree.

Finally \eqref{geo:cut-specialization} identifies this map with
$\spc_{\bnu}\delta_f$.  Dimensional reduction is an isomorphism, so
specialization is injective and the critical groups are even.

\end{proof}

\section{The exact wheel image and the Euler quotient}
\label{geo:wheel-image}

Let $I_{\Gamma,\bnu}\subset R_{\bnu}$ be the homogeneous wheel ideal
of Theorem~\ref{alg:wheel}.  Thus for each labeled lift of a Serre
wheel through the chosen spectral assignment, one imposes the ideal
identifying all its local root coordinates, and intersects these
ideals before taking invariants.  Orthogonal two-variable wheels are
included.  When no wheel is possible the ideal is $R_{\bnu}$.
We recall the ordinary Hall word formula in the convention compatible
with \eqref{geo:ordinary-kernel}:
\begin{equation}\label{geo:hall-kernel}
 \kappa_\alpha(y)=
 \prod_{a<b}(y_b-y_a)^{e_{v_av_b}-\delta_{v_av_b}},
 \qquad
 G\longmapsto\Sym_{\prod_v S_{\nu_v}}(G\kappa_\alpha).
\end{equation}
Although the displayed kernel can have diagonal denominators, its
symmetrization is polynomial.  After clearing the equal-vertex
Vandermonde, the numerator is alternating in every equal-vertex
block and hence divisible by that Vandermonde.
The equality with the wheel ideal proved in Theorem~\ref{alg:wheel}
is the exact equality
\begin{equation}\label{geo:hall-wheel}
 I_{\Gamma,\bnu}
 =\sum_\alpha\im\big(G\mapsto\Sym(G\kappa_\alpha)\big).
\end{equation}

\begin{theorem}\label{geo:critical-wheel-image}
Specialization identifies unframed critical cohomology, in every
dimension vector, with the wheel ideal:
\begin{equation}\label{geo:unframed-identification}
 H^{2*}_{G_{\bnu},\crit}
   (\operatorname{Rep}_{\bnu}(\Gamma),f_{\bnu})
      \xrightarrow[\spc]{\sim}I_{\Gamma,\bnu}\subset R_{\bnu}.
\end{equation}
It intertwines Hall multiplication.  On the cyclic framed moduli it
is injective and has exact image
\begin{equation}\label{geo:stable-image}
 H^{2*}_{\crit}(X_{\bnu,\bm},f)
 \xrightarrow[\spc]{\sim}
 (I_{\Gamma,\bnu}+J_{\bm})/J_{\bm}
 \subset R_{\bnu}/J_{\bm}.
\end{equation}
In particular framed critical cohomology is finite dimensional and even.
\end{theorem}

\begin{proof}
Fix a complete vertex flag of type $\alpha$. The space of
representations strictly lowering this flag is a vector bundle over
the flag variety. Every cyclic word has trace zero on that space,
so its potential is identically zero. Projection to the product of
its one-dimensional diagonal representations is smooth, and the
map forgetting the flag is proper. The critical push-pull class
decorated by a labeled polynomial $G$ is the corresponding Hall
word. Indeed the successive invariant-subspace correspondences
compose to this complete-flag correspondence; smooth base change
and proper composition identify their iterated push-pull maps.
All their one-dimensional diagonal potentials are zero.

Specialization on the flag space is the identity.
Corollary~\ref{geo:specialization-naturality} therefore sends this
critical Hall word to its ordinary push-pull class
$\Sym(G\kappa_\alpha)$. In this formula the arrow blocks give the
numerator Euler factors and the flag tangent bundle gives the
denominator. Equation~\eqref{geo:hall-wheel} proves
$I_{\Gamma,\bnu}\subset\im\spc$.

For the reverse inclusion, \eqref{geo:cut-specialization} places the
specialization image in the image of ordinary cohomology with support
on $f^{-1}(0)$.  Fix a labeled wheel.  Choose its indicated
one-dimensional lines in the vertex spaces, put nonzero maps on the
cycle \eqref{geo:graded-potential}, and put all other arrow blocks
zero.  The value of $f$ is nonzero: for a nonorthogonal wheel the
selected cyclic term is nonzero and all the other terms vanish; for
an orthogonal pair its value is twice the product of its two nonzero
arrows.  A diagonal torus stabilizing this representation equates the
characters on the wheel lines and leaves all complementary line
characters independent.  Pullback from $R_{\bnu}$ to this torus is
therefore precisely the wheel substitution.  Every supported image
class restricts to zero, since the representation lies outside
$f^{-1}(0)$.  Its kernel is the intersection with $R_{\bnu}$ of the
corresponding labeled wheel-diagonal ideal.  Intersecting over all
wheels proves $\im\spc\subset I_{\Gamma,\bnu}$.
Proposition~\ref{geo:epsilon-injectivity} proves injectivity.

These equalities also prove that the critical algebra is generated
by the vertex polynomial classes. To check multiplication, first
multiply an arbitrary critical class by one such vertex class.
The vertex potential is zero. The Thom--Sebastiani map with this
factor is exterior product with its constant complex, equivalently
smooth pullback followed by the Chern decoration. The invariant-line
correspondence has a smooth bundle projection and a proper
flag-forgetting map, so the same naturality proves compatibility
with this multiplication, on either side. Expressing a general
class as a sum of vertex words and applying associativity extends
the compatibility to every Hall product.

Adding the incoming framing is a vector bundle with pulled-back
potential, so its critical cohomology agrees with the unframed group
in unshifted cohomological indexing.  The section in
Proposition~\ref{geo:kirwan-section} makes critical stable restriction
surjective, and \eqref{geo:section-specialization} transfers
injectivity to the stable quotient.  The commutative square with
ordinary stable restriction and
Theorem~\ref{geo:ordinary-kernel-theorem} gives exactly
\eqref{geo:stable-image}.  Its target is finite dimensional and even,
which proves the last assertion.
\end{proof}

\begin{theorem}\label{geo:critical-quotient}
With $\bp=\bm-\bl$ as in \eqref{geo:euler-operator}, there is a natural
graded isomorphism
\begin{equation}\label{geo:quotient-identification}
 Q_{\bnu,\bp}:=
 H^{2*}_{\crit}(X_{\bnu,\bm},f)/\ker D
 \ \simeq\ I_{\Gamma,\bnu}/(I_{\Gamma,\bnu}\cap J_{\bp}).
\end{equation}
The source quotient is assigned its source grading.  Multiplication
by $D$ identifies it with its literal image after the polynomial
degree shift $L=\bl\cdot\bnu$.  Its centered grading is
\begin{equation}\label{geo:centered-grading}
 \deg_{\mathrm{cen}}x=2\deg_{\mathrm{pol}}x-d_{\bp}(\bnu).
\end{equation}
Together with the actual simple-kernel identification of
Theorem~\ref{alg:kernel}, \eqref{geo:quotient-identification} realizes
the local $\ell$-weight space of the corresponding simple module.
\end{theorem}

\begin{proof}
The definition of the ordered kernels immediately gives
$K_{\alpha,\bp}=\Delta_{\bl}K_{\alpha,\bm}$ and hence
\begin{equation}\label{geo:colon-ideal}
 J_{\bp}=(J_{\bm}:\Delta_{\bl}).
\end{equation}
This is an equality of all labeled test kernels, so it remains valid
without a genericity assumption on the roots.  Under
\eqref{geo:stable-image}, $D$ is multiplication by $\Delta_{\bl}$.
Its kernel on the image of $I_{\Gamma,\bnu}$ consists precisely of the
classes represented by $I_{\Gamma,\bnu}\cap J_{\bp}$.
This proves \eqref{geo:quotient-identification}, and multiplication by
$\Delta_{\bl}$ has polynomial degree $L$.
Lemma~\ref{geo:coordinate-kernel} identifies the additive kernels with
the multiplicative local kernels on the same labeled tests.
Theorem~\ref{alg:kernel} identifies those tests with the actual
highest-vector annihilator.  The resulting isomorphism sends each shuffle word to its
evaluation on the highest vector.
Finally \eqref{geo:net-center} gives the stated center.
\end{proof}


\section{Proper critical support and the weight-one action}
\label{geo:properness}

We record the support statement needed for duality and for the
fundamental-module calculation.  It concerns the entire critical
locus of the one-sided framed model used here.

\begin{proposition}\label{geo:proper-support}
For every finite dimension vector, the reduced critical locus
\[
 Z_{\bnu,\bm}=\operatorname{Crit}(f:X_{\bnu,\bm}\to\C)_{\red}
\]
is contained in the projective zero fiber of the cyclic GIT morphism
and is projective.  Scaling the chosen cut arrows with weight one and
all other arrows and framing maps with weight zero induces an
algebraic $\Gm$-action on $X_{\bnu,\bm}$ for which $f$ has weight one.
\end{proposition}

\begin{proof}
We first match the present potential and stability with the simply
framed graded $W_2$ model of \cite[\S\S3.1.5, 3.1.7]{VV24}.
For an orthogonal pair the potential on its two arrow spaces is
$2\Tr(AB)$.  The trace pairing between the rectangular matrix spaces
is perfect, so its two derivatives force $A=B=0$ at every critical
point.  These equations do not involve any other variable.
Removing those zero arrow maps leaves the cyclicity condition
unchanged.  Thus the entire reduced critical locus of the augmented
model is identified with the critical locus after removing these
pairs.  The extra variables remain part of its smooth ambient space
and of the dimension $d_{\bm}$ in \eqref{geo:dimension-form}.

On a genuine Dynkin edge $\{i,j\}$ put $r_{ij}=-c_{ij}>0$.
The two terms of our potential have plus signs.  Choose nonzero
constants $\lambda_i$ so that
$\lambda_i^{r_{ij}}/\lambda_j^{r_{ji}}=-1$ on every edge.
Such constants exist: root each connected Dynkin tree, choose the
constant at its root, and solve for the constant at each new adjacent
vertex by taking a nonzero complex root.  Replace every graded copy
of $\epsilon_i$ by $\lambda_i\epsilon_i$, and then rescale one
cross-color arrow on each edge to normalize the common coefficient.
The resulting two coefficients are $+1$ and $-1$, according to the
chosen orientation.  These diagonal linear changes preserve the
gauge action, the spectral grading, and stability.  The potential
has become the signed $W_2$ potential.

Here is an explicit framing and grading dictionary.  Take the vector
space dual of every arrow map and every framing map, keeping the
vertex index $k$ on the dual vector space:
\[
 \widehat V_{i,k}=V_{i,k}^{\vee},\qquad
 \widehat W_{i,k-d_i}=W_{i,k}^{\vee}.
\]
An arrow from $(i,k)$ to $(j,k-b_{ij})$ becomes an arrow from
$(j,k-b_{ij})$ to $(i,k)$, of grade $b_{ij}$.
An $\epsilon_i$-arrow becomes one of grade $2d_i$, and the framing
becomes an outgoing map of grade $-d_i$.  These are the grades of
\cite[(3.3)]{VV24}.  Transposition reverses each cyclic word;
cyclically moving its factors gives the same signed $W_2$ expression.
An incoming cyclic representation becomes an outgoing stable one:
a subrepresentation of the dual annihilated by the outgoing framing
has annihilator containing the entire framing-generated original
representation, and is therefore zero.  The converse follows by
annihilators in the same way.  This is an isomorphism of the relevant
smooth moduli after the changes just described.  In particular the
comparison involves the single outgoing framing of $W_2$.

Proposition 5.8 of \cite{VV24} places the entire graded $W_2$ critical
locus in its projective nilpotent fiber.  The nilpotence behind this
statement is useful here.  In its Jacobian algebra the cross-color
arrow ideal is nilpotent in finite Dynkin type, and the homogeneous
central element $\sum_i\epsilon_i^{t/d_i}$ has degree $2t>0$, where
$t=\max_i d_i$.  On a finitely supported graded module this element,
and hence each $\epsilon_i$, is nilpotent.  These facts make the whole
augmentation ideal act nilpotently; see \cite[\S3.2]{VV24}.
Thus all nonconstant cycle invariants vanish.  The graded critical
representation maps to the origin of the affine quotient.  Reversing
the dictionary and restoring the zero orthogonal arrows proves the
same assertion for $X_{\bnu,\bm}$.  Its reduced critical locus is
closed in a projective fiber, which proves projectivity.

Finally each term of \eqref{geo:graded-potential} contains exactly one
cut arrow, including each quadratic orthogonal term.  The cut action
therefore gives $f(u\cdot x)=u f(x)$.  For $u\ne0$, multiplying an
arrow by a nonzero scalar leaves invariant-subspace conditions
unchanged.  It preserves cyclicity, commutes with the gauge action,
and descends to the smooth quotient.  This proves the stated
algebraic action.
\end{proof}

\begin{corollary}\label{geo:whole-critical-lefschetz}
Put $d=d_{\bm}(\bnu)$.  The whole critical cohomology of
$X_{\bnu,\bm}$ is pure and satisfies hard Lefschetz about cohomological
degree $d$.  More precisely, for the first Chern class $h$ of an ample
line bundle restricted to $Z_{\bnu,\bm}$,
\[
 h^r:H^{d-r}_{\crit}(X_{\bnu,\bm},f)
          \xrightarrow{\sim}H^{d+r}_{\crit}(X_{\bnu,\bm},f)
 \qquad(r\ge0),
\]
after the corresponding Tate twist.  Forgetting Tate twists, Verdier
duality gives a perfect pairing
\begin{equation}\label{geo:critical-pairing}
 H^k_{\crit}(X_{\bnu,\bm},f)\otimes
 H^{2d-k}_{\crit}(X_{\bnu,\bm},f)\longrightarrow\Q.
\end{equation}
\end{corollary}

\begin{proof}
The variety is smooth quasi-projective, and the potential has positive
weight one under the cut action.  Euler's identity gives $f=0$ on
its critical locus.  Proposition~\ref{geo:proper-support} shows that
this entire locus is projective, so it is a projective union of
complete critical components.  These are the hypotheses of
\cite[Corollary 3.2]{DMSZ15}; weight one removes the nonunipotent
monodromy summand.  The result gives purity of the global critical
cohomology and hard Lefschetz.  In our convention
$\Phi_f(\Q_X[d])$ is perverse, so the center is $d$, the dimension
of the smooth ambient quotient.  This also retains the contributions
of the orthogonal quadratic arrow pairs.

Normalized vanishing cycles commute with Verdier duality.
Since $X$ is smooth of dimension $d$,
$\mathbb D(\Phi_f\Q_X)=\Phi_f\Q_X[2d]$ after forgetting Tate twists.
The support is proper, so ordinary and compactly supported
hypercohomology agree.  Verdier duality consequently yields
\eqref{geo:critical-pairing}.
\end{proof}

\section{Flavor parameters and homogeneous generation}
\label{geo:flavor-generation}

We retain an external torus $A=\Gm$ acting on the framing spaces and
trivially on the unframed arrow representation.  Choose also
$A$-representations $U_v$ of dimensions $l_v$, and replace the Euler
bundle by
\[
 \cE_{\bl,A}=\bigoplus_v\Hom(U_v,\cV_v).
\]
If the weights of $U_v$ are $c'_{v,r}$, its equivariant Euler class is
\begin{equation}\label{geo:flavor-euler}
 \Delta_{\bl,A}=
 \prod_{v,a}\prod_{r=1}^{l_v}(y_{v,a}-c'_{v,r}a),
 \qquad \deg a=\deg y_{v,a}=1.
\end{equation}
Let
\[
 H_A=H^{2*}_{A,\crit}(X_{\bnu,\bm},f),\qquad
 D_A=\Delta_{\bl,A}\cup-,\qquad
 \cQ_A=H_A/\ker D_A.
\]
These definitions retain all flavor factors, including factors which
coincide at $a=0$.

\begin{theorem}\label{geo:homogeneous-generation}
For every finitely supported dimension vector there is a homogeneous
surjection of $\Q[a]$-modules
\begin{equation}\label{geo:homogeneous-surjection}
 I_{\Gamma,\bnu}[a]\longrightarrow H_A.
\end{equation}
The modules $H_A$ and $\cQ_A$ are finite free over $\Q[a]$.
The unframed critical Hall algebra is generated by its homogeneous
vertex classes.  Its framed module, and the quotient by the Euler
kernel, are generated by their dimension-zero vacuum.
These statements hold with the source grading on $\cQ_A$.
\end{theorem}

\begin{proof}
Since $A$ acts trivially on the unframed representation space,
Theorem~\ref{geo:critical-wheel-image} identifies its equivariant
critical cohomology with $I_{\Gamma,\bnu}\otimes\Q[a]$.
Adding the incoming framing is an equivariant vector bundle with
pulled-back potential.  Unshifted critical cohomology is invariant
under that pullback.  Proposition~\ref{geo:kirwan-section}, with
flavor group $A$, makes stable restriction surjective.
Every map has cohomological degree zero and is $\Q[a]$-linear, so this
is precisely the homogeneous surjection
\eqref{geo:homogeneous-surjection}.

The ordinary critical group is finite dimensional and even by
Theorem~\ref{geo:critical-wheel-image}.  The equivariant spectral
sequence therefore degenerates: both the base $H^*(BA)$ and the fiber
critical cohomology are even.  Homogeneous lifts of a basis of the
fiber give a free basis over $\Q[a]$ by the same degree induction as
in the Leray--Hirsch argument.  There are no local-system twists since
$A$ is connected.  Thus $H_A$ is finite free.  Multiplication by
\eqref{geo:flavor-euler} is homogeneous, and
$\cQ_A\simeq\im D_A$ as ungraded $\Q[a]$-modules.
Its image is a submodule of the free module $H_A$, so is torsion free;
finite torsion-free modules over the polynomial ring in one variable
are free.  The grading gives a homogeneous basis, obtained for example
by homogeneous lifts of a basis modulo $a$ and degree induction.

For generation, \eqref{geo:hall-wheel} states that every polynomial
wheel class is a sum of Hall words in vertex polynomials.  Each vertex
polynomial has its critical lift, and injectivity in
Theorem~\ref{geo:critical-wheel-image} shows that these words are the
critical classes themselves.  Tensoring with $\Q[a]$ gives the equivariant assertion.

The Hall action on cyclic framed moduli uses an unframed invariant
kernel and an old cyclic framed quotient.  At old dimension zero the
flag is unique.  The correspondence then consists exactly of adding
an incoming framing to the unframed representation and restricting to
the cyclic open.  Its action on the vacuum is therefore the map
\eqref{geo:homogeneous-surjection}; there is no further framing Euler
factor.  Surjectivity proves framed vacuum generation.

Finally, on a Hall correspondence with unframed kernel $\mathcal U$
and old framed quotient $\mathcal V'$, the tautological exact sequence
$0\to\mathcal U\to\mathcal V\to\mathcal V'\to0$ gives
\[
 e(\cE_{\bl,A}(\mathcal V))
   =e(\cE_{\bl,A}(\mathcal U))
        e(\cE_{\bl,A}(\mathcal V')).
\]
Let $b$ be an unframed Hall class of dimension vector $\boldsymbol u$,
and write $F_b$ for its convolution action. The projection formula gives
\begin{equation}\label{geo:euler-hall-identity}
 D_{\bnu+\boldsymbol u,A}F_b(x)=
 F_{\,b\,e(\cE_{\bl,A}(\mathcal U))}
                       (D_{\bnu,A}x).
\end{equation}
If $D_{\bnu,A}x=0$, the right side vanishes, so the Euler kernel is
Hall stable.  Passing to the quotient preserves homogeneous vacuum
generation.
\end{proof}

\begin{corollary}\label{geo:strict-lifts}
Let $R=\Q[[a]]$ and $\cQ_R=\cQ_A\otimes_{\Q[a]}R$, with the completed
total-degree filtration $F^d$.  Every class in $F^d\cQ_R$ has a wheel
representative of total degree at least $d$.  Moreover
\begin{equation}\label{geo:filtration-saturation}
 F^d\cQ_R\cap a\cQ_R=aF^{d-1}\cQ_R.
\end{equation}
These statements concern the source quotient grading. The natural map
\[
              \cQ_A/a\cQ_A\longrightarrow
                        H_{\crit}^{2*}(X,f)/\ker D_0
\]
is a homogeneous surjection; it need not be an isomorphism.
\end{corollary}

\begin{proof}
Choose a homogeneous free basis $e_1,\ldots,e_s$ of $\cQ_A$ with
degrees $d_j$.  Homogeneity and surjectivity of
\eqref{geo:homogeneous-surjection}, followed by the quotient map,
provide homogeneous lifts $f_j\in I_{\Gamma,\bnu}[a]$ of the same
degrees.  Every element of $F^d\cQ_R$ has a unique expression
\[
 \sum_j a^{\max(0,d-d_j)}u_j(a)e_j,\qquad u_j(a)\in R.
\]
Replacing $e_j$ by $f_j$ gives the required representative.
The same expression shows that divisibility by $a$ lowers the total
filtration index by exactly one, which proves
\eqref{geo:filtration-saturation}.  Reduction of the source quotient
maps onto the quotient by the reduced Euler kernel, since an element
of $\ker D_A$ reduces to an element of $\ker D_0$.
\end{proof}

\section{Coulomb realizations of ordinary modules}\label{cou:section}

In this section $\g$ has type $G_2$ or $F_4$, and $q\in\C^\times$ is
not a root of unity.  We construct an action of a fixed Coulomb algebra on
every ordinary tensor product of fundamental modules, including tensor
products with coincident spectral parameters.  We shall subsequently use
the same action on their simple subquotients.

We number the vertices with the long vertices first.  Thus
\begin{align}\label{cou:cartan-data}
 C_{G_2}&=\begin{pmatrix}2&-1\\-3&2\end{pmatrix},\qquad
 (d_1,d_2)=(3,1),\notag\\
 C_{F_4}&=\begin{pmatrix}
 2&-1&0&0\\-1&2&-1&0\\0&-2&2&-1\\0&0&-1&2
 \end{pmatrix},\quad (d_1,d_2,d_3,d_4)=(2,2,1,1).
\end{align}
Put $q_i=q^{d_i}$ and $b_{ij}=d_ic_{ij}=d_jc_{ji}$.  A framing zero at
$a\in\C^\times$ will mean a factor $1-a/u$ of a framing polynomial.
This convention distinguishes a framing zero from the underlying flavor
coordinate, which differs from it by $q_i^2$ at vertex $i$.

\subsection{The ordinary current map over the flavor ring}

Let $\ba=(a_i)$ be the gauge ranks, $\bl=(l_i)$ the framing ranks, and
assume that
\begin{equation}\label{cou:zero-shift}
 l_i=\sum_jc_{ji}a_j.
\end{equation}
Let $S$ be a complex algebra carrying invertible framing coordinates
$z_{i,t}$, $1\leq t\leq l_i$.  We write $\cC_S$ for the base change to
$S$ of the equivariant $K$-theoretic Coulomb algebra with symmetrizers
and with these ranks.  More precisely, it is the convolution algebra of
the classical BFN space, using the determinant covers of the gauge
groups and fixing the loop-rotation parameter to be $q$.  The BFN space
parametrizes a gauge lattice together with a matter section regular in
both the original and the new lattice.  Its convolution is defined by
composing lattice modifications and pulling back the common matter
section.  We use the construction of \cite[\S\S3.1--3.2]{VV25}, including
its finite-dimensional approximations.  All the specializations used
below are taken in this algebra before considering its modules.

The gauge coefficient ring is
\begin{equation}\label{cou:gauge-ring}
 B_S=S\bigl[w_{i,t}^{\pm1},D_i^{\pm1}\bigr]^{\prod_i\mathfrak S_{a_i}}
       \big/\left(D_i^2-\prod_tw_{i,t}\right)_{i\in I}.
\end{equation}
The symmetric groups fix the $D_i$.  Set
\[
 W_i(u)=\prod_{t=1}^{a_i}(1-w_{i,t}/u),\qquad
 Z_i(u)=\prod_{t=1}^{l_i}(1-q_i^2z_{i,t}/u).
\]
The Cartan current in the geometric map is the rational function
\begin{equation}\label{cou:cartan-rational}
 \Psi_i(u)=\left(\prod_jD_j^{c_{ji}}\right)Z_i(u)
 \frac{\displaystyle\prod_{j\ne i}\prod_{r=1}^{-c_{ji}}
          W_j\bigl(q_j^{-c_{ji}-2r}u\bigr)}
      {W_i(u)W_i(q_i^{-2}u)}.
\end{equation}
An empty product is $1$.  Its expansions at infinity and zero are the
positive and negative Cartan series.  Equality of these two expansions
as rational functions follows either from this formula or from the
geometric positive and negative $A$-polynomials: their additional power
of $u$ is $l_i-\sum_jc_{ji}a_j$, which is zero by
\eqref{cou:zero-shift}.

The minuscule classes give the root currents.  In the conventions of
\cite[Theorem~7.1 and \S7(i)]{FT19} and
\cite[(4.2.8), Lemmas~5.1.1--5.1.2]{VV25}, the ordinary root generators
before imposing the central relation are
\[
 e_{i,r}=E_{i,r}^{\mathrm{geom}}/(q_i-q_i^{-1}),\qquad
 f_{i,r}=F_{i,r}^{\mathrm{geom}}/(q_i-q_i^{-1}),\qquad r\in\Z.
\]
The flavor coordinate in the first convention is $q_i$ times the
coordinate $z_{i,t}$ used here.  We use the extended simply connected
current presentation, in which the leading positive and negative
Cartan coefficients are independent units, and impose the ordinary
relation after the following calculation.

\begin{lemma}\label{cou:central-normalization}
In $\cC_S$ one has
\begin{equation}\label{cou:central-product}
 \Psi_i(\infty)=\prod_jD_j^{c_{ji}},\qquad
 \Psi_i(0)=K_i\prod_jD_j^{-c_{ji}},\qquad
 K_i=q_i^{2l_i-2a_i}\prod_{j\ne i}q_j^{-a_jc_{ji}}
                         \prod_{t=1}^{l_i}z_{i,t}.
\end{equation}
Suppose $\kappa_i\in S^\times$ satisfy $\kappa_i^2K_i=1$.  Multiplying
$\Psi_i^\pm$ and every $e_{i,r}$ by $\kappa_i$, while leaving every
$f_{i,r}$ unchanged, gives an ordinary
quantum loop algebra homomorphism
\begin{equation}\label{cou:ordinary-map}
 \Phi_S:\U\otimes_\C S\longrightarrow\cC_S.
\end{equation}
\end{lemma}

\begin{proof}
The value at infinity is immediate.  At zero the two denominator
factors at vertex $i$ contribute $D_i^4q_i^{2a_i}$ before including the
leading determinant.  For a neighboring vertex $j$, the sum of the
shifts in its numerator factors is
\[
 \sum_{r=1}^{-c_{ji}}(c_{ji}+2r)=-c_{ji}.
\]
The powers of the gauge determinants therefore give the inverse of
the determinant at infinity.  The remaining scalar is the stated
$K_i$.  The sign is
$(-1)^{l_i-\sum_{j\ne i}c_{ji}a_j}=(-1)^{2a_i}=1$.
Thus the product of the leading positive and negative coefficients
is $K_i$ in the whole coefficient ring.

The substitution
\[
  \varphi_i^\pm\longmapsto \kappa_i\varphi_i^\pm,\qquad
  e_{i,r}\longmapsto \kappa_ie_{i,r},\qquad
  f_{i,r}\longmapsto f_{i,r}\quad(r\in\Z)
\]
preserves every relation of the extended current presentation.  Indeed,
the Cartan--root and Serre relations are homogeneous under it, and
both sides of each $e_{i,r},f_{i,s}$ commutator are multiplied by $\kappa_i$.
The leading central product in the image becomes $\kappa_i^2K_i=1$.
Consequently the map factors through the ordinary presentation.
\end{proof}

We shall need surjectivity before specializing a deformation of the
framing parameters.  The following argument also gives precise control
of the gauge coefficients at a numerical value of $q$.

\begin{proposition}\label{cou:surjection}
For the paired framings constructed below, the map
\eqref{cou:ordinary-map} is surjective over their Laurent parameter
rings, and after every base change of those rings.  Its image contains
the whole ring $B_S$.
\end{proposition}

\begin{proof}
Both Cartan matrices in \eqref{cou:cartan-data} have determinant $1$.
The leading Cartan coefficients and their inverses therefore recover
all $D_i^{\pm1}$ by the integral inverse of $C^{\mathsf T}$, with the
known scalar units $\kappa_i$ removed.  To recover the remaining gauge
coefficients, put $s_{j,n}=\sum_tw_{j,t}^n$ for $n>0$, and let
$h_{i,n}$ be the coefficient of $u^{-n}$ in
\[
 n\log\!\left(\frac{\Psi_i(u)}{\Psi_i(\infty)Z_i(u)}\right).
\]
The logarithm is expanded at infinity.  Each of its coefficients is a
finite polynomial in the current coefficients already obtained.
Put $h_n=(h_{i,n})_{i\in I}$ and $s_n=(s_{j,n})_{j\in I}$.
Formula \eqref{cou:cartan-rational} gives
\begin{equation}\label{cou:quantum-cartan}
  h_n=B(q^n)s_n,
 \qquad B_{ii}(x)=1+x^{2d_i},\quad
 B_{ij}(x)=-\sum_{r=1}^{-c_{ji}}x^{d_j(c_{ji}+2r)}\quad(i\ne j).
\end{equation}
In the present numbering the matrices and their determinants are
\begin{equation}\label{cou:det-g}
\begin{gathered}
 B_{G_2}(x)=\begin{pmatrix}
 1+x^6&-x^3-x-x^{-1}\\-x^3&1+x^2
 \end{pmatrix},
 \qquad\det B_{G_2}(x)=x^8-x^4+1=\Phi_{24}(x).
\end{gathered}
\end{equation}
\begin{equation}\label{cou:det-f}
\begin{gathered}
 B_{F_4}(x)=\begin{pmatrix}
 1+x^4&-x^2&0&0\\
 -x^2&1+x^4&-1-x^2&0\\
 0&-x^2&1+x^2&-x\\
 0&0&-x&1+x^2
 \end{pmatrix},\\
 \det B_{F_4}(x)=x^{12}-x^6+1=\Phi_{36}(x).
\end{gathered}
\end{equation}
Here $\Phi_n$ denotes the $n$th cyclotomic polynomial.  These
determinants at $q^n$ are nonzero complex numbers.  Inverting them and
using Newton identities recovers every coefficient of $W_j(u)$.
The inverse of its last coefficient is already supplied by $D_j^{-2}$.
This proves that $B_S$ is in the image without inverting any flavor
function.

Retain the flavor ring in the generation argument of
\cite[\S4.2.2]{VV25}. Its magnetic filtration is indexed by dominant
magnetic charges $\gamma=(\gamma_{i,r})$. Each graded piece is the
Laurent coefficient ring invariant under the stabilizer of $\gamma$,
multiplied by the structure-sheaf class of the corresponding magnetic
stratum.  In the abelian algebra,
one removes a positive coordinate of a nonzero charge by a positive
unit step.  The framing factor in this product is $1$: the two steps
have the same sign in that coordinate and the intermediate lattice
does not cross the framing hyperplane.  If there is no positive
coordinate, one removes a negative coordinate by a negative unit step;
the same statement holds.  This is also obtained directly by comparing
the three intersections of matter lattices in
\cite[(3.2.21), (3.2.25)]{VV25}.

The coefficients in these one-step products are consequently Laurent
polynomials in gauge variables and in $q$, with no flavor factor.
The no-common-zero argument of \cite[Lemma~4.2.4]{VV25}, at a fixed
non-root-of-unity $q$, supplies a Bezout identity in the gauge Laurent
ring.  More explicitly, for the set of positive coordinates
$\Gamma=\{(i,r):\gamma_{i,r}>0\}$ the coefficients involve only
the variables indexed by $\Gamma$.  Every factor has the form
$1-q^e w_{i,r}w_{j,s}^{-1}$.  For the finite-type orientations in
\cite[\S2.2.1]{VV25}, all the exponents $e$ have the same weak
sign, and exponent zero can occur at only one vertex.
At a putative common zero, choose a vanishing factor at every
coordinate.  Following the resulting arrows in the finite set
$\Gamma$ produces a cycle.  Since the graph has no loops and
zero exponents occur at only one vertex, that cycle has a
nonzero sum of exponents of the same sign.  Multiplying its
equalities would give $q^e=1$ for a nonzero integer $e$, a
contradiction.  The sign-reversed argument handles all-negative
charges.  The Laurent Nullstellensatz now supplies the Bezout
identity in the gauge ring.  This identity is unchanged after
tensoring with the flavor ring.
Induction on the magnetic charge generates the abelian magnetic
classes.  The nil-Hecke operators and the full spherical idempotent
used in \cite[Lemmas~4.2.3 and 4.2.5]{VV25} are flavor equivariant, so
the same induction generates the spherical algebra over that ring.
This proves that $B_S$ and the dressed minuscule classes generate
$\cC_S$ before flavor specialization.

For a minuscule charge at vertex $i$, the partial invariant ring is
generated over the full invariant ring by the one selected root $w$.
It satisfies the total characteristic polynomial
$\prod_t(T-w_{i,t})$.  Since the constant coefficient of this polynomial
is a unit, $w^{-1}$ is also a polynomial in $w$ with coefficients in
$B_S$.  All its powers occur among the integer current modes, up to
invertible determinant and numerical factors.  They therefore supply
all minuscule dressings.  The determinant cover only base changes
this assertion.  Hence the ordinary current image contains every
generator of $\cC_S$, which proves surjectivity.  Each step was an
identity over the flavor ring, so it remains valid after base change.
\end{proof}

\begin{lemma}\label{cou:fixed-q-faithfulness}
The geometric difference-operator realization of $\cC_S$ is injective
at every numerical non-root-of-unity $q$ used here.  In particular,
polynomial bimodule identities established in that realization hold
in the specialized Coulomb algebra.
\end{lemma}

\begin{proof}
The magnetic filtration and its associated graded description
\cite[(3.2.17)]{VV25} are obtained by the equivariant affine-cell
filtration.  They commute with the present specializations: each
piece is an invariant Laurent coefficient ring times one magnetic
class. At fixed $q$ these pieces are free over the external flavor
coefficient ring and torsion-free with respect to the gauge Laurent
Euler factors used in torus localization. Such a denominator preserves
the nonzero leading magnetic term of a nonzero element. Gauge
localization is therefore injective.  Torus localization identifies
the resulting algebra with its difference-operator image, as in
\cite[\S3.2.4 and \S5.1]{VV25}.  This proves the assertion directly at
the specialized parameter.
\end{proof}

\subsection{Six fundamental cup modules}

We next specify all finite data needed to construct the fundamental
modules.  The unfolded diagrams are $D_4$ and $E_6$.  Number their
vertices by the following ordered lists of pairs, and define their
edges by the displayed sets:
\begin{equation}\label{cou:unfolded-data}
\begin{array}{c|c|c}
 &\text{unfolded vertices }1,2,\ldots&\text{edges}\\ \hline
 D_4&(1,2),(1,4),(1,6),(2,1)&
       \{\{1,4\},\{2,4\},\{3,4\}\}\\
 E_6&(1,2),(1,4),(2,2),(2,4),(3,2),(4,1)&
       \{\{1,4\},\{2,3\},\{3,5\},\{4,5\},\{5,6\}\}.
\end{array}
\end{equation}
Write $\widetilde C$ for the simply laced Cartan matrix determined by
this graph, and $\varpi_c$ for its fundamental weights.  If an original
gauge root has color $i$ and value $q^k$, its unfolded color is
$(i,k\bmod 2d_i)$, with residues represented by $1,\ldots,2d_i$.
Its weighted-flag height is
\begin{equation}\label{cou:height}
 h=k+\sigma_i,\qquad
 \sigma=(0,-2)\text{ in }G_2,\quad
 \sigma=(0,0,0,0)\text{ in }F_4.
\end{equation}
The short-vertex shift in $G_2$ is the change of grading in
\cite[(3.1.72)]{VV25}.  A framing zero $q^K$ at vertex $i$ gives a
red strand of color $(i,K-2d_i\bmod 2d_i)$ at height
$K-2d_i+\sigma_i$.

Set $H=12$ in type $G_2$ and $H=18$ in type $F_4$.  For the fundamental
of color $a$ we use framing zeros
\begin{equation}\label{cou:paired-framing}
 q^{s-d_a},\qquad q^{s+d_a+H},
\end{equation}
both at color $a$.  Table~\ref{cou:red-table} gives $s$, the red
heights, and their unfolded colors.  In its last two columns are the
folded and unfolded gauge ranks.  The red weights are
$\lambda_L=\varpi_{c_L}$ and $\lambda_R=\varpi_{c_R}$.

\begin{table}[htbp]
\centering\small
\begin{tabular}{cccrrcc}
\toprule
Type, $a$&$s$&$(c_L,c_R)$&$\ell_L$&$\ell_R$&$\ba$&$\widetilde\ba$\\
\midrule
$G_2,1$&1&$(2,2)$&$-8$&10&$(4,2)$&$(1,2,1,2)$\\
$G_2,2$&0&$(4,4)$&$-5$&9&$(6,4)$&$(2,2,2,4)$\\
$F_4,1$&0&$(1,2)$&$-6$&16&$(4,6,4,2)$&$(2,2,3,3,4,2)$\\
$F_4,2$&0&$(3,4)$&$-6$&16&$(6,12,8,4)$&$(3,3,6,6,8,4)$\\
$F_4,3$&1&$(5,5)$&$-2$&18&$(8,16,12,6)$&$(4,4,8,8,12,6)$\\
$F_4,4$&0&$(6,6)$&$-3$&17&$(4,8,6,4)$&$(2,2,4,4,6,4)$\\
\bottomrule
\end{tabular}
\caption{The red pairs and gauge ranks.}\label{cou:red-table}
\end{table}

We give a complete rule for the highest loading.  Read the admissible
blocks in decreasing height.  A color $c=(i,r)$ occurs at height $h$
if $h-\sigma_i\equiv r\pmod{2d_i}$.  The allowed interval is
\begin{equation}\label{cou:red-interval}
 \ell_L<h\leq\ell_R.
\end{equation}
The right endpoint is included because a black strand at a red
height precedes the red in the weighted convention.  At a common
height read in decreasing order of $(\epsilon'_i,i,r)$, where
\[
 \epsilon'=(1,0)\text{ for }G_2,\qquad
 \epsilon'=(0,1,0,1)\text{ for }F_4.
\]
Initialize the Dynkin-label vector $v$ to the unit vector at $c_R$.
At each allowed block $c$ set
\begin{equation}\label{cou:greedy}
 A_{h,c}=\max(0,v_c),\qquad
 v_j\longleftarrow v_j-A_{h,c}\widetilde C_{jc}\quad\text{for all }j.
\end{equation}
A zero power is an actual step of this rule and is retained.

Tables~\ref{cou:greedy-g} and \ref{cou:greedy-f} give every allowed
power.  A dash means that the height lies outside
\eqref{cou:red-interval}.  For $D_4$ there is one allowed color at
each height, printed in its own column.  For $E_6$, at even heights
the entries are triples in the color order $(4,5,2)$ if
$h\equiv0\pmod4$ and $(3,5,1)$ if $h\equiv2\pmod4$; at odd heights
the single color is $6$.

\begin{table}[htbp]
\centering\small
\begin{tabular}{rrrr}
\toprule
$h$&color&$a=1$&$a=2$\\\midrule
$10$&$2$&$1$&$--$\\
$9$&$4$&$1$&$1$\\
$8$&$1$&$1$&$1$\\
$7$&$4$&$0$&$0$\\
$6$&$3$&$1$&$1$\\
$5$&$4$&$1$&$1$\\
$4$&$2$&$1$&$2$\\
$3$&$4$&$0$&$1$\\
$2$&$1$&$0$&$1$\\
$1$&$4$&$0$&$0$\\
$0$&$3$&$0$&$1$\\
$-1$&$4$&$0$&$1$\\
$-2$&$2$&$0$&$0$\\
$-3$&$4$&$0$&$0$\\
$-4$&$1$&$0$&$0$\\
$-5$&$4$&$0$&$--$\\
$-6$&$3$&$0$&$--$\\
$-7$&$4$&$0$&$--$\\
\bottomrule
\end{tabular}
\caption{The complete greedy powers for $G_2$.}\label{cou:greedy-g}
\end{table}

\begin{table}[htbp]
\centering\small
\begin{tabular}{rrrrr}
\toprule
$h$&$a=1$&$a=2$&$a=3$&$a=4$\\\midrule
$18$&$--$&$--$&$(0,1,0)$&$--$\\
$17$&$--$&$--$&$1$&$1$\\
$16$&$(0,0,1)$&$(1,1,0)$&$(1,1,0)$&$(0,1,0)$\\
$15$&$0$&$1$&$0$&$0$\\
$14$&$(1,1,0)$&$(1,1,1)$&$(2,1,1)$&$(1,0,0)$\\
$13$&$1$&$0$&$1$&$0$\\
$12$&$(1,1,0)$&$(2,1,1)$&$(2,2,2)$&$(1,1,1)$\\
$11$&$0$&$1$&$1$&$1$\\
$10$&$(1,0,1)$&$(2,2,1)$&$(3,2,1)$&$(1,1,1)$\\
$9$&$0$&$1$&$1$&$0$\\
$8$&$(1,1,1)$&$(2,1,1)$&$(3,2,1)$&$(2,1,0)$\\
$7$&$1$&$0$&$1$&$1$\\
$6$&$(1,1,0)$&$(2,1,1)$&$(2,1,2)$&$(1,1,1)$\\
$5$&$0$&$1$&$0$&$0$\\
$4$&$(1,0,0)$&$(1,1,1)$&$(2,1,1)$&$(1,0,1)$\\
$3$&$0$&$0$&$1$&$0$\\
$2$&$(0,0,1)$&$(1,0,0)$&$(1,1,0)$&$(1,1,0)$\\
$1$&$0$&$0$&$0$&$1$\\
$0$&$(0,0,0)$&$(0,0,0)$&$(0,0,0)$&$(0,0,0)$\\
$-1$&$0$&$0$&$0$&$0$\\
$-2$&$(0,0,0)$&$(0,0,0)$&$--$&$(0,0,0)$\\
$-3$&$0$&$0$&$--$&$--$\\
$-4$&$(0,0,0)$&$(0,0,0)$&$--$&$--$\\
$-5$&$0$&$0$&$--$&$--$\\
\bottomrule
\end{tabular}
\caption{The complete greedy powers for $F_4$.}\label{cou:greedy-f}
\end{table}

Let $e_c$ denote the standard basis vector at a vertex $c$, and let
$w_0$ be the longest Weyl group element of the unfolded root system.
All entries and their endpoints can be checked using only subtraction
of columns of the graph matrix \eqref{cou:unfolded-data}.  The endpoint
in each column is $-e_{c_L}$.  The number $n_{i,k}$ of gauge roots at
$q^k$ is, by definition, the entry $A_{k+\sigma_i,(i,k\bmod2d_i)}$.
Thus the two tables specify the gauge multisets completely, including
repeated roots.  Summing their entries by original or unfolded color
gives Table~\ref{cou:red-table}, and direct multiplication gives
\begin{equation}\label{cou:finite-zero-shift}
 C^{\mathsf T}\ba=2e_a,\qquad
 \widetilde C\widetilde\ba=e_{c_L}+e_{c_R}.
\end{equation}
In particular $\lambda_L=-w_0\lambda_R$, as is also read directly
from the $D_4$ and $E_6$ diagrams.

Here is a further check of the rational highest weight, stated in a
form that makes its verification finite.  Work in the free abelian
group on symbols $[i,k]$.  Start with
$[a,s-d_a]+[a,s+d_a+H]$.  For each of the $n_{j,k}$ roots subtract
$[j,k]+[j,k+2d_j]$ and add
\begin{equation}\label{cou:divisor-rule}
 \sum_{i:c_{ji}<0}\sum_{r=1}^{-c_{ji}}
              [i,k+d_jc_{ji}+2d_jr].
\end{equation}
For each of the six columns the result is
$[a,s-d_a]-[a,s+d_a]$.  This follows by inserting the full lists in
the two tables and canceling equal symbols.  Formula
\eqref{cou:divisor-rule} is exactly the divisor of
\eqref{cou:cartan-rational}.  Hence on the prescribed loading
\begin{equation}\label{cou:fundamental-divisor}
 \frac{\Psi_i(u)}{\Psi_i(\infty)}=
 \begin{cases}
 (1-q^{s-d_a}/u)/(1-q^{s+d_a}/u),&i=a,\\
 1,&i\ne a.
 \end{cases}
\end{equation}
The leading determinant and normalizing exponents are
listed in Table~\ref{cou:normalizer-table}.  If
$S_j=\sum_k k n_{j,k}$, they are respectively
$t_i=\frac12\sum_jc_{ji}S_j$ and
$\tau_i=d_i\delta_{ia}-t_i$.  In particular $\kappa_i=q^{\tau_i}$ gives
the ordinary highest constant $q_i^{\delta_{ia}}$.  Evaluating
\eqref{cou:fundamental-divisor} at zero gives
$\kappa_i\Psi_i(0)=q_i^{-\delta_{ia}}$ on this loading.  Since
$K_i$ is independent of the loading by
Lemma~\ref{cou:central-normalization}, this also verifies
$\kappa_i^2K_i=1$ for the displayed normalizers.

\begin{table}[htbp]
\centering\small
\begin{tabular}{ccc}
\toprule
Type, $a$&$(t_i)_i$&$(\tau_i)_i$\\\midrule
$G_2,1$&$(1,4)$&$(2,-4)$\\
$G_2,2$&$(-12,12)$&$(12,-11)$\\
$F_4,1$&$(9,-4,3,0)$&$(-7,4,-3,0)$\\
$F_4,2$&$(0,1,6,0)$&$(0,1,-6,0)$\\
$F_4,3$&$(0,-12,18,0)$&$(0,12,-17,0)$\\
$F_4,4$&$(0,-6,4,9)$&$(0,6,-4,-8)$\\
\bottomrule
\end{tabular}
\caption{Leading Cartan constants and ordinary normalization.}
\label{cou:normalizer-table}
\end{table}

\begin{proposition}\label{cou:fundamental-cups}
For each row of Table~\ref{cou:red-table}, the algebra $\cC_\C$ with
the indicated gauge ranks and paired framing has a finite-dimensional
simple module whose pullback by $\Phi_\C$ is the ordinary fundamental
module $L(Y_{a,q^s})$.
\end{proposition}

\begin{proof}
We first construct the actual module of the unfolded flag algebra.
Mirror the horizontal strand convention of \cite[\S2.1]{VV25} to
the convention of \cite[\S7.1]{Webster17}.  The ordered red pair is
then $(\lambda,\lambda^*)=(\lambda_R,\lambda_L)$.  Put
$\beta=\lambda+\lambda^*$ in the positive root lattice and take the
weight-zero tensor-product algebra $T=T^{\lambda,\lambda^*}_0$.
Let $J$ be generated by the idempotents with a black strand beyond
the second red strand in this convention.  The first-red violating
relation already excludes black strands on the other outside region.
Black diagrams between the red strands give a surjection
\begin{equation}\label{cou:cup-quotient}
 R^\lambda(\beta)\twoheadrightarrow T/J.
\end{equation}
To check surjectivity, cut a diagram at a height where a black strand
crosses a red.  The diagram factors through an excluded idempotent
and is zero in the quotient.  Every surviving diagram is therefore
a black cyclotomic diagram between the red strands.  Its first-red
relation is the defining cyclotomic relation, proving the claim.

The cyclotomic algebra in \eqref{cou:cup-quotient} is the lowest-weight
block of $V_\lambda$.  It has one simple isomorphism class because
that weight space is one dimensional.  The extremal divided-power
construction of \cite[Proposition~7.2]{Webster17} gives a simple
projective in this block.  A finite-dimensional algebra with one
simple class whose simple is projective is a matrix algebra.  On the
other hand, \cite[Lemma~7.3]{Webster17} supplies a nonzero cup simple
annihilated by every idempotent generating $J$.  Thus $T/J\ne0$, and
the surjection from the matrix algebra is an isomorphism.  This
identifies the actual between-red restriction of the cup.

Read this lowest-weight cyclotomic module from the right red in the
original convention.  At an extremal weight $v$, the maximal
nonzero lowering divided power of color $c$ is
$\max(0,\langle v,\alpha_c^\vee\rangle)$.  If the pairing is
negative the extremal vector is already at the bottom of that
string; if it is zero both directions vanish.  A maximal nonzero
step gives the next extremal simple projective.  The rule
\eqref{cou:greedy} is exactly this construction, and the displayed
endpoint is the lowest weight.  The corresponding loading is
therefore supported and nonzero.  Any lexicographically larger
loading has a first block where it takes more than the maximal
nonzero power following the same prefix.  That prefix is zero, so
the whole loading is zero.  This proves that the displayed loading
is the unique greatest supported loading.  The inclusion of every
zero block in the tables makes this comparison over all admissible
loadings.

We now pass from complete words to weighted flags.  A block of
multiplicity $A$ uses $F_c^A$ in the complete-word description and
$F_c^{(A)}$ in the extremal construction.  The complete-to-weighted
flag map has fiber $\Flag(\C^A)$; for several blocks these fibers
multiply.  The flag inside a common-height block is unrestricted:
the strict lowering condition and the prescribed even-before-odd
order force a same-height arrow to be zero before choosing these
additional flags.  The matrix factor in
\cite[(2.1.35)--(2.1.42)]{VV25} consequently has size
$\prod A!$, equal to the dimension of the cohomology of these
fibers.  The multiplicity of the divided-power extremal projective
is one, so the greatest \emph{weighted} idempotent space is one
dimensional.

It remains to check that the positive currents annihilate this
space.  Let $\Gamma_i\subset\Spec B_\C\times\Spec B_\C$ be the
finite closed correspondence replacing one root of color $i$ by
$q_i^2$ times that root, with the compatible change of determinant.
It is the symmetric quotient of the finite union of the labeled
shift graphs.  In the difference realization, a positive current
is a sum of terms $f_t(w)\mathbf D_{i,t}^{-1}$.  Every element of
the ideal of $\Gamma_i$, acting by the two coefficient-ring
actions, annihilates these terms.  By
Lemma~\ref{cou:fixed-q-faithfulness}, it annihilates the current in
the actual algebra.  If two generalized gauge characters are
outside $\Gamma_i$, one element of that ideal has a nonzero value
there.  On their finite generalized block it is an invertible
scalar plus a nilpotent.  The corresponding current block is
therefore zero.  This proves the support assertion also when roots
within a block coincide.

The relation
$\mathbf D_{i,t}^{-1}w_{i,t}=q_i^{-2}w_{i,t}\mathbf D_{i,t}^{-1}$
shows that a positive current takes a source root to $q_i^2$ times
that root.  It strictly raises its height and hence the loading.
It thus kills the greatest loading.  There are only finitely many
allowed heights and a fixed total number of black strands.  The
cup restriction has finite complete-word spaces, so its nonzero
weighted even corner is finite dimensional and simple.  The
completed equivalence of \cite[Theorem~4.3.1]{VV25} gives an actual
finite-dimensional $\cC_\C$-module from this corner.

Finally, the ordinary map is surjective by
Proposition~\ref{cou:surjection}.  Its pullback is therefore simple,
and the greatest line is an ordinary highest $\ell$-weight line.
Equations \eqref{cou:fundamental-divisor} and
\eqref{cou:central-product}, with Table~\ref{cou:normalizer-table},
identify that $\ell$-weight with $Y_{a,q^s}$.  Since the simple
module is generated by this line, the ordinary Cartan--root
relations make its leading Cartan operators semisimple with
type-$1$ eigenvalues.  The classification by highest $\ell$-weight
proves the assertion.
\end{proof}

Every gauge rank in Table~\ref{cou:red-table} is even.  If all gauge
roots and framing coordinates of one row are multiplied by
$b\in\C^\times$, the chosen determinant is multiplied by
$b^{a_i/2}$, and
\begin{equation}\label{cou:spectral-normalization}
 K_i(b)=b^{2\delta_{ia}}K_i(1),\qquad
 \kappa_i(b)=b^{-\delta_{ia}}\kappa_i(1).
\end{equation}
These are Laurent functions of $b$.  We thus obtain the ordinary
fundamental $L(Y_{a,bq^s})$ for every $b$, with a Laurent-algebraic
ordinary current map.  Products of paired framings have the product
normalization, since the constants $K_i$ multiply.

\subsection{Separated parameters and completed convolution}

Fix finitely many of the preceding fundamental factors, indexed by
$r$, with spectral scalars $b_r$, and first suppose that
\begin{equation}\label{cou:separated}
 b_r/b_s\notin q^\Z\quad(r\ne s).
\end{equation}
Use the sums of their gauge ranks and the unions of their framing
multisets to define the total Coulomb algebra.  Let $\Omega_r$ be the
collection of all integral loadings with the ranks and residue
refinements of the $r$th factor.  Let $\Omega$ be their product,
regarded as loadings of the total multiset.  Completions below are
two-sided completions at these gauge characters, retaining every
source and target idempotent.

\begin{lemma}\label{cou:determinant-branch}
At these completed loadings, the determinant cover of the total
gauge group admits a compatible branch on which
$D_i=\prod_rD_{i,r}$.  The branch is compatible with every current
correspondence between the loadings in $\Omega$.
\end{lemma}

\begin{proof}
The total monic characteristic polynomial factors uniquely into
its prescribed spectral clusters in the completed coefficient
ring.  This is Hensel factorization: the resultants between distinct
cluster polynomials have nonzero constant terms by
\eqref{cou:separated}.  Each cluster determinant is a unit.  A
square root of a unit in this complete local complex algebra is
uniquely determined by its nonzero constant term, since the
derivative of $X^2-d$ at either root is nonzero.

At a loading $\gamma$, prescribe the constant term
\begin{equation}\label{cou:det-branch-value}
 D_{i,r}(\gamma)=b_r^{a_{i,r}/2}
             q^{\frac12\sum_kkn_{i,r,k}}.
\end{equation}
The even ranks make the power of $b_r$ integral.  Fix the
half-character convention for the powers of $q$ once and for all.
A current replacing $k$ by $k+2d_i$, or by $k-2d_i$, multiplies
this value by $q_i$, or by $q_i^{-1}$, respectively.  These are the
determinant commutation factors of the minuscule correspondence.
A closed path returning to the same multiset returns to the same
sum of exponents, so these choices are consistent on every path.
Uniqueness of the formal square roots extends the consistency to
the completed rings on both sides of a correspondence.  Prescribing
the global constant as their product gives $D_i=\prod_rD_{i,r}$.
\end{proof}

\begin{proposition}\label{cou:separated-factorization}
With the branch of Lemma~\ref{cou:determinant-branch}, there is an
isomorphism of completed convolution algebras
\begin{equation}\label{cou:completed-product}
 \widehat{\cC}_{\Omega}\ \simeq
 \widehat{\bigotimes_r\widehat{\cC}_{r,\Omega_r}}.
\end{equation}
It respects every source and target idempotent and multiplies the
Cartan rational functions of the factors.
\end{proposition}

\begin{proof}
We give the fixed-point and multiplication argument, including its
dependence on the spectral scalars.  At any finite lattice bound,
a coefficient of a gauge, matter, or framing map from cluster $s$
to cluster $r$ can be fixed by the defining semisimple operator only
if
\[
 (b_r/b_s)q^n=1
\]
for an integer $n$.  The symmetrizers change the integer $n$ and do
not change the scalar ratio; this follows directly from the
ramified loop characters in \cite[(3.1.60), (3.1.64)]{VV25}.
Thus there are no fixed maps between different clusters.

The splitting of fixed lattices holds as a statement of schemes.
Between two lattice bounds, the loop quotient is finite dimensional.
The semisimple operator has disjoint eigenvalue sets for distinct
clusters.  Polynomial spectral projectors decompose every invariant
submodule functorially, over any test algebra, into its cluster
summands.  Consequently the fixed lattice functor is the product of
the factor functors.  The decompositions commute with increasing
the lattice bounds.  The fixed gauge centralizer, matter bundles,
framing bundles, and classical BFN correspondences are therefore
products.  In particular for every pair $\gamma,\eta\in\Omega$,
\begin{equation}\label{cou:fixed-products}
 \cZ_{\gamma,\eta}=\prod_r\cZ_{\gamma_r,\eta_r}.
\end{equation}
The threefold correspondence and all projections defining
convolution split in the same way.

Within one cluster the scalar $b_r$ cancels from every fixed
equation, leaving its integral weighted-flag model.  Simultaneous
scalar multiplication of the gauge and framing spaces acts trivially
on the internal Hom spaces and flags.  Translating that scalar
character from $b_r$ to $1$ in the completed equivariant coefficient
ring changes only the scalar parameter.  Formula
\eqref{cou:det-branch-value} supplies its lift to the determinant
cover.  Thus the reduction to the integral model is compatible with
equivariant $K$-theory, in addition to the underlying fixed varieties.

Apply semisimple equivariant localization to each finite
approximation.  Cross-cluster normal characters give Euler factors
$1-(b_r/b_s)q^n$, whose constant terms are nonzero, hence units in
the chosen completion.  Localization only requires the given
semisimple element and its diagonalizable Zariski closure.  It
places no restriction on the absolute value of $q$ and requires
no integral cocharacter containing every $b_r$.

To check multiplication, write the restriction as
$r_\Omega=\sigma^*\iota_*^{-1}$, as in
\cite[\S4.3.2]{VV25}.  If $r_2$ is torus localization of the
original correspondence and $r_1$ is torus localization of its
fixed correspondence, the Cartesian base-change diagram for the
threefold correspondence gives
\[
 r_2=r_1r_\Omega.
\]
The two torus localizations are algebra homomorphisms, and $r_1$
is injective.  The last assertion follows on every pair of blocks
from the product of the factor embeddings.  It follows that
$r_\Omega$ is multiplicative.  The moving cross-cluster normal
factors have already entered $\iota_*^{-1}$ in this identity.
Thus their Euler factors are part of the multiplication comparison.

Finally apply the modified equivariant Riemann--Roch map and
Borel--Moore Kunneth to \eqref{cou:fixed-products}.  This gives
\eqref{cou:completed-product} on each pair of source and target
blocks, compatibly with convolution.  The determinant comparison
holds on the branch already chosen.  Equivalently, the product
of the factor determinant covers maps to the restricted total
cover by a finite central isogeny acting trivially on the varieties;
on the chosen branch it gives the same rational equivariant
cohomology.  The Hensel factorization of the characteristic
polynomial is unique at the unordered total multiset, so it
introduces no permutation factor.  Total $W_i$, $D_i$, and $Z_i$
are the products of their factor counterparts.  Formula
\eqref{cou:cartan-rational} proves the asserted compatibility of
Cartan currents.
\end{proof}

\subsection{Descent through collisions}

\begin{theorem}\label{cou:ordinary-action}
Let $m=\prod_{r=1}^NY_{a_r,A_r}$ be any dominant monomial in type
$G_2$ or $F_4$, with arbitrary $A_r\in\C^\times$.  Associate to its
$r$th factor the spectral translate of
Table~\ref{cou:red-table} with $b_r=A_rq^{-s_r}$.  For the total
gauge ranks and paired framing obtained by their sums and unions,
respectively, there is a surjective ordinary map
\[
 \Phi_m:\U\twoheadrightarrow\cC_m.
\]
Every ordered ordinary tensor product
$\bigotimes_rL(Y_{a_r,A_r})$ factors through this map.  Its simple
subquotients factor through the same map.  These assertions also
hold over the Laurent deformation family separating the factors
and over its completion at any prescribed collision.
\end{theorem}

\begin{proof}
First assume \eqref{cou:separated}.  The external tensor product
of the finite cup modules is a finite-dimensional smooth module
of the right side of \eqref{cou:completed-product}.  It has finite
idempotent support.  Transporting it by that isomorphism and
restricting to the uncompleted algebra gives an actual
$\cC_m$-module $N$.  This restriction is well defined: a word of
finitely many minuscule generators has finitely many root-shift
possibilities from any fixed source block.  The finite support
therefore uses only finite sums.

The Cartan compatibility in
Proposition~\ref{cou:separated-factorization} gives
\begin{equation}\label{cou:separated-character}
 \chq N=\prod_r\chq L(Y_{a_r,A_r}).
\end{equation}
The leading normalized Cartan operators are tensor products of
the leading operators of the factors.  They are semisimple with
type-$1$ eigenvalues.  Fundamental modules are special, and all
variables of their characters lie in their spectral $q$-orbit
\cite[Theorem~2.11 and Corollary~2.13]{Her06}.  Distinct factors
in the present situation have disjoint such orbits.  Hence the
right side of \eqref{cou:separated-character} has just one
dominant monomial, namely $m$, with coefficient $1$: a negative
power in any factor cannot be canceled by another factor.
Every finite-dimensional simple composition factor contributes
its dominant highest monomial with positive multiplicity.  Thus
both $N$ and the ordinary tensor product have a single composition
factor, of multiplicity one.  They are isomorphic to the ordinary
simple of highest monomial $m$.  The actual ordinary tensor
therefore factors through $\Phi_m$ at separated parameters.

For arbitrary initial parameters, put
\[
 S=\C[u,u^{-1}],\qquad b_r(u)=b_r(1)u^{c_r},
\]
where the integers $c_r$ are pairwise distinct.  The spectral
automorphism on each fundamental representation and the ordinary
coproduct give an $S$-free tensor module $M_S$ whose action matrices
are Laurent polynomials in $u$.  The framing coordinates and the
normalization units \eqref{cou:spectral-normalization} are likewise
Laurent monomials.  Proposition~\ref{cou:surjection} gives a
surjection $\Phi_S:\U\otimes S\twoheadrightarrow\cC_S$ over this
same ring.

For each pair $r\ne s$, the failure of separation at a complex
point $u_0$ means
$u_0^{c_r-c_s}\in b_s(1)b_r(1)^{-1}q^\Z$.  The union of the
excluded sets is countable.  Its complement is infinite and
Zariski dense in $\C^\times$.  If $x\in\ker\Phi_S$, its
specialization annihilates $M_{u_0}$ at every point of this
complement by the separated construction.  Every entry of its
action matrix on $M_S$ is a Laurent polynomial, and vanishes at
all these points.  It is identically zero.  We have proved
\[
 (\ker\Phi_S)M_S=0.
\]
Surjectivity now gives an action of $\cC_S$ on the actual tensor
family.  Specializing at $u=1$ proves the assertion at the
prescribed collision, in any order of the factors.  Completing
at $u=1$, or making the formal substitution $u=\exp(a)$, gives
the corresponding action over $\C[[a]]$.

More generally, if a surjective algebra map $U_R\to A_R$ acts
on an $R$-torsion-free lattice whose generic action factors
through $A_R\otimes_R\operatorname{Frac}(R)$, its kernel kills
the lattice.  This proves the same assertion for every ordinary
stable lattice in the tensor family used later.  Right exactness
under specialization ensures that the special fiber is an
$A_R/\mathfrak mA_R$-module.  Finally, a $U$-stable subspace of a
module factoring through a surjection $U\to A$ is $A$-stable.
The assertion for all simple subquotients follows.
\end{proof}

The generalized gauge support of a subquotient is contained in
that of the tensor module.  Indeed, the finite gauge spectral
projectors are polynomials in the coefficient ring, obtained
by Chinese remainders; their functors are exact.  Moreover the
highest gauge multiset is determined by the ordinary Cartan
series and the fixed framing: the logarithmic inversion
\eqref{cou:quantum-cartan} recovers all its power sums and hence
its characteristic polynomials.  Thus the highest ordinary
line and its gauge loading agree in every construction above.

\section{The weighted-flag algebra and geometric currents}
\label{cou:flag-section}

We record the precise comparison needed for the filtrations in
the subsequent sections.  In particular, the comparison keeps
the ordinary quantum loop action just constructed.

\subsection{The fixed weighted-flag model}

Fix a gauge loading and a framing semisimple character.
For a single spectral component, the unfolded vector space is
graded by the colors in \eqref{cou:unfolded-data}.  Orient each
edge from parity $0$ to parity $1$, using the lifted parities
$\epsilon'$ specified before \eqref{cou:greedy}.  A weighted
flag is an increasing sequence $F^{\leq h}$ of unfolded-graded
subspaces, equal to $0$ for $h\ll0$ and to the whole gauge
space for $h\gg0$.  A flag of type $\gamma$ has graded block
dimensions prescribed by its root multiplicities, at the
heights \eqref{cou:height}.  Let $\Flag_\gamma$ be its partial
flag variety.  Write the unfolded representation space as
\[
 N=\bigoplus_c\Hom(\widetilde W_c,\widetilde V_c)
    \oplus\bigoplus_{c\to d}\Hom(\widetilde V_c,\widetilde V_d),
\]
and denote a point by $(A,B)$.  Give the framing space its
red-height grading.  Then set
\[
 X_\gamma=\{(A,B,F):F\in\Flag_\gamma,
 B(F^{\leq h})\subset F^{<h},\quad
 A(\widetilde W^h)\subset F^{\leq h}\text{ for every }h\}.
\]
These are linear equations in the representation coordinates
at each flag.  Thus $X_\gamma$ is a vector
bundle over $\Flag_\gamma$.  Its proper projection to $N$ is
denoted by $\pi_\gamma$.  The unfolded gauge group is
$\underline G=\prod_c\GL(\widetilde V_c)$.

The weighted-flag convolution algebra is
\begin{equation}\label{cou:flag-algebra}
 \cA^{\mathrm{fl}}=
 \bigoplus_{\gamma,\eta}
 \Ext^*_{\underline G}\left(
  (\pi_\eta)_*\C_{X_\eta}[\dim X_\eta],
  (\pi_\gamma)_*\C_{X_\gamma}[\dim X_\gamma]\right).
\end{equation}
Its product is convolution on
$X_\gamma\times_NX_\eta$.  The indicated shifts are the
selfdual normalization and fix its grading.  We use the locally
unital algebra, with idempotents $e_\gamma$ for all loadings.
Source coefficient classes act on the right of a correspondence,
and target coefficient classes act on the left.

\begin{proposition}\label{cou:flag-comparison}
The semisimple completion of the Coulomb algebra acting in
Theorem~\ref{cou:ordinary-action} is identified with the completion
of \eqref{cou:flag-algebra} by localization and the modified
Riemann--Roch map.  Write this isomorphism as $\Theta$.  It
preserves source and target idempotents and the completed gauge
coefficient rings.  For arbitrary finitely many multiplicative
spectral components the fixed spaces are products of these unfolded models, and their
convolution algebra is the corresponding completed tensor product.
\end{proposition}

\begin{proof}
For the selected integral component this is the construction of
\cite[Theorem~4.3.1 and Lemmas~4.3.2--4.3.5]{VV25}.  The fixed
loop stabilizer is identified, by evaluation at $1$, with the
parabolic preserving the weighted flag.  Its action on a graded
quotient factors through the corresponding Levi factor.  The
fixed matter equations give exactly the vector bundles
$X_\gamma$ and their common-flag correspondences.  The modified
Chern character of \cite[Lemma~2.1.2]{VV25} is multiplicative for
their convolution.

We check the component extension.  The parity vectors for
gauge roots are $(0,1)$ in type $G_2$ and $(0,0,0,1)$ in type
$F_4$.  Components can be represented by roots $bq^k$ with
these parities, with $b$ taken modulo $q^{2\Z}$.  A one-step
spectral translate gives the other parity class.  Same-color
gauge shifts are $2d_i$; on an edge the parity of the fixed
matter shift is the difference of the displayed parities.
Consequently gauge, matter, and framing fixed equations do
not connect distinct components.  Their lattice functors and
all their twofold and threefold correspondences split, by
the spectral-projector argument of
Proposition~\ref{cou:separated-factorization}.  Inside a component
the common scalar cancels from the equations.  The moving normal
Euler factors between components have nonzero constants and
are units in its semisimple completion.  The same
$r_2=r_1r_\Omega$ argument proves multiplicativity, followed by
modified Riemann--Roch and Borel--Moore Kunneth.  The determinant
branches are selected by Lemma~\ref{cou:determinant-branch}.
This proves the asserted extension, including the two parity
classes inside one $q^\Z$-orbit.
\end{proof}

\subsection{The minuscule current object}

Consider a negative current that takes one color-$i$ root from
$q^k$ to $q^{l}$, where $l=k-2d_i$.  A scalar spectral translate
does not change the following calculation.  Denote this transition
by $v$.  If the source block dimensions at $k,l$ are $\alpha,\beta$,
the target dimensions are $\alpha-1,\beta+1$.  Since the same
unfolded color has heights congruent modulo $2d_i$, there is no
intermediate step of that color between them.  The common flag
refinement $S_v$ is the projective bundle of lines in the source
block of dimension $\alpha$.  The moved line determines the target
flag uniquely.  Equivalently $S_v$ is the projective bundle of
one-dimensional quotients of the target block of dimension
$\beta+1$.

Let $Z_v$ consist of the common-refinement flags together with
the representation maps satisfying both weighted flag conditions.
It is a constant-rank vector bundle over $S_v$.  To see the
constant rank, the unfolded group is transitive on the
common-refinement flag base, and the two flag conditions are
equivariant linear equations in the representation coordinates.
Thus $Z_v$ is smooth.  It has a closed incidence embedding
\[
 j:Z_v\hookrightarrow X_t\times_NX_s.
\]
Write $L$ for the moved line and $x=c_1(L)$ in the logarithmic
line convention for the current.  The source and target of this
embedding are labeled $s$ and $t$ throughout.

The geometric negative and positive current objects are the
tautological line sheaves on the classical minuscule spaces:
\begin{equation}\label{cou:classical-current}
 \mathscr Q_i=(V_{i,\cO}/L')\otimes\cO_{\cR_{\omega_{i,1}}},
 \qquad
 \mathscr S_i=(L'/V_{i,\cO})\otimes\cO_{\cR_{\omega_{i,-1}}}.
\end{equation}
Here $V_{i,\cO}=V_i[[z_i]]$, $L'$ is the gauge lattice, and
$\cR_{\omega_{i,\pm1}}$ is the classical BFN stratum for the positive
or negative first minuscule coweight at $i$.
Integer current modes are powers of these line
sheaves, with fixed determinant and numerical factors.  This
description determines the localization of the actual current
class.

\begin{lemma}\label{cou:minuscule-localization}
On a finite approximation, let $B$ be the minuscule Grassmannian,
$T_B$ its ambient matter vector bundle, and $E\subset T_B$ the
classical minuscule BFN subbundle.  For a fixed component
$B_0\subset B$, write
\[
 T_B|_{B_0}=T_0\oplus T_m,\qquad E|_{B_0}=E_0\oplus E_m
\]
for fixed and moving parts.  The localized class is
\begin{equation}\label{cou:koszul-unit}
 r_\gamma[\cO_E]=[\cO_{E_0}]
 \frac{\lambda_{-1}((T_m/E_m)^*)}
      {\lambda_{-1}(N_{B_0/B}^*)}.
\end{equation}
The quotient in this formula is a regular unit at the selected
semisimple completion.
\end{lemma}

\begin{proof}
The minuscule Grassmannian $B$ is a projective space.  The
classical minuscule space is the constant-rank subbundle
$E_b=N_\cO\cap gN_\cO$, as in
\cite[(3.1.23)--(3.1.26)]{VV25}.  Proper naturality of localization
allows the current class, supported on this minuscule piece, to
be localized on $B$.  The inclusion $B_0\hookrightarrow B$ is
a regular immersion of smooth varieties.  Since $E$ is flat
over $B$, this first restriction gives $\cO_{E|_{B_0}}$, and
inverting fixed pushforward contributes exactly
$\lambda_{-1}(N_{B_0/B}^*)^{-1}$.

Put $Q=T_B|_{B_0}/E|_{B_0}$.  Its fixed and moving parts are
$Q_0=T_0/E_0$ and $Q_m=T_m/E_m$.  Resolve $\cO_E$, as a sheaf
on the ambient bundle, by the Koszul complex of $Q^*$ and pull
it back to $T_0$.  The fixed equations $Q_0^*$ remain a regular
sequence and resolve $\cO_{E_0}$.  The moving equations become
zero and contribute the exterior class $\lambda_{-1}(Q_m^*)$.
This is also the derived pullback in the BFN construction:
associativity of derived tensor products gives
\[
 \cO_E\mathbin{\otimes^L_{\cO_{\mathfrak R}}}
 \left(\cO_{\mathfrak R}\mathbin{\otimes^L_{\cO_{T_B}}}\cO_{T_0}\right)
 \simeq\cO_E\mathbin{\otimes^L_{\cO_{T_B}}}\cO_{T_0}.
\]
Here $\mathfrak R$ is the derived BFN space, and the original
class is the classical sheaf \eqref{cou:classical-current}.
Combining the base and ambient restrictions proves
\eqref{cou:koszul-unit}.  Every Euler factor in the displayed
quotient is moving, hence has the form $1-\xi\exp(\eta)$ with
$\xi\ne1$.  In the integral component $\xi=q^b$, $b\ne0$;
between different components its constant is likewise nontrivial.
It is therefore a unit.  Both normal quotients are finite
dimensional.  Enlarging the finite approximation adds common
free directions, which cancel from the formula.
\end{proof}

\begin{proposition}\label{cou:current-object}
Let $F_{v,r}$ denote the local negative current tested against
the $r$th power of the logarithm of its spectral variable,
centered at the source root, for $r\geq0$.  Under $\Theta$ one has
\begin{equation}\label{cou:current-object-formula}
 \Theta(F_{v,r})=j_*\bigl([Z_v]x^rU_v\bigr),\qquad
 U_v\in\widehat H^*_{\underline G}(Z_v)^\times.
\end{equation}
The unit $U_v$ has nonzero constant term and is independent of
$r$.  The analogous statement holds for the reversed positive
correspondence with its own unit.
\end{proposition}

\begin{proof}
The fixed classical subbundle $E_0$ in
Lemma~\ref{cou:minuscule-localization}, induced from its flag
parabolic to $\underline G$, is exactly the common-refinement
incidence $Z_v$.  A line in the source block is the entire
minuscule datum, so there is no additional component or covering
degree in this identification.  Its representation equations
are the two flag conditions defining $Z_v$.

Use the symmetric modified Riemann--Roch convention. Let $\chi_v$
denote the product of the restricted line and determinant characters.
The image of the localized current is the incidence fundamental class
times
\begin{equation}\label{cou:rr-unit}
 U_v=\chi_v\,
 \CH\!\left(
 \frac{\lambda_{-1}((T_m/E_m)^*)}{\lambda_{-1}(N_{B_0/B}^*)}
 \right)
 \frac{\operatorname{Td}(TZ_v)}
 {\sqrt{p_s^*\operatorname{Td}(TX_s)\,
             p_t^*\operatorname{Td}(TX_t)}}.
\end{equation}
Here $p_s:Z_v\to X_s$ and $p_t:Z_v\to X_t$ are the source and
target projections, and $\CH$ is the completed equivariant Chern character; the
fixed mode-independent powers of the tautological line are
included in the first factor.  This convention sends the
diagonal structure sheaf to the diagonal fundamental class
and preserves convolution.  Every Todd series has constant
term $1$, the line and determinant characters have nonzero
constant terms, and the middle factor is a unit by
Lemma~\ref{cou:minuscule-localization}.

An integer current mode differs from the next by multiplication
by the character of the single tautological line.  At a fixed
spectral center, change from its Laurent character to its
logarithm.  Center the line character first, including the
$q_i^2$ shift between the displayed current variable and that
character.  The local test of order $r$ is then precisely
$x^r$.  All remaining factors in \eqref{cou:rr-unit} are
independent of $r$.  Riemann--Roch for the closed incidence
embedding proves \eqref{cou:current-object-formula}.
\end{proof}

\subsection{Source coefficients and the exact inverse}

\begin{lemma}\label{cou:source-cohomology}
The equivariant cohomology of $Z_v$ is generated by the Chern
classes of the source flag blocks and the moved-line class $x$.
Every homogeneous class has a polynomial lift in these
generators of the same nonnegative degree.
\end{lemma}

\begin{proof}
The map $Z_v\to S_v$ is a vector bundle, and
$S_v\to\Flag_s$ is the projective bundle of lines in the
source block of rank $\alpha$.  Hence
\begin{equation}\label{cou:projective-cohomology}
 H^*_{\underline G}(Z_v)=H^*_{\underline G}(S_v)
 =H^*_{\underline G}(\Flag_s)[x]\big/
       (x^\alpha-c_1x^{\alpha-1}+\cdots+(-1)^\alpha c_\alpha),
\end{equation}
with the chosen line convention.  The flag coefficient ring
is the polynomial ring in the Chern classes of its graded
blocks.  The target block classes add no generators: removing
$L$ from the source block divides its total Chern polynomial
by $1+x$, and adding $L$ to the lower block multiplies that
block polynomial by $1+x$.  The relation in
\eqref{cou:projective-cohomology} is homogeneous.  This proves
the lifting assertion degree by degree.
\end{proof}

These source coefficients are actual completed gauge Cartan
elements.  The characteristic polynomial of $W_i$ has a unique
Hensel factorization by distinct spectral centers; its
denominators are differences of distinct constants.  This
separates blocks, including blocks of repeated roots, without
separating the individual roots of a repeated block.  Writing
the roots of a block as $q^k\exp(x_b)$ identifies their symmetric
logarithmic polynomials with its Chern classes.  This is the
coefficient-ring part of $\Theta$.

The polynomial $W_i(u)$ satisfies the following identity in the
Coulomb algebra:
\begin{equation}\label{cou:cartan-current-identity}
 W_i(u)F_j(z)=F_j(z)W_i(u)
 \left(\frac{1-q_i^{-2}z/u}{1-z/u}\right)^{\delta_{ij}}.
\end{equation}
Indeed, a negative minuscule shift acts on one root, and its
delta function is evaluated after that shift.  The relation
$\mathbf D_{i,t}w_{i,t}=q_i^2w_{i,t}\mathbf D_{i,t}$ together
with that delta function gives the displayed ratio.  Other
colors commute.  Lemma~\ref{cou:fixed-q-faithfulness} transfers
the identity from difference operators to $\cC_S$ itself.

Iterating \eqref{cou:cartan-current-identity} through a negative
word gives, for the logarithmic power sum of order $r>0$ in
the block $(i,k)$, the operator
\begin{equation}\label{cou:chern-operator}
 P^{\mathrm{top}}_{i,k,r}
 +\sum_{\text{lowering variables at }(i,k+2d_i)}y_b^r
 -\sum_{\text{lowering variables at }(i,k)}y_b^r.
\end{equation}
This is an identity on words and survives every module quotient.
At the undeformed highest line $P^{\mathrm{top}}_{i,k,r}=0$.
In a formal spectral deformation $q^k\exp(c_ba)$ its value is
$\sum_b(c_ba)^r$.  Thus every expression in
\eqref{cou:chern-operator} has polynomial degree $r$ when
$\deg a=\deg y_b=1$.  Newton identities give the actual
degree-$r$ action of the block Chern class $c_r$.  This
calculation includes the nilpotent Cartan part of the action.

\begin{corollary}\label{cou:sided-inverse}
Write, degree by degree in the incidence coefficient ring,
\[
 U_v^{-1}=\sum_{a\geq0}x^ap_a,
\]
where $p_a$ are completed source Cartan classes, using
Lemma~\ref{cou:source-cohomology}.  Then the homogeneous
decorated minuscule class satisfies the exact identity
\begin{equation}\label{cou:inverse-formula}
 j_*([Z_v]x^r)=\sum_{a\geq0}\Theta(F_{v,r+a})p_a.
\end{equation}
The source coefficients occur on the right.  On every
finite-dimensional smooth module the identity is a finite
operator identity.
\end{corollary}

\begin{proof}
The projection formula for $j$ identifies right multiplication
by $p_a$ with pullback from the source.  Multiplying
\eqref{cou:current-object-formula} by these coefficients and
summing cancels $U_v$, giving \eqref{cou:inverse-formula}.
Every homogeneous term of $U_v^{-1}$ has a polynomial lift
of nonnegative degree by \eqref{cou:projective-cohomology}.
The local Cartan maximal ideals act nilpotently on a
finite-dimensional smooth module.  Its current action also
depends on finitely many jets at finitely many reduced spectral
centers.  Chinese remainders realize each needed logarithmic
jet by a finite linear combination of Laurent current modes.
Consequently both the coefficient expansions and the local
mode expansions truncate on the module.  The same argument
for the reversed correspondence uses its own source ring.
\end{proof}

\subsection{The geometric degree}

Let $n_j(h)$ be the number of current gauge roots at $q^h$
of color $j$, and let $w_i(h)$ be the multiplicity of the
underlying flavor coordinate $q^h$.  Thus a framing zero at
$q^K$ contributes to $w_i(K-2d_i)$.  For the move at
$(i,k)$ define
\begin{equation}\label{cou:current-pole}
 p'_{i,k}=n_i(k)+n_i(k-2d_i)-w_i(k-2d_i)
 -\sum_{j:c_{ji}<0}\sum_{b=1}^{-c_{ji}}
                   n_j(k-b_{ij}-2d_jb).
\end{equation}
It is the pole order at this center of the current Cartan
rational function \eqref{cou:cartan-rational}; cancellation
of numerator and denominator zeros is included in this signed
order.

\begin{proposition}\label{cou:degree}
In the selfdual Ext grading of \eqref{cou:flag-algebra},
\begin{equation}\label{cou:degree-formula}
 \deg j_*([Z_v]x^r)=2r+1-p'_{i,k}.
\end{equation}
The reversed minuscule correspondence has the same degree.
\end{proposition}

\begin{proof}
The degree of its undecorated fundamental class is
\[
 \dim X_s+\dim X_t-2\dim Z_v.
\]
The source-to-target change in flag dimension is
$\alpha-\beta-1$, while the common-refinement flag has
dimension $\dim\Flag_s+\alpha-1$.  The flag contribution to
the displayed difference is therefore $1-\alpha-\beta$.

We compute the remaining contribution from the linear matter
conditions.  In a basis compatible with the common refinement,
each coordinate of a representation map is independently
allowed or required to be zero.  The contribution of a
coordinate to
$\rk X_s+\rk X_t-2\rk Z_v$, after removing the flag
dimensions, is $1$ precisely when it is allowed in exactly
one flag condition.  It is zero when it is allowed in both
or forbidden in both.  A framing coordinate changes its
condition exactly at height $k-2d_i$, giving
$w_i(k-2d_i)$.  A coordinate between color $j$ and the moved
line changes its condition precisely when its source root
has exponent
\[
 k-b_{ij}-2d_jb,\qquad 1\leq b\leq-c_{ji}.
\]
These are the $-c_{ji}$ ramified matter shifts in
\eqref{cou:cartan-rational}.  Summing the changed coordinates
therefore gives
\[
 w_i(k-2d_i)+\sum_{j:c_{ji}<0}\sum_{b=1}^{-c_{ji}}
                              n_j(k-b_{ij}-2d_jb).
\]
Explicitly, the changed neighboring exponents are as follows,
in the numbering of Section~\ref{cou:section}:
\begin{center}
\begin{tabular}{ccl}
\toprule
Type & Moved color $i$ & Neighbor $j$ and exponents\\
\midrule
$G_2$ & $1$ & $j=2:\ k+1,\ k-1,\ k-3$\\
$G_2$ & $2$ & $j=1:\ k-3$\\
$F_4$ & $1$ & $j=2:\ k-2$\\
$F_4$ & $2$ & $j=1:\ k-2;\quad j=3:\ k,\ k-2$\\
$F_4$ & $3$ & $j=2:\ k-2;\quad j=4:\ k-1$\\
$F_4$ & $4$ & $j=3:\ k-1$\\
\bottomrule
\end{tabular}
\end{center}
For example, at the long $G_2$ node the three values are
$k-(-3)-2b$ for $b=1,2,3$. The height correction
$\sigma_2=-2$ converts the unfolded flag inequalities to these
original root exponents. The other rows follow by the same substitution
in $k-b_{ij}-2d_jb$.

Adding the flag contribution, with
$\alpha=n_i(k)$ and $\beta=n_i(k-2d_i)$, gives
$1-p'_{i,k}$.  A power $x^r$ has cohomological degree $2r$.
This proves \eqref{cou:degree-formula}.  Upon reversing the
step the two multiplicities become $\alpha-1,\beta+1$;
their sum is unchanged, as are the framing and neighboring
terms.  The same formula therefore applies to the reverse
correspondence.
\end{proof}

The remaining input for a filtered action is the lower
degree bound for the local negative insertion and positive
removal operators.  Together with
\eqref{cou:chern-operator}, those bounds will imply that
every decorated minuscule class acts with its geometric
degree.


\section{Euler quotients and integral standard modules}
\label{fil:section}

In this section and through Section~\ref{sec:closure}, we treat
the types \(G_2\) and \(F_4\).
We use the polynomial realization of the negative shuffle algebra.  In
particular, the arrows between orthogonal colours that occur in this
realization are retained in the bilinear form \(E\).  Their quadratic
contribution cancels from the centered grading, as explained at the
end of Section~\ref{std:bb-section}.
All completions in this section are completions of a specified finite
module over a specified Noetherian coefficient ring.

\subsection{The intrinsic quotient}

Fix a dominant monomial \(n\), and write its highest Cartan functions as
\(\bpsi=(\psi_i(z))_{i\in I}\).  At each spectral vertex \(v\), let
\(m_v\) and \(l_v\) be the pole and zero multiplicities in the framing
presentation of \(\bpsi\).  We retain a pole and a zero separately if
they have the same centre.  Thus
\[
 p_v=m_v-l_v,\qquad
 d_p(\nu)=p\cdot\nu+E(\nu,\nu)-\nu\cdot\nu .
\]
Here \(E(\nu,\mu)=\sum_{v\to w}\nu_v\mu_w\), and
\(\nu\cdot\mu=\sum_v\nu_v\mu_v\).
We abbreviate \(E(\nu,e_v)\) and \(E(e_v,\nu)\) to
\(E(\nu,v)\) and \(E(v,\nu)\), respectively.
The polynomial degree of a logarithmic Chern root is one.  The
corresponding cohomological degree is two.

Let \(\cH_\nu\) be the critical cohomology group of the framed stable
space in dimension \(\nu\), with the unshifted convention of the
preceding geometric construction.  Multiplication by the numerator
Euler class is the homogeneous map
\begin{equation}\label{fil:euler-fixed}
 D_\nu=\prod_v\prod_{b=1}^{\nu_v}y_{v,b}^{\,l_v},
 \qquad L_\nu=\deg D_\nu=l\cdot\nu .
\end{equation}
Set
\begin{equation}\label{fil:intrinsic-definition}
 N(n)_\nu=\cH_\nu/\ker D_\nu,\qquad
 N(n)=\bigoplus_\nu N(n)_\nu .
\end{equation}
We give the quotient the grading of the \emph{source} of \(D_\nu\).
The map to \(\im D_\nu\), with its inherited grading, raises polynomial
degree by \(L_\nu\). The decreasing normalized filtration is
\begin{equation}\label{fil:intrinsic-filtration}
 F^hN(n)_\nu
 =\bigoplus_{2d-d_p(\nu)\geq h}N(n)_\nu[d].
\end{equation}
The bracket \([d]\) denotes polynomial degree \(d\), not a shift of a
complex.
For a finite filtered module \(V=\bigoplus_\mu V_\mu\) whose filtration
respects its generalized \(\ell\)-weight spaces, we use
\begin{equation}\label{fil:graded-character-definition}
 \cht\gr_F V
 =\sum_{\mu,h}\dim\bigl(F^hV_\mu/F^{h+1}V_\mu\bigr)t^h\mu.
\end{equation}
Here \(\mu\) is the monomial of the corresponding \(\ell\)-weight.

\begin{proposition}\label{fil:intrinsic-simple}
Evaluation on an ordinary highest vector induces an isomorphism
\[
 N(n)\xrightarrow{\ \sim\ }L(n),\qquad [F]\longmapsto Fv_n .
\]
In particular, \eqref{fil:intrinsic-filtration} is a finite, exhaustive,
separated filtration on the actual ordinary simple module.  It respects
the generalized \(\ell\)-weight decomposition, and its highest line has
degree zero.
\end{proposition}

\begin{proof}
The homogeneous evaluation in
Theorem~\ref{geo:homogeneous-generation} maps the additive wheel ideal
onto \(\cH_\nu\).  Theorem~\ref{geo:critical-quotient} identifies its
Euler source quotient with the quotient by all local ordered residue
tests.  The local completion assertion of
Theorem~\ref{alg:wheel} identifies this map with the completed numerical
small-shuffle evaluation.  The elementwise assertion of
Theorem~\ref{alg:kernel} says precisely that a negative shuffle element
annihilates \(v_n\) if and only if its evaluation is killed by \(D_\nu\).
It therefore identifies the two quotients, with the displayed map on
every word.  Surjectivity is also immediate from highest generation.
The quotient map transports the ordinary action.

The simple \(L(n)\) is finite dimensional.  Each homogeneous quotient in
\eqref{fil:intrinsic-definition} is finite dimensional, and only
finitely many are nonzero by the preceding identification.  Its grading
is bounded on both sides.  The direct sum filtration is therefore finite.
The sector \(\nu=0\) is \(\C\), its Euler class is one, and
\(d_p(0)=0\).  The spectral decomposition used in evaluation is the
generalized Cartan decomposition, which proves the remaining assertions.
\end{proof}

\subsection{A framing deformation and its source quotient}

Write
\[
 m=\prod_{r=1}^NY_{i_r,b_r}.
\]
Choose pairwise distinct integers \(c_r\), and move the two framing
parameters belonging to factor \(r\) together by
\(b_r\mapsto b_r\exp(c_ra)\).  Let
\[
 R_0=\C[a],\qquad R=\C[[a]],\qquad K=\C((a)).
\]
The framing torus \(A=\Gm\) acts on both framing lines of factor
\(r\) with weight \(c_r\), and \(a\) is its equivariant parameter.
Although the multiplicative parameters are formal, their logarithmic
framing weights are the homogeneous linear functions \(c_ra\).
The homogeneous equivariant groups of
Theorem~\ref{geo:homogeneous-generation} will be denoted
\(\cH_{0,\nu}\).  They are finite graded free \(R_0\)-modules.  Define
\begin{equation}\label{fil:family-euler}
 D_{0,\nu}=
 \prod_{v,b}\ \prod_{\substack{r\\
                   \text{numerator frame of }r\text{ at }v}}
       (y_{v,b}-c_ra),
 \qquad
 \cQ_{0,\nu}=\cH_{0,\nu}/\ker D_{0,\nu}.
\end{equation}
The degree of \(D_{0,\nu}\) is again \(L_\nu=l\cdot\nu\).
Put \(H_{R,\nu}=\cH_{0,\nu}\otimes_{R_0}R\),
\(Q_{R,\nu}=\cQ_{0,\nu}\otimes_{R_0}R\), and
\(D_{R,\nu}=D_{0,\nu}\otimes_{R_0}R\). We write
\[
 \cH_0=\bigoplus_\nu\cH_{0,\nu},\qquad
 \cQ_0=\bigoplus_\nu\cQ_{0,\nu},\qquad
 H_R=\bigoplus_\nu H_{R,\nu},\qquad
 D_R=\bigoplus_\nu D_{R,\nu}.
\]

\begin{lemma}\label{fil:free-quotient}
The module \(\cQ_{0,\nu}\) is finite graded free over \(R_0\).
Formation of its source quotient commutes with completion:
\[
 Q_{R,\nu}=H_{R,\nu}/\ker D_{R,\nu}
 \simeq\im D_{R,\nu},
\]
where the last isomorphism has polynomial degree \(L_\nu\).
If \(e_1,\ldots,e_s\) is a homogeneous \(R_0\)-basis of
\(\cQ_{0,\nu}\), of degrees \(d_1,\ldots,d_s\), then its polynomial
filtration after completion is
\begin{equation}\label{fil:free-filtration}
 F_{\pol}^dQ_{R,\nu}
 =\bigoplus_{j=1}^s
       a^{\max(0,d-d_j)}R e_j .
\end{equation}
In particular,
\begin{equation}\label{fil:strict-specialization}
 F_{\pol}^dQ_{R,\nu}\cap aQ_{R,\nu}
       =aF_{\pol}^{d-1}Q_{R,\nu}.
\end{equation}
\end{lemma}

\begin{proof}
The image of the homogeneous map \(D_{0,\nu}\) is a finite torsion-free
module over the principal ideal domain \(\C[a]\).  It is therefore
free.  The source quotient is isomorphic to this image, with the
indicated degree shift.  Since its kernel is homogeneous, a homogeneous
basis can be chosen.  Flatness of completion preserves the kernel,
image, and quotient of the finite-module map.  Expanding a vector in
the homogeneous basis proves \eqref{fil:free-filtration}.  Intersecting
each summand with \(aR e_j\) gives
\[
 a^{\max(1,d-d_j)}R e_j
   =a\,a^{\max(0,d-1-d_j)}R e_j,
\]
which proves \eqref{fil:strict-specialization}.
\end{proof}

\begin{proposition}\label{fil:integral-action}
The direct sum \(Q_R=\bigoplus_\nu Q_{R,\nu}\) is a finite free
\(R\)-module.  It carries the actual ordinary \(U_R=R\otimes_\C\U\)
action with highest Cartan function \(\bpsi(a)\), and is generated by
its highest vector.
\end{proposition}

\begin{proof}
First fix the ordinary root degree.  Only finitely many spectral
dimension vectors occur in the framed stable space of this degree.
Indeed, a cyclic representation of total dimension \(d\) is generated
by framing paths of length at most \(d-1\): the spans of paths of
length at most \(j\) form an increasing sequence, and equality at one
stage implies equality thereafter.  Before stabilization their
dimensions strictly increase.  There are finitely many framing
grades and arrow shifts, so this bounds the set of grades that can
occur.

For this finite set of sectors, the homogeneous wheel evaluation and
the local numerical completion theorem give simultaneous surjectivity
of negative-word evaluation modulo \(a\).  To see that this is a
simultaneous assertion, take the invariant Laurent coordinate ring
of the fixed ordinary root degree.  Distinct reduced coloured
multisets give comaximal maximal ideals.  The Chinese remainder
theorem prescribes the required finite local jets at all of them.
Each resulting small-shuffle element is a finite linear combination
of negative words.  Choose finitely many such words whose evaluated
images span the special fibre.  Their determinant on a chosen basis
has nonzero reduction, so their images generate the completed finite
free module over \(R\).  The same conclusion applies to its source
quotient \(Q_R\).

Let \(V_R(\bpsi(a))\) be the ordinary highest \(\ell\)-weight induced
module, with its distinguished highest vector.  Negative evaluation
defines a surjection
\[
 f_R:V_R(\bpsi(a))\longrightarrow Q_R .
\]
At this stage the map is a map for the negative algebra and the
Cartan algebra.  Put \(\mathcal N_R=\ker f_R\).
We identify its generic kernel through the equivariant residue
presentation. For a fixed sector let
\[
 B_{R,\nu}=H_A^{2*}(X_{\nu,m})\otimes_{\C[a]}R,
\]
and let \(I_{\nu,R}\) be the \((a,\mathbf y)\)-adic completion
of \(I_{\Gamma,\nu}\otimes_\C\C[a]\). Write \(J_{m(a)}\)
for the kernel of ordinary completed evaluation into \(B_{R,\nu}\).
Proposition~\ref{geo:ordinary-flavor-kernel} identifies this kernel
with the ordered tests retaining all factors \(y-c_ra\).
The unframed critical image is \(I_{\nu,R}\), since \(A\) acts
trivially on the unframed arrow space. The equivariant section of
Proposition~\ref{geo:kirwan-section} makes critical stable
restriction surjective and transfers injectivity of specialization
to the stable quotient. Thus its specialization image is precisely
\(\ev_R(I_{\nu,R})\subset B_{R,\nu}\).

Specialization commutes with Chern multiplication. If
\(\Delta_{l(a)}\) denotes the product in
\eqref{fil:family-euler}, the ordered kernels give
\[
 J_{p(a)}=(J_{m(a)}:\Delta_{l(a)}),\qquad
 \ker\bigl(I_{\nu,R}\longrightarrow Q_{R,\nu}\bigr)
       =I_{\nu,R}\cap J_{p(a)}.
\]
Indeed, a representative \(F\) maps to zero exactly when
\(\Delta_{l(a)}\ev_R(F)=0\) in \(B_{R,\nu}\).
This proves the deformed kernel identity for the same word
evaluation that defines \(f_R\).

The ordinary modules \(B_{R,\nu}\) for the finitely many sectors
are finite free over \(R\). Their labeled flag integrations span
a finite \(R\)-submodule of the dual of their direct sum.
Choose finitely many tests generating that submodule and apply
them after multiplication by \(\Delta_{l(a)}\).
Flat localization \(R\to K\) commutes with their common kernel.
Over \(K\), the parameters \(b_r\exp(c_ra)\) belong to distinct
\(q^\Z\)-orbits: an equality of two such parameters up to a
power of \(q\) would first identify their constant terms and
then force \(c_r=c_s\). Chinese remainders separate the finitely
many resulting pole centers and prescribe their finite jets,
with repeated poles at one center kept in a single local algebra.
On these jets the change \(z=x\exp(y)\) of
Lemma~\ref{geo:coordinate-kernel} identifies the deformed additive
tests with the multiplicative tests for \(\bpsi(a)\).
All localization here is performed on finite modules; the local
jet calculations use only finitely many inverses in \(K\).

The generic ordinary tensor product is absolutely simple.
To see this algebraically, first use the Laurent family with
parameters \(b_ru^{c_r}\). Choose a complex value of \(u\)
for which their \(q^\Z\)-orbits are disjoint. At this value the
product of the fundamental characters has a unique dominant
monomial, of coefficient one, so the tensor product is simple.
Burnside's theorem supplies finitely many ordinary algebra words
whose matrices form a basis of its full endomorphism algebra.
The determinant of these word matrices is a nonzero Laurent
polynomial in \(u\). It remains nonzero under the embedding
\(\C(u)\hookrightarrow K\), \(u\mapsto\exp(a)\), and after
extension to an algebraic closure of \(K\). The generic word
matrices therefore generate the full matrix algebra there.

The characteristic-zero version of Theorem~\ref{alg:kernel}
now identifies the generic deformed residue kernel with the
highest-vector annihilator of this tensor product. Equality
descends from the algebraic closure to \(K\) by faithful scalar
extension. Hence \(\mathcal N_R\otimes_RK\) is stable under the
full ordinary algebra.

If \(h\in\mathcal N_R\) and \(u\in U_R\), then
\(f_R(uh)\) vanishes after extension to \(K\).  The target is
torsion-free, so \(f_R(uh)=0\) already over \(R\).  Thus
\(\mathcal N_R\) is \(U_R\)-stable, and the full ordinary action
descends to \(Q_R\).  This construction proves all ordinary current
relations on the quotient at once.

The generic tensor product is finite dimensional.  Outside its
finitely many ordinary root degrees the torsion-free source
quotients become zero over \(K\), and hence are zero over \(R\).
Together with the finiteness in each root degree proved above, this
shows that \(Q_R\) is finite free.  The construction makes it
generated by its highest vector.
\end{proof}

The choice of a logarithmic Euler representative in
\eqref{fil:family-euler} does not change this construction.
An actual multiplicative Euler factor equals its linear factor
times a formal unit.  The product of the numerator units is an
invertible multiplication operator on the target cohomology.
It has the same kernel as the linear Euler product.  Units in
labelled shuffle kernels are instead absorbed into arbitrary labelled
test functions before symmetrization.  These are two different
operations, each valid in its stated module.

\begin{theorem}\label{fil:standard-lattice}
Order the fundamental factors in decreasing spectral order within
each \(q^\Z\)-orbit, and let \(M_R(m)\) be their ordinary ordered
tensor product with the above framing deformation.  There is a
highest-vector-preserving \(U_R\)-isomorphism
\begin{equation}\label{fil:lattice-isomorphism}
 Q_R\xrightarrow{\ \sim\ }M_R(m).
\end{equation}
Consequently the actual ordinary standard module is
\begin{equation}\label{fil:standard-fibre}
 Q(m):=Q_R/aQ_R
   =H_R/(\ker D_R+aH_R)\simeq M(m).
\end{equation}
Its polynomial filtration is the grading of
\(\cQ_0/a\cQ_0\), and its normalized sector degree is
\(2d-d_p(\nu)\).
\end{theorem}

\begin{proof}
For \(a=0\), the decreasingly ordered tensor product of fundamental
modules is highest \(\ell\)-weight; the ordering criterion of
\cite[Section~6.3]{CM05} permits equal spectral exponents.  Factors
in distinct \(q^\Z\)-orbits can be placed in any order.  Hence a
finite set of negative words applied to its highest vector spans
its special fibre.  The tensor product is finite free over \(R\),
so Nakayama's lemma gives
\[
 M_R(m)=U_Rv_{\mathrm{top}}.
\]
Proposition~\ref{fil:integral-action} gives the identical equality
for \(Q_R\).  Over \(K\) both modules are the same simple
highest \(\ell\)-weight module.  Normalize its isomorphism by
sending highest vector to highest vector.  In that common
\(K\)-module both lattices are literally \(U_Rv_{\mathrm{top}}\).
They are therefore equal. The highest-sector Euler class is one,
which fixes this normalization.

Reduction of this integral isomorphism proves
\eqref{fil:standard-fibre}.  Lemma~\ref{fil:free-quotient} proves
the assertion about the quotient filtration.
\end{proof}

\begin{remark}\label{fil:birth-classes}
The fibre in \eqref{fil:standard-fibre} retains classes that appear
in positive powers of the deformation parameter in the Euler image.
It need not be \(\im D_0\).  For example, multiplication by \(a\)
on \(R\) has source quotient \(R\), and its special fibre has
dimension one; the image of its specialization is zero.  The
isomorphism \(Q_R\simeq\im D_R\), including its sectorwise Euler
degree shift, is the appropriate comparison throughout.
\end{remark}

\section{Current operators and local filtrations}
\label{fil:currents-section}

\subsection{The positive formula in all integer modes}

We first work in the rational shuffle realization.  Put
\[
 \zeta^{\rat}_{ij}(z)=\frac{z-q^{-b_{ij}}}{z-1},
 \qquad c_j=q_j-q_j^{-1},\qquad a_{ij}=q^{-b_{ij}}.
\]
The hypothesis on \(q\) implies \(c_j\ne0\).  If \(v\) is an ordinary
highest vector, its Cartan series are the expansions at infinity and
zero of the same rational function \(\psi_j(z)\):
\[
 \psi_j(z)|_\infty=\sum_{h\geq0}\psi^+_{j,h}z^{-h},
 \qquad
 \psi_j(z)|_0=\sum_{h\geq0}\psi^-_{j,h}z^h.
\]
Both expansions are regular at their indicated endpoints.
For a Laurent polynomial \(g(z)=\sum_dg_dz^d\), write
\(e_j[g]=\sum_dg_de_{j,d}\).

\begin{proposition}\label{fil:positive-residue}
On an actual highest-generated module obtained by the evaluation
above, the positive action is
\begin{equation}\label{fil:positive-rational}
 e_j[g](Fv)=\frac1{c_j}
 \sum_{\substack{p\text{ finite}\\p\ne0}}
 \Res_{z=p}
 \frac{g(z)F(z,\mathbf z)\psi_j(z)}
      {\prod_b\zeta^{\rat}_{ji_b}(z/z_b)}
 \frac{\dd z}{z}\,v .
\end{equation}
The sum is over all finite nonzero poles over the rational function
field of the remaining variables.  In particular it includes moving
poles.  The formula holds for all integer modes and includes the
contraction \((\varphi^+_{j,0}-\varphi^-_{j,0})/c_j\) at mode zero.
Its complete value is a finite small-shuffle expression.
\end{proposition}

\begin{proof}
The ordinary Cartan--negative commutation factor is
\begin{equation}\label{fil:cartan-factor}
 R_{ij}(z)=
 \frac{\zeta^{\rat}_{ij}(z)}
      {\zeta^{\rat}_{ji}(z^{-1})}
       =\frac{a_{ij}-z}{1-a_{ij}z}.
\end{equation}
Its two expansions have coefficients
\[
\begin{array}{ll}
 \rho^+_{ij}(0)=a_{ij},&
 \rho^+_{ij}(h)=(a_{ij}^2-1)a_{ij}^{h-1}\quad(h>0),\\
 \rho^-_{ij}(0)=a_{ij}^{-1},&
 \rho^-_{ij}(h)=(a_{ij}^{-2}-1)a_{ij}^{-(h-1)}
                       \quad(h>0).
\end{array}
\]
Taking a fixed coefficient gives finite identities
\begin{align}
 \varphi^+_{j,k}f_{i,s}
   &=\sum_{h=0}^k\rho^+_{ij}(h)
               f_{i,s+h}\varphi^+_{j,k-h},\label{fil:plus-recurrence}\\
 \varphi^-_{j,k}f_{i,s}
   &=\sum_{h=0}^k\rho^-_{ij}(h)
               f_{i,s-h}\varphi^-_{j,k-h}.\label{fil:minus-recurrence}
\end{align}
These are the ordinary double relations, in the conventions of
\cite[Equations~(2.6)--(2.11), (3.7), (3.10), and (3.13)]{Neg25a}.

Consider a word \(W=f_{i_1,s_1}\cdots f_{i_N,s_N}\), with all
\(s_b\in\Z\), and a positive mode \(e_{j,d}\), with \(d\in\Z\).
Contract the occurrence \(k\) with \(i_k=j\), and set \(h_k=d+s_k\).
For \(h_k>0\) its contribution is
\(c_j^{-1}\varphi^+_{j,h_k}\); for \(h_k<0\) it is
\(-c_j^{-1}\varphi^-_{j,-h_k}\); for \(h_k=0\) it is
\(c_j^{-1}(\varphi^+_{j,0}-\varphi^-_{j,0})\).
Move this Cartan coefficient through the suffix \(b>k\), using
\eqref{fil:plus-recurrence} or \eqref{fil:minus-recurrence}.
For \(h_k>0\), the resulting contribution is
\begin{equation}\label{fil:word-positive-sum}
 \frac1{c_j}
 \sum_{\substack{u_b\geq0\ (b>k)\\\sum_{b>k}u_b\leq h_k}}
 \left(\prod_{b>k}\rho^+_{i_bj}(u_b)\right)
 \psi^+_{j,h_k-\sum u_b}\,W^+_{k,\mathbf u}v,
\end{equation}
where \(W^+_{k,\mathbf u}\) is obtained by deleting the \(k\)-th
letter and replacing \(s_b\) by \(s_b+u_b\) in its suffix.
For \(h_k<0\) use a minus sign, \(\rho^-\), \(\psi^-\), and
the replacements \(s_b-u_b\).  For \(h_k=0\), the deleted word is
multiplied by
\begin{equation}\label{fil:zero-mode}
 \frac1{c_j}\left(
 \psi^+_{j,0}\prod_{b>k}a_{i_bj}
 -\psi^-_{j,0}\prod_{b>k}a_{i_bj}^{-1}\right).
\end{equation}
Every sum in this calculation is finite.

The shuffle representative of \(W\) is
\[
 F_W=\Sym_{\mathrm{col}}
 \left[\prod_bz_b^{s_b}
       \prod_{b<c}\zeta^{\rat}_{i_ci_b}(z_c/z_b)\right],
\]
where symmetrization is a sum.  Distinguish a variable \(z\) of
colour \(j\).  Partition the assignments in this symmetrization
according to the occurrence sent to \(z\).  Pair factors from
earlier letters cancel against the denominator below, and factors
from later letters give \eqref{fil:cartan-factor}.  This proves
\begin{align}
 \frac{F_W(z,\mathbf z)}
      {\prod_b\zeta^{\rat}_{ji_b}(z/z_b)}
  =\sum_{k:i_k=j}\Sym_{\mathrm{remaining}}
  \left[
  z^{s_k}\prod_{b\ne k}z_b^{s_b}
  \prod_{\substack{b<c\\b,c\ne k}}
       \zeta^{\rat}_{i_ci_b}(z_c/z_b)
  \prod_{b>k}R_{i_bj}(z_b/z)
  \right].                                      \label{fil:word-partition}
\end{align}
Multiply by \(z^d\psi_j(z)\).
The constant coefficient at infinity of the \(k\)-th term is
\eqref{fil:word-positive-sum}, before the factor \(c_j^{-1}\);
it is zero if \(h_k<0\).
Its constant coefficient at zero is the analogous negative sum
and is zero if \(h_k>0\).  If \(h_k=0\), the difference of the
two constant coefficients is exactly \eqref{fil:zero-mode}.
Thus
\[
 e_{j,d}(F_Wv)=\frac1{c_j}
  (\CT_\infty-\CT_0)
  \left[
  \frac{z^dF_W(z,\mathbf z)\psi_j(z)}
       {\prod_b\zeta^{\rat}_{ji_b}(z/z_b)}
  \right]v .
\]
The residue theorem on the projective line over the field of the
remaining variables turns this into \eqref{fil:positive-rational}.
The endpoint calculation is already a finite word calculation;
hence the full residue sum belongs to the small shuffle image.
Repeated colours give no extra factor: the assignments of \(n_j\)
variables split into \(n_j\) groups with \((n_j-1)!\) assignments,
exactly the sum over contracted occurrences.
Finally, linearity gives every Laurent polynomial test.
\end{proof}

For the polynomial realization, use the augmentation
\(F^{\pol}=D_{\boldsymbol n}F^{\rat}\) and the kernel
\eqref{eq:pol-zeta}.
For a distinguished colour-\(j\) variable,
\[
 \frac{D_{\boldsymbol n+e_j}(z,\mathbf z)}{D_{\boldsymbol n}(\mathbf z)}
 =\prod_{i_b>j}(1-z/z_b)
  \prod_{i_b<j}(1-z_b/z)
 =\prod_b
   \frac{\zeta^{\pol}_{ji_b}(z/z_b)}
        {\zeta^{\rat}_{ji_b}(z/z_b)} .
\]
Multiplying \eqref{fil:positive-rational} by the target augmentation
therefore gives the exact identity
\begin{equation}\label{fil:positive-polynomial}
 e_j[g]F^{\pol}
  =\frac1{c_j}\sum_{\substack{p\text{ finite}\\p\ne0}}
       \Res_{z=p}
       \frac{g(z)F^{\pol}(z,\mathbf z)\psi_j(z)}
            {\prod_b\zeta^{\pol}_{ji_b}(z/z_b)}
       \frac{\dd z}{z}.
\end{equation}
At coincident orthogonal colours the augmentation is a nonunit.
Its factor remains in the denominator and contributes to the
augmented-arrow pole count.

\subsection{An integral formula for a whole pole cluster}

\begin{lemma}\label{fil:cluster-residue}
Let \(B=\C[[a,y_1,\ldots,y_s]]\), \(J=(a,\mathbf y)\), and let
\(s_1,\ldots,s_H\in J\) be homogeneous linear forms, with repetitions
allowed.  For \(H>0\), set \(D(x)=\prod_{h=1}^H(x-s_h)\).
If \(P(x)=\sum_{k\geq0}P_kx^k\in B[[x]]\), then the whole-cluster
residue is the convergent integral expression
\begin{equation}\label{fil:complete-symmetric}
 \mathcal R_D(P)
    =\sum_{k\geq H-1}P_k
           h_{k-H+1}(s_1,\ldots,s_H),
\end{equation}
where \(h_j\) is the complete homogeneous symmetric polynomial.
It agrees with the sum of residues of \(P(x)\dd x/D(x)\) at all
roots whenever this sum is computed after splitting the roots.
If \(P\) has total degree at least \(d\), its residue has total
degree at least \(d-H+1\).  Thus an input error of order \(N\)
loses at most \(H-1\) orders.  For \(H=0\) a regular cluster has
zero residue.
\end{lemma}

\begin{proof}
The formal expansion
\[
 \frac1{D(x)}
   =x^{-H}\prod_{h=1}^H(1-s_h/x)^{-1}
   =x^{-H}\sum_{j\geq0}h_j(\mathbf s)x^{-j}
\]
shows that \eqref{fil:complete-symmetric} is its coefficient of
\(x^{-1}\) after multiplication by \(P\).
A monomial \(x^ka^\alpha\mathbf y^\beta\) contributes a
homogeneous term of degree
\[
  k+\alpha+|\beta|-H+1
\]
or contributes zero.  Only finitely many monomials can affect any
fixed bounded output degree.  This proves convergence and the
degree bound.  For distinct roots the coefficient formula is the
residue theorem, and for repeated roots it follows either by the
same theorem with multiplicities or by polynomial specialization.
All coefficients of the displayed formula are polynomial in the
\(s_h\).  No difference \(s_h-s_{h'}\) has been inverted.
\end{proof}

For example, for a polynomial \(P\),
\[
 \mathcal R_{(x-ca)(x-y)}(P)
       =\frac{P(ca)-P(y)}{ca-y}.
\]
The right side is a polynomial divided difference.  It is defined
when \(ca-y\) is not a unit, including at \(ca=y\), by
\eqref{fil:complete-symmetric}.

For the transition from lowering sector \(\nu+e_v\) to \(\nu\),
write \(v=(j,s)\), \(z=bq^s\exp(x)\) within its \(q^\Z\)-orbit.
The remaining variables at vertex \(w\) are
\(z_{w,b}=b_wq^{s_w}\exp(y_{w,b})\).
The notation \(b_w\) here denotes the orbit representative.
In a fixed cluster all framing poles are \(x=c_ra\) and all moving
poles are \(x=y_{w,b}\).  Factoring
\eqref{fil:positive-polynomial} gives, up to a nonzero scalar,
\begin{equation}\label{fil:factored-positive}
 \frac{x^rF(x,\mathbf y,a)\,N(x,\mathbf y,a)}
      {D(x,\mathbf y,a)}\,
 U(x,\mathbf y,a)\dd x ,
\end{equation}
where \(U\) is a regular formal unit and
\begin{equation}\label{fil:positive-orders}
 \deg D=m_v+E(\nu,v),\qquad
 \deg N=l_v+\nu_v .
\end{equation}
The same-vertex reciprocal Vandermonde contributes the \(\nu_v\)
numerator factors.  The incoming-arrow denominator includes the
orthogonal-colour factors from the augmentation.  A coinciding
numerator and denominator framing remains in \(N/D\); neither is
moved to \(U\).  Every other exponential difference is its linear
difference times a unit.  Since \(\dd z/z=\dd x\), there is no
additional Jacobian degree.
Lemma~\ref{fil:cluster-residue} consequently gives the polynomial
lower degree
\begin{equation}\label{fil:positive-bound}
 \Delta^+_v(r,\nu)
   =r-p_v-E(\nu,v)+\nu_v+1 .
\end{equation}
For fixed source and target, the loss in a Taylor approximation is
bounded by \(\max(m_v+E(\nu,v)-1,0)\).  In particular, this bound
does not depend on the modes or lengths of word representatives.

\subsection{Homogeneous lifts and simultaneous approximation}

The next lemma is the strictness needed to apply this estimate to a
quotient class.

\begin{lemma}\label{fil:strict-lifts}
Every class of polynomial filtration at least \(d\) in \(Q_{R,\nu}\)
has a lift in the completed additive wheel ideal with no monomial
of total degree below \(d\).  At \(a=0\) the same assertion holds
for \(N(n)_\nu\).
\end{lemma}

\begin{proof}
The geometric evaluation is a homogeneous surjection
\[
 I_{\Gamma,\nu}[a]\longrightarrow\cH_{0,\nu}
          \longrightarrow\cQ_{0,\nu}
\]
by Theorem~\ref{geo:homogeneous-generation}.  Every homogeneous
class therefore has a homogeneous polynomial lift of the same
degree: take that degree component of any lift.  Choose homogeneous
lifts \(f_j\) of the basis \(e_j\) in
Lemma~\ref{fil:free-quotient}.  A vector in
\eqref{fil:free-filtration} has an expansion
\[
  \sum_j a^{\max(0,d-d_j)}u_j(a)e_j,\qquad u_j(a)\in R.
\]
The expression with \(f_j\) in place of \(e_j\) is the required
lift.  It has no total degree below \(d\).  The same component
argument proves the assertion on the finite fixed fibre.
\end{proof}

\begin{lemma}\label{fil:simultaneous-jets}
Fix an ordinary root degree.  Let \(\mathcal T\) be a finite set
of reduced coloured multisets.  At every \(\gamma\in\mathcal T\)
prescribe a completed additive wheel element.  For every \(N\)
these elements can be matched, modulo order \(N\) in their joint
\((a,\mathbf y)\)-adic expansions, by one numerical small-shuffle
element over \(R\).  It is a finite \(R\)-linear combination of
actual negative words.

For a fixed finite set of target sectors, \(\mathcal T\) can be
chosen to contain every possible source multiset contributing to
a positive-current residue, whether or not that source is in the
support of the module.  This choice is independent of the mode
indices of the approximating words.
\end{lemma}

\begin{proof}
Let
\[
 A=\C[z_{i,b}^{\pm1}]^{\prod_iS_{n_i}},
\]
and let \(S\subset A\) be the numerical polynomial small-shuffle
image.  It is an \(A\)-ideal: multiplication by an invariant
Laurent polynomial can be performed on the unrestricted labelled
input of every symmetrization.  Same-colour alternating
divisibility shows that these maps land in \(A\).  In particular
\(S\) is a finite module over the Noetherian ring \(A\).
Set \(A_R=R\otimes_\C A\) and \(S_R=R\otimes_\C S\).
For the maximal ideal
\(\mathfrak m_\gamma=(a,\text{invariant coordinates at }\gamma)\),
let \(\Gamma_\gamma\) be the spectral quiver of the multiset
\(\gamma\), let \(I_{\Gamma_\gamma}\) be its wheel ideal in the
corresponding dimension vector, and let
\(P_\gamma=\prod_v\mathfrak S_{\nu_v(\gamma)}\) be the stabilizer
of a labeled representative. Theorem~\ref{alg:wheel} gives
\begin{equation}\label{fil:completed-wheel}
 S_R\,\widehat{(A_R)_{\mathfrak m_\gamma}}
   =I_{\Gamma_\gamma}
         \C[[a,\mathbf y]]^{P_\gamma}.
\end{equation}
The completion on the left is taken \emph{after} adjoining \(R\);
it is the joint completion.  Flatness of this completion applies
because \(S_R\) is finite.

The kernel comparison in
\eqref{fil:completed-wheel} can be read on the labelled finite
cover.  It splits over the finite orbit of labelings of \(\gamma\).
On one component every word map has the form
\[
  G\longmapsto\Sym_{P_\gamma}(G\kappa_\sigma U_\sigma),
\]
where \(\kappa_\sigma\) is the homogeneous additive word kernel
and \(U_\sigma\) is a regular unit.  Multiplication by
\(U_\sigma\) is an automorphism of the labelled input, regardless
of whether that unit is invariant.  Hence the completed image
equals the completed additive word image.  The latter is the
wheel ideal by Theorem~\ref{alg:wheel}.  This argument retains
every nonunit linear factor in \(\kappa_\sigma\).

The ideals for distinct \(\gamma\) have distinct reductions modulo
\(a\), and so their powers are pairwise comaximal.
The Chinese remainder theorem for the finite module \(S_R\),
together with \eqref{fil:completed-wheel}, gives the required
simultaneous approximation.  If it is carried out using labelled
polynomials directly, choose the labelled jets to order \(N+V\),
where \(V\) is the total same-vertex Vandermonde degree.
Alternating division then leaves order at least \(N\).
The integer \(V\) is fixed by the root degree.  This is the only
additional loss in that explicit construction.

For the second assertion, examine
\eqref{fil:positive-polynomial}.  Given a target multiset, every
finite nonzero pole centre comes either from a pole of a highest
Cartan function or from a fixed \(q\)-shift of one of its remaining
roots.  There are finitely many such possibilities.  Adjoin each
of them to that target, for every possible contracted colour.
The resulting finite list contains all source multisets of any
positive residue.  The input is a Laurent polynomial and hence
introduces no further finite nonzero pole; arbitrary integer
mode indices introduce poles only at zero or infinity.  Thus
neither this list nor its maximal pole order changes with the
approximating words.
\end{proof}

\begin{proposition}\label{fil:positive-on-quotient}
The local logarithmic positive current with test \(x^r\), \(r\geq0\),
acts on \(Q_R\) with polynomial lower degree
\eqref{fil:positive-bound}.  It is the already constructed ordinary
action on \(Q_R\).  After specialization the corresponding finite
current jet acts on \(Q(m)\) and on every intrinsic simple \(N(n)\)
with the same bound.
\end{proposition}

\begin{proof}
Take a class in \(F_{\pol}^dQ_{R,\nu+e_v}\).
Lemma~\ref{fil:strict-lifts} gives a wheel lift \(F\) of total degree
at least \(d\). In Lemma~\ref{fil:simultaneous-jets}, include every
occupied source sector of this ordinary root degree and every possible
source multiset for the retained target sectors. Prescribe \(F\) at
its selected source multiset and zero at every other multiset in this
finite list. Let \(F_N\) be the resulting finite word approximants.

Their evaluations converge to the given class in the \(a\)-adic
topology.  Indeed, on a finite free equivariant module the degree
topology and the \(a\)-adic topology are equivalent: the positive
Chern ideal is nilpotent modulo \(a\), so for some \(M\) its
\(M\)-th power sends the module into \(a\) times the module;
conversely multiplication by \(a\) has positive degree.
Every fixed ordinary current is \(R\)-linear on a finite free
module, and is therefore continuous.

For a Laurent test \(g\), its action on \(F_N\) is precisely
\eqref{fil:positive-polynomial}, by
Proposition~\ref{fil:positive-residue}.  At a retained target,
each possible source cluster belongs to the chosen finite list.
The uniform loss in Lemma~\ref{fil:cluster-residue} shows that
the local residue values converge to the cluster residue of the
prescribed lift.  Thus this limiting residue is the actual
ordinary action, by continuity.  This first proves the
completed-class formula for ordinary Laurent tests.

Now approximate the logarithmic test \(x^r\) in its selected
cluster, and zero in the other clusters, by Laurent polynomials.
The Chinese remainder theorem is used only between distinct
reduced nonzero centres.  Roots with the same reduction remain
together in one cluster.  Applying the preceding uniform estimate
to homogeneous lifts of a finite \(R\)-basis gives convergence of
these operators in \(\End_R(Q_R)\).  Their limit still belongs to
the actual ordinary operator image: that image is an \(R\)-submodule
of the finite free module \(\End_R(Q_R)\), hence is finite and
\(a\)-adically closed.  Formula \eqref{fil:factored-positive}
and Lemma~\ref{fil:cluster-residue} give the required lower degree.
The filtration steps are closed by
\eqref{fil:free-filtration}, so the estimate passes to the limit.

Two different lifts give the same value, since both limiting
constructions recover the action defined in
Proposition~\ref{fil:integral-action}.
On a fixed finite fibre, positive local Chern coordinates are
nilpotent.  A sufficiently high finite jet of the test is then
exact.  The same proof, without \(a\), applies to
Proposition~\ref{fil:intrinsic-simple}.
\end{proof}

\section{The fixed convolution grading}
\label{fil:convolution-section}

\subsection{Normalized current degrees}

The local negative current is particularly explicit.  For
\(v=(i,s)\) and \(w=(j,t)\), put
\[
 K_{vw}(x,y)=
       \zeta^{\pol}_{ji}(q^{t-s}e^{y-x})
\]
when the vertices belong to the same orbit; for different orbits
include the quotient of their orbit representatives in the argument.
Every factor has a unique decomposition
\begin{equation}\label{fil:negative-factor}
 K_{vw}(x,y)
     =c_{vw}(y-x)^{E(v,w)-\delta_{vw}}U_{vw}(x,y),
 \qquad c_{vw}\ne0,\qquad U_{vw}(0,0)=1.
\end{equation}
Here \(E(v,w)\) is the number of arrows from \(v\) to \(w\).
This is read directly from \eqref{eq:pol-zeta}.
If \(i=j\) and \(s=t\), its order is \(-1\); if the corresponding
arrow is resonant, its order is \(1\); every other factor has
order zero.  Distinct orbits give only the last case.
No \(c_{vw}\) vanishes at the fixed numerical non-root-of-unity
parameter.

For a local polynomial \(F\) in sector \(\nu\), the insertion with
test \(x^r\) is
\begin{equation}\label{fil:negative-insertion}
 \mathcal F_{v,r}F
  =\sum_{a=1}^{\nu_v+1}x_{v,a}^r
       F(\mathbf x\setminus x_{v,a})
       \prod_{\substack{(w,b)\text{ old}}}
          K_{vw}(x_{v,a},x_{w,b}).
\end{equation}
In each summand the old variables at \(v\) are all the variables
other than \(x_{v,a}\).  The permutations of these old variables
cancel the shuffle normalization, giving exactly this sum and no
additional factorial.  After a common Vandermonde denominator is
introduced, the numerator is alternating in the variables of
each repeated vertex.  Its divisibility by the Vandermonde holds
in each homogeneous degree.  Thus the displayed operator is a
polynomial formal series, and \eqref{fil:negative-factor} gives
its lower polynomial degree
\begin{equation}\label{fil:negative-bound}
 \Delta^-_v(r,\nu)=r+E(v,\nu)-\nu_v .
\end{equation}
The same finite-jet argument as in
Lemma~\ref{fil:simultaneous-jets}, with a fixed Vandermonde loss,
shows that this is the actual local ordinary negative operator
on every quotient under consideration.

\begin{proposition}\label{fil:current-bounds}
For either sign, the normalized filtration degree of the local
current with logarithmic test \(x^r\) is bounded below by
\begin{equation}\label{fil:normalized-current-degree}
 \delta(v,r)=2r+1-p'_v,\qquad
 p'_v=p_v+E(\nu,v)-E(v,\nu),
\end{equation}
where \(p'_v\) is the signed pole order of the current ordinary
Cartan function.  The formula applies to the transition
\(\nu\leftrightarrow\nu+e_v\) and to both its directions.
\end{proposition}

\begin{proof}
There is no loop at a spectral vertex.  Consequently
\begin{equation}\label{fil:centre-difference}
 d_p(\nu+e_v)-d_p(\nu)
   =p_v+E(\nu,v)+E(v,\nu)-2\nu_v-1 .
\end{equation}
Twice \eqref{fil:negative-bound} minus
\eqref{fil:centre-difference} equals
\(2r+1-p_v-E(\nu,v)+E(v,\nu)\).
Twice \eqref{fil:positive-bound} plus
\eqref{fil:centre-difference} gives the same number.
The Cartan commutation factor
\eqref{fil:cartan-factor} gives exactly the indicated change of
the signed pole divisor after lowering by \(\nu\).
Since \(E(v,v)=0\), that divisor is unchanged by reversing this
single \(v\)-step.
\end{proof}

\subsection{Intrinsic sectors in a larger framing}

Fix a standard \(M(m)\) and its actual action of the Coulomb algebra
\(\cC_m\) from Theorem~\ref{cou:ordinary-action}.
An ordinary simple constituent \(L(n)\) inherits this same action:
the ordinary current map is surjective by
Proposition~\ref{cou:surjection}.  We transport it to the intrinsic
quotient \(N(n)\) using Proposition~\ref{fil:intrinsic-simple}.

Let \(W_i(u)\) be its gauge-root polynomial, normalized by its
nonzero constant roots, and let \(\eta_{i,k}\) be the multiplicity
of the roots \(bq^k\) on the highest line of \(L(n)\).
We suppress the orbit index where it does not affect a formula.
The actual Cartan root-movement identity is
\begin{equation}\label{fil:root-movement}
 W_i(u)F_j(z)=F_j(z)W_i(u)
       \left(\frac{1-q_i^{-2}z/u}{1-z/u}\right)^{\delta_{ij}} .
\end{equation}
The branch of the determinant cover is the one in the ordinary
current map of Theorem~\ref{cou:ordinary-action}.

\begin{lemma}\label{fil:inherited-idempotents}
The intrinsic lowering sector \(\nu\) corresponds to the inherited
gauge-root multiplicities
\begin{equation}\label{fil:gauge-multiplicities}
 a_{i,k}(\nu)=\eta_{i,k}
                 +\nu_{i,k+2d_i}-\nu_{i,k}.
\end{equation}
This correspondence is injective on finite-support dimension
vectors.  Hence the intrinsic filtration respects the actual
degree-zero idempotents of the larger weighted convolution algebra.
\end{lemma}

\begin{proof}
Iterate \eqref{fil:root-movement} through a negative word.
Each lowering at \(z=bq^k\) removes a gauge root at \(bq^k\)
and adds one at \(bq^{k-2d_i}\), giving
\eqref{fil:gauge-multiplicities}.
If two finite-support arrays give the same multiplicities, their
difference \(h\) satisfies
\(h_{i,k+2d_i}=h_{i,k}\).
A nonzero such periodic array has infinite support.
Thus \(h=0\).
Every occupied block of \(L(n)\) is reached by a negative word,
since this module is highest generated.  Its intrinsic sectors
and inherited occupied idempotents consequently coincide.
\end{proof}

\begin{lemma}\label{fil:chern-operators}
On the intrinsic quotient with its inherited larger action, every
source block Chern class of polynomial degree \(r\) has filtration
degree at least \(2r\).  The same assertion holds on the standard
family before specialization when \(\deg a=1\).  On the fixed
fibres the ordinary Cartan zero modes remain semisimple of type
one on the associated graded.
\end{lemma}

\begin{proof}
Separate the distinct reduced gauge-root centres by Hensel
factorization of \(W_i(u)\).  The relevant resultants are units;
roots within one repeated block are not separated.
For \(r>0\), logarithmic power sums in the block at \(bq^k\)
give from \eqref{fil:root-movement} the operator identity
\begin{equation}\label{fil:log-power-sums}
 P_{i,k,r}
  =P^{\mathrm{top}}_{i,k,r}
    +\sum_{\text{lowerings at }(i,k+2d_i)}y_b^r
    -\sum_{\text{lowerings at }(i,k)}y_b^r .
\end{equation}
This is first an identity on negative words and therefore on
their actual quotient.  On an intrinsic fixed fibre the top
logarithmic roots are zero, so \(P^{\mathrm{top}}_{i,k,r}=0\).
In the deformed standard family they are the corresponding sums
of \((c_ba)^r\).  In either case the right side has polynomial
degree \(r\).  Newton identities express all block Chern classes
as polynomials of the asserted degree in these power sums.
This proves an operator statement, including the nilpotent
Chern parts.

For the last assertion, sum \eqref{fil:log-power-sums} over all
vertices and all finitely many occupied orbits.  The lowering
terms telescope.  At \(a=0\) the total positive logarithmic power
sums therefore vanish.  If
\(N_i=\sum_k\nu_{i,k}\), the nonzero constant part of the gauge
determinant, computed from
\eqref{fil:gauge-multiplicities}, is
\begin{equation}\label{fil:determinants}
 \det V_i|_\nu=(\det V_i)_{\mathrm{top}}q_i^{-2N_i}.
\end{equation}
Its logarithmic nilpotent part is zero.  On the prescribed square
root branch, \(D_i=(\det V_i)^{1/2}\) consequently acts as
\(D_i^{\mathrm{top}}q_i^{-N_i}\).
The ordinary current normalization expresses \(k_i\) as a fixed
framing constant times \(\prod_jD_j^{c_{ji}}\).
Symmetrizability then gives
\begin{equation}\label{fil:type-one}
 k_i|_\nu
    =k_i^{\mathrm{top}}\,
                  q_i^{-\sum_jc_{ij}N_j}.
\end{equation}
The vanishing logarithmic relations used above are homogeneous
Chern relations and pass to the associated graded.
Thus \eqref{fil:type-one}, with no nilpotent part, holds there
as well.
\end{proof}

For a dimension vector \(\tau\), write
\(A^\tau=\prod_{i,k}A_{i,bq^k}^{\tau_{i,k}}\), taking the product
over all parameter orbits. If \(n=mA^{-\tau}\), then
\[
 p^n_v=p^m_v+E(\tau,v)-E(v,\tau),
\]
and direct expansion gives
\begin{equation}\label{fil:centre-translation}
 d_{p^m}(\tau+\mu)-d_{p^m}(\tau)-d_{p^n}(\mu)
       =2\bigl(E(\mu,\tau)-\mu\cdot\tau\bigr).
\end{equation}
The compatible degree in
\eqref{fil:normalized-current-degree} instead uses the pole
order of the actual current Cartan function.  That function is
the same in the inherited action and in the intrinsic realization.

\subsection{Homogeneous operators}

For an ordinary constituent $L(n)$ of $M(m)$, let
\[
 \theta_n:N(n)\xrightarrow{\sim}L(n),\qquad [F]\longmapsto Fv_n,
\]
be the word-evaluation isomorphism of
Proposition~\ref{fil:intrinsic-simple}. The surjection in
Theorem~\ref{cou:ordinary-action} gives its inherited $\cC_m$-action.
Explicitly, for $c\in\cC_m$ choose $u\in\U$ mapping to $c$ and set
\[
                         c\cdot v=\theta_n^{-1}(u\theta_n(v)).
\]
The kernel of $\U\to\cC_m$ kills $L(n)$, so this formula is
independent of the chosen preimage. Hence completed algebra identities
act on the same ordinary module used in the intrinsic filtration.

Use the fixed completed comparison
\[
 \Theta:\widehat{\cC_m}
     \xrightarrow{\sim}\widehat{\cA^{\mathrm{fl}}}
\]
of Proposition~\ref{cou:flag-comparison}.
For either sign, Proposition~\ref{cou:current-object} identifies a
local actual current with
\begin{equation}\label{fil:object-current}
 \Theta(T_{v,r})=j_*([Z_v]x^rU_v),\qquad U_v(0)\ne0 .
\end{equation}
The unit is independent of \(r\).
The homogeneous fundamental class \([Z_v]\) has actual Ext
degree \(1-p'_v\), by Proposition~\ref{cou:degree}.

\begin{proposition}\label{fil:unit-transfer}
Every homogeneous minuscule class
\(j_*([Z_v]x^r)\), of either sign, acts on the above finite
fibres with its Ext-degree filtration bound.
The statement applies simultaneously to the standard and to the
intrinsic filtrations of all its actual simple constituents.
\end{proposition}

\begin{proof}
The source-cohomology description
of Lemma~\ref{cou:source-cohomology} expresses the
incidence ring using the source block Chern classes and the
moved-line class \(x\), with a homogeneous monic
projective-bundle relation.  Every homogeneous part of
\(U_v^{-1}\) therefore has a polynomial lift in those classes.
Write, degree by degree,
\[
 U_v^{-1}=\sum_{h\geq0}x^hp_h .
\]
The projection formula, with the source coefficient on the right,
gives the exact identity of
Corollary~\ref{cou:sided-inverse}:
\begin{equation}\label{fil:sided-inverse}
 j_*([Z_v]x^r)
       =\sum_{h\geq0}\Theta(T_{v,r+h})\,p_h .
\end{equation}
For the reversed positive incidence, its own source ring is used
in the same calculation.

If a homogeneous part of \(p_h\) has polynomial degree \(b\),
Lemma~\ref{fil:chern-operators} and
Proposition~\ref{fil:current-bounds} give the bound
\[
  \delta(v,r+h)+2b
       =\delta(v,r)+2(h+b)\geq\delta(v,r).
\]
This is exactly the degree of \(j_*([Z_v]x^r)\).
The series acts by a finite sum on a finite continuous module:
the positive central ideal acts nilpotently, and the moved-line
powers reduce by the monic projective-bundle relation.
Equivalently, a summand with $h+b$ sufficiently large has lower
degree bound $\delta(v,r)+2(h+b)$ beyond the finite filtration
range and acts by zero. Thus passage from the completed identity to this
operator bound is valid on the stated finite fibres.
The extra top gauge roots of a lower simple have zero logarithm,
and \eqref{fil:log-power-sums} already includes their ranks.
Finally, its intrinsic \(p'_v\) is the same ordinary Cartan
divisor as in \eqref{fil:object-current}, by
\eqref{fil:centre-translation} and the preceding discussion.
\end{proof}

\begin{theorem}\label{fil:same-algebra}
There is one finite-dimensional \(\Z\)-graded quotient
\(\cA_S\) of the known weighted-flag convolution algebra on which
the standard \(Q(m)\), all its ordinary simple constituents, and
their associated graded modules are defined.
For every homogeneous \(a\in(\cA_S)_d\),
\begin{equation}\label{fil:compatibility}
 aF^hV\subseteq F^{h+d}V
\end{equation}
for the standard filtration and every intrinsic simple
filtration.  The degree-zero idempotents retain their ordinary
generalized \(\ell\)-weight sectors.
\end{theorem}

\begin{proof}
Let \(S\) be the finite union of the gauge-root supports of
\(Q(m)\) and its finitely many ordinary simple constituents.
By Lemma~\ref{fil:inherited-idempotents}, it is also the union
of their intrinsic supports.  In the locally unital graded
algebra \(\cA^{\mathrm{fl}}\), first kill the homogeneous trace
ideal
\[
 J_{\mathrm{out}}
       =\sum_{\gamma\notin S}
              \cA^{\mathrm{fl}}e_\gamma\cA^{\mathrm{fl}} .
\]
Then take the corner with \(e_S=\sum_{\gamma\in S}e_\gamma\).
Each of the stated modules factors through this quotient,
since every \(e_\gamma\) outside \(S\) acts by zero.
Taking the trace quotient first also kills paths that leave
the retained support and return to it.

For fixed \(\gamma,\eta\), the diagonal unfolded gauge group has
finitely many orbits on \(\Flag_\gamma\times\Flag_\eta\), indexed by
relative positions. Above each orbit, the two flag conditions on
matter maps define a constant-rank vector bundle. Its stabilizer is
an intersection of two parabolic subgroups. The unipotent radical
does not change equivariant cohomology, and its Levi factor is a
product of general linear groups. The corresponding Weyl-invariant
polynomial ring is finite over \(H^*_{\underline G}(\mathrm{pt})\).
The finite stratum filtration and Noetherianity therefore show that
\[
 H_*^{\underline G,\BM}(X_\gamma\times_NX_\eta)
\]
is finite over this central ring. The geometric dimension bound
bounds the selfdual Ext degrees below; equivariant parameters have
positive degree.

There are finitely many retained pairs, so their corner is bounded
below and finite over the positively graded central equivariant
ring. The trace quotient preserves both properties. Choose a common
power of the positive central ideal annihilating the finitely many
continuous modules under consideration, and take that homogeneous
quotient. It is the required finite-dimensional graded algebra
\(\cA_S\).

\paragraph{The ordinary current map on the finite quotient.}
Let $\cC_m$ be the specialized Coulomb algebra associated with
$M(m)$, let $B$ be its gauge coefficient ring, and let $\mathfrak m_\gamma\subset B$ be the maximal ideal of a
gauge loading $\gamma$, including the chosen determinant branch.
Its semisimple completion has the two-sided blocks
\cite[(4.2.18)--(4.2.20)]{VV25}
\[
 \widehat{\cC}_m(\gamma,\eta)
 =\varprojlim_k\cC_m/
   (\mathfrak m_\gamma^k\cC_m+\cC_m\mathfrak m_\eta^k).
\]
The algebra $\cC_m$ acts on their direct sum by multipliers.
Indeed, coefficient elements are diagonal, and a minuscule generator
has only finitely many possible target loadings from a fixed source,
and finitely many possible source loadings for a fixed target.
The same is true of every finite word.  Proposition~\ref{cou:surjection}
therefore makes the matrix of every $c\in\cC_m$ row and column finite,
with its $(\gamma,\eta)$ entry equal to the image of $c$ in the
displayed two-sided completion.  Multiplication of these matrices is
the multiplication defining the completed algebra.

Each flag coefficient ring is finite over the central equivariant
ring. Modulo the positive central ideal it is finite dimensional and
graded, so a power of its positive augmentation ideal lies in that
central ideal. These two adic topologies are consequently cofinal.
The homogeneous quotient map from the flag algebra to $\cA_S$ extends
to each completed block, because its kernel contains the chosen power
of the positive central ideal.  Denote this map, after transport by
$\Theta$, by $\pi$.  For $c\in\cC_m$, define
\[
 \rho_S(c)=\sum_{\gamma,\eta\in S}
       \pi\bigl(c_{\gamma,\eta}\bigr)\in\cA_S.
\]
This is a finite sum.  It is a unital algebra homomorphism: in the
block formula for a product, a term with an intermediate loading
outside $S$ belongs to the completed trace ideal generated by that
idempotent and has zero image.  The remaining terms give precisely
$\rho_S(c)\rho_S(c')$.  The completed trace ideal is killed because
$\pi$ is continuous and kills its algebraic generators.

We prove surjectivity on the algebra itself.  On the block $\gamma$,
$\Theta$ carries $\mathfrak m_\gamma$ to power series with zero
constant term in the positive-degree Chern classes of its source
flag blocks.  Since $\cA_S$ is finite dimensional and graded, there is
one integer $M$ such that these $M$th powers act by zero on both sides
of every retained block of $\cA_S$.  For example, if all nonzero
degrees of $\cA_S$ lie between $d_-$ and $d_+$, it suffices to take
$2M>d_+-d_-$.  Thus one common order of coefficient jets suffices on the quotient
algebra.

Consequently the image of an element of
$\widehat{\cC}_m(\gamma,\eta)$ depends only on its class modulo
$\mathfrak m_\gamma^M\cC_m+\cC_m\mathfrak m_\eta^M$.
By the definition of that completed block, this class is represented
by some $c\in\cC_m$.  Distinct retained loadings give distinct
maximal ideals of $B$: their colored multisets of gauge roots differ,
or their determinant branches differ.  The fixed spectral
representatives and the non-root-of-unity condition ensure that the
weighted-flag label is determined by this multiset.  Chinese
remainders therefore give elements $p_\gamma\in B$ such that
\[
 p_\gamma\equiv\delta_{\gamma,\delta}
           \pmod{\mathfrak m_\delta^M}\qquad(\delta\in S).
\]
They satisfy $\rho_S(p_\gamma)=e_\gamma$.  Thus
$p_\gamma c p_\eta$ represents the required element of the
$(\gamma,\eta)$ block and has zero image in every other retained
block.  Summing over the finitely many retained pairs proves that
$\rho_S$ is surjective.  Composition with the ordinary current map
gives the surjection
\begin{equation}\label{fil:ordinary-finite-map}
 \U\xrightarrow{\Phi_m}\cC_m
          \xrightarrow{\rho_S}\cA_S.
\end{equation}
Its action on $Q(m)$ and on each intrinsic $N(n)$ is the action already
constructed: their smooth completed actions factor through the same
trace and central-power ideals, and the block maps above are the
extensions of those actions.  Hence this is one ordinary current map
on the quotient algebra, available also on every associated graded
module subsequently constructed.

\paragraph{Filtration bounds.}
The retained idempotents preserve each filtration by
Lemma~\ref{fil:inherited-idempotents}. Source Chern classes satisfy
the required bounds by Lemma~\ref{fil:chern-operators}, and decorated
minuscule classes satisfy them by Proposition~\ref{fil:unit-transfer}.


Let $\mathscr D\subseteq\cA_S$ be generated by the retained
idempotents, the homogeneous source-block Chern classes, and all
homogeneous decorated minuscule fundamental classes of both signs.
For an ordinary integer current mode, its blockwise image under
$\Theta$ is the full incidence class of
Proposition~\ref{cou:current-object}.  Its coefficient is the unit
$U_v$ times the corresponding Laurent-line character, expanded in
the logarithmic moved-line class.  By
Lemma~\ref{cou:source-cohomology}, every homogeneous coefficient has
a polynomial lift in that class and the source Chern classes.
The projection formula puts the latter coefficients on the right.
Only finitely many homogeneous terms can survive in $\cA_S$, so each
ordinary root-current image belongs to $\mathscr D$.  The same
argument uses the source of the reversed incidence for a positive
current.  The Cartan coefficients, including their zero modes and
the prescribed determinant branch, are completed coefficient-ring
expressions; they likewise truncate to polynomials in source Chern
classes and constants on the retained idempotents.  Their images
also belong to $\mathscr D$.

The surjective map $\U\to\cA_S$ gives $\mathscr D=\cA_S$.
Since these generators are homogeneous, a homogeneous element of
\(\cA_S\) is a sum of words of its own degree. The filtration bounds
add under composition and are preserved under such sums. This proves
\eqref{fil:compatibility}, and the induced action on each associated
graded follows from that identity.
\end{proof}

\begin{corollary}\label{fil:graded-simple}
For every ordinary simple constituent \(L(n)\) of the fixed
standard, \(\gr N(n)\) is a graded lift of the same simple for
the algebra \(\cA_S\).  Its ordinary highest line has degree zero.
\end{corollary}

\begin{proof}
We use the following elementary fact.  If \(A\) is a
finite-dimensional \(\Z\)-graded complex algebra, \(V\) is a
simple ungraded \(A\)-module, and \(F\) is a finite exhaustive
separated filtration satisfying
\[
                       A_dF^hV\subseteq F^{h+d}V,
\]
then \(\gr_FV\) is isomorphic to \(V\) as an ungraded
\(A\)-module.

Indeed, the Jacobson radical of \(A\) is homogeneous: scaling
degrees gives algebra automorphisms that preserve the radical.
The radical kills \(V\) and therefore also kills \(\gr_FV\).
Put \(B=A/\rad(A)\).  The grading action of \(\C^\times\)
preserves the finite set of central primitive idempotents of
the semisimple algebra \(B\).  Since \(\C^\times\) is connected,
each of these idempotents is fixed and has degree zero.
Choose degree-zero representatives in \(A\) of these central
idempotents. The block idempotent acting
as one on \(V\) also acts as one on \(\gr_FV\), and every other
block idempotent acts as zero on both.
If the selected block is \(M_d(\C)\), then \(\dim V=d\).
The associated graded has the same dimension \(d\) and is a
module for the same matrix block, so it is that block's unique
simple module.  This proves the fact.

Apply it to \(A=\cA_S\) and \(V=N(n)\), using
Theorem~\ref{fil:same-algebra}.  The ordinary restriction of
\(N(n)\) is \(L(n)\), and the same surjective ordinary current
map acts on its associated graded.  Thus \(\gr N(n)\) is a
graded lift of that same simple.  Its highest line is the
one-dimensional degree-zero line of
Proposition~\ref{fil:intrinsic-simple}.
\end{proof}

\subsection{Leading Hall operators and the standard highest line}

Let \(\mathsf H_{v,r}\) denote the homogeneous Hall insertion of
a line at \(v\) decorated by its \(r\)-th Chern power.
In the conventions used here its flag is
\[
 0\longrightarrow U\longrightarrow V
             \longrightarrow V'\longrightarrow0,
 \qquad \dim U=e_v,
\]
where \(U\) is unframed and \(V'\) is the old framed quotient.

\begin{lemma}\label{std:leading-hall}
In the transition \(\nu\to\nu+e_v\), the lowest polynomial
degree part of \(\mathcal F_{v,r}\) is a nonzero scalar multiple
of \(\mathsf H_{v,r}\), namely
\begin{equation}\label{std:hall-formula}
 C_{v,\nu}\sum_{a=1}^{\nu_v+1}x_{v,a}^r
   F(\mathbf x\setminus x_{v,a})
 \frac{\displaystyle
   \prod_{v\to w}\prod_{\substack{b\text{ old}\\\text{at }w}}
                    (x_{w,b}-x_{v,a})}
      {\displaystyle\prod_{b\ne a}(x_{v,b}-x_{v,a})},
 \qquad
 C_{v,\nu}=\prod_wc_{vw}^{\,\nu_w}\ne0 .
\end{equation}
For the actual homogeneous minuscule class,
\[
 \gr_{\delta(v,r)}j_*([Z_v]x^r)
      =U_v(0)^{-1}C_{v,\nu}\,\mathsf H_{v,r}.
\]
\end{lemma}

\begin{proof}
Substitute \eqref{fil:negative-factor} into
\eqref{fil:negative-insertion}.
The degree-zero parts of all \(U_{vw}\) are one, giving exactly
\eqref{std:hall-formula}.  Every other term has strictly
larger total degree.  Alternating divisibility by the common
Vandermonde holds degree by degree, so taking the lowest degree
commutes with its cancellation.

For the geometric Hall map, preservation of the subrepresentation
\(U\) imposes the vanishing of
\(\Hom(U_v,V'_w)\) for each arrow \(v\to w\).
Its Euler factor is the numerator of
\eqref{std:hall-formula}.  The tangent bundle of the
Grassmannian of lines contributes
\(\Hom(U_v,V'_v)\), giving the denominator.  The incoming
framing component into \(U\) is free, so it gives no Euler
numerator.  The extension pullback is smooth and the flag
pushforward is proper.  Their usual localization formula is
therefore the sum in \eqref{std:hall-formula}.
Critical specialization is injective and compatible with
these two maps by
Theorem~\ref{geo:homogeneous-generation}; hence the same
identity holds on the actual critical Hall module.

Finally apply \eqref{fil:sided-inverse}.
Its summand of minimal normalized degree is the constant
term \(h=b=0\), whose coefficient is \(U_v(0)^{-1}\).
Every other homogeneous summand has degree at least two higher.
Combining this with
Proposition~\ref{fil:current-bounds} gives the second assertion.
\end{proof}

\begin{lemma}\label{std:euler-hall-stability}
The homogeneous Euler kernel is preserved by Hall insertions.
In particular, the Hall action on \(\cH_0\) descends to
\(\cQ_0\) and then to \(\cQ_0/a\cQ_0\).
\end{lemma}

\begin{proof}
On the Hall flag the tautological exact sequence gives
an exact sequence for the numerator framing bundles.
Write \(\cE_{\bl,A}(U)\) for the numerator framing bundle
evaluated on the unframed kernel \(U\), as in
\eqref{geo:euler-hall-identity}.
Its Euler classes satisfy
\[
 D_{\nu+e_v}|_{\Flag}
      =e(\cE_{\bl,A}(U))\,D_\nu|_{\Flag}.
\]
The projection formula consequently gives, for every
decoration \(b\),
\begin{equation}\label{std:euler-hall-identity}
 D_{\nu+e_v}\,\mathsf H_b(h)
       =\mathsf H_{b\,e(\cE_{\bl,A}(U))}(D_\nu h).
\end{equation}
Thus \(D_\nu h=0\) implies
\(D_{\nu+e_v}\mathsf H_b(h)=0\).
The operators and the Euler kernels are homogeneous over
\(\C[a]\), so both quotient and specialization preserve the
assertion.
\end{proof}

\begin{theorem}\label{std:graded-cyclicity}
As a module for the same algebra \(\cA_S\) of
Theorem~\ref{fil:same-algebra}, the associated graded standard
\(\gr Q(m)\) is generated by its degree-zero highest line under
the homogeneous negative minuscule classes.
Its semisimple head consists of one copy of the simple with
ordinary highest monomial \(m\), with that highest line in
degree zero.
\end{theorem}

\begin{proof}
The equivariant Hall-vacuum generation of
Theorem~\ref{geo:homogeneous-generation} generates \(\cH_0\)
over \(\C[a]\) by homogeneous Hall words.
Lemma~\ref{std:euler-hall-stability} passes this generation to
the source quotient \(\cQ_0\).
Equation~\eqref{fil:strict-specialization} identifies its
specialized grading with the associated graded of the actual
standard \(Q(m)\).  Hence Hall words generate that graded
space from its vacuum.

By Lemma~\ref{std:leading-hall}, each such word is a nonzero
scalar multiple of the corresponding word in the homogeneous
negative minuscule classes, after fixing its intermediate
sectors with the degree-zero idempotents.
The scalar is the product of the nonzero constants on its
successive transitions.  Therefore those known homogeneous
classes generate \(\gr Q(m)\) from the same vacuum, whose
normalized degree is zero.

Every simple quotient must contain the image of this generating
\(m\)-line.  If its ordinary highest monomial is \(n\), all
its \(\ell\)-weights belong to the support of \(Q(m)\), so
\(n\leq m\).  Since it also contains the monomial \(m\),
highest generation of the simple gives \(m\leq n\).
Thus \(n=m\).  The \(m\)-space of the standard has dimension
one, so the semisimple head contains only one such copy.
The degree of its highest line fixes the displayed shift.
\end{proof}



\section{The grading and duality of simple modules}\label{sec:normalization}

Fix an ordered standard module \(M(m)\). We use the finite graded
flag algebra of Theorem~\ref{fil:same-algebra} for this standard and
its ordinary simple constituents, with their intrinsic filtrations.

\subsection{Ordinary labels of the finite graded algebra}

Throughout this section we use the single finite graded algebra
\(\cA_S\) of Theorem~\ref{fil:same-algebra}, and the surjective ordinary
current homomorphism \eqref{fil:ordinary-finite-map}.
For each retained spectral idempotent \(e_\gamma\), let
\(n_\gamma\) be the ordinary monomial obtained by substituting
its constant gauge roots and the fixed determinant branch in the
Cartan rational functions.  These constants occur on the actual
standard or one of its ordinary constituents, so \(n_\gamma\)
is an ordinary type~\(1\) \(\ell\)-weight.

\begin{lemma}\label{norm:simple}
Every ungraded simple \(\cA_S\)-module \(V\) restricts, through
\eqref{fil:ordinary-finite-map}, to a simple finite-dimensional type~\(1\)
quantum loop module.  Its ordinary \(q\)-character is
\[
        \chq V=\sum_{\gamma\in S}
                        \dim(e_\gamma V)\,n_\gamma.
\]
If several idempotents determine the same monomial, their
dimensions are added in this formula.
\end{lemma}

\begin{proof}
Surjectivity of \eqref{fil:ordinary-finite-map} implies simplicity of the
ordinary restriction.  It remains to check its Cartan action;
the same calculation applies to every surviving simple.

Fix \(\gamma\).  In its completed gauge coefficient ring,
the maximal ideal at the chosen constant roots is generated
by positive logarithmic Chern coordinates.  Their images
in \(e_\gamma\cA_Se_\gamma\) are homogeneous of strictly
positive degree and commute.  Since \(\cA_S\) is finite
dimensional and graded, the ideal they generate in this
commutative coefficient subalgebra is nilpotent: a product
of sufficiently many positive-degree coordinates has degree
larger than every nonzero graded piece of \(\cA_S\).
Denote this nilpotent ideal by \(J_\gamma\).

Every coefficient of a Cartan current commutes with
\(e_\gamma\).  The fixed Cartan rational formula, expanded
at zero or infinity, expresses its restriction to this block
as its constant value plus an element of \(J_\gamma\).
The same nilpotent ideal works for all Cartan coefficients.
Consequently \(e_\gamma V\) is a simultaneous generalized
\(\ell\)-weight space with generalized eigencharacter
\(n_\gamma\), or is contained in that space when different
retained idempotents have the same constant character.
In particular all eigenvalues of each \(k_i\) are integral
powers of \(q_i\).

The Cartan zero modes on this simple restriction are semisimple.  Write the multiplicative Jordan
decomposition on \(V\) as \(k_i=s_i u_i\), with \(s_i\)
semisimple and \(u_i\) unipotent.  The relations
\[
 k_i x_{j,r}^{\pm}k_i^{-1}
       =q_i^{\pm c_{ij}}x_{j,r}^{\pm}
\]
make every root generator an eigenvector of
\(\operatorname{Ad}(k_i)\).  In the multiplicative Jordan
decomposition of this adjoint operator, its unipotent
part \(\operatorname{Ad}(u_i)\) fixes every such eigenvector.
It fixes the Cartan generators as well.  Thus \(u_i\)
commutes with the entire ordinary quantum loop action.
Schur's lemma makes \(u_i\) scalar; being unipotent, it
equals the identity.  Hence all \(k_i\) are semisimple,
and their already determined eigenvalues prove type~\(1\).
The displayed \(q\)-character formula follows from the
simultaneous generalized eigenspaces just identified.
\end{proof}

\subsection{The positive basic presentation}

Write the normalized ordinary flag complexes as
\begin{equation}\label{eq:ordinary-flag-decomp}
 \cL_\gamma=(\pi_\gamma)_*\C_{X_\gamma}[\dim X_\gamma]
    =\bigoplus_{\alpha,d}\IC_\alpha[-d]\otimes V^d_{\gamma,\alpha}.
\end{equation}
The sources $X_\gamma$ are smooth and the maps $\pi_\gamma$ are proper.
The decomposition theorem gives the displayed finite decomposition in
each finite corner. The $\IC_\alpha$ are distinct simple perverse
complexes, with their local systems included in the index $\alpha$.
Let $G$ be their direct sum. Perverse orthogonality and Schur's lemma give
\begin{equation}\label{eq:positive-basic}
 B=\Ext^*(G,G),\qquad B_{<0}=0,
             \qquad B_0=\bigoplus_\alpha\C e_\alpha.
\end{equation}
The additive categories generated by all shifts of the $\cL_\gamma$
and by all shifts of the $\IC_\alpha$ agree. Their Ext algebras are
therefore graded Morita equivalent. The spaces
$V^d_{\gamma,\alpha}$ occur in the projective generator of this
equivalence. They account for the internal grading of a geometric
simple and must be retained when discussing composition shifts.

Under this equivalence the trace ideal and the central-power ideal
defining $\cA_S$ induce homogeneous ideals. They can impose positive-degree
relations in addition to deleting basic idempotents. Quotienting a
nonnegative algebra by a homogeneous ideal nevertheless preserves a
nonnegative grading and semisimple degree zero. We obtain a finite
basic algebra $B'$ graded Morita equivalent to $\cA_S$, with
\begin{equation}\label{eq:positive-basic-finite}
 B'_{<0}=0,\qquad B'_0=\bigoplus_{\alpha\in\Lambda}\C e_\alpha,
                          \qquad\rad B'=B'_{>0}.
\end{equation}
A surviving simple has the same geometric multiplicity spaces in
\eqref{eq:ordinary-flag-decomp}; the extra positive relations can
change extensions but do not change those spaces.

\subsection{Bar invariance and the highest-line normalization}

Verdier duality fixes every $\cL_\gamma$, since the flag source is smooth
and the map is proper. It permutes the simple perverse labels by
$\alpha\mapsto\alpha^*$ and gives
\begin{equation}\label{eq:Verdier-multiplicity}
             V^d_{\gamma,\alpha^*}\cong
                          (V^{-d}_{\gamma,\alpha})^*.
\end{equation}
Verdier duality fixes each normalized flag complex and hence each
geometric idempotent. It preserves the homogeneous trace ideal in
Theorem~\ref{fil:same-algebra}. On the equivariant coefficient ring
it preserves degree and the augmentation ideal; a possible sign on
a Chern class preserves the same ideal. It therefore preserves the
chosen power of that ideal and descends to \(\cA_S\).
Thus the basic simple at degree zero with label $\alpha$ is dual to
the basic simple at degree zero with label $\alpha^*$.

\begin{proposition}\label{norm:bar}
For every dominant monomial $n$ in type $G_2$ or $F_4$, the intrinsic
character
\begin{equation}\label{eq:intrinsic-character}
                         S_t(n)=\cht\gr N(n)
\end{equation}
is the character of the degree-zero normalized basic simple. In
particular,
\begin{equation}\label{eq:intrinsic-bar}
       \overline{S_t(n)}=S_t(n),\qquad
                         S_t(n)|_{t=1}=\chq L(n).
\end{equation}
Its highest monomial has coefficient one in degree zero.
\end{proposition}

\begin{proof}
Work in any fixed standard module in which $L(n)$ occurs; one may
take $M(n)$. Equation~\eqref{eq:Verdier-multiplicity} makes the
ungraded geometric-idempotent dimensions of the two normalized basic
simples equal. By the current comparison these dimensions are their
ordinary $q$-characters. Lemma~\ref{norm:simple} makes both pullbacks simple type~$1$
modules with these characters. Injectivity of the $q$-character map
therefore fixes their label.
Thus $\alpha^*=\alpha$ on the relevant simples, and their geometric
graded characters are palindromic.

The highest $\ell$-weight space is one-dimensional. If its degree in
the normalized basic simple is $r$, its coefficient is $t^r$.
Palindromy forces $r=0$. Every graded lift of the same ungraded
$\cA_S$-simple is a shift of this lift, because its basic version is a
one-dimensional simple concentrated in a single degree. By
Corollary~\ref{fil:graded-simple}, $\gr N(n)$ is such a lift, and its intrinsic
highest line has degree zero. Hence the shift is zero. This proves
bar invariance and the normalization. The specialization follows
from the actual isomorphism $N(n)\cong L(n)$ and the dimensions of
the finite filtration.
\end{proof}

The construction is independent of the larger standard module used to
realize $L(n)$. Indeed the intrinsic filtration is defined using the
ordinary highest rational function of $n$; the comparison with a larger
framing proves compatibility with its inherited action, and the
highest-line normalization removes the possible shift. This normalization applies in every standard module containing
$L(n)$.

\section{Exceptional fundamental characters}\label{sec:fundamental}

\begin{theorem}\label{thm:fundamental}
For $\g$ of type $G_2$ or $F_4$, every $i\in I$, and every
$a\in\C^\times$, one has
\begin{equation}\label{eq:fund-calibration}
                 S_t(Y_{i,a})=F_t(Y_{i,a}).
\end{equation}
\end{theorem}

\begin{proof}
The ordinary $G_2$ fundamental characters at the short and long nodes
have respectively $7$ and $15$ distinct monomials, all of multiplicity one
\cite[Lemma~3.1]{LM13}. Every nonzero intrinsic graded coefficient is
therefore one-dimensional. Proposition~\ref{norm:bar} forces it to be
$1$. For the formal fundamental character, each coefficient lies in
$\N[t^{\pm1}]$ by \cite[Theorem~7.5]{Her05}, and its value at $t=1$
is the corresponding ordinary multiplicity by
\cite[Theorem~6.2]{Her04}. A nonnegative Laurent polynomial with
value one at $1$ is a single monomial $t^r$.
The bar invariance of \eqref{eq:fund-formal} forces $r=0$.
Coefficients with value zero vanish. This proves the assertion in
type $G_2$.

For type $F_4$ use Hernandez's numbering
\begin{equation}\label{eq:F4-Cartan}
 C=\begin{pmatrix}
 2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2
 \end{pmatrix},\qquad(d_1,d_2,d_3,d_4)=(1,1,2,2).
\end{equation}
This is the numbering of Section~\ref{cou:section} reversed by
\(i\mapsto5-i\).
The known fundamental characters have the following multiplicities
\cite[Sections~7--8]{Her05}, \cite{HS}:
\begin{center}
\begin{tabular}{crrr}
\toprule
Node & Distinct monomials & Ordinary dimension & Multiplicity-two monomials\\
\midrule
1 & 26 & 26 & 0\\
2 & 283 & 299 & 16\\
3 & 1532 & 1703 & 171\\
4 & 53 & 53 & 0\\
\bottomrule
\end{tabular}
\end{center}
There are no other multiplicities. The formal coefficient of a thin
monomial is $1$, and that of each of the $187$ remaining monomials is
$t+t^{-1}$. The full list of the latter monomials, together with the
calculation needed below, is recorded in Appendix~\ref{app:F4}.

We may translate the spectral parameter and take the highest monomial
to be $Y_{i,1}$ in integer spectral notation. Write a monomial in this
character uniquely as
\[
                        n=Y_{i,1}\prod_{j,s}A_{j,s}^{-\nu_{j,s}}.
\]
The signed pole divisor of \eqref{eq:highest-rational} has a positive
entry at $(i,1+d_i)$ and a negative entry at $(i,1-d_i)$. Thus
\begin{equation}\label{eq:F4-center}
 L=\nu_{i,1-d_i},\qquad
 d_{\bp}(\bnu)=\nu_{i,1+d_i}-\nu_{i,1-d_i}
   +\sum_{(j,s)\to(k,t)}\nu_{j,s}\nu_{k,t}
   -\sum_{j,s}\nu_{j,s}^2.
\end{equation}
For every multiplicity-two monomial, the division and the integer
calculation in Appendix~\ref{app:F4} give
\begin{equation}\label{eq:F4-finite-input}
                          L=0,\qquad d_{\bp}(\bnu)\equiv1\pmod2.
\end{equation}

The geometric quotient in this sector is consequently the whole
critical cohomology: the negative-framing Euler operator has rank
zero and is the identity. Its total dimension is two by
Theorem~\ref{alg:kernel} and the critical quotient theorem. The whole
critical support is projective, as in the projectivity statement for
the spectral $W_2$ model \cite[Proposition~5.8]{VV24}. Scaling a cut
direction gives an algebraic $\C^\times$-action of potential weight
one; it preserves cyclic stability. The critical Hard Lefschetz
theorem therefore applies to the whole critical support
\cite[Corollary~3.2]{DMSZ15}. With perverse-normalized vanishing
cycles its center is the smooth ambient dimension. Since $L=0$,
this is precisely $d_{\bp}(\bnu)$ in
\eqref{eq:F4-center}. The injective specialization into ordinary
cohomology also shows that the cohomology is even.

These facts determine the two-dimensional graded
coefficient completely. Its centered exponents are odd by
\eqref{eq:F4-finite-input}, and it is palindromic by
Proposition~\ref{norm:bar}. It has the form $t^r+t^{-r}$ with $r$ a
positive odd integer. If $r>1$, the Lefschetz isomorphism from the
lower nonzero degree to the upper one factors through a zero
intermediate degree, since cup product raises cohomological degree
by two. This is impossible. Hence $r=1$, giving $t+t^{-1}$.
Equivalently, the two-dimensional Lefschetz representation is the
irreducible representation of highest weight one; two trivial
summands have the wrong parity.

For a thin monomial, the graded coefficient is one-dimensional and
Proposition~\ref{norm:bar} gives $1$. Monomials absent from the
ordinary character have zero-dimensional local quotients and
contribute nothing. We have therefore identified every coefficient
with that of $F_t(Y_{i,1})$. Spectral translation proves
\eqref{eq:fund-calibration} for every $a$.
\end{proof}




\section{Restriction quotients and the standard character}
\label{std:bb-section}

We now calculate the grading of the actual standard obtained in
Theorem~\ref{fil:standard-lattice}.  The calculation uses ordinary
restriction to a closed union of complete attracting strata.
The coefficient complex on that closed locus is the restriction
of the ambient critical complex.

\subsection{A restriction-quotient theorem}

Let \(A=\Gm\) act on a smooth quasiprojective variety \(M\),
preserving a regular function \(f\).  Suppose that the relevant
critical support \(C=\crit(f)\) is projective and has an
\(A\)-linearized ample line bundle.  Write
\(\Phi=\Phi_f\Q_M\), with the unshifted convention
\(\Phi_0\Q=\Q\) used above.
For an ambient fixed component \(F_\alpha\subset M^A\), set
\[
 C_\alpha=C\cap F_\alpha,\qquad
 \Phi_\alpha=\Phi_{f|F_\alpha}\Q_{F_\alpha}.
\]
At a fixed point the derivative of an invariant function
vanishes on every nonzero tangent weight, so
\(C_\alpha=\crit(f|F_\alpha)\).
Let \(S_\alpha\subset C\) be its critical attracting stratum,
with inclusion \(\eta_\alpha\) and limit map
\(p_\alpha:S_\alpha\to C_\alpha\).
Denote by \(r_\alpha^+\) the rank of the positive tangent
weight part of \(T_M|_{F_\alpha}\).
We assume that \(H^*(C_\alpha,\Phi_\alpha)\) is even for
every \(\alpha\).

Let \(Z\subset C\) be closed, invariant, and a union of whole
strata \(S_\alpha\).  Let \(i:Z\hookrightarrow C\) and
\(j:U=C\setminus Z\hookrightarrow C\).  Call a fixed block
good if its stratum belongs to \(Z\).  Let
\[
 H_A=H_A^*(C,\Phi),\qquad R_0=H_A^*(\mathrm{pt})=\Q[a],
 \quad \deg_{\pol}a=1.
\]
Coefficients may subsequently be extended to \(\C\).
Let \(D:H_A\to H_A\) be \(R_0\)-linear and homogeneous of
polynomial degree \(L\).
Assume that after inverting \(a\), its action on ordinary
fixed-point restriction is invertible on every good block
and zero on every bad block.

\begin{theorem}\label{std:restriction-quotient}
The restriction map is a homogeneous surjection between finite
graded free \(R_0\)-modules, and
\begin{equation}\label{std:kernel-equality}
 \ker D
   =\ker\bigl(H_A^*(C,\Phi)\longrightarrow
                         H_A^*(Z,i^*\Phi)\bigr).
\end{equation}
Consequently there are natural graded isomorphisms
\begin{equation}\label{std:restriction-image}
 H_A/\ker D\simeq H_A^*(Z,i^*\Phi),
 \qquad
 (\im D)[-L]\simeq H_A^*(Z,i^*\Phi).
\end{equation}
Here a polynomial shift \([b]\) raises generator degrees by
\(b\).  If
\[
 h_\alpha(u)=
       \sum_k\dim H^{2k}(C_\alpha,\Phi_\alpha)u^k ,
\]
then
\begin{equation}\label{std:restriction-hilbert}
 \operatorname{Hilb}\bigl((H_A/\ker D)/a;u\bigr)
       =\sum_{\alpha\text{ good}}
                      u^{r_\alpha^+}h_\alpha(u).
\end{equation}
These assertions commute with completion at \(a=0\).
\end{theorem}

\begin{proof}
The hyperbolic localization theorem for vanishing cycles
gives the sheaf-level identity
\begin{equation}\label{std:hyperbolic}
 (p_\alpha)_!(\eta_\alpha)^*\Phi
       \simeq
       \Phi_\alpha[-2r_\alpha^+](-r_\alpha^+).
\end{equation}
In the present convention this follows by commuting vanishing
cycles with hyperbolic localization and then applying the
smooth constant-sheaf Bia\l{}ynicki--Birula calculation;
compatibility with the closed critical support gives the
displayed formula on \(C_\alpha\).
These are the complex-level statements of
\cite[Lemma~3.15, Proposition~6.2, Equation~(6.8), and
Theorem~7.1]{Des25}. If $r_\alpha^-$ is the negative tangent
rank, then $\dim M-\dim F_\alpha=r_\alpha^++r_\alpha^-$.
Removing the ambient shifts from the perverse-normalized formula
changes its cohomological shift to
\[
 (\dim F_\alpha-\dim M)-(r_\alpha^+-r_\alpha^-)
                         =-2r_\alpha^+,
\]
as in \eqref{std:hyperbolic}. The Tate twist does not
affect the cohomological Hilbert polynomial.
Since \(C_\alpha\) is proper, it gives
\begin{equation}\label{std:stratum-cohomology}
 H_c^k(S_\alpha,\Phi|_{S_\alpha})
       \simeq H^{k-2r_\alpha^+}(C_\alpha,\Phi_\alpha).
\end{equation}
Thus each stratum has even compactly supported cohomology
with these coefficients.

Choose a sufficiently high very ample power of the linearized
line bundle.  Its sections embed \(C\) equivariantly into
\(\mathbb P(V)\), where \(V=\bigoplus_\lambda V_\lambda\)
is a finite weight decomposition.  The closed sets
\[
 C\cap\mathbb P\left(\bigoplus_{\mu\geq\lambda}V_\mu\right)
\]
give a finite filtration by weights.  Each successive
difference is a union of attracting strata with one limiting
weight.  If several fixed components have that weight, their
inverse images under the limit map are open and closed in
this difference, and cohomology splits as a direct sum.
Intersect this filtration with \(Z\) and with \(U\).
The assumption that \(Z\) is a union of whole strata ensures
that these differences are still unions of whole good or
whole bad strata.  Compact-support excision and
\eqref{std:stratum-cohomology}, by induction on this finite
filtration, give
\[
 H_c^{\mathrm{odd}}(Z,i^*\Phi)
      =H_c^{\mathrm{odd}}(U,j^*\Phi)=0.
\]
They also give, since \(Z\) is proper,
\begin{equation}\label{std:ordinary-good-polynomial}
 \operatorname{Hilb}H^*(Z,i^*\Phi)
      =\sum_{\alpha\text{ good}}
                     u^{r_\alpha^+}h_\alpha(u).
\end{equation}
Evenness is used here for bad as well as good fixed blocks.

The closed/open triangle
\[
 j_!j^*\Phi\longrightarrow\Phi
           \longrightarrow i_*i^*\Phi\xrightarrow{+1}
\]
therefore gives short exact sequences in even cohomological
degrees:
\begin{equation}\label{std:ordinary-restriction-sequence}
 0\longrightarrow H_c^*(U,j^*\Phi)
   \longrightarrow H^*(C,\Phi)
   \xrightarrow{\rho}H^*(Z,i^*\Phi)
   \longrightarrow0 .
\end{equation}
Compact support in the first term follows from properness of
\(C\).  In particular, \(H^*(C,\Phi)\) is also even.

Apply the same triangle on finite Borel approximations.
For each of its three terms the Borel spectral sequence has
\[
 E_2^{p,q}=H^p(BA)\otimes H^q(\text{ordinary term}).
\]
Both indices are even.  A differential of odd length has
target in an odd column, and one of even length has target in
an odd row.  All differentials vanish.  The base \(BA\) is
simply connected, so no nonconstant local system enters this
formula.  Lifting a homogeneous ordinary basis gives a free
graded \(R_0\)-basis.  Taking the stable range of the Borel
approximations gives an exact sequence of finite graded free
modules
\begin{equation}\label{std:equivariant-restriction-sequence}
 0\longrightarrow H_{c,A}^*(U,j^*\Phi)
  \longrightarrow H_A^*(C,\Phi)
  \xrightarrow{\rho_A}H_A^*(Z,i^*\Phi)
  \longrightarrow0 .
\end{equation}
Compact support here is in the variety direction of the
Borel approximation.  Reduction modulo \(a\) recovers
\eqref{std:ordinary-restriction-sequence}.

Write \(k:C^A\hookrightarrow C\) and
\(k_\alpha:C_\alpha\hookrightarrow C\) for the inclusions.
Let \(K_0=\Q(a)\).  Equivariant localization, naturally
applied to \(\rho_A\), uses ordinary restriction of \(\Phi\)
to \(C^A\):
\[
 H_A\otimes K_0
   \simeq
 H_A^*(C^A,k^*\Phi)\otimes K_0.
\]
On a good fixed block, restriction to \(Z\) is the identity;
on a bad block its target is zero.  Therefore
\[
 (\ker\rho_A)\otimes K_0
  =\bigoplus_{\alpha\text{ bad}}
        H_A^*(C_\alpha,k_\alpha^*\Phi)\otimes K_0
  =(\ker D)\otimes K_0 .
\]
Both kernels are saturated submodules of the finite free
module \(H_A\).  For \(\ker\rho_A\) this follows from the
freeness of the quotient in
\eqref{std:equivariant-restriction-sequence}.
For \(\ker D\) it follows because its quotient is
isomorphic to \(\im D\), a torsion-free submodule of \(H_A\).
A saturated submodule is the intersection of its fraction-field
span with \(H_A\).  The equality of those spans therefore
proves \eqref{std:kernel-equality}.
The map
\[
 \rho_A(h)\longmapsto Dh
\]
is now well defined, bijective, and homogeneous of degree \(L\).
This proves \eqref{std:restriction-image}.
Reduction modulo \(a\), together with
\eqref{std:ordinary-good-polynomial}, proves
\eqref{std:restriction-hilbert}.
Flatness of completion preserves each finite-module exact
sequence and the actual graded identifications.
\end{proof}

In the geometric Euler case the map in
\eqref{std:restriction-image} has a direct construction.
If \(D=e_A(E)\), and \(Z\) is the zero set of an equivariant
section of \(E\), that section is nowhere zero on \(U\).
Thus \(D|_U=0\), and the projection formula gives
\[
 D\,j_!(\alpha)=j_!\bigl((D|_U)\alpha\bigr)=0.
\]
Multiplication by \(D\) factors through \(\rho_A\)
integrally.  On good fixed blocks its factor is invertible
after localization, so the kernel of the factored map is
torsion.  Its source is free, hence that kernel is zero.
This gives the same isomorphism for the section under consideration.

\subsection{Whole good strata for ordered fundamental factors}

We verify the hypotheses above for the Euler quotient
\eqref{fil:family-euler}.  It is convenient to use the
cocyclic form of the framed geometry, obtained by dualizing
every vector space and every arrow.  Spectral vertex labels
are retained.

Write \(m_r=Y_{i_r,bq^{s_r}}\) in one orbit, with
\[
 s_1\leq\cdots\leq s_N.
\]
For several orbits choose any total order extending these
individual orders.  Terms in different orbits have disjoint
support.  The positive and negative framing vertices of
factor \(r\) are
\[
 v_r^+=(i_r,s_r+d_{i_r}),\qquad
 v_r^-=(i_r,s_r-d_{i_r}).
\]
Let \(I_r^+\) and \(I_r^-\) be their graded injective modules
in the critical representation category. The arrow
\(\epsilon_{i_r}:v_r^-\to v_r^+\) in the cocyclic quiver induces a
morphism of injective modules
\[
 \varphi_r:I_r^+\longrightarrow I_r^-,
 \qquad K_r=\ker\varphi_r .
\]
Put \(I^+=\bigoplus_rI_r^+\), \(K=\bigoplus_rK_r\).
The critical support is described by submodules
\(V\subset I^+\) of the prescribed dimension.
The proper equivariant comparison with this injective
Grassmannian is the one in \cite[Proposition~5.8]{VV24}.
Its underlying proper bijection suffices for the support
and limiting arguments below.

\begin{lemma}\label{std:strict-support}
The fundamental kernel \(K_r\) is supported in spectral
grades strictly greater than \(s_r\).  In particular,
\begin{equation}\label{std:kernel-support}
 (K_r)_{v_r^-}=0,\qquad
 (K_s)_{v_r^-}=0\quad(r<s).
\end{equation}
Moreover, for every finite-dimensional critical module \(V\),
\begin{equation}\label{std:injective-adjunction}
 \Hom(V,I_r^-)\simeq(V_{v_r^-})^*.
\end{equation}
\end{lemma}

\begin{proof}
The graded multiplicities of the fundamental kernel are
\[
 \dim(K_r)_{(j,s_r+u)}=
 \begin{cases}
 \widetilde c_{j i_r}(u),&0\leq u\leq r_{\g}h^\vee,\\
 0,&\text{otherwise},
 \end{cases}
 \qquad r_{\g}=\max_i d_i .
\]
This follows from \cite[Equation~(5.1), Lemma~5.3, and
Equation~(5.3)]{FM21}. Proposition~4.2(4) of that work
gives zero coefficient at \(u=0\), proving the strict support
assertion.
If \(r<s\) in the same orbit, then
\(s_r-d_{i_r}<s_s\); if they belong to different orbits,
their supports are disjoint.  This proves
\eqref{std:kernel-support}.
Finally \(I_r^-\) is the coinduced injective at the indicated
vertex, so evaluation of a morphism on that vertex gives
the adjunction \eqref{std:injective-adjunction}.
\end{proof}

Choose distinct cocyclic flavour weights
\[
 c_1^{\mathrm{out}}>\cdots>c_N^{\mathrm{out}}.
\]
The two framings of factor \(r\) have the same weight.
An attracting limit of a submodule \(V\subset I^+\) is a
direct sum \(\bigoplus_rU_r\), with \(U_r\subset I_r^+\).
Call this allocation good when \(U_r\subset K_r\) for every
\(r\).
Let \(Z\) be the locus \(V\subset K\).

\begin{proposition}\label{std:whole-strata}
The locus \(Z\) is closed and is the union of all whole good
critical attracting strata.  Goodness is a condition on an
entire fixed allocation.  For that allocation it is equivalent
to
\begin{equation}\label{std:good-dimension}
 (\nu_r)_{v_r^-}=0\quad\text{for all }r.
\end{equation}
\end{proposition}

\begin{proof}
Let \(F_rI^+=\bigoplus_{s\leq r}I_s^+\).
The filtration of an attracting submodule by \(V\cap F_rI^+\)
has associated graded \(U_r\).  Suppose first that its limit
is good.  The component map \(\varphi_r|_V\) vanishes on
\(V\cap F_rI^+\): its projection to \(I_r^+\) is \(U_r\),
which lies in \(K_r\), and the earlier summands have zero
\(r\)-component.
Thus it factors through \(V/(V\cap F_rI^+)\).
At the vertex \(v_r^-\), the dimension of this quotient is
\[
 \sum_{s>r}\dim(U_s)_{v_r^-}=0
\]
by \eqref{std:kernel-support}.
The injective adjunction
\eqref{std:injective-adjunction} implies that the factored
map is zero.  This holds for every \(r\), so all components
of \(\bigoplus_r\varphi_r\) vanish on \(V\), and \(V\subset K\).

Conversely, \(Z\) is closed and flavour invariant.  The
attracting limit of any point of \(Z\) therefore remains in
\(Z\); its summands lie in their corresponding \(K_r\).
This proves the equality with the union of whole good
strata.
It can be checked on the reduced underlying supports through
the proper equivariant Grassmannian comparison: an equivariant
proper bijection carries each submodule limit to the same
critical limit.

Closedness also has an intrinsic algebraic description on the
framed moduli space. Let \(W_r\) be the one-dimensional framing
space, let \(j_r:\cV_{v_r^+}\to W_r\) be its outgoing framing,
and let \(\epsilon_{i_r}:\cV_{v_r^-}\to\cV_{v_r^+}\) be the
corresponding arrow in the cocyclic convention.
Their composition is a section of
\(\Hom(\cV_{v_r^-},W_r)\), given by
\[
 j_r\epsilon_{i_r}:\cV_{v_r^-}\longrightarrow W_r.
\]
On the critical locus it is exactly \(\varphi_r|_V\) under
coinduction.  Its simultaneous zero set has underlying
closed set \(Z\).  The vector bundle and its section are
therefore defined on the moduli space itself.

If \((\nu_r)_{v_r^-}=0\), the adjunction shows that every
submodule of that dimension in \(I_r^+\) is killed by
\(\varphi_r\), and hence lies in \(K_r\).
If \((\nu_r)_{v_r^-}>0\), no such submodule can lie in
\(K_r\), by \eqref{std:kernel-support}.
This proves \eqref{std:good-dimension} and proves that
goodness does not cut out a proper part of a fixed block.
\end{proof}

\begin{lemma}\label{std:euler-good-bad}
After inverting the equivariant parameter \(a\), the Euler
operator is invertible on every good fixed block and is zero
on every bad fixed block, acting on the ordinary restriction
of the ambient critical complex.
\end{lemma}

\begin{proof}
At a fixed decomposition \(V=\bigoplus_sV^s\), the numerator
bundle decomposes into the pieces
\(\Hom(\cV^s_{v_r^-},W_r)\).
If \(r\ne s\), this bundle has nonzero flavour weight
\(c_r^{\mathrm{out}}-c_s^{\mathrm{out}}\).
Its Euler class is a nonzero power of
\((c_r^{\mathrm{out}}-c_s^{\mathrm{out}})a\) times one plus
a nilpotent positive Chern class, and is invertible after
inverting \(a\).  Positive ordinary cohomology acts
nilpotently on the bounded cohomology of the proper fixed
support, also with the restricted coefficient complex.

For a self piece \(r=s\), condition
\((\nu_r)_{v_r^-}=0\) means that the bundle has rank zero
and Euler class one.  If its rank is positive, the canonical
section \(s_r\) is nowhere zero on the whole fixed support:
a zero would give a submodule of \(K_r\) with a nonzero
component at \(v_r^-\), contradicting
Lemma~\ref{std:strict-support}.
This section is equivariant of weight zero.  It splits off
an equivariantly trivial line locally, with a vector bundle
quotient of rank one less.  The top equivariant Chern class
is consequently zero.
This assertion applies on the reduced support even if the
critical scheme is singular.  In a local trivialization,
one component of a nowhere-zero section is invertible,
giving the locally free quotient just used.
It follows that the self Euler factor acts by zero on
every coefficient complex restricted to this support.
Combining the self and cross factors proves the result.
\end{proof}

All fixed critical blocks, including the bad blocks, have
even critical cohomology.  Indeed, the fixed ambient variety
is the product of the one-framing ambient varieties, and
the restricted potential is the sum of their potentials.
Thom--Sebastiani and the evenness in
Theorem~\ref{geo:homogeneous-generation} prove the assertion.
The critical support is projective, and the ambient GIT
ample bundle restricts to an ample linearized bundle on it.
Proposition~\ref{std:whole-strata} and
Lemma~\ref{std:euler-good-bad} therefore verify every
hypothesis of Theorem~\ref{std:restriction-quotient}.
In particular, in source degree,
\begin{equation}\label{std:actual-standard-restriction}
 \cQ_{0,\nu}
     \simeq H_A^*(Z_\nu,i_\nu^*\Phi).
\end{equation}
This identifies the retained Euler quotient, including its
classes that survive only through the integral family.

\subsection{The tangent shift and the quantum-torus exponent}

We keep track of the chamber and the degree centre.
Vertexwise duality from the incoming model to the cocyclic
model sends
\begin{equation}\label{std:duality-sign}
 y^{\mathrm{out}}=-y^{\mathrm{in}},\qquad
 c_r^{\mathrm{out}}=-c_r^{\mathrm{in}}.
\end{equation}
It reverses arrows but does not replace a spectral label \(s\)
by \(-s\).  Thus the chosen decreasing cocyclic flavour weights
are obtained by choosing the increasing incoming deformation
weights \(c_r^{\mathrm{in}}=-c_r^{\mathrm{out}}\) in
\eqref{fil:family-euler}.  The integral construction required
only their distinctness.  The incoming Euler factor transforms
as
\[
 y^{\mathrm{in}}-c_r^{\mathrm{in}}a
      =c_r^{\mathrm{out}}a-y^{\mathrm{out}},
\]
which is the Euler factor of
\(\Hom(\cV^{\mathrm{out}}_{v_r^-},W_r^{\mathrm{out}})\).
If the variables in both conventions are renamed to the
same symbols instead, their Euler polynomials differ by
the scalar \((-1)^{L_\nu}\).  Neither this scalar nor the
coordinate change \(a\mapsto-a\) changes a polynomial
degree or inverts the character variable \(t\).

Let \(\bm^{(r)}\), \(\bl^{(r)}\), and
\(\bp^{(r)}=\bm^{(r)}-\bl^{(r)}\) be the framing vectors
of factor \(r\), and let
\(\nu=\sum_r\nu_r\) be a fixed allocation.
Define
\begin{align}
 \beta^m_{rs}
  &=\bm^{(r)}\cdot\nu_s
      +E(\nu_r,\nu_s)-\nu_r\cdot\nu_s,\label{std:beta-m}\\
 \beta^p_{rs}
  &=\beta^m_{rs}-\bl^{(r)}\cdot\nu_s.\label{std:beta-p}
\end{align}
The form \(E\) is still that of the incoming computational
quiver.  The actual cocyclic quiver is its opposite.

\begin{lemma}\label{std:tangent-ranks}
In the chosen cocyclic chamber the positive and negative
ambient tangent ranks at the fixed allocation are
\begin{equation}\label{std:tangent-formula}
 r^+=\sum_{r<s}\beta^m_{rs},\qquad
 r^-=\sum_{r<s}\beta^m_{sr}.
\end{equation}
On a good allocation,
\begin{equation}\label{std:cross-euler-degree}
 \bl^{(r)}\cdot\nu_s=0\quad(r\leq s),\qquad
 L_\nu=\sum_{r<s}\bl^{(s)}\cdot\nu_r .
\end{equation}
\end{lemma}

\begin{proof}
The cocyclic flavour action sends an outgoing framing \(j\)
to \(A(t)j\).  Gauge action sends it to \(jg^{-1}\).
To return a fixed representative to itself one therefore
uses the compensating gauge matrix
\[
 g(t)=\bigoplus_rt^{c_r^{\mathrm{out}}}
                         \id_{V^r}.
\]
An arrow variation from \(V^s\) to \(V^r\), and a framing
variation from \(V^s\) to \(W^r\), have weight
\(c_r^{\mathrm{out}}-c_s^{\mathrm{out}}\).
For \(r<s\) it is positive.
The framing contribution has dimension
\(\bm^{(r)}\cdot\nu_s\).
Because the arrows are reversed in the cocyclic model, their
contribution is \(E(\nu_r,\nu_s)\).
The gauge tangent contribution to be subtracted is
\(\nu_r\cdot\nu_s\).
Stability makes the gauge tangent map injective; the quotient
tangent sequence is exact in each weight.
Thus the positive block rank is precisely
\(\beta^m_{rs}\).
Reversing \(r,s\) gives the negative block.
Summing proves \eqref{std:tangent-formula}.

For a good allocation \(U_s\subset K_s\).
The strict support assertion
\eqref{std:kernel-support}, including the case \(r=s\),
gives \((\nu_s)_{v_r^-}=0\) for \(r\leq s\).
Each \(\bl^{(r)}\) is the numerator framing at this vertex.
This proves the first assertion of
\eqref{std:cross-euler-degree}.  Expanding
\(L_\nu=\sum_{r,s}\bl^{(r)}\cdot\nu_s\)
then gives the second.
\end{proof}

For a good factor allocation, let $C_{r,\nu_r}$ be the critical support
of the $r$th fundamental framed model and let $\Phi_{r,\nu_r}$ be the
restriction of its unshifted vanishing-cycle complex to that support. Set
\[
 h_{r,\nu_r}(u)
   =\sum_k\dim H^{2k}(C_{r,\nu_r},
                               \Phi_{r,\nu_r})u^k,
 \qquad d_r=d_{\bm^{(r)}}(\nu_r).
\]
Its self numerator Euler bundle has rank zero.
Hence this is also the polynomial Hilbert function of
the corresponding intrinsic fundamental quotient, and
its centre \(d_r\) equals \(d_{\bp^{(r)}}(\nu_r)\).
Let $\gr_{\pol}$ denote the associated graded for the uncentered
polynomial-degree filtration. Equation~\eqref{fil:strict-specialization}
identifies $\gr_{\pol}Q(m)$ with $\cQ_0/a\cQ_0$. Thom--Sebastiani and
\eqref{std:restriction-hilbert} give
\begin{equation}\label{std:uncentred-standard}
 h_{\nu}(u):=
 \operatorname{Hilb}\bigl((\gr_{\pol}Q(m))_\nu;u\bigr)
  =\sum_{\substack{\nu=\sum_r\nu_r\\
                   \text{good allocations}}}
       u^{\sum_{r<s}\beta^m_{rs}}
                 \prod_r h_{r,\nu_r}(u).
\end{equation}
The ambient dimension expands as
\[
 d_{\bm}(\nu)
    =\sum_rd_r+
       \sum_{r<s}(\beta^m_{rs}+\beta^m_{sr}),
 \qquad d_{\bp}(\nu)=d_{\bm}(\nu)-L_\nu .
\]
Using \eqref{std:cross-euler-degree}, multiplication by the
normalizing centre and the substitution \(u=t^2\) yield
\begin{equation}\label{std:centred-standard}
 t^{-d_{\bp}(\nu)}h_\nu(t^2)
   =\sum_{\text{good allocations}}
      t^{\sum_{r<s}(\beta^p_{rs}-\beta^p_{sr})}
      \prod_r\left[t^{-d_r}h_{r,\nu_r}(t^2)\right].
\end{equation}
Every exponent in this formula is determined by the
ambient tangent sequence and the source Euler quotient.

\begin{lemma}\label{std:gamma-identity}
For \(M_r=m_rA^{-\nu_r}\), the quantum-torus form in the
conventions of the introduction satisfies
\begin{equation}\label{std:gamma-formula}
 \gamma(M_r,M_s)-\gamma(m_r,m_s)
       =2(\beta^p_{rs}-\beta^p_{sr}).
\end{equation}
\end{lemma}

\begin{proof}
The root-monomial identities for the torus form are
\begin{equation}\label{std:gamma-roots}
 \gamma(m_r,A^{\nu_s})
       =-2\bp^{(r)}\cdot\nu_s,\qquad
 \gamma(A^{\nu_r},A^{\nu_s})
       =2\bigl(E(\nu_r,\nu_s)-E(\nu_s,\nu_r)\bigr).
\end{equation}
They follow by substituting a fundamental monomial and a
root monomial into the inverse-Cartan definition of
\(\gamma\); equivalently they are
\cite[Section~2.6, Equation~(5)]{FQ26} in the present
spectral notation.  Bilinearity and skew symmetry give
\begin{align*}
 \gamma(M_r,M_s)-\gamma(m_r,m_s)
   &=-\gamma(m_r,A^{\nu_s})
       -\gamma(A^{\nu_r},m_s)
       +\gamma(A^{\nu_r},A^{\nu_s})\\
   &=2\bigl(\bp^{(r)}\cdot\nu_s
            -\bp^{(s)}\cdot\nu_r
            +E(\nu_r,\nu_s)-E(\nu_s,\nu_r)\bigr).
\end{align*}
The symmetric terms \(\nu_r\cdot\nu_s\) cancel in
\(\beta^p_{rs}-\beta^p_{sr}\), proving the formula.
\end{proof}

\begin{theorem}\label{std:standard-character}
For every dominant monomial \(m\), the filtered actual
standard constructed above satisfies
\begin{equation}\label{std:bar-standard}
 \cht\gr Q(m)=\overline{E_t(m)}.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:fundamental}, the centered fundamental
polynomials
\[
 \sum_{\nu_r\text{ good}}
       t^{-d_r}h_{r,\nu_r}(t^2)\,
                        m_rA^{-\nu_r}
\]
are the fundamental characters \(F_t(m_r)\).
Formula \eqref{std:centred-standard}, together with
\eqref{std:gamma-formula} and the product convention
\(M*N=t^{\gamma(M,N)/2}MN\), therefore says that
\[
 \cht\gr Q(m)
   =t^{-\frac12\sum_{r<s}\gamma(m_r,m_s)}
          F_t(m_1)*\cdots*F_t(m_N).
\]
This is the product in increasing spectral order, normalized
so that its highest monomial has coefficient one.
The definition of \(E_t(m)\) uses decreasing spectral order
\cite[Section~2.7, Equation~(7)]{FQ26}.
The bar is an anti-involution, each \(F_t(m_r)\) is
bar invariant, and the normalized highest coefficient is
preserved.  Its bar is consequently the increasing product
displayed above, proving \eqref{std:bar-standard}.

Equal spectral exponents introduce no choice: the fundamental
characters at the same exponent commute, as in the statement
following Equation~(7) of \cite{FQ26}.
Distinct \(q^\Z\)-orbits have no arrows or root-torus pairings
between them, so the same formulas factor over their finitely
many components.  Thus the computation applies to arbitrary
nonzero complex spectral parameters.
\end{proof}

The quadratic pairs introduced by the polynomial augmentation
give the same centered formula.  Here is the cancellation
explicitly.  For an orthogonal pair the extra arrow spaces
are \(T=\Hom(V_i,V_j)\) and its dual, and their potential is
the nondegenerate trace pairing \(2\Tr(AB)\).
If \(\rk T=h\), its critical equations are \(A=B=0\);
on this locus stability is exactly stability for the reduced
quiver.  In a neighborhood of the critical support the added
normal directions are therefore the vector bundle
\(T\oplus T^*\) with this hyperbolic quadratic function.
The relative quadratic Thom calculation identifies its
unnormalized critical complex with the reduced complex
shifted by \([-2h](-h)\).  Thus it multiplies the polynomial
Hilbert series by \(u^h\).  The smooth ambient dimension has
increased by \(2h\), and the numerator Euler bundle has not
changed.  With \(u=t^2\), the additional factor \(t^{2h}\)
is cancelled by the additional centering factor \(t^{-2h}\).
This applies to each orthogonal pair and to every fixed
allocation.  Equivalently, their two opposite arrows change
the symmetric part of \(E\) but not the antisymmetric form
in \eqref{std:gamma-roots}.
One may therefore use the augmented computational quiver
consistently throughout the formulas or its reduced
geometric model.


\section{Canonical triangularity and the main theorem}\label{sec:closure}

\begin{proposition}\label{can:positive-triangle}
For every dominant monomial $m$ in type $G_2$ or $F_4$, there are
polynomials $p_{m,n}(t)\in t\N[t]$, only finitely many nonzero, such that
\begin{equation}\label{eq:positive-triangle}
 \overline{E_t(m)}=S_t(m)+\sum_{n<m}p_{m,n}(t)S_t(n).
\end{equation}
\end{proposition}

\begin{proof}
Let $Q(m)$ be the special fiber of the source-quotient standard lattice.
Its ordinary action is the action on $M(m)$, and its associated graded
is a module for the same finite graded algebra $\cA_S$ as the intrinsic
simple subquotients. Each spectral idempotent $e_\gamma$ has degree
zero, so
\[
 \dim e_\gamma\gr Q(m)=\dim e_\gamma Q(m).
\]
Lemma~\ref{fil:chern-operators} gives a type~$1$ ordinary action on
$\gr Q(m)$. The Cartan calculation of Lemma~\ref{norm:simple}
then gives $\chq(\gr Q(m))=\chq M(m)$.
Additivity and injectivity of the $q$-character imply equality of their
ordinary composition multiplicities. Thus every graded simple
composition factor is a shift of an intrinsic constituent lift,
by Corollary~\ref{fil:graded-simple} and
Proposition~\ref{norm:bar}.
The homogeneous generation theorem
\ref{std:graded-cyclicity} says that $\gr Q(m)$ is generated by its
degree-zero highest line and has the unique simple head $L(m)$.
Theorem~\ref{std:standard-character} gives
\begin{equation}\label{eq:graded-standard-final}
                       \cht\gr Q(m)=\overline{E_t(m)}.
\end{equation}

Pass to the finite basic algebra $B'$ of
\eqref{eq:positive-basic-finite}. Proposition~\ref{norm:bar} identifies
the intrinsic character of each simple with its normalized basic
character; its highest line has degree zero. Thus the simple head of
the basic module corresponding to $\gr Q(m)$ is the basic simple
with label $m$ at external degree zero. Lift a basis vector of this
head homogeneously and apply $e_m$ to the lift. It defines a graded
map from $B'e_m$ to the basic module which is surjective modulo
the radical, hence surjective by graded Nakayama.
The degree-zero part of $B'e_m$ is just its head,
and its radical is concentrated in positive degree. Every other
graded composition factor of its quotient consequently occurs in a
strictly positive external shift. These shifts give the coefficients
$p_{m,n}(t)\in t\N[t]$.

All composition labels satisfy $n\leq m$, because their highest
monomials occur in the ordinary standard character. The coefficient
of $m$ in that character is one, already accounted for by the head,
so the remaining labels satisfy $n<m$. There are finitely many
composition factors. Transferring the character equality back
through the graded Morita equivalence retains the internal spaces
$V^d_{\gamma,\alpha}$ of \eqref{eq:ordinary-flag-decomp} and proves
\eqref{eq:positive-triangle}.
\end{proof}

\begin{theorem}\label{can:identification}
For type $G_2$ or $F_4$ and every dominant monomial $m$, the intrinsic
graded character belongs to the quantum Grothendieck ring and equals
its canonical element:
\begin{equation}\label{eq:canonical-identification}
                            S_t(m)=L_t(m)\in\K.
\end{equation}
\end{theorem}

\begin{proof}
We use induction on the nonnegative integer $h(m)$ in
\eqref{eq:height}. The trivial monomial has $S_t(1)=E_t(1)=1$.
Suppose the assertion has been established at all smaller heights.
Every lower dominant label in \eqref{eq:positive-triangle} has
smaller height by \eqref{eq:height-drop}. Since $\K$ is bar-stable,
that equality first gives
\[
 S_t(m)=\overline{E_t(m)}-
                      \sum_{n<m}p_{m,n}(t)S_t(n)\in\K.
\]
Now use the bar invariance of Proposition~\ref{norm:bar} and apply
bar to \eqref{eq:positive-triangle}. We obtain
\begin{equation}\label{eq:negative-triangle}
 S_t(m)=E_t(m)-\sum_{n<m}p_{m,n}(t^{-1})S_t(n).
\end{equation}
By induction, every lower $S_t(n)$ has an expansion in the standard
basis with leading term $E_t(n)$ and all remaining coefficients in
$t^{-1}\Z[t^{-1}]$. The coefficient $p_{m,n}(t^{-1})$ belongs to
this same ideal. Substituting those finite expansions into
\eqref{eq:negative-triangle}, and using transitivity of the monomial
order, yields
\[
 S_t(m)-E_t(m)\in\sum_{n<m}t^{-1}\Z[t^{-1}]E_t(n).
\]
Together with $S_t(m)\in\K$ and bar invariance, these are exactly
the three properties in \eqref{eq:canonical-definition}.
Uniqueness proves \eqref{eq:canonical-identification}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
In types $G_2$ and $F_4$, Theorem~\ref{can:identification} and the
actual intrinsic realization give
\[
 L_t(m)|_{t=1}=S_t(m)|_{t=1}=\chq L(m).
\]
Every numerical step used only that $q$ is not a root of unity.
In particular the determinant inversions
\eqref{eq:cartan-determinants} and the corresponding Coulomb
Cartan inversions have only root-of-unity zeros.
The local wheel-ideal comparison in Theorem~\ref{alg:wheel}
holds at each such numerical parameter, and
Theorem~\ref{fil:standard-lattice} identifies the lattices before
collision specialization.

Suppose the spectral support of $m$ meets several
$q^\Z$-orbits. The spectral quiver is the disjoint union of the
corresponding quivers. Local resonance factors between different
orbits are units and the skew form between them is zero. The
constructions of the actual action, finite filtrations, ordinary
flag algebra, and standard character therefore apply to their
finite union. A lower dominant label introduces no new orbit, by
the argument following \eqref{eq:height-drop}. The same finite-height
induction proves the equality for arbitrary parameters. This
argument applies to the entire finite union of spectral orbits.

For simply-laced types the statement is the theorem of Nakajima
\cite{Nak01,Nak04}; for the remaining classical types it is
\cite[\S2.5 and Theorem~2.13]{FQ26}. To spell out the passage from
integral spectral notation, write $m=\prod_Om_O$ over its occupied
$q^\Z$-orbits. The skew form between distinct orbits vanishes, so
$E_t(m)=\prod_OE_t(m_O)$. The product $\prod_OL_t(m_O)$ belongs to
$\K$ and is bar-invariant. Multiplying the finite triangular
expansions shows that its highest term is $E_t(m)$ and that every
other standard-basis coefficient lies in $t^{-1}\Z[t^{-1}]$.
Canonical uniqueness therefore gives
\[
                         L_t(m)=\prod_OL_t(m_O).
\]
The ordinary spectral decomposition
$L(m)=\bigotimes_OL(m_O)$ \cite[\S2.5]{FQ26} now transfers
the one-orbit specialization to $m$. These cases and the two
exceptional cases exhaust the finite Dynkin types, proving the theorem.
\end{proof}

\begin{corollary}\label{can:geometric-realization}
In types $G_2$ and $F_4$, the coefficient of a monomial $n$ in $L_t(m)$
is the centered Hilbert polynomial of the corresponding intrinsic
critical-cohomology quotient of the actual module $L(m)$. In
particular it belongs to $\N[t^{\pm1}]$, is invariant under
$t\mapsto t^{-1}$, and its value at $t=1$ is the dimension of the
$\ell$-weight space $L(m)_n$.
\end{corollary}

\begin{proof}
The intrinsic construction assigns to the local sector with dimension
vector $\bnu$ its source-quotient filtration, centered by
$d_{\bp}(\bnu)$. Its graded dimensions are nonnegative integers and
their sum is $\dim L(m)_n$. Proposition~\ref{norm:bar} gives
palindromy, and Theorem~\ref{can:identification} identifies this
character with $L_t(m)$.
\end{proof}



\appendix
\section{A finite certificate for the nonthin \texorpdfstring{$F_4$}{F4} coefficients}
\label{app:F4}

We give the complete finite input to \eqref{eq:F4-finite-input}. The
numbering is \eqref{eq:F4-Cartan}, and the highest monomial is $Y_{i,1}$.
In the tables below $j_s$ denotes $Y_{j,s}$, and products are ordinary
commutative products. The row number is the zero-based number of the
monomial in the complete fundamental table \cite{HS}; the monomial itself
is included so the certificate is independent of those row numbers.
Every displayed monomial has ordinary multiplicity two. There are $16$
rows for node $2$ and $171$ rows for node $3$; nodes $1$ and $4$ have no
such rows. The last column is $d_{\bp}(\bnu)$. In every row
$\nu_{i,1-d_i}=0$.

Here is the full division procedure for verifying the table. Represent a
Laurent monomial by its finite exponent array $u_{j,s}$, and start with
\[
 r_{j,s}=\delta_{j,i}\delta_{s,1}-u_{j,s},\qquad \nu_{j,s}=0.
\]
Take the smallest $s$ with a nonzero entry of $r$. For each color $j$
at that index, put $a=r_{j,s}$, set
$\nu_{j,s+d_j}\leftarrow\nu_{j,s+d_j}+a$, and subtract $a$ times
the exponent array of $A_{j,s+d_j}$ from $r$. Repeat until $r=0$.
For the four root monomials used here the exponent arrays are explicitly
\begin{align*}
 A_{1,s}&=1_{s-1}1_{s+1}2_s^{-1},\\
 A_{2,s}&=2_{s-1}2_{s+1}1_s^{-1}3_s^{-1},\\
 A_{3,s}&=3_{s-2}3_{s+2}2_{s-1}^{-1}2_{s+1}^{-1}4_s^{-1},\\
 A_{4,s}&=4_{s-2}4_{s+2}3_s^{-1}.
\end{align*}
The unique lowest-degree factor of $A_{j,s+d_j}$ is $j_s$; all its
other factors have larger spectral degree. Each step is thus forced,
and it verifies uniqueness as well as the factorization. In every
listed case all the extracted $a$ are nonnegative and the remainder
is zero after finitely many steps. Finally evaluate
\begin{equation}\label{app:center-calculation}
 \nu_{i,1+d_i}-\nu_{i,1-d_i}-\sum_{j,s}\nu_{j,s}^2
 +\sum_{j,k=1}^4\sum_s\nu_{j,s}\nu_{k,s-d_jc_{jk}}.
\end{equation}
The final sum includes $j=k$ and $c_{jk}=0$, exactly as required by
the augmented spectral quiver.

For example, the first node-$2$ row is
$1_{10}2_7 2_9^{-1}2_{11}^{-1}4_6$. Its nonzero root exponents,
all equal to one, are
\[
 (1,3),(1,7),(2,2),(2,4),(2,6),(2,8),(2,10),
 (3,4),(3,6),(3,8),(4,6).
\]
Formula~\eqref{app:center-calculation} gives $9$, and the exponent at
$(2,0)$ is zero. This illustrates the same deterministic calculation
used in every row. As checks on the enumeration, the node-$2$ centers
have distribution
\[
 9^2,\ 11^2,\ 13^2,\ 15^1,\ 17^2,\ 19^1,\ 21^2,\ 23^1,\ 25^2,\ 29^1,
\]
where $d^r$ means that the value $d$ occurs $r$ times. Every center
in both tables is odd. The corresponding calculation on the full
fundamental support has $1894$ terminating divisions, $L=0$
in every row, and $1707$ thin coefficients. Only the
$187$ displayed rows and the known ordinary multiplicities are
needed in Theorem~\ref{thm:fundamental}.


\Needspace{10\baselineskip}
\subsection{The fundamental at node 2}
\begingroup\footnotesize
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{0.95}
\begin{longtable}{r>{\raggedright\arraybackslash}p{0.78\textwidth}r}
\toprule No. & Monomial & $d_{\bp}$\\\midrule\endfirsthead
\toprule No. & Monomial (continued) & $d_{\bp}$\\\midrule\endhead
\bottomrule\endfoot
70 & $1_{10}\,\allowbreak 2_{7}\,\allowbreak 2_{9}^{-1}\,\allowbreak 2_{11}^{-1}\,\allowbreak 4_{6}$ & 9\\
87 & $1_{12}^{-1}\,\allowbreak 2_{7}\,\allowbreak 2_{9}^{-1}\,\allowbreak 4_{6}$ & 9\\
89 & $1_{10}\,\allowbreak 2_{7}\,\allowbreak 2_{9}^{-1}\,\allowbreak 2_{11}^{-1}\,\allowbreak 3_{8}\,\allowbreak 4_{10}^{-1}$ & 11\\
105 & $1_{12}^{-1}\,\allowbreak 2_{7}\,\allowbreak 2_{9}^{-1}\,\allowbreak 3_{8}\,\allowbreak 4_{10}^{-1}$ & 11\\
109 & $1_{10}\,\allowbreak 2_{7}\,\allowbreak 3_{12}^{-1}$ & 13\\
120 & $1_{8}\,\allowbreak 1_{10}\,\allowbreak 2_{9}^{-1}\,\allowbreak 3_{8}\,\allowbreak 3_{12}^{-1}$ & 17\\
124 & $1_{12}^{-1}\,\allowbreak 2_{7}\,\allowbreak 2_{11}\,\allowbreak 3_{12}^{-1}$ & 13\\
142 & $2_{7}\,\allowbreak 2_{13}^{-1}$ & 15\\
143 & $1_{8}\,\allowbreak 1_{12}^{-1}\,\allowbreak 2_{9}^{-1}\,\allowbreak 2_{11}\,\allowbreak 3_{8}\,\allowbreak 3_{12}^{-1}$ & 17\\
151 & $1_{10}^{-1}\,\allowbreak 1_{12}^{-1}\,\allowbreak 2_{11}\,\allowbreak 3_{8}\,\allowbreak 3_{12}^{-1}$ & 21\\
155 & $1_{8}\,\allowbreak 2_{9}^{-1}\,\allowbreak 2_{13}^{-1}\,\allowbreak 3_{8}$ & 19\\
168 & $1_{10}^{-1}\,\allowbreak 2_{13}^{-1}\,\allowbreak 3_{8}$ & 23\\
173 & $1_{8}\,\allowbreak 2_{11}\,\allowbreak 2_{13}^{-1}\,\allowbreak 3_{12}^{-1}\,\allowbreak 4_{10}$ & 21\\
188 & $1_{10}^{-1}\,\allowbreak 2_{9}\,\allowbreak 2_{11}\,\allowbreak 2_{13}^{-1}\,\allowbreak 3_{12}^{-1}\,\allowbreak 4_{10}$ & 25\\
193 & $1_{8}\,\allowbreak 2_{11}\,\allowbreak 2_{13}^{-1}\,\allowbreak 4_{14}^{-1}$ & 25\\
206 & $1_{10}^{-1}\,\allowbreak 2_{9}\,\allowbreak 2_{11}\,\allowbreak 2_{13}^{-1}\,\allowbreak 4_{14}^{-1}$ & 29\\
\end{longtable}
\endgroup

\subsection{The fundamental at node 3}
\begingroup\footnotesize
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{0.95}
\begin{longtable}{r>{\raggedright\arraybackslash}p{0.78\textwidth}r}
\toprule No. & Monomial & $d_{\bp}$\\\midrule\endfirsthead
\toprule No. & Monomial (continued) & $d_{\bp}$\\\midrule\endhead
\bottomrule\endfoot
64 & $1_{3}\,\allowbreak 1_{9}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 4_{5}$ & 7\\
90 & $1_{5}^{-1}\,\allowbreak 1_{9}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 4_{5}$ & 9\\
91 & $3_{5}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{5}$ & 13\\
93 & $1_{3}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 4_{5}$ & 7\\
96 & $1_{3}\,\allowbreak 1_{9}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{9}^{-1}$ & 9\\
117 & $1_{5}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 4_{5}$ & 9\\
125 & $1_{5}^{-1}\,\allowbreak 1_{9}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{9}^{-1}$ & 11\\
126 & $3_{5}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{9}^{-1}$ & 15\\
128 & $1_{3}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{9}^{-1}$ & 9\\
138 & $1_{3}\,\allowbreak 1_{9}\,\allowbreak 2_{6}\,\allowbreak 3_{11}^{-1}$ & 11\\
152 & $1_{5}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{8}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{9}^{-1}$ & 11\\
159 & $2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{5}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}$ & 17\\
162 & $1_{3}\,\allowbreak 1_{7}\,\allowbreak 1_{9}\,\allowbreak 2_{8}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 15\\
165 & $1_{3}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{10}\,\allowbreak 3_{11}^{-1}$ & 11\\
166 & $1_{5}^{-1}\,\allowbreak 1_{9}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 3_{11}^{-1}$ & 13\\
194 & $1_{5}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{10}\,\allowbreak 3_{11}^{-1}$ & 13\\
208 & $1_{5}^{-1}\,\allowbreak 1_{7}\,\allowbreak 1_{9}\,\allowbreak 2_{4}\,\allowbreak 2_{8}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 17\\
209 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{9}^{-1}$ & 17\\
220 & $1_{3}\,\allowbreak 2_{6}\,\allowbreak 2_{12}^{-1}$ & 13\\
221 & $1_{3}\,\allowbreak 1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 15\\
237 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}$ & 19\\
238 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 23\\
251 & $1_{5}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{6}\,\allowbreak 2_{12}^{-1}$ & 15\\
252 & $1_{3}\,\allowbreak 1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 19\\
253 & $1_{5}^{-1}\,\allowbreak 1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 17\\
257 & $1_{3}\,\allowbreak 1_{7}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}$ & 17\\
281 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 3_{5}\,\allowbreak 3_{9}^{-1}$ & 17\\
289 & $1_{5}^{-1}\,\allowbreak 1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 21\\
294 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}$ & 25\\
296 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}$ & 21\\
298 & $1_{3}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}$ & 21\\
300 & $1_{5}^{-1}\,\allowbreak 1_{7}\,\allowbreak 2_{4}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}$ & 19\\
303 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{5}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 23\\
320 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{5}$ & 19\\
332 & $1_{3}\,\allowbreak 1_{7}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 19\\
351 & $1_{5}^{-1}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}$ & 23\\
353 & $1_{7}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{7}$ & 25\\
359 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{5}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}$ & 27\\
361 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}$ & 25\\
362 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{5}$ & 21\\
368 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{11}^{-1}$ & 29\\
370 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{11}^{-1}$ & 25\\
382 & $1_{3}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 23\\
384 & $1_{5}^{-1}\,\allowbreak 1_{7}\,\allowbreak 2_{4}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 21\\
394 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}$ & 21\\
399 & $1_{3}\,\allowbreak 1_{7}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{13}^{-1}$ & 23\\
414 & $1_{9}^{-1}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{7}$ & 29\\
422 & $1_{5}^{-1}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 25\\
428 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}$ & 29\\
431 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{11}^{-1}$ & 29\\
432 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}$ & 23\\
436 & $1_{7}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 27\\
438 & $1_{7}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}$ & 27\\
461 & $1_{3}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{13}^{-1}$ & 27\\
463 & $1_{5}^{-1}\,\allowbreak 1_{7}\,\allowbreak 2_{4}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{13}^{-1}$ & 25\\
469 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 4_{11}^{-1}$ & 25\\
495 & $1_{5}^{-1}\,\allowbreak 1_{9}^{-1}\,\allowbreak 2_{4}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{13}^{-1}$ & 29\\
498 & $1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{10}^{-1}\,\allowbreak 4_{7}$ & 25\\
500 & $1_{9}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}$ & 31\\
502 & $1_{7}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 29\\
505 & $1_{9}^{-1}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 31\\
512 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{7}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{11}^{-1}$ & 33\\
514 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{8}\,\allowbreak 4_{11}^{-1}$ & 27\\
526 & $1_{7}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 4_{13}^{-1}$ & 31\\
528 & $1_{7}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{11}^{-1}$ & 31\\
564 & $2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{7}$ & 35\\
575 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{7}$ & 31\\
577 & $1_{9}^{-1}\,\allowbreak 2_{8}^{2}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 33\\
581 & $1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 31\\
583 & $1_{9}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 4_{11}^{-1}$ & 35\\
586 & $1_{7}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 33\\
591 & $1_{9}^{-1}\,\allowbreak 2_{6}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{5}\,\allowbreak 4_{13}^{-1}$ & 35\\
622 & $1_{7}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 33\\
648 & $2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 41\\
656 & $2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 37\\
657 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{7}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 37\\
666 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 33\\
669 & $1_{7}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 37\\
672 & $1_{9}^{-1}\,\allowbreak 2_{8}^{2}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 37\\
694 & $1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{12}\,\allowbreak 3_{13}^{-1}$ & 35\\
696 & $1_{9}^{-1}\,\allowbreak 2_{8}^{2}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 37\\
729 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-2}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 39\\
755 & $1_{9}^{-1}\,\allowbreak 2_{8}^{2}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 41\\
764 & $2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 41\\
765 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 37\\
767 & $2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 43\\
768 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 39\\
770 & $1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{6}\,\allowbreak 2_{14}^{-1}$ & 37\\
771 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{12}\,\allowbreak 3_{7}\,\allowbreak 3_{13}^{-1}$ & 41\\
772 & $3_{7}\,\allowbreak 3_{13}^{-1}$ & 45\\
815 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-2}\,\allowbreak 3_{11}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 45\\
818 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-2}\,\allowbreak 3_{9}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 45\\
822 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{7}$ & 43\\
834 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{7}\,\allowbreak 4_{9}$ & 41\\
840 & $2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{11}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 47\\
844 & $2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 47\\
845 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 3_{11}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 43\\
849 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 43\\
907 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-2}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 51\\
911 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{7}$ & 49\\
916 & $2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 51\\
920 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 49\\
928 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}\,\allowbreak 4_{7}\,\allowbreak 4_{13}^{-1}$ & 47\\
930 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{9}\,\allowbreak 4_{11}^{-1}$ & 47\\
950 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 47\\
953 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{11}^{-1}\,\allowbreak 4_{9}$ & 45\\
979 & $1_{11}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 4_{7}$ & 49\\
981 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 55\\
998 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{14}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{7}$ & 51\\
1001 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 4_{11}^{-1}\,\allowbreak 4_{13}^{-1}$ & 53\\
1005 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 55\\
1016 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}\,\allowbreak 3_{9}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{11}^{-1}$ & 55\\
1017 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{9}$ & 49\\
1026 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 51\\
1044 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{13}^{-1}$ & 49\\
1058 & $1_{11}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 55\\
1060 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 61\\
1074 & $1_{11}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 4_{7}$ & 49\\
1075 & $1_{13}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 4_{7}$ & 51\\
1076 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{9}$ & 51\\
1078 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 57\\
1079 & $1_{11}\,\allowbreak 2_{8}\,\allowbreak 2_{14}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}$ & 59\\
1085 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 57\\
1096 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{14}\,\allowbreak 3_{9}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{11}^{-1}$ & 57\\
1101 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{11}\,\allowbreak 4_{13}^{-1}$ & 55\\
1123 & $1_{13}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 4_{7}$ & 51\\
1137 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{9}$ & 57\\
1138 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{9}$ & 61\\
1140 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{14}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}$ & 65\\
1146 & $1_{11}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{13}^{-1}$ & 59\\
1154 & $1_{11}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 55\\
1155 & $1_{13}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 57\\
1156 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{11}\,\allowbreak 4_{13}^{-1}$ & 57\\
1158 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 63\\
1160 & $1_{13}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{12}\,\allowbreak 2_{14}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}$ & 61\\
1177 & $1_{7}\,\allowbreak 1_{9}\,\allowbreak 3_{15}^{-1}$ & 59\\
1191 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 1_{15}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 65\\
1193 & $1_{13}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}\,\allowbreak 4_{11}^{-1}$ & 57\\
1204 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 2_{12}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{11}\,\allowbreak 4_{13}^{-1}$ & 63\\
1205 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{13}^{-1}$ & 67\\
1209 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}\,\allowbreak 2_{14}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}$ & 67\\
1231 & $1_{11}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 3_{13}^{-1}$ & 59\\
1232 & $1_{13}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{12}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{13}^{-1}$ & 61\\
1239 & $1_{7}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{10}\,\allowbreak 3_{15}^{-1}$ & 61\\
1256 & $1_{13}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{12}\,\allowbreak 3_{13}^{-1}$ & 61\\
1263 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 2_{12}\,\allowbreak 2_{14}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}$ & 71\\
1276 & $1_{7}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}\,\allowbreak 3_{15}^{-1}$ & 67\\
1277 & $1_{15}\,\allowbreak 2_{8}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}$ & 65\\
1278 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 67\\
1284 & $1_{9}\,\allowbreak 1_{11}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 65\\
1288 & $1_{9}^{-1}\,\allowbreak 1_{11}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{10}\,\allowbreak 3_{15}^{-1}$ & 67\\
1305 & $1_{9}\,\allowbreak 1_{15}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}$ & 71\\
1310 & $1_{17}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{14}^{-1}$ & 65\\
1311 & $1_{9}\,\allowbreak 1_{13}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{12}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 67\\
1312 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{12}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 71\\
1317 & $1_{9}^{-1}\,\allowbreak 2_{8}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{11}\,\allowbreak 3_{15}^{-1}$ & 73\\
1346 & $1_{11}^{-1}\,\allowbreak 1_{13}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{12}\,\allowbreak 3_{9}\,\allowbreak 3_{13}^{-1}$ & 71\\
1348 & $1_{11}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{9}$ & 75\\
1349 & $2_{10}^{-1}\,\allowbreak 2_{12}^{-1}\,\allowbreak 3_{9}\,\allowbreak 3_{11}\,\allowbreak 3_{15}^{-1}$ & 81\\
1356 & $1_{9}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{10}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{9}$ & 71\\
1360 & $1_{9}\,\allowbreak 1_{15}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{11}$ & 75\\
1387 & $1_{11}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{9}$ & 75\\
1392 & $3_{11}\,\allowbreak 3_{13}^{-1}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{11}$ & 85\\
1394 & $1_{11}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{11}$ & 79\\
1397 & $1_{9}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{11}$ & 75\\
1398 & $1_{9}\,\allowbreak 1_{15}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 4_{15}^{-1}$ & 79\\
1424 & $1_{11}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 3_{13}^{-1}\,\allowbreak 4_{11}$ & 79\\
1431 & $3_{11}\,\allowbreak 3_{15}^{-1}\,\allowbreak 4_{15}^{-1}$ & 89\\
1433 & $1_{11}^{-1}\,\allowbreak 1_{15}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 2_{16}^{-1}\,\allowbreak 4_{15}^{-1}$ & 83\\
1436 & $1_{9}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{15}^{-1}$ & 79\\
1452 & $1_{11}^{-1}\,\allowbreak 1_{17}^{-1}\,\allowbreak 2_{10}\,\allowbreak 2_{12}\,\allowbreak 2_{14}^{-1}\,\allowbreak 4_{15}^{-1}$ & 83\\
\end{longtable}
\endgroup


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\end{thebibliography}
\end{document}