Proof. We first specify the correspondences and their degrees, then prove the identity by a faithful localization calculation.

Proof. Choose a \(\nu^m\)-semi-invariant \(f\), with \(m>0\), that is nonzero at \(\sigma_\infty\). Its evaluation on \(\sigma\) is a global section of a positive power of \(\mathcal L_\nu\), nonzero at infinity. This proves nonnegativity of the degree.

If the degree is zero, this section is nowhere zero. Thus \(\sigma\) is everywhere semistable and determines a morphism \[\gamma:\mathbb P^1\longrightarrow (T^*N)/\!/_{\nu}G.\] Every invariant polynomial evaluated on \(\sigma\) is a regular function on \(\mathbb P^1\), hence constant. The image of \(\gamma\) therefore lies in one fiber of the projective morphism to the affine quotient \((T^*N)/\!/G\). A sufficiently divisible GIT polarization is relatively ample and pulls back to a positive power of \(\mathcal L_\nu\). Its degree is zero, so \(\gamma\) is constant: an ample line bundle has positive degree on the image of a nonconstant morphism from \(\mathbb P^1\). These are the usual character-linearized affine GIT constructions; see (King 1994, sec. 2).

The constant value is the stable quotient point represented by \(\sigma_\infty\). Its fiber in the semistable locus is the single orbit \(G\sigma_\infty\). Consequently \(\sigma\) is a section of the associated orbit bundle \(\mathcal P\times_G(G\sigma_\infty)\). Since \(G\sigma_\infty\simeq G\) equivariantly, this section trivializes \(\mathcal P\). Freeness makes the trivialization unique once the coordinates of the section are prescribed to be \(\sigma_\infty\); it agrees with the specified frame at infinity.

In the lattice presentation, compare this global trivialization with the fixed trivialization off zero. Their transition is a regular map \(\mathbb A^1_{t^{-1}}\to G\) equal to \(1\) at infinity. It is the loop \(h\) in (19), and the lattice of regular sections at zero is \(hG(\mathcal O)\). ◻

Proof. Differentiating in \(y\) gives \(\operatorname{pr}_{L_+}n=s\). Differentiating in the section variable \(n\in E_b\) says that \(p+y\) annihilates \(E_b\) in the residue pairing. The annihilator is the displayed truncated dual lattice. These conditions are zeros of algebraic bundle maps and hence define a closed locus.

This derivative calculation gives a support bound even when \(B\) is singular. Off \(\Sigma_B\), a derivative in one of the vector-bundle fiber coordinates is nonzero. The function is affine linear in that coordinate; where its coefficient is invertible, replacing the coordinate by \(W_B\) gives a product chart over the unchanged base. The constant sheaf has zero vanishing cycles in this chart.

A bounded lattice in each general linear factor specifies a modification of the trivial vector bundle at zero, determined by finite jets. A point of \(\Sigma_B\) gives two global sections on \(\mathbb P^1\) for the resulting principal bundle. In the off-zero frame they are the polynomials \(n\) and \(p+y\) in \(t^{-1}\), with values \(s\) and \(p\) at infinity. The lattice conditions are exactly their extendability at zero: the discarded \(t^l\) tails already lie in the corresponding lattices. Their value at infinity is the stable pair \((s,p)\). Lemma 9 now excludes negative character-line degree and, in degree zero, forces the entire topological class to be zero. This proves the last assertion. ◻

Proof. First, every point of \(\Sigma_Z\) lies over \(U\). By Lemma 9, its principal bundle is trivial, with a based loop \(h\) as in (19). A vector in the kernel of \(E_b\to L_+\) is a negative polynomial section of its associated \(N\)-bundle, extending at zero and vanishing at infinity. That associated bundle is trivial, so the vector is zero. The two spaces have dimension \(r\), and the projection is an isomorphism.

Over \(U\), write \(E_b\) as the graph of a regular linear map \(A_b:L_+\to L_-\). Thus \(n=a+A_ba\) for \(a\in L_+\). Define the linear functional \[\ell_{b,p}(a)=\mathop{\mathrm{Res}}\langle p,A_ba\rangle.\] Via \(P_-=L_+^*\), Equation (20) becomes \[ W_Z=\langle y,a-s\rangle+\ell_{b,p}(a). \tag{27}\] Its fiber critical equations have the unique solution \(a=s\), \(y=-\ell_{b,p}\), depending regularly on \((b,s,p)\). Consequently \(\Sigma_Z\) is the graph \(\Gamma\simeq U\times D\). It is closed in all of \(M_Z\) because the equations defining \(\Sigma_Z\) are closed and every solution has already been shown to lie over \(U\).

It remains to compute the critical value. Apply Lemma 9 to this solution. In the off-zero frame, \[n=hs,\qquad p+y=hp.\] Invariance of the pairing gives \(\langle hp,hs\rangle=\langle p,s\rangle\), a constant series. Hence \[W_{\mathrm{crit}} =-\langle[t^{-1}](hp),s\rangle.\] Write \(h=1+\xi t^{-1}+O(t^{-2})\), where \(\xi\in\mathfrak g\). The coefficient \([t^{-1}](hp)\) is the infinitesimal dual action \(\xi\cdot p\). Therefore \[ W_{\mathrm{crit}} =-\langle\xi\cdot p,s\rangle =\langle p,\xi\cdot s\rangle =\langle\mu(s,p),\xi\rangle=0. \tag{28}\] Thus the regular, equivariant translations \(a'=a-s\), \(y'=y+\ell_{b,p}\) turn (27) into the fixed split form \[W_Z=\langle y',a'\rangle\] over \(U\times D\).

The vanishing cycles of this form on \(L_+\oplus L_+^*\) are one copy of \(\mathbb C[-2r]\) at the origin. The external-product theorem applies to the constant sheaf of \(U\times D\) even if \(U\) is singular (Massey 2001, Theorem, pp. 1–2 of the arXiv preprint). The splitting is global and equivariant. Its structure group acts as \(g\oplus g^{-T}\), so the quadratic vanishing-cycle line has trivial equivariant monodromy. Choosing its identification with \(\mathbb C\) once gives compatible identifications over the whole base.

Lemma 10 says that the vanishing cycles are zero off the closed graph \(\Gamma\). The product calculation therefore gives the first isomorphism in (25) on all of \(M_Z\). In particular, there are no further cycles over \(Z\setminus U\): every point there fails at least one of the fiber equations. The same computation at \(1\) has \(A_1=0\), \(\ell_{1,p}=0\), and the same \(2r\) quadratic variables, giving the second isomorphism.

The map \(\rho_Z\) is restriction to the base lattice. To identify its test as a morphism, not just its source and target, put \(B=U\times D\) and consider the maps \[D\xrightarrow{\,j\,}U\times D\xrightarrow{\,\pi\,}D, \qquad j(d)=(1,d),\quad \pi(b,d)=d.\] These maps are \(\mathsf K\)-equivariant and satisfy \(\pi j=\mathrm{id}\). Their ordinary cohomological pullbacks satisfy \(j^*\pi^*=\mathrm{id}\). On the same base \(B\), apply the natural external-product identification to the morphism \(\mathbb C_B\to Rj_*\mathbb C_D\), with zero function on \(B\) and the fixed split quadratic on \(L_+\oplus L_+^*\). Since \(j\) is closed, proper-direct-image compatibility identifies the target with the quadratic calculation over \(D\). One common choice of generator of the quadratic line then identifies the resulting morphism with \(\mathbb C_B[-2r]\to Rj_*\mathbb C_D[-2r]\). Taking ordinary equivariant cohomology gives exactly Equation (26). This uses naturality in a sheaf morphism and proper direct image, not commutation with an arbitrary nonsmooth pullback. The identity \(j^*\pi^*=\mathrm{id}\) supplies the \(R\)-linear splitting, without any properness requirement on \(\pi\). ◻

Proof of Proposition 8. The assertion is immediate if \(D=\varnothing\), so assume \(D\) is nonempty. Let \(u\in R\) be homogeneous and zero in \(B_F\). Proposition 3 expresses \(u\) in \(A_F\) as a finite sum of positive-length words in nontrivial dressed positive and negative minuscule generators, with coefficients absorbed into the dressings. Taking the full topological-degree-zero part, every word has total topological degree zero.

The character-line degrees of its steps sum to zero. Choose a step with nonpositive character-line degree and, by commutativity, put it first. Its topological degree is nonzero. Apply Lemma 6, with a common neutral endpoint bound \(Z\). The morphism \(u\rho_Z\) is a sum of compositions, each factoring through the first-step object \(f_{B,*}\mathbb C_{S_B}\), with the prescribed shifts. By Lemma 10, the test of every such object is zero. It follows that \[\mathsf T(u\rho_Z)=0.\]

The degree calculation in Lemma 6 identifies the degree of this morphism with \(\deg u\), including when intermediate lattice components are nonneutral. The test commutes with shifts, and Lemma 11 places the same quadratic shift on its source and target. Its \(R\)-linearity therefore gives the equality of graded maps \[(u|_D)\smile j^*=0.\] Apply this equality to \(\pi^*(1)\). Since \(j^*\pi^*(1)=1\), we obtain \(u|_D=0\) in \(H_{\mathsf K}^*(D)\).

The free stable-locus action identifies \(H_{\mathsf K}^*(D)=H_F^*(X)\), compatibly with the coefficient map. We have proved \[\ker(R\to B_F)\subseteq\ker(R\to H_F^*(X)).\] Both rings are generated by their coefficient images. For \(B_F\) this follows from Proposition 3; for \(H_F^*(X)\) it follows from Kirwan surjectivity (McGerty and Nevins 2018) and its equivariant extension recalled above. The displayed inclusion induces the graded \(R\)-algebra surjection (18). ◻

Proof. Choose, for each color, an identification of its \(l_i\) positive slots with \(l_i\) of its \(v_i\) gauge slots. On the remaining \(v_i-l_i\) slots put \(s(d)=0\); call these the zero slots and write their set as \(\Omega_m^0\). Define a coweight by \[(\lambda_m)_d=s(d)-1\qquad(d\in\Omega).\] The word “positive” refers to the part size, so at a rank-three vertex the sizes \((2,1,0)\) give the coweight \((1,0,-1)\). The coweight \(\lambda_m\) has topological degree zero, since the sum of its coordinates at vertex \(i\) is \(\sum_s s m_{i,s}-v_i=0\). Its stabilizer is \[\mathcal W_{\lambda_m} =\mathcal W_m\times\prod_i S_{v_i-l_i}.\] The last factors permute the zero slots.

By the degree convention in Proposition 3, the half cohomological degree of \(r_{\lambda_m}\) is the polynomial degree of its Laurent coefficient. For a remaining arrow \(i\to j\), that degree is \[\sum_{a=1}^{v_i}\sum_{b=1}^{v_j} \max(s(ia)-s(jb),0) =v_iv_j-\sum_{a=1}^{v_i}\sum_{b=1}^{v_j} \min(s(ia),s(jb)),\] because \(\sum_a s(ia)=v_i\). The framing contribution at \(i\) is \(w_i\sum_{d\in\Omega_m^+,\ i(d)=i}(s(d)-1)=w_i(v_i-l_i)\). Summing gives \[ \delta_m:=\deg_q r_{\lambda_m} =C(\mathbf v)-D_m+\mathbf w\cdot(\mathbf v-\mathbf l). \tag{33}\] This also proves integrality and nonnegativity. The formula includes remaining loops: their ordered pairs of gauge slots give precisely the diagonal term with \(c_{ii}=2n_{ii}\).

For \(-1\leq L\leq\sum_i v_i\), define \(\mathcal F_{-1}=0\) and, for \(L\geq0\), let \(\mathcal F_L\subseteq B_0\) be the sum of the images of \[\mathsf S_{\lambda_m}\bigl(P^{\mathcal W_{\lambda_m}}\bigr), \qquad m\in\mathcal M(\mathbf v),\quad |l|\leq L.\] Here and below an orbit sum denotes its image in the quotient when it is viewed in \(B_0\). This is a finite filtration by graded vector subspaces. The array \(m_{i,1}=v_i\) has \(\lambda_m=0\), so the last stage contains the image of all coefficients. Those coefficients generate \(B_0\), by Proposition 3; hence \(\mathcal F_{\sum_i v_i}=B_0\). Exhaustion is asserted in the quotient, and requires no assertion about spanning the entire topological-degree-zero part of \(A_0\) before quotienting. For later use denote the symbol map of each orbit type by \[\Phi_m:P^{\mathcal W_{\lambda_m}} \longrightarrow\mathcal F_{|l|}/\mathcal F_{|l|-1},\qquad p\longmapsto[\mathsf S_{\lambda_m}(p)].\] It adds the half-degree shift \(\delta_m\).

We first eliminate dressings in the zero slots. For each color \(i\) and \(k\geq1\), define an ambient Laurent-ring derivation by \[D_{i,k}(p e^\lambda) =p e^\lambda\sum_a\lambda_{ia}x_{ia}^k, \qquad D_{i,k}(p)=0\quad(p\in\mathbb C(x_{ia})).\] Additivity of the displayed multiplier in \(\lambda\) proves the Leibniz rule. This derivation commutes with \(\mathcal W\), preserves the algebra in Equation (29), and preserves every topological homogeneous summand. It therefore preserves the ideal defining \(B_0\). Its induced derivation on \(B_0\) is zero, since it kills the coefficient generators. Applying it to an arbitrarily dressed orbit sum gives \[ \mathsf S_{\lambda_m}\left( p\left[ \sum_{\substack{d\in\Omega_m^+\\i(d)=i}} (s(d)-1)x_d^k -\sum_{\substack{z\in\Omega_m^0\\i(z)=i}}x_z^k \right]\right)=0\quad\text{in }B_0 \tag{34}\] for every \(p\in P^{\mathcal W_{\lambda_m}}\). The invariant dressing ring is the tensor product of the symmetric-polynomial rings on the positive type blocks and the zero blocks. Power sums generate each zero-block ring over \(\mathbb C\). Equation (34), including its arbitrary multiplier \(p\), replaces each such power sum by the indicated positive-slot polynomial. Thus positive-slot dressings alone span the image of this orbit type. These replacements preserve degree.

Fix now a nonempty \(S\subseteq\Omega_m^+\), and split the array of sizes on all gauge slots as \(s=a+b\), by putting \(b_d=s(d)\) on \(S\) and \(b_d=0\) elsewhere. Set \[\alpha=a-1,\qquad \beta=b.\] The topological degree of \(\beta\) is a nonzero vector with nonnegative entries, so every product \[\mathsf S_\alpha(p_a)\mathsf S_\beta(p_b)\] vanishes in \(B_0\). The dressings here may be arbitrary polynomials invariant under their respective coweight stabilizers, by Proposition 12.

To analyze the terms of this product, fix \(a\) and match it with a colorwise permutation \(b'\) of \(b\). The resulting coweight is \(t-1\), where \(t=a+b'\) has nonnegative entries and \(\sum_{i(d)=i}t_d=v_i\). It therefore determines another array in \(\mathcal M(\mathbf v)\). Writing \(A=\operatorname{supp}(a)\) and \(B'=\operatorname{supp}(b')\), its length is \[|\operatorname{supp}(t)| =|A|+|B'|-|A\cap B'|=|l|-|A\cap B'|.\] Every overlap term belongs to \(\mathcal F_{|l|-1}\). For a disjoint matching, the positive sizes of \(t\) are the multiset union of those of \(a\) and \(b\) at each color. Hence its array is exactly \(m\); no other orbit type occurs at length \(|l|\).

For the chosen disjoint decomposition, the excess coefficient in \(r_\alpha r_\beta/r_{\lambda_m}\) is \(q_S\) up to a nonzero constant sign. Here is the calculation for each weight. For an arrow from slot \(d\) to slot \(e\), its excess exponent is \[(a_d-a_e)_+ +(b_d-b_e)_+ -(a_d+b_d-a_e-b_e)_+.\] This is zero unless one slot belongs to \(\operatorname{supp}(a)\) and the other to \(S\). In that case it is \(\min(s(d),s(e))\). For the framing weight \(-x_d\), the excess is \[(a_d-1)_+ +(b_d)_+ -(a_d+b_d-1)_+,\] which equals one on \(S\) and zero elsewhere. These give exactly the exponents in Equation (31). With the weight conventions above, the sign is \((-1)^{\epsilon_S}\), where \[\epsilon_S= \sum_{d\in S}w_{i(d)}+ \sum_{\substack{d\in S\\e\in\Omega_m^+\setminus S}} n_{i(d)i(e)}\min(s(d),s(e)).\] This integer depends only on the numbers of selected and unselected copies of each type. In particular the sign is constant on the averaging orbit considered next.

It remains to check the full span of the leading dressing relations. Let \[H=\mathcal W_{\lambda_m},\qquad H_S=\mathcal W_\alpha\cap\mathcal W_\beta.\] Then \(H_S\subseteq H\). Its blocks are the selected and unselected copies of each positive type, together with the zero block at every color. All disjoint decompositions of the fixed representative \(\lambda_m\) into translates of \(\alpha\) and \(\beta\) form exactly the set \(H/H_S\). The coefficient of its leading orbit sum in the product above is \((-1)^{\epsilon_S}\operatorname{Tr}_{H_S}^H(q_Sp_ap_b)\). Since the sign is nonzero and all other terms have lower length, we obtain \[ \Phi_m\bigl(\operatorname{Tr}_{H_S}^H(q_Sp_ap_b)\bigr)=0, \qquad \operatorname{Tr}_{H_S}^H(f)=\sum_{h\in H/H_S}h(f). \tag{35}\]

Choose \(p_a\) to depend only on the positive-size blocks of \(a\) and \(p_b\) only on the positive-size blocks of \(b\). Such choices are allowed by their coweight stabilizers. Their products span the \(H_S\)-invariant polynomial ring on the positive slots, since that ring is the tensor product of the symmetric-polynomial rings on the selected and unselected copies of every type. Moreover \(q_S\) is \(H_S\)-invariant. For an arbitrary \(f\in P_m\), Reynolds averaging first over \(H_S\) and then over \(H\) consequently shows that \[\operatorname{Av}_H(q_S f) =\operatorname{Av}_H\bigl(q_S\operatorname{Av}_{H_S}(f)\bigr)\] is a linear combination of the relations in Equation (35). All normalizing factors are nonzero constants in \(\mathbb C\). Averaging the generators \(q_S\) with arbitrary multipliers spans \(J_m\cap P_m^{\mathcal W_m}\), because \(J_m\) is stable under \(\mathcal W_m\) and averaging is exact in characteristic zero. Therefore all invariant cut-ideal relations annihilate the symbol of this orbit type.

Combining the zero-slot elimination with these cut relations gives, for every \(L\), a surjection of graded vector spaces \[ \bigoplus_{\substack{m\in\mathcal M(\mathbf v)\\|l|=L}} (P_m/J_m)^{\mathcal W_m}[\delta_m] \twoheadrightarrow \mathcal F_L/\mathcal F_{L-1}. \tag{36}\] Here \([\delta_m]\) means that polynomial half degree \(d\) is placed in half cohomological degree \(d+\delta_m\). Exactness of invariants identifies the source with the invariant dressing quotient just obtained. Summing its Hilbert-series bound over the finite exhaustive filtration proves Equation (32). ◻

Proof. We first specify the moment-level comparison. Give \(N\) a Hermitian metric invariant under \(G_c=\prod_iU(V_i)\), and equip \(T^*N\) with its flat hyperkähler structure. The hyperkähler moment map has one real and one complex component. Under the trace identification of the central real dual with \(\mathbb R^{Q_0}\), let \(\eta=(1,\ldots,1)\) be the level corresponding to the determinant character. The GIT–moment-map correspondence identifies \(X_{\det}(v,w)\) with the compact-group quotient at real level \(\eta\) and complex level zero. Rotation of the three moment-map components carries this quotient to the one at real level zero and complex level \(c\eta\), for a common nonzero normalization constant \(c\). At real level zero, affine Kempf–Ness identifies the latter quotient with \[\mu_{\mathbb C}^{-1}(c\eta)/\!/G.\] These are identifications of underlying topological spaces; they do not assert that rotation is holomorphic in the original complex structure. The comparison uses the direct rotation on the flat representation (Hitchin et al. 1987, sec. 3(D)) and the topological affine Kempf–Ness theorem recalled in Section 2. The quiver GIT description is explained in (Nakajima 1994); the flat-representation argument just given includes loops. Multiplying the cotangent coordinate by \(c^{-1}\) is an algebraic isomorphism from this affine quotient to \[M_\eta(v,w)=\mu_{\mathbb C}^{-1} ((\mathrm{id}_{V_i})_i)/\!/G,\] because the cotangent moment map is linear in that coordinate. Thus \(M_\eta(v,w)\) and \(X_{\det}(v,w)\) have the same ordinary cohomology, including simultaneous emptiness.

Hausel’s formula for \(M_\eta(v,w)\) explicitly allows loops and multiple arrows (Hausel 2006, Equation (9)). We record its grading and all factors needed here. For partitions \(\lambda^i\) and \(\lambda^j\), write \[\langle\lambda^i,\lambda^j\rangle =\sum_{a,b}\min(\lambda_a^i,\lambda_b^j),\qquad E_Q(\lambda) =\sum_{a:i\to j\in Q_1}\langle\lambda^i,\lambda^j\rangle -\sum_i\langle\lambda^i,\lambda^i\rangle.\] Let \(m(\lambda)_{i,s}\) be the number of parts of size \(s\) in \(\lambda^i\), and set \[\delta(v,w)=\sum_{a:i\to j\in Q_1}v_iv_j -\sum_i v_i^2+w\cdot v.\] The nonempty smooth variety has complex dimension \(2\delta(v,w)\). With \(q=t^2\), where \(t\) is the ordinary cohomological variable in Hausel’s formula, that formula reads \[ \sum_v q^{-\delta(v,w)}P_{M_\eta(v,w)}(q)z^v = \frac{\displaystyle\sum_{\lambda} z^{|\lambda|} q^{-E_Q(\lambda)-\sum_i\langle\lambda^i,(1^{w_i})\rangle} /(q)_{m(\lambda)}} {\displaystyle\sum_{\lambda} z^{|\lambda|}q^{-E_Q(\lambda)}/(q)_{m(\lambda)}}. \tag{45}\] Here \(\lambda\) ranges over all multipartitions, including the empty one. In particular, the left side uses ordinary cohomology. Its normalization can also be checked by Poincaré duality: if \(u\) is the arithmetic variable, a smooth variety of complex dimension \(2\delta\) satisfies \(P_c(u)=u^{2\delta}P(u^{-1})\). The normalized count \(u^{-\delta}P_c(u)\) therefore becomes \(q^{-\delta}P(q)\) after setting \(u=q^{-1}\), as in Equation (45).

Now write \(m=m(\lambda)\). Removing one loop at each vertex cancels the vertex self-pairing in \(E_Q\), and therefore \[E_Q(\lambda)=D_m, \qquad \sum_i\langle\lambda^i,(1^{w_i})\rangle =\sum_i w_i l_i=w(m), \qquad \delta(v,w)=C(v)+w\cdot v.\] Consequently the right side of Equation (45) is exactly the ratio in Equation (43). Taking its \(z^v\) coefficient and multiplying by \(q^{C(v)+w\cdot v}\) proves Equation (44).

The right side is a finite polynomial before this identification: Lemma 15 makes each \(h_m\) finite, there are only finitely many multipartitions of \(v\), and the shifts are nonnegative integers. Explicitly, \[C(v)-D_m =\tfrac12\sum_{i,j,s,t}c_{ij} \bigl(st-\min(s,t)\bigr)m_{i,s}m_{j,t}\geq0, \qquad w\cdot(v-l)=\sum_{i,s}w_i(s-1)m_{i,s}\geq0.\] The diagonal entries \(c_{ii}\) are even and the off-diagonal terms pair, which also verifies integrality. ◻

Proof. Proposition 8 gives the displayed graded \(R_0\)-algebra surjection. Proposition 13 and Proposition 18 bound the dimension of each graded piece of its source from above by that of its target. Surjectivity gives the reverse inequality. Its kernel therefore vanishes in every degree. Since the bijection is the original coefficient-compatible algebra map, this is an isomorphism over \(R_0\) and an equality of the two coefficient ideals, with the quotient rings retaining nilpotents. The same argument includes the zero ring in the empty case. ◻

Proof. We specify the operator algebra and the sign convention. Let \(D_{ia}w_{jb}=(w_{jb}+\hbar\delta_{ij}\delta_{ab})D_{ia}\), with all the \(D_{ia}\) commuting. Allow coefficients in \[S[w_{ia},(w_{ia}-w_{ib}+r\hbar)^{-1}: a\ne b,\ r\in\mathbb Z].\] This ring is preserved by every shift \(D_{ia}^{\pm1}\). For subsets \(A_i\subseteq\{1,\ldots,d_i\}\), write \(w_A=(w_{ia})_{a\in A_i,i\in\Gamma_0}\). Define \[\begin{align*} E_{\mathbf n,g} &=\sum_{|A_i|=n_i} g(w_A) \frac{\displaystyle\prod_{a:i\to j} \prod_{\alpha\in A_i,\ \beta\notin A_j} (w_{i\alpha}-w_{j\beta}+u_a)} {\displaystyle\prod_i \prod_{\alpha\in A_i,\ \beta\notin A_i} (w_{i\alpha}-w_{i\beta})} \prod_{i,\alpha\in A_i}D_{i\alpha},\\ F_{\mathbf n,g} &=\sum_{|A_i|=n_i} g(w_A-\hbar) \frac{\displaystyle\prod_{a:j\to i} \prod_{\alpha\in A_i,\ \beta\notin A_j} (w_{i\alpha}-w_{j\beta}-u_a)} {\displaystyle\prod_i \prod_{\alpha\in A_i,\ \beta\notin A_i} (w_{i\alpha}-w_{i\beta})} \prod_{i,\alpha\in A_i}D_{i\alpha}^{-1}. \end{align*}\] These are the fundamental formulas (Jindal and Neguţ 2026, Equations (78)–(79)). Let \(\mathcal C_S\) be the \(S\)-subalgebra generated by these operators and the polynomials symmetric in \(w\) separately at each vertex.

We construct the maps first over \(K\), then prove that their polynomial halves land in \(\mathcal C_S\). Use Tsymbaliuk’s maps on the full trigonometric shuffle algebra and its opposite (Tsymbaliuk 2023, Theorem 5.7, Equations (93) and (96)), with no framing, and make the formal substitutions \[q=e^{\varepsilon\hbar},\qquad t_a=e^{\varepsilon u_a},\qquad W_{ia}=e^{\varepsilon w_{ia}}.\] Here \(\varepsilon\) is an auxiliary formal variable, not \(\hbar\). The multiplicative shift in the source satisfies \(D^{\mathrm{trig}}_{ia}W_{ia}=q^{-1}W_{ia}D^{\mathrm{trig}}_{ia}\). Thus its inverse becomes our positive shift \(D_{ia}\), while it becomes \(D_{ia}^{-1}\) in the negative map. The leading coefficient of the trigonometric shuffle kernel is \[\left(\frac{x-\hbar}{x}\right)^{\delta_{ij}} \prod_{a:i\to j}[-(x+u_a)] \prod_{a:j\to i}(x+\hbar-u_a).\] It is precisely the kernel for \(\star\) in Equation (51): the incoming-arrow signs have been removed from the kernel of (Jindal and Neguţ 2026, Equation (31)). This distinction is necessary. For the quiver consisting of one arrow \(i\to j\), the untwisted product gives \(e_{(1,0),1}*e_{(0,1),1} =-e_{(1,1),z_i-z_j+u_a}\), whereas the corresponding fundamental operator product has a plus sign.

Here are the order calculations that justify taking leading coefficients, including for loops and repeated gauge slots. Put \[\kappa(\mathbf n)=\sum_{a:i\to j}n_in_j,\qquad \ell_i=\#\{a:i\to i\},\qquad L_i^+=\sum_{a:i\to j}d_j-d_i+1,\qquad L_i^-=\sum_{a:j\to i}d_j-d_i+1.\] Lift a polynomial \(E\) of dimension \(\mathbf n\) to \(\varepsilon^{\kappa(\mathbf n)} E((X_{ia}-1)/\varepsilon)\). The cross-kernel order is \(\kappa(\mathbf n+\mathbf m)-\kappa(\mathbf n)-\kappa(\mathbf m)\); thus the leading source product is \(\star\). In an operator summand, let \(m_{ia}\) be the length of the string assigned to gauge slot \((i,a)\), so \(\sum_a m_{ia}=n_i\), and set \(P_2=\sum_{i,a}\ell_i m_{ia}^2\). The three prefactors in Tsymbaliuk’s formulas have orders \[\begin{array}{c|c} \text{factor}&\varepsilon\text{-order}\\ \hline \text{single-entry factors }Y\text{ or }Y' &L^\pm\cdot\mathbf n-\sum_i\ell_i n_i\\ \text{inverse kernels within one string} &-P_2+\sum_i\ell_i n_i\\ \text{factors between distinct slots} &-\kappa(\mathbf n)+P_2 . \end{array}\] Together with the lifted input, their total order is \(L^\pm\cdot\mathbf n\), independently of the composition. Multiply the operator image of the lift by \(\varepsilon^{-L^\pm\cdot\mathbf n}\) and take its \(\varepsilon^0\) coefficient to define the normalized leading map. The remaining denominators have nonzero leading coefficients over \(K(w)\). There are only finitely many compositions. Consequently an input with zero normalized leading coefficient also has zero normalized leading output. The construction is independent of the lift. Since \(L^\pm\cdot\mathbf n\) is additive, the normalized leading maps preserve multiplication. Multiplying them in dimension \(\mathbf n\) by \(\hbar^{-|\mathbf n|}\) preserves that property.

The use of formal parameters in this argument requires no analytic specialization theorem. For real \(\hbar<u_a<0\) and small positive \(\varepsilon\), the substitutions satisfy Tsymbaliuk’s parameter inequalities. The monomial factors involving logarithms of torus parameters are expanded near \(W=1\); they have leading coefficient one and rational Taylor coefficients in \(\hbar,u_a,w_{ia}\). Each coefficient of the multiplication identities is therefore a rational identity valid on this nonempty real open set, hence valid over \(K(w)\). This justifies the formal leading maps, denoted \(\Psi_+\) and \(\Psi_-\).

For the distinguished elements the maps are especially simple: \[ \Psi_+(e_{\mathbf n,g}) =(-1)^{L^+\cdot\mathbf n+\kappa(\mathbf n)}E_{\mathbf n,g}, \qquad \Psi_-(f_{\mathbf n,g})=F_{\mathbf n,g}. \tag{52}\] Indeed, the same-color factor in \(e_{\mathbf n,g}\) vanishes on every additive string of length at least two. Only selected subsets of distinct slots remain. Its diagonal factors contribute \(\hbar^{|\mathbf n|}\), which cancels the normalization \(\hbar^{-|\mathbf n|}\); its off-diagonal factors cancel the same-color denominators in the operator formula. The factors for arrows between two selected slots cancel as well. In the positive formula the single-entry factors contribute \((-1)^{\sum_i n_i(L_i^+-\ell_i)}\), and the internal-arrow cancellations contribute \((-1)^{\kappa(\mathbf n)-\sum_i\ell_i n_i}\), giving the stated sign. In the opposite formula the selected variables are \(w_A-\hbar\); the incoming-arrow factors cancel without these signs. This proves both formulas, and proves that the image is zero when \(\mathbf n\not\leq\mathbf d\).

By (Jindal and Neguţ 2026, Theorem 2.4), the distinguished elements generate \(\mathcal S_S^+\) over \(S\). The twist preserves their word spans, and the opposite statement follows by reversing words. Equation (52) therefore proves that both polynomial halves land in \(\mathcal C_S\). Although we constructed the maps over \(K\), their restrictions are thus defined over \(S\); no inverse of a parameter is needed in the identities in \(\mathcal C_S\).

Finally specialize \(\hbar=0\), \(w_{ia}=x_{ia}\), and \(u_a=-f_a\). Every allowed denominator becomes \(x_{ia}-x_{ib}\), with \(a\ne b\), so this defines a homomorphism to \(\operatorname{Frac}(P_F)[X_*(T)]\), sending each \(D_{ia}\) to the monomial for its standard slot coweight. For a selected-variable dressing \(g\), also denoting its specialization by \(g\), direct comparison with Proposition 3 gives \[M^+_{\mathbf n,g} =(-1)^{\sum_{a:i\to j}n_i(d_j-n_j)} E_{\mathbf n,g}\big|_{\mathrm{specialization}}, \qquad M^-_{\mathbf n,g} =(-1)^{\sum_i n_i(d_i-n_i)} F_{\mathbf n,g}\big|_{\mathrm{specialization}}.\] The first sign comes from the outgoing matter factors; the second comes from reversing the minuscule tangent weights. The coefficient roots themselves are unchanged. Thus every generator of \(\mathcal C_S\) specializes to an element of \(A_{\Gamma,\mathbf d,F}\), and so does every image of the polynomial halves. Only this homomorphism is needed; no injectivity or flatness after specialization is asserted. ◻

Proof. Polynomial Hall-algebra input. Use the independent polynomial ring \(S\) defined above. This is the equivariant coefficient ring \(H^*(BT_{\mathrm{univ}};\mathbb Q)\) for \(T_{\mathrm{univ}}=(\mathbb C^\times)^{1+|\Gamma_1|}\), with independent rotation and arrow characters. It lies in the geometric parameter setting of (Jindal and Neguţ 2026, Equation (30) and Remark 2.6); no extension to arbitrary field-valued exponential parameters is needed. On the tripled quiver use weights \(-u_a\), \(u_a-\hbar\), and \(\hbar\) on an original arrow, its reverse, and an added loop, respectively. These choices satisfy all the genericity hypotheses of (Jindal and Neguţ 2026, sec. 2.2); in particular there is no nonzero nonnegative integral relation among \(u_a,\hbar-u_a\), and \(u_a\notin\mathbb Z\hbar\) for every original loop.

We use the following polynomial-form results of that source: the loop-nilpotent Hall algebra identifies with its shuffle half (Jindal and Neguţ 2026, Proposition 3.7); its BPS descendants generate over \(S\) (Jindal and Neguţ 2026, Theorem 3.10, Equation (109), and Corollary 3.15); and the BPS comparison and Kac formula show that the nonzero dimension-vector support of these generators is \(\Delta_\Gamma\) (Jindal and Neguţ 2026, Proposition 3.14 and §3.10). Here the Kac identity retains the full cohomological grading: the coefficient of \(t^{d/2}\) is \((-1)^d\dim H^d\). A zero Kac polynomial therefore forces every graded piece of the corresponding BPS space to vanish. Proposition 3.14 of that source transfers this vanishing to the loop-nilpotent BPS space. The concrete twist in Equation (51) preserves these homogeneous word spans. Here polynomial form refers to equivariant parameters over \(\mathbb Q\), not integral cohomology.

Apply the generation statement to \(e_{\mathbf b,g}\) or \(f_{\mathbf b,g}\), and retain dimension-vector degree \(\mathbf b\). All nonconstant Hall degrees are nonnegative, so the result is a finite \(S\)-linear combination of products of BPS descendants with degrees in \(\Delta_\Gamma\) summing to \(\mathbf b\). Applying Lemma 25 gives the corresponding finite identity in \(\mathcal C_S\). Its coefficients lie in \(S\); no parameter from \(S\) is inverted in this identity. Every factor has degree at most \(\mathbf b\leq\mathbf d\); no assertion about vanishing of entire Hall components outside this bound is needed.

Specialization. Now extend scalars to \(\mathbb C\) and apply the specialization homomorphism of Lemma 25. Each finite identity becomes an identity in \(A_{\Gamma,\mathbf d,F}\). The signs in the images of the distinguished elements are nonzero constants and can be absorbed in its coefficients. This operation is performed on the abstract polynomial-form algebra and its map, not on the ambient polynomial representation of the shuffle algebra. An element whose ambient representative is divisible by \(\hbar\) need not belong to \(\hbar\) times the polynomial shuffle half, and is therefore not discarded before applying the map.

General dressings. It remains to allow an arbitrary stabilizer-invariant dressing \(p\). At each vertex, the polynomial algebra symmetric separately in the selected and complementary variables is generated by the polynomials symmetric in all variables together with those symmetric in the selected variables. Indeed, each complementary power sum is the total power sum minus the selected power sum. Thus \(p\) is a finite sum of coefficient polynomials times selected-variable dressings \(g\). The already proved identities for \(g\) give Equation (53). ◻

Proof. First, \(\Gamma\) has an indecomposable complex representation \(x\) of dimension \(\boldsymbol\gamma\). For completeness, the required passage from Kac support to characteristic zero can be made by constructibility. A representation is decomposable over an algebraic closure exactly when its endomorphism algebra contains an idempotent other than \(0\) and \(1\). The equations for such idempotents, together with these two open exclusions, define a constructible incidence condition over \(\mathbb Z\). Chevalley’s theorem therefore gives a constructible locus of geometrically indecomposable representations. A nonzero Kac polynomial gives points of this locus in infinitely many residue characteristics. Its generic fiber cannot be empty, since a constructible subset of \(\mathop{\mathrm{Spec}}\mathbb Z\) that omits the generic point is finite. Base change of the nonempty characteristic-zero locus to \(\mathbb C\) gives \(x\).

The following trace-annihilator argument is the lifting criterion of Crawley–Boevey (Crawley-Boevey 2001, Lemma 3.1 and Theorem 3.3); we recall it before passing to the real moment level. Every endomorphism \(e\) of \(x\) has the form \(c\mathrm{id}+n\), where \(n\) is nilpotent. Indeed, its generalized eigenspaces are subrepresentations, so indecomposability permits only one eigenvalue. Consequently \[\tau_{\boldsymbol\theta}(e) =c\,\boldsymbol\theta\cdot\boldsymbol\gamma=0.\] The cotangent moment map with \(x\) fixed is the transpose of the infinitesimal orbit map: \[N_{\boldsymbol\gamma}^*\longrightarrow \mathop{\mathrm{Lie}}(G_{\boldsymbol\gamma})^*, \qquad y\longmapsto\bigl(\xi\longmapsto y(\xi x)\bigr).\] Its image is the annihilator of \(\mathop{\mathrm{End}}_\Gamma(x)=\ker(\xi\mapsto\xi x)\). Hence the complex moment fiber at \(\tau_{\boldsymbol\theta}\) is nonempty, as is the fiber at any common scalar multiple of that value.

This fiber is a closed invariant affine subset, since the moment value is central. The closure of any orbit in it contains an orbit closed in the ambient affine representation. By the zero-character affine Kempf–Ness theorem, that orbit meets \(\mu_{\mathbb R}^{-1}(0)\); see (King 1994, Theorem 6.1 and Corollary 6.2). Thus there is a point with hyperkähler moment value \((0,\tau_{\boldsymbol\theta},0)\), using the real and imaginary parts of the complex moment map and adjusting the common scalar to the chosen normalization.

The quaternionic rotation on the flat cotangent representation commutes with the compact gauge group and rotates its three moment maps. A single rotation sends this value to \((\tau_{\boldsymbol\theta},0,0)\) (Hitchin et al. 1987, sec. 3(D)). This proves the asserted nonemptiness. The simultaneous scalar acts trivially on all arrow spaces and their duals, proving the last statement. ◻

Proof. Lemma 2 allows us to suppose that \(F\) scales the individual arrows and framing rows. Adding or removing the gauge-trivial cotangent summands in that lemma preserves both regularity and all the fixed quotients under consideration. Write \[\nu((g_i))=\prod_i\det(g_i)^{\nu_i}.\] Adjoin one vertex \(\infty\) of dimension one and \(w_i\) arrows \(i\to\infty\), keeping the original arrows. Denote this unframed quiver by \(\widehat Q\) and its dimension vector by \(\mathbf d=(\mathbf v,1)\). Its representation is the original \(N\), after a choice of basis in each framing space.

There is a group isomorphism \[G\times\mathbb C^\times\longrightarrow G_{\mathbf d}, \qquad ((g_i),c)\longmapsto((cg_i),c).\] The second factor acts trivially on \(N\). The BFN space of triples and its convolution therefore split, giving \[ A_{\widehat Q,\mathbf d,F} \cong A_F\otimes_\mathbb C\mathbb C[\xi,z,z^{-1}]. \tag{54}\] Here \(\xi\) is the coefficient parameter of the extra scalar and \(z\) is its fundamental Laurent monopole, of cohomological degree zero. This \(z\) is unrelated to the degree-two added-loop mass in Section 7. In the original vertex coordinates, \(z\) has topological charge \(\mathbf d\): it translates every gauge coweight entry, including the entry at \(\infty\), by one.

Extend the character by \[\widehat\nu_i=\nu_i\quad(i\in Q_0), \qquad \widehat\nu_\infty=-\sum_i\nu_i v_i.\] Then \(\widehat\nu\cdot\mathbf d=0\), and under the displayed group isomorphism \(\widehat\nu\) is \(\nu\) on \(G\) and trivial on the extra scalar. In particular, both \(\xi\) and \(z^{\pm1}\) have \(\widehat\nu\)-weight zero. Taking the cocharacter fixed quotient in Equation (54) consequently gives \[\bigl(A_{\widehat Q,\mathbf d,F}\bigr)_{\mathrm{fix},\widehat\nu} \cong \bigl(A_F\bigr)_{\mathrm{fix},\nu} \otimes_\mathbb C\mathbb C[\xi,z,z^{-1}],\] where the subscript denotes the quotient by nonzero cocharacter weights. This extension is faithfully flat.

Let \(M_{\mathbf b',p}^{+}\) be a nontrivial positive minuscule generator of \(A_F\), and put \(\mathbf b=(\mathbf b',0)\). If \(\widehat x_{ia}\) denotes an enlarged gauge Chern root, the group isomorphism above identifies the original root with \(x_{ia}=\widehat x_{ia}-\xi\). Let \(p^{\uparrow}\) be the polynomial obtained from \(p\) by these substitutions, leaving every flavor parameter unchanged. The original generator is then the positive minuscule class \(M_{\mathbf b,p^{\uparrow}}^{+}\). Central translation also expresses it using the complementary negative coweight: \[ M_{\mathbf b,p^{\uparrow}}^{+} =z\,M_{\mathbf d-\mathbf b,p^{\uparrow}}^{-}. \tag{55}\] The two coweights have the same Weyl stabilizer, so the same polynomial \(p^{\uparrow}\) is an admissible dressing on both sides. It need not depend only on the variables selected by the complementary negative coweight; Lemma 26 allows all stabilizer-invariant dressings. The equality follows directly from Equation (9): all matter weights pair trivially with the central coweight, and central translation identifies the minuscule orbits and their equivariant tangent bundles. Equivalently it follows from convolution with the invertible central class. Thus it holds with compatible fundamental-class conventions; rescaling minuscule generators changes only a nonzero constant.

Suppose that this generator has nonzero image in the cocharacter fixed quotient. Apply Lemma 26 in the enlarged quiver to \(M_{\mathbf b,p^{\uparrow}}^{+}\). Any word containing a factor of nonzero \(\widehat\nu\)-weight vanishes in that quotient. A nonzero image therefore requires a decomposition \[\mathbf b=\sum_a\boldsymbol\alpha_a, \qquad \boldsymbol\alpha_a\in\Delta_{\widehat Q}, \qquad \widehat\nu\cdot\boldsymbol\alpha_a=0.\] Since \(z\) is an invertible element of weight zero, Equation (55) implies that \(M_{\mathbf d-\mathbf b,p^{\uparrow}}^{-}\) also survives. Applying the negative half of the same lemma yields \[\mathbf d-\mathbf b=\sum_b\boldsymbol\beta_b, \qquad \boldsymbol\beta_b\in\Delta_{\widehat Q}, \qquad \widehat\nu\cdot\boldsymbol\beta_b=0.\] Both sums are nonempty: \(\mathbf b\ne0\), while its \(\infty\)-entry is zero and that of \(\mathbf d\) is one.

Together these decompositions express \(\mathbf d\) as a sum of at least two nonzero dimension vectors \(\boldsymbol\gamma_1,\ldots,\boldsymbol\gamma_k\) in \(\Delta_{\widehat Q}\), each orthogonal to \(\widehat\nu\). For each summand choose the point furnished by Lemma 27, with real central parameter \(\widehat\nu\). Use orthogonal decompositions of the vertex spaces and take the direct sum of these points. Moment maps are block diagonal on direct sums, so the resulting point has \[\mu_{\mathbb C}=0,\qquad \mu_{\mathbb R}=\tau_{\widehat\nu}.\] Its stabilizer contains \(k\) independent block scalar circles.

The enlarged reduction is exactly the original framed reduction: the extra diagonal scalar is ineffective, its moment equation is redundant, and \(\widehat\nu\) is trivial on it. After dividing by this scalar, the stabilizer still contains a torus of dimension \(k-1>0\). By the affine Kempf–Ness correspondence at the character level, the resulting point is \(\nu\)-semistable in the original zero complex moment fiber (King 1994, Theorem 6.1 and Corollary 6.2). This contradicts the assumed freeness of every such point. The argument also applies when the stable locus is empty: it would then produce a semistable point where none exists.

Every nontrivial positive minuscule generator therefore vanishes. For a negative generator use the identity \(M_{\mathbf b,p^{\uparrow}}^{-} =z^{-1}M_{\mathbf d-\mathbf b,p^{\uparrow}}^{+}\) and interchange the two halves in the argument; it vanishes as well. By Proposition 3, the ideal they generate is the ideal generated by the nonzero full topological-charge spaces. It is therefore contained in the ideal generated by the nonzero \(\nu\)-weight spaces. For \(m\ne0\), the \(\nu\)-weight space \((A_F)_m\) is a sum of nonzero topological-charge spaces. This gives the opposite inclusion. The ideals coincide, proving the proposition. If all gauge ranks are zero there are no such generators and the same conclusion is immediate. ◻