# Opers, critical level and the Beilinson–Drinfeld construction *Draft. Self-checked by the writing AI. Original mathematical exposition: CC0 1.0.* An oper is a connection with a maximally transverse Borel reduction. The reduction produces scalar differential equations and gives concrete affine families of local systems. At critical level, an affine Lie algebra's center is described by functions on opers. Localization assigns to a global oper a system of differential equations on the bundle stack with a tensor-compatible Hecke eigenproperty. We construct the adjoint-semisimple principal slice, prove its unique gauge normal form in ordinary families, and derive the coordinate action and global affine oper space. We also prove the homogeneous invariant ring over the stated characteristic-zero field, identify its degrees with the principal heights and construct the weighted Kostant section. A separate scalar proof treats \(PGL_n\) through jets. We also prove the Schwarzian rule, algebraic \(PGL_2\) irreducibility and the critical \(\mathfrak{sl}_2\) invariant. Section 3.2 extends that calculation to the entire rank-one polynomial vacuum center and its coordinate-equivariant ordinary disc-oper comparison. Section 3.3 proves the general current-jet invariant algebra, PBW symbol bound, ordinary-family base change and vacuum commutativity; it constructs every basic type A lift and proves the whole type A polynomial vacuum algebra. Section 3.4 proves the exact quantum coordinate laws and the coordinate-equivariant ordinary disc-oper comparison for every type A factor, including nilpotent parameter families. Section 3.5 proves reductive vacuum factorization, the whole type A reductive polynomial algebra and its framed central coefficient comparison, and the exact algebraic distinction between central frames and gauges. Section 3.6 constructs the smooth affine completion, proves its full type A center and ordinary punctured-disc coordinate comparison, and computes the complete abelian center at every fixed form. Section 3.7 constructs the intrinsic critical Poisson vertex structure and jointly continuous completed bracket for every finite-dimensional affine datum, proves the ordered Miura injection and its actual type A affine-to-boson Poisson comparison, and identifies the full type A oper Poisson bracket. Section 3.8 constructs the ordinary principal matrix Hamiltonian reduction in type A, proves its regular-jet and Laurent gauge quotients and actual classical Miura reduction, and identifies its complete scalar Adler bracket and coordinate action. Section 3.9 proves the ordinary principal classical reduction for the general specified graded semisimple datum: its affine PVA normalizer, polynomial regular-jet invariants, cofinal full Laurent quotient, all-mode reduced Poisson bracket and full ordinary coordinate action, including the quadratic Virasoro and Schwarzian laws at the chosen invariant form. The quantum-center and center/oper-Poisson comparison in other types, exact dual-form normalization, derived/global reduction, localization, quantization and fundamental local equivalence remain unproved; their precise statements below retain their complete scope. The classical calculations use a fixed smooth projective connected curve over an algebraically closed characteristic-zero field, \(g\ge2\). The regular singular example separately uses the punctured projective line. The classical center and eigenobject statements in §§4–5 have their stated complex semisimple or simply connected hypotheses. The type A center, Poisson and framed reductive coefficient comparisons in §§3.3–3.8, and the general principal classical reduction at its fixed structural data in §3.9, hold over every characteristic-zero field and ordinary coefficient algebra at their stated foundations. The full eigenobject theorem and intrinsic/global reductive extensions remain unproved. Connections are algebraic de Rham connections; localization uses left D-modules. Ordinary-family assertions concern the fixed curve. Analytic comparisons and full derived moduli retain separate foundations. ## 1. General oper condition and the full adjoint target For a connected reductive group \(H\), choose a Borel \(B\), maximal torus and simple roots \(\Delta\), and put \[ \mathfrak h_{-1}=\mathfrak b+\sum_{\alpha\in\Delta}\mathfrak h_{-\alpha}. \tag{A8.1} \] An \(H\)-oper is an \(H\)-bundle with connection and Borel reduction whose second fundamental form lies in \(\omega\otimes(\mathfrak h_{-1}/\mathfrak b)\), with every simple-root line component nowhere zero. Positive-root conjugation preserves this quotient subspace, and torus conjugation acts by the root characters, proving frame independence. A central connection is not measured by these components; general reductive and adjoint oper problems must retain that distinction. The full adjoint-group theorem says that, for adjoint semisimple \(H\) and \(g\ge2\), its oper space is an affine space modeled on \[ \bigoplus_jH^0(X,\omega^{d_j+1}), \tag{A8.2} \] where the principal exponents \(d_j\) are retained with multiplicity. Sections 1.1.1–1.1.8 prove the full adjoint construction from the stated root/pinning and group premises, including its finite \(\mathfrak{sl}_2\) decomposition, Drinfeld–Sokolov slice, coordinate gauges and global existence. They also construct the ordinary formal-disc and punctured-disc coefficient functors and coordinate actions. Section 1.2 proves the comparison of principal exponents with invariant-polynomial degrees and the weighted Kostant section, using the earlier invariant-ring proofs at their explicit recursive foundations. Section 2 gives an independent scalar proof for \(PGL_n\), with exponents \(1,\ldots,n-1\). ### 1.1. Drinfeld–Sokolov normal form and global adjoint opers #### 1.1.1. The pinned principal triple and the height grading Let \(k\) be algebraically closed of characteristic zero. Let \(H\) be an adjoint semisimple group, with a pinning \((B,T,(e_i))\). Write \(N\) for the unipotent radical of \(B\), \(\mathfrak g=\operatorname{Lie}H\), \(\mathfrak t=\operatorname{Lie}T\), and \(\Delta=\{\alpha_1,\ldots,\alpha_\ell\}\) for the simple roots. Normalize the negative simple-root vectors by \[ [e_i,f_j]=\delta_{ij}h_i,\qquad h_i=\alpha_i^\vee . \tag{DS.A1} \] A coroot in a Lie-algebra formula means the differential of that cocharacter. The algebraic root and group foundations used here are explicit: the reduced crystallographic root datum of the pinned group; its one-dimensional root-space decomposition and bracket rules; the simple coroot basis of \(\mathfrak t\); the displayed pinning relations; the semidirect product \(B=N\rtimes T\) and ordered positive-root subgroup coordinates on \(N\); and a finite-dimensional algebraic representation realizing \(H\) as a closed subgroup of a matrix group. For an adjoint group, \(X^*(T)\) is the root lattice. These are prerequisites concerning algebraic groups over this \(k\). A root-space proof restricted to complex Lie algebras does not by itself establish their full field generality. The arguments below prove the principal triple, the needed module decomposition and the gauge theorem from these premises, for every adjoint semisimple \(H\). Put \[ h=2\rho^\vee:=\sum_{\beta\in\Phi^+}\beta^\vee =\sum_i c_i h_i,\qquad f=\sum_i f_i,\qquad e=\sum_i c_i e_i. \tag{DS.A2} \] Every \(c_i\) is a positive integer. Here is the root-system argument, including the normalization of \(h\). The dual root system has simple roots \(\alpha_i^\vee\). Indeed a positive coroot is a positive real combination of these vectors by rescaling the simple-root expansion of its root. Each \(\alpha_i^\vee\) is indecomposable among positive coroots: a decomposition would have all coefficients outside \(i\) zero, forcing proportional positive coroots, which reducedness excludes. The base theorem in [*Root systems and their Weyl groups*, Theorem 2.1 and the dual-system calculation (1.3)](../../RT-LIE/src/RT-LIE-08.md) then makes their expansions integral and nonnegative. The sum defining \(h\) includes each \(h_i\), so \(c_i>0\). A simple reflection permutes positive coroots other than \(h_i\), and takes \(h_i\) to \(-h_i\). To check the permutation, expand a positive coroot other than \(h_i\) in the simple coroot basis. Some coefficient outside \(i\) is positive; reflection changes only the \(i\)-coefficient. Its image is a root with a positive coefficient, hence is positive by the same-sign property. Involutivity gives the permutation. This is the argument of Lemma 4.1 in that same root-system lesson. Therefore \[ s_i(h)=h-2h_i. \] But the reflection formula on \(\mathfrak t\) is \(s_i(x)=x-\alpha_i(x)h_i\). Comparing gives \(\alpha_i(h)=2\). The pinning relations now give \[ [h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h. \tag{DS.A3} \] For \(H\ne1\), these are an actual \(\mathfrak{sl}_2\) triple. The three vectors are nonzero and independent: \(h\) has simple-root values \(2\), and \(e,h,f\) have distinct adjoint-\(h\) weights \(2,0,-2\). This uses no principal-nilpotent classification. For the trivial group the three vectors are zero, and its coefficient functors are treated below. Since the simple roots form a basis of \(X^*(T)\), there is a cocharacter \(\rho^\vee\) with \(\alpha_i(\rho^\vee(a))=a\). Its differential is \(h/2\). For a root \(\beta=\sum_i n_i\alpha_i\), set \(\operatorname{ht}(\beta)=\sum_i n_i\). The adjoint action of \(h\) gives \[ \mathfrak g=\bigoplus_{m=-q}^{q}\mathfrak g_m,\qquad [h,x]=2m x\quad(x\in\mathfrak g_m),\qquad \mathfrak g_0=\mathfrak t,\quad \mathfrak b=\bigoplus_{m\geq0}\mathfrak g_m,\quad \mathfrak n=\bigoplus_{m\geq1}\mathfrak g_m . \tag{DS.A4} \] Here \(q\) is the largest positive-root height when \(H\ne1\). Brackets add heights. In particular \(E=\operatorname{ad}e\) raises height by one and \(F=\operatorname{ad}f\) lowers it by one. Both are nilpotent on this finite graded vector space. #### 1.1.2. A constructive finite \(\mathfrak{sl}_2\) decomposition We need a splitting, not merely a dimension count. The following argument constructs it using only finite linear algebra. Consider a finite-dimensional vector space \(K\) with operators \(E,F,H_0\) satisfying \([H_0,E]=2E\), \([H_0,F]=-2F\), and \([E,F]=H_0\). Suppose \(H_0\) is diagonalizable with integer weights and \(E,F\) move between those weight spaces. Finiteness of the weights makes \(E,F\) nilpotent. If \(Ev=0\) and \(H_0v=nv\), the commutator induction gives \[ H_0F^jv=(n-2j)F^jv,\qquad EF^jv=j(n-j+1)F^{j-1}v. \tag{DS.A5} \] For the second identity the first step is \(EFv=nv\), and the next step is \[ EF^{j+1}v=(FE+H_0)F^jv =\bigl(j(n-j+1)+n-2j\bigr)F^jv =(j+1)(n-j)F^jv. \] If \(a\) is the last index with \(F^av\ne0\), applying this identity at \(a+1\) gives \(0=(a+1)(n-a)F^av\). Characteristic zero implies \(n=a\geq0\). Its string consequently has exactly \(n+1\) independent vectors, with the displayed coefficients. This reproduces the elementary string argument in [*Representations of \(\mathfrak{sl}_2\)*, §1](../../RT-LIE/src/RT-LIE-05.md); the calculation works over our \(k\). We now split these strings off without importing complete reducibility. Induct on \(\dim K\). Let \(n\) be its largest weight. Then \(EK[n]=0\), and the preceding argument makes \(n\geq0\). The submodule \[ W=\bigoplus_{j=0}^{n}F^jK[n] \tag{DS.A6} \] is a direct sum of strings. Each map \(F^j:K[n]\to K[n-2j]\) is injective for \(j\leq n\), since \[ E^jF^j|_{K[n]}= j!\,\frac{n!}{(n-j)!}\operatorname{id}. \] Different \(j\)'s have different weights, and a basis of \(K[n]\) therefore supplies a string basis of \(W\). Set \[ C=H_0^2+2H_0+4FE. \tag{DS.A7} \] This operator commutes with \(E,F,H_0\). For example, \[ [C,E]=2H_0E+2EH_0+4E-4H_0E=0, \] using \(H_0E-EH_0=2E\); the computation with \(F\) is \(-2H_0F-2FH_0-4F+4FH_0=0\). It acts on an \(n\)-string by \(n(n+2)\), first on its highest vector and then on every \(F\)-iterate. The quotient \(K/W\) has smaller dimension and has no weight \(n\). By induction it is a direct sum of strings with highest weights \(\mu0. \tag{IS.F1} \] The \(p_i\) are homogeneous and algebraically independent. Their degrees will be compared with the principal heights by the argument below. The earlier [*The isomorphism theorem and Serre's theorem*](../../RT-LIE/src/RT-LIE-11.md), §§1–6.3, constructs the rational Serre algebra \(\mathfrak g_{\mathbb Q}(A)\) for the finite Cartan matrix. Corollary 6.3 proves that its presentation commutes with field extension, has the prescribed one-dimensional root spaces and has dimension \(\ell+|\Phi|\). Here is why it gives the rational form of this particular pinned Lie algebra. Its generators map to the chosen \(e_i,f_i,h_i\). The Cartan, cross-root and Serre relations follow from the root-string/pinning premises of §1.1.1. These generators span every root space: for a positive nonsimple root \(\beta\), choose a simple root \(\alpha_i\) with \((\beta,\alpha_i)>0\). The root string gives a positive root \(\beta-\alpha_i\) of smaller height, and its bracket with \(e_i\) is nonzero in the one-dimensional \(\beta\)-space. Induction generates every positive root space; the opposite base does the same for the negative spaces. The brackets \([e_i,f_i]\) give the Cartan basis. The resulting map is surjective, and both dimensions are \(\ell+|\Phi|\); it is therefore the pinned isomorphism \[ \mathfrak g_{\mathbb Q}(A)\otimes_{\mathbb Q}k \simeq\mathfrak g. \tag{IS.F2} \] The recursive root-space and root-string construction from the algebraic group remains the explicit structural premise. The rational Serre presentation and its field-extension argument themselves are supplied by the earlier lesson. Write \(\mathfrak g_{\mathbb Q}\) and \(\mathfrak t_{\mathbb Q}\) for these rational spaces. Each Weyl reflection on the Cartan is the rational matrix \(s_i(x)=x-\alpha_i(x)h_i\). In polynomial degree \(d\), infinitesimal invariants are the kernel of a finite rational linear map \[ L_d:\operatorname{Sym}^d(\mathfrak g_{\mathbb Q}^*) \longrightarrow \bigoplus_a\operatorname{Sym}^d(\mathfrak g_{\mathbb Q}^*), \qquad F\longmapsto (x_a\cdot F)_a, \tag{IS.F3} \] where the \(x_a\) form a rational Lie basis. Weyl invariants are likewise the kernel of the finite family of maps \(w-1\) on \(\operatorname{Sym}^d(\mathfrak t_{\mathbb Q}^*)\). Tensoring these finite linear maps with a field is exact, so both kernels commute with extension of scalars. Restriction is a rational map between them. The earlier [*The centre of the enveloping algebra and Harish-Chandra's theorem*](../../RT-LIE/src/RT-LIE-15.md), Theorem 3.1, proves complex Chevalley restriction. Its proof establishes root-exponential invariance, uses an invertible differential of the root-conjugation map for injectivity, and lifts orbit-power sums by traces of finite-dimensional highest-weight representations for surjectivity. Its §§9.1–9.7 prove polynomiality of the Weyl-invariant ring through invariant fundamental characters, completion at the identity, and homogeneous generators. These are the matching polynomial statements, rather than the enveloping-algebra conclusion. Applied to \(\mathfrak g_{\mathbb Q}\otimes\mathbb C\), restriction in each degree is an isomorphism. Its rational kernel and cokernel vanish after tensoring with \(\mathbb C\), hence vanish: a nonzero rational vector stays nonzero after extension of a basis. Tensoring this rational isomorphism with \(k\) proves the restriction part of (IS.F1) for Lie invariants. Polynomiality also descends, with no chosen complex generators assumed rational. Put \(B_{\mathbb Q}=\mathbb Q[\mathfrak t_{\mathbb Q}]^W\), \(J=\bigoplus_{d>0}(B_{\mathbb Q})_d\). Every graded piece is finite-dimensional. Formation of \(J/J^2\) commutes with field extension: in degree \(d\), the square is the image of the finite sum of multiplication maps with positive degrees adding to \(d\). For a polynomial ring on homogeneous generators, this quotient has their classes as a basis. Thus the complex polynomiality proof gives \[ \dim_{\mathbb Q}J/J^2=\ell, \qquad (J/J^2)_d=0\text{ outside finitely many positive degrees}. \tag{IS.F4} \] Choose a homogeneous rational basis of this quotient and homogeneous lifts \(q_1,\ldots,q_\ell\) in \(B_{\mathbb Q}\). They generate the ring. Indeed in degree \(d>0\), subtract the linear combination of lifts matching the indecomposable class. The remainder is a finite sum of products of strictly smaller positive degrees, each already generated by induction. Degree zero consists of rational constants. They are algebraically independent as well. Their degrees are exactly the complex polynomial-generator degrees, with multiplicity, since those degrees are measured by \(J/J^2\). The graded surjection \(\mathbb Q[z_1,\ldots,z_\ell]\to B_{\mathbb Q}\), \(z_i\mapsto q_i\), after complex scalar extension has equal finite dimensions in every degree: both Hilbert series are \(\prod_i(1-t^{D_i})^{-1}\). Its degreewise kernels vanish, hence its rational kernel vanishes. This proves \[ B_{\mathbb Q}=\mathbb Q[q_1,\ldots,q_\ell], \qquad k[\mathfrak t]^W=k[q_1,\ldots,q_\ell]. \tag{IS.F5} \] The inverse rational restriction map lifts each \(q_i\) to a homogeneous rational \(p_i\). Extending to \(k\) proves polynomiality for the Lie-invariant ring. All of this uses scalar extension through \(\mathbb Q\); the field \(k\) requires no chosen embedding into \(\mathbb C\). Finally Lie invariance agrees with group invariance here. If \(F\) is killed by every Lie derivative, a root subgroup \(\exp(te_\alpha)\) fixes it: differentiation of \(F(\operatorname{Ad}(\exp(te_\alpha))X)\) gives zero, and a polynomial in \(t\) with zero derivative is constant in characteristic zero. A torus weight component killed by every Cartan derivative has weight zero, since a nonzero integral character has a nonzero differential in characteristic zero. Thus the torus fixes \(F\) too. For completeness one can check that these identities force invariance on the whole connected group without a pointwise generation assumption. For fixed \(X\), the orbit function \(a\mapsto F(\operatorname{Ad}(a)X)\) on \(H\) has zero derivative in every left-invariant tangent direction, by Lie invariance at \(\operatorname{Ad}(a)X\). Such directions span each tangent space. A rational function on an integral finite-type variety over an algebraically closed characteristic-zero field with differential zero is constant: if it were transcendental, include it in a transcendence basis of the function field; the finite algebraic remainder is separable, and its differential would be a nonzero basis element. An algebraic function-field element over \(k\) lies in \(k\). Hence this orbit function is constant. Connected smooth \(H\) is integral; its irreducible components are disjoint because its regular local rings are domains, so connectedness leaves one. These smooth/group and regular-local facts retain the structural foundations already specified. Evaluation at the identity gives the required group invariance, as a polynomial identity in \(X\) and group coordinates. The converse follows by differentiation. This finishes (IS.F1). The assertion persists over every ordinary \(k\)-algebra \(R\), including nonreduced ones. Group invariants mean the kernel of the algebraic coaction difference \(F\mapsto\operatorname{coact}(F)-F\otimes1\), rather than invariance only under the points \(H(R)\). In each homogeneous degree its domain is finite-dimensional, and its coefficient image lies in a finite-dimensional part of the coaction target. Tensor with \(R\) is exact because \(k\) is a field, so \[ R[\mathfrak g_R]^{H_R} =R\otimes_k k[\mathfrak g]^H =R[p_1,\ldots,p_\ell], \quad R[\mathfrak t_R]^W=R\otimes_k k[\mathfrak t]^W, \tag{IS.F6} \] and restriction remains an isomorphism. Products are componentwise; for the trivial group both rings are the coefficient ring and the generator list is empty. #### 1.2.2. Molien's formula and the two degree identities Continue with the pinned adjoint semisimple \(H\) over an algebraically closed characteristic-zero field \(k\), its Cartan \(\mathfrak t\), Weyl group \(W\), principal triple \((e,h,f)\), and \(V=\ker(\operatorname{ad}e)\). Put \(\ell=\dim\mathfrak t\). The input from §1.2.1 is precisely a graded presentation \[ k[\mathfrak g]^H\xrightarrow{\ \sim\ }k[\mathfrak t]^W =k[P_1,\ldots,P_\ell],\qquad \deg P_i=D_i>0, \tag{IS.A1} \] where the generators are algebraically independent and are also denoted \(P_i\) on \(\mathfrak g\). Here invariance means invariance of the algebraic group coaction. The complete complex-field restriction and polynomiality arguments are [*The centre of the enveloping algebra and Harish-Chandra's theorem*, Theorems 3.1 and 9.1](../../RT-LIE/src/RT-LIE-15.md). The algebraic root/group and characteristic-zero transport premises specified in §1.2.1 are retained; a complex-field theorem is not silently substituted for that transport. We prove the degree comparison and slice restriction from this exact presentation, without a generator-degree classification. First determine which elements of \(W\) are linear reflections. The root datum supplies its rational reflection representation, whose scalar extensions give the real Euclidean root space and the action on \(\mathfrak t^*\). In a simple-root basis the matrices have integer entries. Ranks of their fixed-space equations are therefore the same over \(\mathbf Q\), \(\mathbf R\), and \(k\). A nonidentity real orthogonal element with a fixed hyperplane acts by \(-1\) on its perpendicular line, so it is the orthogonal reflection in that hyperplane. Its hyperplane must be a root hyperplane. Otherwise choose a point on it avoiding all root hyperplanes: their finitely many proper intersections cannot cover it. In rank one there is only the hyperplane \(\{0\}\), which is already the root hyperplane. In higher rank the avoidance assertion follows by taking the product of the finitely many nonzero restricted linear equations; a nonzero real polynomial cannot vanish everywhere. Such an avoided point lies inside a Weyl chamber. An element fixing it preserves that chamber, because it permutes the root-hyperplane arrangement. The chamber stabilizer is trivial by [*Root systems and their Weyl groups*, §3 and Theorem 5.1](../../RT-LIE/src/RT-LIE-08.md). The proof there joins chambers through single-wall crossings, while the root-sign length formula forces a chamber-preserving element to have length zero. Thus our alleged nonidentity reflection would be the identity, a contradiction. Its fixed hyperplane is consequently \(\alpha^\perp\) for a root, and uniqueness of orthogonal reflection gives \(s_\alpha\). Conversely every \(s_\alpha\) is a reflection. Reducedness identifies precisely the pair \(\alpha,-\alpha\) on each root line, so the number of reflections is \[ r_{\mathrm{ref}}=|\Phi|/2=|\Phi^+|. \tag{IS.A2} \] Now consider the homogeneous degree-\(n\) space \(\operatorname{Sym}^n(\mathfrak t^*)\). Its Reynolds operator is \[ \mathcal R_n=\frac1{|W|}\sum_{w\in W}w. \] It is idempotent, has image the invariants, and its trace is the dimension of that image: choose a basis of its image and kernel, on which it is respectively identity and zero. Every \(w\) is diagonalizable over \(k\), since its finite order makes its minimal polynomial divide a separable polynomial \(X^a-1\). If its eigenvalues on \(\mathfrak t^*\) are \(\lambda_1,\ldots,\lambda_\ell\), the monomial basis gives \[ \sum_{n\geq0}\operatorname{Tr}\!\left(w\mid\operatorname{Sym}^n\mathfrak t^*\right)t^n =\prod_{j=1}^{\ell}\left(\sum_{a\geq0}(\lambda_jt)^a\right) =\frac1{\det(1-tw\mid\mathfrak t^*)}. \] Averaging this identity proves Molien's formula, including its coefficients: \[ \operatorname{Hilb}\bigl(k[\mathfrak t]^W,t\bigr) =\frac1{|W|}\sum_{w\in W}\frac1{\det(1-tw)} =\prod_{i=1}^{\ell}(1-t^{D_i})^{-1}. \tag{IS.A3} \] The second equality follows from the monomial basis in the polynomial presentation (IS.A1). Assume \(\ell>0\), put \(s=1-t\), and expand these rational functions as formal Laurent series in \(s\). The identity element contributes \(s^{-\ell}\). A reflection contributes \[ \frac1{(1-t)^{\ell-1}(1+t)} =\tfrac12s^{1-\ell}+O(s^{2-\ell}). \] Every other element fixes a space of dimension at most \(\ell-2\); its denominator therefore gives a pole of order at most \(\ell-2\). Thus the first two coefficients in the group average are \[ \frac1{|W|}s^{-\ell} +\frac{r_{\mathrm{ref}}}{2|W|}s^{1-\ell} +O(s^{2-\ell}). \tag{IS.A4} \] For \(\ell=1\) the remaining group-element sum is empty, and the same expansion holds. On the polynomial side, \[ 1-(1-s)^D=Ds\left(1-\frac{D-1}{2}s+O(s^2)\right), \] so its expansion is \[ \frac1{\prod_iD_i}s^{-\ell} \left(1+\frac12\sum_i(D_i-1)s+O(s^2)\right). \] These are formal algebraic expansions, requiring no analytic limit. Comparing their two coefficients gives \[ \prod_iD_i=|W|,\qquad \sum_i(D_i-1)=r_{\mathrm{ref}}=|\Phi^+|. \tag{IS.A5} \] #### 1.2.3. Dominance of the principal slice Choose the homogeneous basis \(v_1,\ldots,v_\ell\) of \(V\) constructed in §1.1.2, with heights \(d_j\geq1\), and put \[ \Sigma=f+V,\qquad x=f+\sum_j z_jv_j,\qquad w_j=d_j+1. \] On each constructed \(\mathfrak{sl}_2\) string, \(\operatorname{ad}f\) maps every vector except its lowest vector to the next vector, and its image misses exactly the highest line. Taking the sum of those strings gives the full splitting \[ \mathfrak g=[f,\mathfrak g]\oplus V,\qquad \dim V=\ell,\qquad \sum_j d_j=|\Phi^+|. \tag{IS.A6} \] This is the whole-module consequence of the written string decomposition, not an assertion of a new slice theorem. The adjoint torus cocharacter \(\rho^\vee\) acts on height \(m\) by \(a^m\). Hence the action \[ a\star x=a\,\operatorname{Ad}(\rho^\vee(a))x \tag{IS.A7} \] fixes \(f\), preserves \(\Sigma\), and sends \(z_j\) to \(a^{w_j}z_j\). All \(w_j\) are strictly positive. Ordinary degree-\(D_i\) homogeneity and adjoint invariance give \(P_i(a\star x)=a^{D_i}P_i(x)\). Therefore \(p_i(z):=P_i(f+\sum_jz_jv_j)\) is weighted homogeneous of degree \(D_i\). Consider \[ \mu:H\times\Sigma\longrightarrow\mathfrak g,\qquad (g,x)\longmapsto\operatorname{Ad}(g)x . \] Its differential at \((1,f)\) is \[ (X,v)\longmapsto[X,f]+v, \tag{IS.A8} \] which is surjective by (IS.A6). We give the polynomial consequence explicitly. Take one parameter for every positive and negative root subgroup, and torus parameters \(u_i\) through the cocharacter points \(\varpi_i^\vee(1+u_i)\), on the open set where \(1+u_i\ne0\). A product of these root-subgroup and torus maps is a regular map to \(H\) whose differential at zero spans the root vectors and a basis of \(\mathfrak t\). Append the \(\ell\) coordinates of \(\Sigma\). The composed map to \(\mathfrak g\) consequently has the surjective differential (IS.A8). By finite linear algebra choose \(\dim\mathfrak g\) linear combinations of these parameter directions on which that differential is an isomorphism. Restrict to that linear parameter space and to the open set containing zero where the torus denominators remain invertible. Denote the resulting map by \(\Psi\). In its formal coordinates \(q\) at zero, \[ \Psi(q)=f+Aq+\text{terms of total degree at least two}, \qquad A\text{ invertible}. \] If a nonzero polynomial \(F\) on \(\mathfrak g\) vanished after this substitution, take the lowest nonzero homogeneous term \(F_b(Y)\) of \(F(f+Y)\). Its substituted lowest term would be \(F_b(Aq)\), which is nonzero by invertible linear substitution. This contradicts the assumed vanishing. The expansions of the torus denominators exist formally because their constant terms are one. Thus no nonzero polynomial vanishes on \(\mu\)'s image: \(\mu\) is dominant. This argument supplies the density directly. Let \(\chi:\mathfrak g\to\operatorname{Spec}k[P_1,\ldots,P_\ell]\) be the invariant map. It is dominant because the polynomial presentation embeds in \(k[\mathfrak g]\). Moreover \(\chi\mu=\chi|_\Sigma\operatorname{pr}_\Sigma\). If \(Q(p_1,\ldots,p_\ell)=0\), the invariant polynomial \(Q(P_1,\ldots,P_\ell)\) therefore vanishes after \(\mu\), hence is zero by the preceding argument. Algebraic independence in (IS.A1) makes \(Q=0\). We have proved \[ p_1,\ldots,p_\ell\text{ are algebraically independent},\qquad \chi|_\Sigma\text{ is dominant}. \tag{IS.A9} \] #### 1.2.4. The constant Jacobian and a finite polynomial inverse We first justify why (IS.A9) gives a nonzero Jacobian in characteristic zero. Put \(L=k(z_1,\ldots,z_\ell)\) and \(K=k(p_1,\ldots,p_\ell)\). Each \(z_j\) is algebraic over \(K\). Indeed \(\ell+1\) rational functions in \(\ell\) variables are algebraically dependent: clear a common denominator for all monomials of total degree at most \(N\). Their numerators then have degree at most \(cN\) for a fixed \(c\); there are \(\binom{N+\ell+1}{\ell+1}\) such monomials but only \(\binom{cN+\ell}{\ell}\) possible numerator monomials. For sufficiently large \(N\), linear dependence gives a polynomial relation. If \(z_j\) were transcendental over \(K\), the independence of the \(p_i\) would make \(p_1,\ldots,p_\ell,z_j\) independent, contradicting this count. Take the irreducible polynomial of \(z_j\) over \(K\). Its derivative does not vanish at \(z_j\): characteristic zero makes the derivative nonzero, and irreducibility prevents a common factor with the original polynomial. In the free \(L\)-space with basis \(dz_1,\ldots,dz_\ell\), differentiate its identity using the polynomial product rule and rational quotient rule. Its coefficients are rational functions of the \(p_i\), so their differentials are combinations of \(dp_i\). The nonzero derivative coefficient consequently expresses \(dz_j\) as an \(L\)-linear combination of the \(dp_i\). This holds for every \(j\). The \(\ell\) differentials \(dp_i\) thus span this \(\ell\)-dimensional space, and \[ J(z):=\det\left(\frac{\partial p_i}{\partial z_j}\right)\ne0. \tag{IS.A10} \] This is a field and polynomial argument; no smooth-image assertion is used. Each matrix entry in (IS.A10) has weighted degree \(D_i-w_j\), if it is nonzero. Every determinant term has weighted degree \(\sum_iD_i-\sum_jw_j\). Formulas (IS.A5) and (IS.A6) give \[ \sum_iD_i=\ell+|\Phi^+| =\sum_j(d_j+1)=\sum_jw_j . \] Thus \(J\) has weighted degree zero. Positive weights leave only constants in degree zero, so \(J=c\in k^\times\). In particular the differential matrix at \(z=0\) is invertible. A linear term \(z_j\) can occur in \(p_i\) only when \(D_i=w_j\). A nonzero determinant term of that linear matrix therefore pairs each row degree with an equal column weight. This proves, with multiplicities, \[ \{D_1,\ldots,D_\ell\}=\{d_1+1,\ldots,d_\ell+1\}. \tag{IS.A11} \] For \(\ell>0\), reorder both lists by their distinct increasing weights \(a_1<\cdots\cdots>x_n\), and first treat a homogeneous symmetric polynomial. Its leading exponent tuple satisfies \(a_1\geq\cdots\geq a_n\): swapping an increasing adjacent pair would give a larger monomial with the same coefficient. The product \(e_1^{a_1-a_2}e_2^{a_2-a_3}\cdots e_n^{a_n}\) has exactly that leading monomial with coefficient one. Subtracting its leading coefficient times this product strictly decreases the leading monomial among the finitely many monomials of the fixed total degree. Repetition terminates and proves generation. Distinct monomials in the \(e_i\) have distinct leading exponent tuples, because \(a_j=\sum_{i\geq j}b_i\) for the tuple of powers \(b_i\); hence a polynomial relation cannot cancel its largest leading monomial. This proves independence. Separate homogeneous components give (IS.F8) for arbitrary polynomials. Trace zero imposes \(e_1=0\). Averaging by \(1/n!\) makes invariants commute with this quotient: an invariant quotient class has an invariant polynomial lift after averaging. If an invariant polynomial equals \(e_1G\), invariance of \(e_1\) and the domain property force \(G\) invariant, so the invariant kernel is precisely \(e_1k[e_1,\ldots,e_n]\). Consequently \[ \bigl(k[x_1,\ldots,x_n]/(x_1+\cdots+x_n)\bigr)^{S_n} =k[e_2,\ldots,e_n]. \tag{IS.F9} \] Chevalley restriction (IS.F1) lifts these uniquely to \(\mathfrak{sl}_n\); they are the coefficients \((-1)^ie_i\) of its characteristic polynomial. That proves the full invariant-algebra assertion used in (K6.2), with the same matrix normalization as the scalar-oper calculation. The previously derived heights \(1,\ldots,n-1\) give the same degrees, and the Hitchin and oper spaces both have dimension \((n^2-1)(g-1)\). The oper ring has the corresponding weighted filtration. Write \[ E=\bigoplus_{m\geq1}V_m\otimes_k H^0(X,\omega^{m+1}). \] A reference projective connection identifies its affine oper torsor with \(E\), so its polynomial coordinate ring is \(\operatorname{Sym}(E^*)\). Give a linear coordinate dual to the height-\(m\) summand weight \(m+1\), and let \(F_a\) span monomials of weight at most \(a\). Changing the reference translates only the quadratic summand in the direction of \(e\), by (DS.G7). Thus a weight-two coordinate changes by a scalar constant, while all higher-weight coordinates are unchanged. Substitution by this translation and its inverse preserve every \(F_a\) and act identically on the highest-weight symbols. Changing homogeneous bases is a linear change within equal-weight blocks. This proves that the filtration is independent of the reference and such bases, and \[ \operatorname{gr}_F k[\operatorname{Op}_H(X)] \simeq\operatorname{Sym}(E^*). \tag{IS.F10} \] The weighted polynomial isomorphism of the slice with invariant coordinates has a weighted polynomial inverse, as proved in §§1.2.2–1.2.5. Applying its monomials to differential-valued coefficients glues, because the weights in every monomial sum to its target degree. The inverse glues for the same reason. Taking global sections therefore identifies \(E\), as a graded affine scheme, with \(\bigoplus_iH^0(X,\omega^{D_i})\), after the stated generator choices. This is an identification in ordinary families as well. For every ordinary \(k\)-algebra \(R\), the fixed-curve tensor proof recalled in §2.1 gives \[ H^0(X_R,\omega_R^d)=R\otimes_k H^0(X,\omega^d). \] Multiplication of sections is defined by multiplication on each open-set complex and commutes with this scalar extension. Thus both weighted polynomial maps, and their already proved composition identities, commute with ordinary base change, including nonreduced coefficients. This identifies the ring in (IS.F10) with the Hitchin-base ring; the dual-root argument gives the same degrees for the automorphic dual group. The identification is a classical graded-ring construction. An equality with global critically twisted differential operators still requires the full filtered theorem (K6.1). For a connected reductive group, at its explicit Lie/root-datum structure premise, \(\mathfrak g=\mathfrak z\oplus\mathfrak g_{\rm ss}\). The adjoint action is trivial on \(\mathfrak z\) and factors through the adjoint semisimple group on the other summand. Polynomial coordinates on \(\mathfrak z\) therefore contribute degree-one generators, while the semisimple generators are the ones just constructed. Taking the coaction kernel gives the tensor product of these invariant rings, including after every ordinary scalar extension. This invariant-ring observation retains the central connections and central automorphisms in the oper problem; it does not turn a nonadjoint reductive oper stack into the adjoint affine scheme. ## 2. Classical \(PGL_n\)-opers in ordinary families ### 2.1. Exact imported foundations and the elementary line calculation (O0) The curve input in [The GL_1 case as an equivalence of categories](the-gl-1-case-as-an-equivalence-of-categories.md), §1.1.12, formula (R7.3), proves \[ H^1(X,\mathcal F)^\vee\simeq\operatorname{Hom}_X(\mathcal F,\omega) \tag{O0.1} \] for coherent \(\mathcal F\), with the constructed differential trace. Its §§1.1.13–1.1.14 prove the global residue formula and \[ \operatorname{tr}_X[\,d\log g_{ij}\,]=\deg M \tag{O0.2} \] for a line \(M\) with frames \(e_j=e_i g_{ij}\), using ordered Čech differential \((\delta a)_{ij}=a_j-a_i\). The class on the left is the class of \(0\to\omega\otimes M\to J^1M\to M\to0\). The trace uses the displayed frame and Čech signs. The fixed-curve tensor/Čech proof in §1.1.16 of that lesson supplies ordinary arbitrary-base tensor compatibility. Its ordinary Picard input in §1.1 is the smooth projective connected Jacobian \(J=\operatorname{Pic}^0(X)\), with the rigidified Picard construction described there. These inputs retain the explicit recursive coherent-cohomology, local-algebra, projective-scheme and Picard foundations of that lesson; their complete recursive proofs remain required. Here is the line-bundle dimension calculation, so no unproved Riemann–Roch citation is needed for the oper dimension. A rational section of a line \(M\) identifies it with \(\mathcal O_X(D)\) for its finite divisor \(D\); this follows by choosing a rational frame and comparing its valuations with a local DVR frame. Principal divisors have degree zero by the proved residue theorem applied to \(d\log f\). For every closed point \(p\), the local parameter calculation gives \[ 0\longrightarrow\mathcal O_X(D-p)\longrightarrow \mathcal O_X(D)\longrightarrow k_p\longrightarrow0. \tag{O0.3} \] The last map takes the coefficient of the one additional permitted local power. The finite-dimensional cohomology long exact sequence, and vanishing above degree one from the two-affine computation, imply that adding \(p\) increases the Euler characteristic by one. Repeating for positive and negative coefficients of \(D\) gives \[ \chi(M)=\deg M+1-g, \qquad g=\dim_kH^1(X,\mathcal O_X). \tag{O0.4} \] Here \(H^0(X,\mathcal O_X)=k\), proved in §1.1.12 of that earlier lesson. Applying (O0.1) to \(\mathcal O_X\) and to \(\omega\) gives \(h^0(\omega)=g\) and \(h^1(\omega)=1\). Applying (O0.4) to \(\omega\) consequently gives \[ \deg\omega=2g-2. \tag{O0.5} \] A nonzero regular section of a line has an effective divisor, so a line of negative degree has no section. Thus for every \(i\geq2\), duality and (O0.5) give \[ H^1(X,\omega^i)=H^0(X,\omega^{1-i})^\vee=0, \qquad h^0(X,\omega^i)=(2i-1)(g-1). \tag{O0.6} \] Finally, a theta characteristic exists. Choose a line \(M\) of degree \(g-1\), for example \(\mathcal O_X((g-1)p)\). The line \(\omega M^{-2}\) is in \(J(k)\). On the smooth group \(J\), the differential of multiplication by two at the identity is \(2\operatorname{id}\): the differential of the group law adds tangent vectors. Translations give the same invertible differential everywhere. The smooth local Jacobian criterion makes \([2]:J\to J\) étale, hence open. It is proper, since \(J\) is proper and its graph is closed in the product with the separated target, so its image is closed too. The image is nonempty, and connectedness of \(J\) makes it all of \(J\). Every nonempty fibre of this finite-type morphism has a closed \(k\)-point, because \(k\) is algebraically closed. Choose \(N\) with \(N^2\simeq\omega M^{-2}\); then \[ \theta=M\otimes N, \qquad \theta^2\xrightarrow{\sim}\omega. \tag{O0.7} \] This last construction uses the specified ordinary Jacobian foundation and the standard smooth-local Jacobian criterion and proper-image foundation; it is not a proof of Picard representability. From now on fix both \(\theta\) and the displayed square isomorphism. ### 2.2. The intrinsic oper condition (O1) Let \(G=PGL_n\), let \(B\subset G\) be the image of upper triangular matrices, and let \(\mathfrak b\subset\mathfrak g\) be its Lie algebra. A \(G\)-oper is a principal \(G\)-bundle \(P\), a connection \(\nabla\), and a \(B\)-reduction \(P_B\), subject to the following condition. The failure of \(\nabla\) to preserve \(P_B\) is its second fundamental form, a section of \(\omega\otimes(P_B\times^B(\mathfrak g/\mathfrak b))\). Require it to belong to the \(B\)-stable subbundle given by \[ \mathfrak g_{-1}/\mathfrak b =\bigoplus_{i=1}^{n-1}\mathfrak g_{-\alpha_i}, \qquad \mathfrak g_{-1}=\mathfrak b+ \bigoplus_{i=1}^{n-1}kE_{i+1,i}, \tag{O1.1} \] and require every simple-root component to be nowhere zero. The notation in (O1.1) means the quotient and its simple-root characters, not a claim that each negative root vector is fixed by \(B\). Conjugation by an upper triangular matrix preserves \(\mathfrak g_{-1}\); modulo \(\mathfrak b\), the adjacent lower diagonal entries are multiplied by their diagonal characters. This proves that the condition is independent of the adapted frame. Equivalently, after a vector-bundle lift, write the complete flag as \[ 0=F_0\subset F_1\subset\cdots\subset F_n=E, \quad\operatorname{rk}F_i=i. \] Then \[ \nabla(F_i)\subset F_{i+1}\otimes\omega, \qquad \alpha_i:\operatorname{gr}_iE \xrightarrow{\sim}\operatorname{gr}_{i+1}E\otimes\omega \quad(1\leq ii+1, \qquad A_{i+1,i}\in\mathcal O_X^\times. \tag{O1.3} \] This is the coordinate expression of the intrinsic condition, and not its definition. ### 2.3. A normalized vector lift, including its choices (O2) A \(B\)-reduction supplies a vector-bundle lift without a Brauer-group theorem. The map \(B_{GL_n}\to B\) has the group section sending a projective upper triangular matrix to its unique representative whose last diagonal entry is \(1\). Products preserve that condition. Associate to \(P_B\) this upper triangular representation, producing a rank-\(n\) bundle \(E_0\) with its complete flag. Its quotient \(Q=E_0/F_{n-1}\) is canonically \(\mathcal O_X\). Its projectivization is \(P\), because the section composed with \(GL_n\to PGL_n\) is the given inclusion of \(B\). The projective connection determines the maps \(\alpha_i\) in (O1.2). Iterating them, starting with \(Q=\mathcal O_X\), gives \[ \operatorname{gr}_i E_0\simeq\omega^{n-i}, \qquad \det E_0\simeq\omega^{n(n-1)/2}. \tag{O2.1} \] These are actual isomorphisms induced by the oper maps, rather than degree equalities. Put \[ L=\theta^{1-n}, \qquad E=E_0\otimes L. \tag{O2.2} \] Equations (O2.1) and \(\theta^2=\omega\) give a distinguished trivialization of \(\det E\); the quotient of its last flag step is the fixed line \(L\). A connection on \(P(E)\) has a unique lift to \(E\) inducing the zero connection on this trivialized determinant. To verify this directly, take frames whose wedges are the specified determinant frame. Their transition matrices have determinant one. Lift the local \(\mathfrak{pgl}_n\)-valued connection form by the unique trace-zero matrix, using \(\mathfrak{sl}_n\xrightarrow{\sim}\mathfrak{pgl}_n\) in characteristic zero. On an overlap the gauge formula \[ A\longmapsto gAg^{-1}-(\partial_tg)g^{-1} \tag{O2.3} \] preserves trace zero since \(\partial_t\log\det g=0\). Therefore the local lifts glue. Two lifts differ by a scalar one-form, and their determinant connections differ by \(n\) times that form, so the determinant condition makes them equal. This proves existence and uniqueness of the normalized lift. This construction does not identify arbitrary \(SL_n\)-opers with operators on the fixed line \(L\). If two determinant-trivialized vector connections have the same projective connection, local lifts of a projective isomorphism differ on overlaps by scalars. Those scalars are the transition functions of a line \(T\), giving \(E'\simeq E\otimes T\). The determinant identifications give \(T^n\simeq\mathcal O_X\). The scalar differences of the connection forms give a connection on \(T\), and the determinant condition says its \(n\)-th tensor power is the trivial connection. Thus other \(SL_n\) lifts are flat \(\mu_n\) twists. Projectivization removes this extra central choice. The normalized lift in (O2.2), with quotient fixed as \(L\), is the one used below. ### 2.4. Scalar normalization is coordinate independent (O3) A scalar oper for the chosen theta characteristic is a differential operator \[ D:L=\theta^{1-n}\longrightarrow L\otimes\omega^n=\theta^{1+n} \tag{O3.1} \] of order \(n\), with principal symbol \(1\) and subprincipal coefficient zero. Here is a definition and proof of the latter normalization. In a coordinate \(t\) and a local square frame \(\tau^2=dt\), use the frames \(\tau^{1-n}\) and \(\tau^{1+n}\). Write \[ D_t=\partial_t^n+a_{n-1}(t)\partial_t^{n-1} +\cdots+a_0(t). \tag{O3.2} \] The condition is \(a_{n-1}=0\). Square frames exist étale locally. Changing \(\tau\) to \(-\tau\) multiplies source and target frames by the same sign, so the expression is unchanged. For a coordinate change \(t=\varphi(s)\), put \(\lambda=\varphi'(s)\), an invertible function, and \(b=(n-1)/2\). The operator in density frames transforms as \[ D_s=\lambda^{b+1}D_t\lambda^b, \qquad\partial_t=\lambda^{-1}\partial_s. \tag{O3.3} \] For example the input coefficient transforms by \(f_t=\lambda^b f_s\), because its weight is \(-b\); the output weight is \(b+1\). These are the two factors in (O3.3). Half powers can be evaluated after an étale square-root extension; their signs cancel in the formula. Inductively, the two highest terms of the repeated operator are \[ (\lambda^{-1}\partial_s)^n =\lambda^{-n}\partial_s^n -\frac{n(n-1)}2\lambda^{-n}\frac{\lambda'}\lambda \partial_s^{n-1}+\text{terms of order at most }n-2. \tag{O3.4} \] Indeed, left multiplication by \(\lambda^{-1}\partial_s\) differentiates the leading coefficient, contributing \(-n\) to the next coefficient at the \((n+1)\)-st step, so the numbers obey \(c_{n+1}=c_n-n\), \(c_1=0\). Inserting (O3.4) into (O3.3), differentiation of the right factor contributes \(nb\lambda'/\lambda\) to the coefficient of \(\partial_s^{n-1}\). The two contributions cancel since \(nb=n(n-1)/2\). The original \(a_{n-1}\) contributes \(\lambda a_{n-1}(\varphi(s))\). Consequently monicity and vanishing of \(a_{n-1}\) are invariant. These equations work on Zariski coordinate opens even if a square frame is unavailable there. If a chosen frame \(\tau\) obeys \(\tau^2=h\,dt\), then, in the \(\tau\)-frames, the two highest coefficients of a normalized operator are \[ h^{-n}\partial_t^n -\frac{n(n-1)}2h^{-n}\frac{h'}h\partial_t^{n-1}. \tag{O3.5} \] To see this, formally put \(\tau=h^{1/2}(dt)^{1/2}\) and conjugate \(\partial_t^n\) by the source and target frame factors. The leading coefficient is \(h^{-n}\), and the next one is \(n(1-n)h'/(2h)\) times it. Both expressions belong to the original coordinate ring, and the square-root sign cancels. Thus (O3.5), with all further coefficients chosen arbitrarily, gives local algebraic normalized operators on an ordinary Zariski cover. ### 2.5. Equivalence of intrinsic opers and scalar opers (O4) Write \(J^rL\) for the bundle of \(r\)-jets of sections of \(L\), defined by the \((r+1)\)-st infinitesimal neighbourhood of the diagonal. In a coordinate and frame its components are \((f,f',\ldots,f^{(r)})\); characteristic zero allows factorials to be absorbed into this basis. The transition formula is the ordinary chain and product rule. Its triangular highest terms give the exact sequences \[ 0\longrightarrow L\otimes\omega^r\longrightarrow J^rL \longrightarrow J^{r-1}L\longrightarrow0, \qquad \det J^{n-1}L=L^n\otimes\omega^{n(n-1)/2}=\mathcal O_X. \tag{O4.1} \] The last equality uses the specified square isomorphism of \(\theta\), so it gives a fixed determinant frame. **Theorem O4.** The ordinary functor of \(PGL_n\)-opers on the fixed \(X\) is isomorphic to the functor of operators (O3.1) with the stated normalization. It has no nontrivial oper automorphisms. The equivalence uses \(\theta\) to present the functor by scalar operators; its projective output is independent of that choice in the precise sense of O5. **Proof from an intrinsic oper to an operator.** Take its normalized vector lift \((E,F_\bullet,\nabla)\) from O2 and the quotient \(\pi:E\to L\). In a local frame, write \(\nabla=\partial_t+A\) and let the quotient row be \(r_0=\pi\). Recursively define rows \[ r_{j+1}=\partial_t r_j-r_jA. \tag{O4.2} \] For a horizontal vector \(e(t)\), differentiation gives \(\partial_t(r_je)=r_{j+1}e\). Therefore \[ \Phi:E\longrightarrow J^{n-1}L, \qquad e\longmapsto(r_0e,\ldots,r_{n-1}e) \tag{O4.3} \] is intrinsic: it takes the finite horizontal Taylor extension of \(e\) and then the jet of its quotient. Existence and uniqueness of that finite extension can also be checked by solving the recursion \((j+1)e_{j+1}=-\sum_{a+b=j}A_a e_b\) modulo the required jet order. Coordinate and frame changes preserve the horizontal equation and the jet of \(\pi(e)\), so (O4.3) glues algebraically. This finite recursion needs no analytic solution theorem. Induction using (O1.2) shows that \(r_j\) vanishes on \(F_{n-j-1}\). Its induced map on \(\operatorname{gr}_{n-j}E\) is \((-1)^j\) times the product of the \(j\) successive oper maps, expressed in the chosen coordinate. Thus all antidiagonal entries of (O4.3) are invertible and \(\Phi\) is an isomorphism. Its determinant, compared with the fixed determinant frames from (O2.1) and (O4.1), is a universal sign: both determinants are products of exactly those oper maps, and the only additional factors are the row signs and the permutation reversing the graded order. In particular this determinant ratio has zero derivative. Since \(r_0,\ldots,r_{n-1}\) form a row basis, there are unique functions \(a_j\) such that \[ r_n+\sum_{j=0}^{n-1}a_jr_j=0. \tag{O4.4} \] Set \(D_t=\partial_t^n+\sum_{j=0}^{n-1}a_j\partial_t^j\). Its horizontal solution jets are exactly the image under \(\Phi\) of the horizontal vectors. The chain and product rules in (O4.3) show that the operators glue as an operator \(L\to L\omega^n\): their common leading symbol is the identity, and applying each local operator to the same formal quotient section gives the same unique relation. This also follows by transforming (O4.4) and using the row-basis uniqueness. In jet components the horizontal system is \[ \partial_t y=C_Dy, \quad C_D= \begin{pmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&&\ddots&\ddots&\vdots\\ 0&\cdots&0&0&1\\ -a_0&-a_1&\cdots&-a_{n-2}&-a_{n-1} \end{pmatrix}. \tag{O4.5} \] The corresponding connection \(\partial_t-C_D\) has determinant connection form \(a_{n-1}\,dt\). But \(\Phi\) carries the determinant-zero connection on \(E\) to this connection, and its determinant ratio with the fixed jet frame is constant. Thus \(a_{n-1}=0\). We have obtained a normalized scalar oper. **Proof from an operator to an intrinsic oper.** Given normalized \(D\), set \(E_D=J^{n-1}L\) and define its connection by (O4.5). To check gluing rather than assume it, view \(D\) as an \(\mathcal O_X\)-linear map \(J^nL\to L\omega^n\). Its restriction to the top jet kernel \(L\omega^n\) is the identity. Its kernel consequently maps isomorphically to \(J^{n-1}L\). There is an intrinsic holonomic map \(J^nL\to J^1(J^{n-1}L)\): take the first jet of the \((n-1)\)-jet of a section. In coordinates it sends \((f_0,\ldots,f_n)\) to the value components \((f_0,\ldots,f_{n-1})\) and derivative components \((f_1,\ldots,f_n)\). The chain and product rules show that this formula is compatible with coordinate and line-frame transitions, so it defines that intrinsic map. Compose it with the inverse from \(J^{n-1}L\) to the kernel of \(D\). The result is a splitting of the first-jet projection of \(E_D\), whose derivative equations are precisely (O4.5). For a local section \(y\), subtracting this splitting from its first jet gives \(dy-C_Dy\). Every ingredient is intrinsic, so the local connections agree on overlaps. Their trace is zero in the fixed determinant frame by the normalization proved in O3. Give \(E_D\) the flag \[ F_i=\ker(J^{n-1}L\longrightarrow J^{n-1-i}L) \quad(1\leq ib\), as \(ba+[a,b]\). Order words first by length and then lexicographically. A swap decreases the lexicographic word, and a bracket decreases length. This order is well founded, so reduction terminates. Disjoint pair reductions commute. The only overlapping ambiguity is \(abc\), \(a>b>c\). Reducing first the left or right pair and then comparing the lower terms gives \[ [a,[b,c]]-[b,[a,c]]-[[a,b],c]=0, \] by Jacobi. These lower words have smaller order, so induction resolves their reductions. Induction on the first possible disagreement between two reduction sequences now shows that their terminal linear combinations agree: resolve their first steps by the disjoint or overlapping calculation and use the smaller-word induction on the resulting terms. Extending the normal-form map linearly, it kills the defining relations inside every surrounding word and has identity on ordered words. Thus those words span the enveloping algebra and are independent. This proves precisely the ordered basis assertion needed here. Order negative modes before \(K\) and the nonnegative modes. Applying this basis to (K2.2) identifies its vector space with \(U(t^{-1}\mathfrak g[t^{-1}])\); its ordered negative-mode words are a basis. A mode \(x_m\) with \(m\) larger than the sum of the positive indices in a given negative-mode word kills that word applied to \(v\): commute it to the right, observing that every bracket subtracts one of those indices and a central term requires exact equality with a partial sum. This proves smoothness. Therefore formal nonnegative series act by finite sums on every vector, extending the vacuum to the formal affine Lie algebra. Any endomorphism of this induced module is determined by the image of \(v\). Conversely a vector \(w\) killed by all nonnegative modes defines the map \(uv\mapsto uw\); the induction relations make it well defined. Consequently \[ \operatorname{End}_{\widehat{\mathfrak g}}(V_\ell) \simeq V_\ell^{\mathfrak g[[t]]} \tag{K2.3} \] as vector spaces. This argument alone does not prove that its endomorphism algebra is commutative, nor identify it with functions on opers. The extension to formal Laurent series also preserves the Lie bracket. For a vector \(w\), sufficiently high positive tails kill \(w\), and sufficiently high tails kill each of the two fixed vectors obtained by applying either Laurent series to \(w\). Each series has a finite negative part. Choose a common truncation above all those annihilation bounds plus the largest absolute value of any negative mode index occurring in either series. The tails then kill both compositions; their brackets with the other's finite negative part have mode indices above the annihilation bound for \(w\); brackets with positive parts have still higher indices. Central residue terms involve only finitely many matching positive and negative indices and are already retained in the chosen truncation. Thus the bracket identity on \(w\) reduces to the polynomial bracket identity. This proves a representation of the formal affine Lie algebra, rather than only a finite-sum definition of its action. ### 3.1. The critical quadratic vector (K3) Take \([h,e]=2e,[h,f]=-2f,[e,f]=h\) and the form \((e,f)=1,(h,h)=2\), with all other basis pairings zero. In this normalization the Killing form is four times the displayed form: \(\operatorname{ad}h\) has eigenvalues \(2,0,-2\), so its squared trace is \(8\), and invariance fixes the remaining pairings. Thus the critical form, minus half the Killing form, means \(\ell=-2\) in (K2.1). Define \[ Q=\left(\tfrac12h_{-1}h_{-1} +e_{-1}f_{-1}+f_{-1}e_{-1}\right)v. \tag{K3.1} \] This is twice the convention in which the Sugawara vector has coefficients \(1/4,1/2,1/2\). Its degree-two ordered symbol is \(\tfrac12h^2+2ef\), so the ordered basis proves that \(Q\ne0\). We compute all nonnegative-mode actions. The zero modes kill \(Q\). For example, \(e_0\) on its first term gives \(-e_{-1}h_{-1}v-h_{-1}e_{-1}v\); on its last two terms it gives the opposite sum. The \(f_0\) calculation exchanges \(e,f\) and replaces \(h\) by \(-h\). The \(h_0\) calculation cancels the weights \(2,-2\) in each mixed term. For mode one, the contributions from the three summands of (K3.1) are as follows: | Acting mode | \(\tfrac12h_{-1}^2v\) | \(e_{-1}f_{-1}v\) | \(f_{-1}e_{-1}v\) | | --- | --- | --- | --- | | \(e_1\) | \(2e_{-1}v\) | \(\ell e_{-1}v\) | \((\ell+2)e_{-1}v\) | | \(f_1\) | \(2f_{-1}v\) | \((\ell+2)f_{-1}v\) | \(\ell f_{-1}v\) | | \(h_1\) | \(2\ell h_{-1}v\) | \(2h_{-1}v\) | \(2h_{-1}v\) | Each entry follows by commuting that mode to the right until it reaches \(v\). For instance \([e_1,h_{-1}]=-2e_0\), \([e_1,f_{-1}]=h_0+\ell\), and \(h_0e_{-1}v=2e_{-1}v\). Thus for each basis vector \(x\), \[ x_1Q=2(\ell+2)x_{-1}v. \tag{K3.2} \] For mode two, the \(e_2,f_2\) terms end in \(e_0v,f_0v\), hence vanish. For \(h_2\), the first summand contributes zero, and the mixed summands give \(2\ell v\) and \(-2\ell v\). They cancel. For \(m\ge3\), commuting twice through the two negative modes leaves only modes of index \(m-2\ge1\), and no central term can occur since the total index is positive. All such terms kill \(v\). We have checked every nonnegative mode. At \(\ell=-2\), therefore, \(Q\in V_{-2}^{\mathfrak{sl}_2[[t]]}\), and (K2.3) gives an actual affine-module endomorphism. At any other level \(e_1Q\ne0\), since \(e_{-1}v\) is a basis vector and the field has characteristic zero. This proves the central-vector calculation at exactly the critical level. The result just proved is a vacuum invariant and its corresponding endomorphism. A claim that all normally ordered Fourier coefficients are central in a completed enveloping algebra also requires the completion, vertex operations and their compatibility. Section 3.2 supplies all quadratic mode operators, their commutativity and the full rank-one vacuum-center proof. Section 3.6 proves completed centrality and the full type A completed center. Section 3.7 supplies the intrinsic critical Poisson vertex structure and completed bracket, and the full type A oper-Poisson comparison. Chiral/factorization compatibility and quantum lifting in other types retain their separate proof obligations. ### 3.2. The full rank-one vacuum center and its coordinate action #### 3.2.1. All normally ordered quadratic modes We retain the conventions proved in §3, “An affine Lie algebra and its vacuum,” of [Opers, critical level and the Beilinson–Drinfeld construction](opers-critical-level-and-the-beilinson-drinfeld-construction.md): the affine bracket (K2.1), the ordered negative-mode basis, smoothness of the vacuum, and the endomorphism correspondence (K2.3). Here \(k\) has characteristic zero, \(\mathfrak g=\mathfrak{sl}_2(k)\), and \[ [h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h,\qquad B(e,f)=1,\quad B(h,h)=2. \] All other basis pairings are zero. On a module of scalar level \(\ell\), the bracket is \[ [x_m,y_j]=[x,y]_{m+j} +m\ell B(x,y)\delta_{m+j,0}\operatorname{Id}. \] A module is **smooth** if, for every vector \(w\), some \(N\ge0\) satisfies \(x_jw=0\) for every \(x\in\mathfrak g\) and \(j\ge N\). Put \(\theta(j)=0\) for \(j<0\) and \(\theta(j)=1\) for \(j\ge0\). Define \[ :a_jb_q: =\begin{cases}a_jb_q,&j<0,\\ b_qa_j,&j\ge0,\end{cases} \qquad S_n=\sum_{j\in\mathbb Z} \left(\tfrac12:h_jh_{n-j}: +:e_jf_{n-j}:+:f_je_{n-j}:\right). \tag{RC.M1} \] These formulas define actual linear operators on every smooth module. If \(w\) has annihilation bound \(N\), a term with \(j\ge N\) has a rightmost mode killing \(w\). A term with \(j<0\) and \(n-j\ge N\) also kills \(w\). Thus any cutoff \(L\ge N+|n|+1\) retains all nonzero terms on \(w\). Taking a common bound for finitely many vectors proves linearity and permits the finite comparisons below. The tensor \[ C=e\otimes f+f\otimes e+\tfrac12h\otimes h =\sum_a a\otimes a^\vee \tag{RC.M2} \] uses the basis \(a=e,f,h\) and its \(B\)-dual basis \(a^\vee=f,e,h/2\). It is invariant under the diagonal adjoint action. For \(e\), its derivative is \(e\otimes h+h\otimes e-e\otimes h-h\otimes e=0\). For \(f\), it is \(-h\otimes f-f\otimes h+f\otimes h+h\otimes f=0\). For \(h\), the terms are \(2e\otimes f-2e\otimes f-2f\otimes e+2f\otimes e=0\). Linearity proves invariance for every \(x\). We compute the normal-order boundary explicitly. Set \(q=n-j\), \(D=[x_m,a_j]\), and \(E=[x_m,b_q]\), including their scalar central terms. Normal ordering a scalar times a mode means ordinary multiplication. Since \(:a_jb_q:=a_jb_q-\theta(j)[a_j,b_q]\), Jacobi gives \[ [x_m,:a_jb_q:] =Db_q+a_jE-\theta(j)\bigl([D,b_q]+[a_j,E]\bigr). \] The Lie-mode part of \(D\) has index \(j+m\); its scalar part commutes with \(b_q\). Subtracting \(:Db_q:=Db_q-\theta(j+m)[D,b_q]\) and \(:a_jE:=a_jE-\theta(j)[a_j,E]\) therefore gives the exact identity \[ \begin{split} [x_m,:a_jb_{n-j}:] ={}&:[x_m,a_j]b_{n-j}: +:a_j[x_m,b_{n-j}]:\\ &+\bigl(\theta(j+m)-\theta(j)\bigr) [[x,a]_{j+m},b_{n-j}]. \end{split} \tag{RC.M3} \] This is an identity of operators for each fixed index \(j\), including the scalar central terms. Sum (RC.M3) over the three dual-basis pairs in (RC.M2). The Lie-mode parts of the first two terms cancel. Indeed, reindex the first sum by \(i=j+m\). For each resulting index \(i\), their tensor coefficient is \[ \sum_a [x,a]\otimes a^\vee+a\otimes[x,a^\vee]=0, \] and both terms use the same normal-order rule with first index \(i\) and second index \(m+n-i\). Their scalar central parts instead give \[ m\ell\sum_a \left(B(x,a)a^\vee_{m+n} +B(x,a^\vee)a_{m+n}\right) =2m\ell x_{m+n}. \tag{RC.M4} \] The first scalar contribution occurs at \(j=-m\); the second occurs at \(j=m+n\). The dual-basis identity proves the last equality. For the remaining boundary term, expand the affine bracket: \[ \sum_a [[x,a]_{j+m},a^\vee_{n-j}] =(A_x)_{m+n} +(j+m)\ell t_x\delta_{m+n,0}\operatorname{Id}, \quad A_x=\sum_a [[x,a],a^\vee],\quad t_x=\sum_a B([x,a],a^\vee). \] Here is the full finite Lie-algebra calculation: \[ \begin{array}{c|ccc|c|c} x&[[x,e],f]&[[x,f],e]&[[x,h],h/2]&A_x&t_x\\ \hline e&0&2e&2e&4e&0\\ f&2f&0&2f&4f&0\\ h&2h&2h&0&4h&2-2=0 \end{array} \qquad A_x=4x,\quad t_x=0. \tag{RC.M5} \] For the \(e\) and \(f\) rows, each pairing defining \(t_x\) is zero because the form pairs only \(e\) with \(f\) and \(h\) with \(h\). Thus even when \(m+n=0\), no scalar boundary term survives. Only finitely many indices cross the dividing point \(0\): \[ \begin{array}{c|c|c|c} m&\text{indices }j\text{ with a boundary term} &\theta(j+m)-\theta(j)& \sum_j\bigl(\theta(j+m)-\theta(j)\bigr)\\ \hline m>0&-m,-m+1,\ldots,-1&1&m\\ m=0&\varnothing&0&0\\ m<0&0,1,\ldots,-m-1&-1&m \end{array} \tag{RC.M6} \] The table shows the mechanism: each boundary index contributes \(4x_{m+n}\) with the displayed sign. The boundary sum is \(4m x_{m+n}\). For clarity about tails, fix a vector \(w\). Choose a cutoff containing the supports of (RC.M1) on both \(w\) and \(x_mw\), the supports of the two normally ordered Lie-bracket families in (RC.M3), the two scalar indices in (RC.M4), and the finite boundary interval (RC.M6). Each family is finite on \(w\), by the same rightmost-mode argument used for (RC.M1), with total index \(m+n\) for the Lie-bracket families. Inside this finite union all the preceding computations are ordinary finite algebra. Reindexing adds only terms already zero on \(w\). We have proved, for every pair of integers \(m,n\), \[ \boxed{[x_m,S_n]=2(\ell+2)m\,x_{m+n}.} \tag{RC.M7} \] At the critical level \(\ell=-2\), every \(S_n\) commutes with every mode. It also preserves each annihilation bound: if \(x_jw=0\) for \(j\ge N\), then \(x_jS_nw=S_nx_jw=0\) for those same indices. Consequently it commutes with a formal Laurent current as well: that current has a finite negative part, and its sufficiently high positive tail kills both \(w\) and \(S_nw\). The desired identity reduces to the already proved finite mode identities. The quadratic operators commute with each other. Let \(S_p^{(L)}\) be a finite partial sum of (RC.M1). Every one of its terms is a product of modes, so it commutes with \(S_n\) at the critical level. Because \(w\) and \(S_nw\) have a common annihilation bound, choose \(L\) with \(S_p^{(L)}w=S_pw\) and \(S_p^{(L)}S_nw=S_pS_nw\). Then \[ S_nS_pw =S_nS_p^{(L)}w =S_p^{(L)}S_nw =S_pS_nw, \qquad [S_n,S_p]=0. \tag{RC.M8} \] This proves commutativity of actual operators with a common finite tail cutoff. #### 3.2.2. Translated vacuum vectors and their symbols The translation operator on the vacuum can also be constructed directly. On the polynomial affine Lie algebra define \[ D(x_m)=-m x_{m-1},\qquad D(K)=0. \tag{RC.M9} \] It is a derivation. The Lie-mode part of \([D(x_m),y_j]+[x_m,D(y_j)]\) is \(-(m+j)[x,y]_{m+j-1}\), which equals the Lie-mode part of \(D([x_m,y_j])\). The possible central difference is \[ -m(m+j-1)B(x,y)\delta_{m+j-1,0}K=0. \] The Leibniz rule therefore induces a derivation of the enveloping algebra: applying it to a relation \(xy-yx-[x,y]\) gives another such relation. The left ideal defining the vacuum is stable. Its generators are \(K-\ell\) and \(x_m\) for \(m\ge0\); the derivative of \(K-\ell\) and of \(x_0\) is zero, and \(D(x_m)=-m x_{m-1}\) still has nonnegative index when \(m\ge1\). Hence \(D\) descends to a linear operator \(T\) on \(V_\ell\) satisfying \[ Tv=0,\qquad [T,x_m]=-m x_{m-1}. \] No vertex-algebra translation axiom is needed for this construction. For \(n\ge-1\), each summand of \(S_n\) has a nonnegative rightmost mode and kills \(v\). If \(n=-r-2\), with \(r\ge0\), precisely the terms with both indices negative can survive. Thus \[ S_nv=0\quad(n\ge-1),\qquad Q_r:=S_{-r-2}v =\sum_{\substack{i,j\ge1\\i+j=r+2}} \left(\tfrac12 h_{-i}h_{-j} +e_{-i}f_{-j}+f_{-i}e_{-j}\right)v. \tag{RC.M10} \] This is a finite sum with \(r+1\) index pairs, at every scalar level. In particular \(Q_0=Q\) of (K3.1). An alternative exact expression records the lower ordered term. Since the two indices are negative, their bracket has no central part, and \(f_{-i}e_{-j}=e_{-j}f_{-i}-h_{-(i+j)}\). Reindexing the finite mixed sum gives \[ Q_r= \left[ \sum_{i=1}^{r+1} \left(\tfrac12h_{-i}h_{-(r+2-i)} +2e_{-i}f_{-(r+2-i)}\right) -(r+1)h_{-(r+2)} \right]v. \tag{RC.M11} \] The \(r+1\) commutators produce both the minus sign and the factor \(r+1\). On negative modes, \(D(x_{-i})=i x_{-i-1}\), so \(D^a(x_{-1})=a!x_{-a-1}\). Applying the Leibniz rule \(r\) times to a two-letter negative-mode word gives \[ \frac{T^r}{r!}(x_{-1}y_{-1}v) =\sum_{a+b=r} \frac{D^a(x_{-1})}{a!} \frac{D^b(y_{-1})}{b!}v =\sum_{a+b=r}x_{-a-1}y_{-b-1}v. \] Apply this identity to the three terms of \(Q\). It proves \[ \boxed{Q_r=\frac{T^rQ}{r!},\qquad TQ_r=(r+1)Q_{r+1}.} \tag{RC.M12} \] There is a compatible operator identity on the entire vacuum. In differentiating a normal product, the only possible change of the first-mode ordering occurs at \(j=0\). Its derivative has coefficient \(-j=0\). Consequently \[ [T,:a_jb_{n-j}:] =-j:a_{j-1}b_{n-j}: -(n-j):a_jb_{n-j-1}:. \] This also follows by subtracting the two theta expressions in the proof of (RC.M3): the boundary coefficient is \(-j(\theta(j-1)-\theta(j))=0\). All the families are finite on a fixed vector, and on its image under \(T\), because the vacuum is smooth. Reindexing the first family by \(i=j-1\), the coefficients become \(-(i+1)\) and \(-(n-i)\). Therefore \[ [T,S_n]=-(n+1)S_{n-1}. \tag{RC.M13} \] At \(n=-r-2\), applying this formula to \(v\) reproduces the second identity in (RC.M12). Filter the vacuum by negative-word length. The ordered basis proved in §3 identifies its associated graded space with \(\operatorname{Sym}(t^{-1}\mathfrak g[t^{-1}])\): swapping generators changes a word by a shorter word, and ordered monomials remain independent in each graded piece. Let \(e_i,f_i,h_i\) denote the symbols of \(e_{-i},f_{-i},h_{-i}\), respectively, for \(i\ge1\). The degree-two symbol of the finite vector \(Q_r\) is exactly \[ C_r:=\sigma_2(Q_r) =\sum_{\substack{i,j\ge1\\i+j=r+2}} \left(\tfrac12h_i h_j+2e_i f_j\right). \tag{RC.M14} \] The \(h_1h_{r+1}\) coefficient is nonzero: it is \(1/2\) for \(r=0\) and \(1\) for \(r\ge1\). Hence every \(Q_r\) is nonzero. Here the length degree is always two; the sum of the positive mode indices is \(r+2\). These two gradings are distinct. The first two coefficients display the translation and ordered correction concretely: \[ \begin{array}{c|c|c} r&Q_r& C_r\\ \hline 0& \bigl(\tfrac12h_{-1}^{\,2}+2e_{-1}f_{-1}-h_{-2}\bigr)v& \tfrac12h_1^{\,2}+2e_1f_1\\[2pt] 1& \bigl(h_{-1}h_{-2} +2e_{-1}f_{-2}+2e_{-2}f_{-1}-2h_{-3}\bigr)v& h_1h_2+2e_1f_2+2e_2f_1 \end{array} \tag{RC.M15} \] The linear correction disappears in the length-two symbol, while the mode-index sum increases by one under translation. At \(\ell=-2\), the operators \(A_r=S_{-r-2}\) are affine-module endomorphisms, and their vacuum images are \(Q_r\). They are the endomorphisms attached to those vectors by (K2.3), since an endomorphism is determined by the image of the cyclic vector \(v\). Moreover \(S_n=0\) on the critical vacuum for \(n\ge-1\): it commutes with all modes and kills \(v\), hence kills every vector generated from \(v\). Equation (RC.M8) proves that all products of the \(A_r\) commute. It therefore defines a homomorphism \[ k[z_0,z_1,\ldots]\longrightarrow \operatorname{End}_{\widehat{\mathfrak{sl}}_2}(V_{-2}), \qquad z_r\longmapsto A_r. \tag{RC.M16} \] Its construction proves a commuting algebra of vacuum endomorphisms. Algebraic independence, exhaustion of all vacuum endomorphisms, a center inside a specified completed enveloping algebra, and identification with functions on opers require further proofs. The finite normal-order calculation and translation construction above use only the affine-module foundations already proved in §3. #### 3.2.3. The complete rank-one current-jet invariant algebra Let \(k\) be a field of characteristic zero. Use \[ e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad f=\begin{pmatrix}0&0\\1&0\end{pmatrix},\qquad h=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad [h,e]=2e,\quad[h,f]=-2f,\quad[e,f]=h. \tag{RC.J1} \] For \(N\ge1\), put \[ S_N=k[a_i,b_i,c_i:0\le i0). \tag{RC.J10} \] Then \(q_i=c_i\). Any polynomial relation among the \(q_i\) would specialize to the same relation among the independent variables \(c_i\), so all its coefficients would be zero. Every step also works over an ordinary \(k\)-algebra \(R\). The coordinate substitution is an algebraic recursion, \(b_0\) remains a non-zero-divisor in a polynomial ring over \(R\), and all nonzero integers remain units. Differentiating polynomial and Laurent monomials therefore has exactly the same kernel calculation, even when \(R\) has nilpotents. The specialization in (RC.J10) gives independence over \(R\) as well. \(\square\) The following table records the successive kernels in the proof. The two triangular matrices have diagonal entries \(b_0\) and \(-2b_0\), respectively; the last operator kills every nonzero Laurent exponent. Its rows refer to (RC.J5), (RC.J7)–(RC.J9), and the \(h_0\) calculation. | Operation | Remaining coordinate ring | Exact operator argument | | --- | --- | --- | | Invert the leading upper-right coefficient | \(k[b_0^{\pm1},b_1,\ldots,b_{N-1},a_0,\ldots,a_{N-1},q_0,\ldots,q_{N-1}]\) | Solve for the lower-left coefficients by (RC.J6). | | Take the common kernel of the \(f\)-currents | \(k[b_0^{\pm1},b_1,\ldots,b_{N-1},q_0,\ldots,q_{N-1}]\) | The triangular matrix in (RC.J7) has diagonal \(b_0\). | | Take the kernel of the positive \(h\)-currents | \(k[b_0^{\pm1},q_0,\ldots,q_{N-1}]\) | The triangular matrix in (RC.J9) has diagonal \(-2b_0\). | | Take the remaining \(h_0\) kernel | \(k[q_0,\ldots,q_{N-1}]\) | The operator \(-2b_0\partial_{b_0}\) removes every nonzero Laurent exponent. | The intermediate kernels need only the displayed \(f\) and \(h\) operators. The final ring is invariant under every current operator, including the \(e\)-currents, by the trace calculation. The table also holds with coefficient ring \(R\). **All coefficients and formal currents.** Put \[ S_\infty=\bigcup_{N\ge1}S_N =k[a_i,b_i,c_i:i\ge0],\qquad q(t)=a(t)^2+b(t)c(t)=\sum_{i\ge0}q_it^i. \] The inclusions preserve the previously defined \(q_i\). Each \(S_N\) is stable under the current derivations, and every \(x t^r\) with \(r\ge N\) acts by zero on it. Every polynomial belongs to some \(S_N\), so it is invariant under \(\mathfrak{sl}_2[t]\) exactly when it is invariant in that finite truncation. It follows that \[ S_\infty^{\mathfrak{sl}_2[t]} =S_\infty^{\mathfrak{sl}_2[[t]]} =k[q_0,q_1,q_2,\ldots]. \tag{RC.J11} \] The formal-current action is defined on each polynomial by its finite truncation. Thus formal sums create no convergence condition or additional invariant. Algebraic independence of the infinite list means independence of every finite sublist, already proved by (RC.J10). **The PBW symbol identification.** In the vacuum module of §3, the ordered negative-mode words give \[ \operatorname{gr}_{\rm PBW}V_\ell =\operatorname{Sym}\bigl(t^{-1}\mathfrak{sl}_2[t^{-1}]\bigr). \tag{RC.J12} \] This is the associated-graded conclusion of the complete ordered-word proof in §3, also proved for arbitrary Lie algebras in [*The universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem*, Theorem 2.1](../../RT-LIE/src/RT-LIE-13.md). The filtration counts negative-mode letters. For \(m\ge0\), commute \(x_m\) through an ordered word. If the first bracket with \(y_{-j}\) is a negative mode, it replaces one letter by one letter. A central bracket removes that letter, and a nonnegative resulting mode must either kill the vacuum or be commuted through another negative letter, using at least one further bracket and producing a term with fewer letters. Hence the action preserves the PBW filtration, and its degree-preserving symbol is the derivation \[ x_m\cdot y_{-j}= \begin{cases} [x,y]_{-(j-m)},&j>m,\\ 0,&j\le m. \end{cases} \tag{RC.J13} \] In particular, this symbol action is independent of the level \(\ell\). Use the invariant form \(B(e,f)=1\), \(B(h,h)=2\), with the remaining basis pairings zero, to identify a negative-mode symbol with a coordinate function by \[ y_{-j}\longmapsto\bigl(A(t)\longmapsto B(y,A_{j-1})\bigr). \tag{RC.J14} \] It is a linear isomorphism on the generator spaces, since \(B\) is nondegenerate, and hence an isomorphism of their polynomial algebras. Its indexing is also the residue identity \(\operatorname{Res}_{t=0}B(y t^{-j},A(t))\,dt=B(y,A_{j-1})\). Invariance and symmetry of \(B\) give \[ B([x,y],A_{i-m})=B(y,[A_{i-m},x]). \] Thus (RC.J13) becomes exactly the coefficient action (RC.J3), including the shift \(i\mapsto i-m\). In coordinates, \[ a_i=\tfrac12h_{-(i+1)},\qquad b_i=f_{-(i+1)},\qquad c_i=e_{-(i+1)}. \tag{RC.J15} \] Consequently the complete associated-graded invariant algebra is \[ \bigl(\operatorname{gr}_{\rm PBW}V_\ell\bigr)^{\mathfrak{sl}_2[[t]]} =k[q_0,q_1,\ldots], \quad q_i=\tfrac14\sum_{j=0}^i h_{-(j+1)}h_{-(i-j+1)} +\sum_{j=0}^i f_{-(j+1)}e_{-(i-j+1)}. \tag{RC.J16} \] All products in this formula are commutative PBW symbols. The vector \(Q\) of §3.1 has symbol \(2q_0\), agreeing with its stated normalization. Each \(q_i\) has PBW degree two and loop grade \(i+2\), when \(y_{-j}\) has loop grade \(j\). In particular, every actual vacuum invariant has a highest PBW symbol in the ring (RC.J16). Lifting these symbols to vacuum invariants requires the further filtered argument; it is not part of the equality (RC.J16). **Finite kernels and ordinary coefficient rings.** The coefficient-extension assertion can also be checked without localization. Give \(a_i,b_i,c_i\) polynomial degree one and loop grade \(i+1\). For each pair \((p,L)\), the space \(E_{p,L}\) of polynomials of degree \(p\) and loop grade \(L\) is finite-dimensional: only variables of weight at most \(L\) and finitely many monomials can occur. This holds in each \(S_N\) and in \(S_\infty\). The derivation \(D_{x,m}\) preserves \(p\) and lowers \(L\) by \(m\); all \(m\ge L\) act by zero. The invariant component is therefore the kernel of the finite linear map \[ E_{p,L}\longrightarrow \bigoplus_{x\in\{e,h,f\}}\ \bigoplus_{0\le mm,\\0,&j\leq m, \end{cases} \qquad m\geq0. \tag{RC.P2} \] The jet calculation of §3.2.3 uses precisely this action, via the displayed invariant pairing. Its polynomial invariants are the quadratic coefficients \[ C_r=\sum_{i+j=r+2}\left(\tfrac12h_{-i}h_{-j}+2e_{-i}f_{-j}\right), \qquad i,j\geq1, \quad r\geq0. \tag{RC.P3} \] These are algebraically independent. Indeed every finite initial list is the nonzero scalar multiple of the independent jet coefficients proved there; a relation in the infinite list would already involve a finite initial list. Put \(Q_r=S_{-r-2}v\), as in §3.2.2. Its leading symbol is \(C_r\). Let \(A_r\) be its corresponding affine-module endomorphism. The operator calculation identifies \(A_r=S_{-r-2}\) on the vacuum, and proves that these endomorphisms commute. Thus there is an algebra map \[ \Theta:k[z_0,z_1,\ldots]\longrightarrow \operatorname{End}_{\widehat{\mathfrak{sl}}_2}(V_{-2}), \qquad z_r\longmapsto A_r. \tag{RC.P4} \] Here the polynomial ring consists of finite polynomials, rather than a completion. The leading symbol of a product of the \(A_r\), evaluated at the vacuum, is the product of its corresponding \(C_r\). To see this without an infinite-mode multiplication argument, write \(Q_r=P_rv\) with \(P_r\) a polynomial in negative modes. For any negative-mode word \(Pv\), the induced-module formula (K2.3) gives \(A_r(Pv)=P P_rv\). The ordered basis and its length filtration give \(\sigma(PP_r)=\sigma(P)C_r\). Repeating proves the product assertion. In particular a nonzero polynomial of highest total degree \(d\) in the \(z_r\) has leading PBW symbol its degree-\(d\) part evaluated on the \(C_r\), in filtration degree \(2d\). Independence in (RC.P3) makes that symbol nonzero. Hence (RC.P4) is injective. For surjectivity, let \(w\) be any vector killed by every nonnegative mode, with PBW degree \(d\). Its leading symbol is invariant under (RC.P2), so §3.2.3 expresses it as a polynomial in finitely many \(C_r\). Each \(C_r\) has PBW degree two. If \(d>0\), the homogeneous degree-\(d\) symbol therefore forces \(d\) to be even and is a homogeneous polynomial of total degree \(d/2\) in those coefficients. Apply the same polynomial in the \(A_r\) to \(v\) and subtract it from \(w\). The result is again an invariant vector and has strictly smaller PBW degree. For \(d=0\) it is a scalar multiple of \(v\). Induction terminates after finitely many steps and expresses every \(w\) as a finite polynomial in the \(A_r\), applied to \(v\). Formula (K2.3) then proves surjectivity. We have obtained the entire algebra \[ \boxed{\operatorname{End}_{\widehat{\mathfrak{sl}}_2}(V_{-2}) =k[A_0,A_1,\ldots].} \tag{RC.P5} \] It is commutative, its generators are independent, and the written argument excludes additional vacuum endomorphisms. This assertion holds over every ordinary \(k\)-algebra \(R\). One can verify the base-change step by finite equations. Give a negative mode \(x_{-j}\) energy \(j\), and \(v\) energy zero. The energy-\(E\) space is finite-dimensional: only modes with \(j\leq E\) occur, and the number of length-weighted ordered words of total energy \(E\) is finite. The action of \(x_m\) has energy \(-m\). Thus all \(m>E\) kill that space, and its invariant subspace is the kernel of finitely many linear maps, using a basis of \(\mathfrak{sl}_2\) and \(0\leq m\leq E\). Tensoring that finite kernel with \(R\) is exact over the field \(k\). The vacuum is the direct sum of its energy spaces and every vector has finite energy support; its invariants therefore commute with this scalar extension. The induced-module endomorphism argument also holds over \(R\), since the ordered negative-mode basis is free over \(R\). The product identities in (RC.P4) are identities of operators, so \[ \operatorname{End}_{\widehat{\mathfrak{sl}}_{2,R}}(V_{-2,R}) =R\otimes_k k[A_0,A_1,\ldots] =R[A_0,A_1,\ldots]. \tag{RC.P6} \] Invariants here are killed by all modes, including the formal nonnegative series whose action was proved in §3. They are not defined by testing a selected set of reduced points. No derived invariant or completed-enveloping-algebra assertion follows merely from (RC.P6). #### 3.2.5. The coordinate action and the projective-connection coefficient We identify the exact coordinate action on this polynomial algebra. For \(l\geq-1\), the vector field \(t^{l+1}\partial_t\) gives a derivation of the affine algebra by \[ D_l(x_m)=m x_{m+l},\qquad D_l(K)=0. \tag{RC.C1} \] The affine bracket is preserved: the possible central contribution to the derivation identity is \(m(m+n+l)B(x,y)\delta_{m+n+l,0}K=0\). The nonnegative-mode subalgebra is preserved too; at \(l=-1\) the only apparently exceptional index has coefficient \(m=0\). Therefore \(D_lv=0\) extends (RC.C1) to the vacuum. Its commutator with translation is \([D_l,T]=(l+1)D_{l-1}\) for \(l\geq0\), whereas \(D_{-1}=-T\). The initial quadratic vector has the following actions, computed directly from its three terms: \[ \begin{array}{c|ccccc} l&-1&0&1&2&\geq3\\ \hline D_lQ_0&-Q_1&-2Q_0&0&6v&0. \end{array} \tag{RC.C2} \] For \(l=1\), the first differentiated factor is a zero mode. Commuting it through the second factor produces the three bracket terms of the invariant quadratic tensor, whose sum is zero. For \(l=2\), those first factors are mode one. Each commutator then has a zero-mode bracket, which kills \(v\), and a central term. The pairings from the three summands are \(1,1,1\), so their total is \(-3\ell v=6v\) at \(\ell=-2\). Second differentiated factors already kill \(v\). For \(l\geq3\), the remaining index is positive and no central term is possible, so every term vanishes. The first two entries follow from translation and the energy of \(Q_0\). Since \(TQ_r=(r+1)Q_{r+1}\), the commutator with \(T\) and (RC.C2) give, for every \(r\geq0\), \[ D_lQ_r= -\mathbf1_{r\geq l}(r+l+2)Q_{r-l} +(l^3-l)\delta_{r,l-2}v, \qquad l\geq-1. \tag{RC.C3} \] The indicated term has an allowed index whenever its indicator is one. Here is the induction including its boundary terms. For \(l\geq0\), apply \[ D_lQ_{r+1} =\frac1{r+1}\bigl(TD_lQ_r+(l+1)D_{l-1}Q_r\bigr). \] When both ordinary terms occur, their coefficient is the negative of \[ \frac{(r+l+2)(r-l+1)+(l+1)(r+l+1)}{r+1} =r+l+3. \] At \(r=l-1\), only the second ordinary term occurs and gives \(-2(l+1)Q_0\), the required new boundary. Translation kills the scalar term. The scalar term from \(D_{l-1}Q_r\) occurs at \(r=l-3\) and has coefficient \((l+1)(l-1)l=l^3-l\) after division by \(r+1=l-2\). Cases with \(l\leq2\) at this scalar boundary are already the initial entries in (RC.C2). For \(l=-1\), \(D_{-1}=-T\) gives the formula directly. These observations prove (RC.C3) for all indices. The induced action on endomorphisms is by conjugation, and on their vacuum vectors is the displayed action. Indeed an affine derivation preserving the induction relations acts on \(V\); conjugating an affine-module map by a coordinate substitution is again an affine-module map. Infinitesimally, its commutator with \(D_l\) has vacuum image \(D_lQ_r\). Thus (RC.C3) gives the derivation on the generators \(A_r\) of (RC.P5). By §§1.1.8 and 2.5, an ordinary \(PGL_2\)-oper on the formal disc has the unique coefficient \[ \partial_t^2-q(t),\qquad q(t)=\sum_{r\geq0}q_rt^r. \] Its full ordinary coefficient functor is represented by the polynomial ring \(k[q_0,q_1,\ldots]\). For a coordinate substitution \(t=\phi(s)\), the scalar calculation in §2.9 gives \[ \mathcal P_\phi(q)(s) =\phi'(s)^2q(\phi(s))-\tfrac12\{\phi,s\}, \qquad \{\phi,s\}=\frac{\phi'''}{\phi'}- \frac32\left(\frac{\phi''}{\phi'}\right)^2. \tag{RC.C4} \] To match the action on functions with current substitution, specify the contravariance explicitly. Current substitution sends \(xF(t)\) to \(xF(\phi(t))\). On the polynomial oper ring its corresponding action sends a function \(f\) to \(f\circ\mathcal P_{\phi^{-1}}\). At \(\phi(t)=t+\epsilon t^{l+1}\), \(\epsilon^2=0\), this action on the coefficients is \[ \delta_lq_r= -\mathbf1_{r\geq l}(r+l+2)q_{r-l} +\tfrac12(l^3-l)\delta_{r,l-2}. \tag{RC.C5} \] In fact the inverse is \(t-\epsilon t^{l+1}\); substituting it in (RC.C4) gives \(-t^{l+1}q'-2(l+1)t^lq+\tfrac12(l+1)l(l-1)t^{l-2}\) as the first variation. Reading its coefficient proves (RC.C5). This determines the normalization: \[ A_r\longmapsto2q_r. \tag{RC.C6} \] The factor two is the factor between our quadratic vector (K3.1) and the half-Casimir normalization. Formulas (RC.C3) and (RC.C5) agree under (RC.C6), including the scalar Schwarzian term. These infinitesimal identities determine the full ordinary coordinate action, also over nonreduced coefficients. We give the integration step. A continuous coordinate substitution over an ordinary \(R\) has nilpotent constant coefficient and invertible linear coefficient, and its inverse was constructed in §1.1.8. It factors as a nilpotent translation followed, under substitution composition, by a pointed substitution. Translation acts on the vacuum by the finite exponential \(\exp(a_0D_{-1})\); it is finite because \(a_0\) is nilpotent. A scaling \(t\mapsto at\) acts on \(Q_r\) by \(a^{-r-2}\), exactly the action on \(2q_r\) in (RC.C6). Every pointed substitution with linear coefficient one is a successively determined product of \(\exp(c_nt^{n+1}\partial_t)\), \(n\geq1\). To prove existence, first match the coefficient of \(t^2\) using \(n=1\); composing with its inverse removes that coefficient. At step \(n\), all lower coefficients have already been removed, and the exponential has leading term \(t+c_nt^{n+1}\), so one uniquely chosen \(c_n\) removes the next coefficient. For any fixed coefficient only finitely many steps contribute, so the product and its inverse define the desired continuous substitution. The denominators in these exponentials are units over a characteristic-zero coefficient field. On the energy-\(E\) vacuum space, \(D_n\) decreases energy by \(n\), hence is locally nilpotent, and all \(n>E\) act by zero. Thus the same product acts by finitely many terms on every vacuum vector. The corresponding finite-jet polynomial operators on an oper coefficient have the same local nilpotence, as seen directly in (RC.C5). Equality of the infinitesimal derivations therefore gives equality of these exponentials; scaling and nilpotent translation give the remaining factors. This proves full equivariance from (RC.C3), rather than assuming that a Lie-algebra equality integrates. Current substitution preserves the central line. For completeness, its residue identity is \[ \operatorname{Res}_t\phi'(t)F(\phi(t)) =\operatorname{Res}_uF(u),\qquad F\in R((u)). \tag{RC.C7} \] It suffices to check Laurent monomials, coefficient by coefficient, using the finite nilpotent substitution of §1.1.8. For \(u^a\) with \(a\ne-1\), the substituted differential is \(d(\phi^{a+1}/(a+1))\) and has residue zero. For \(a=-1\), write \(\phi=(t-b)u(t)\), with \(b\) nilpotent and \(u(t)\) a power-series unit. The nilpotent root and this factorization follow from the inverse-coordinate construction there; dividing by \(t-b\) is finite coefficientwise. Then \(\phi'/\phi=1/(t-b)+u'/u\). The first term has residue one, by its finite negative-power expansion, while the second is a power series. This proves (RC.C7), and hence invariance of the cocycle \(\operatorname{Res}((dF)G)B(x,y)\). The vacuum relations are preserved because substitution preserves \(R[[t]]\). Its induced action and the oper action obey the same substitution composition law, so the preceding integration proves equivariance for every continuous ordinary coordinate substitution. The calculation can be read as the following commuting square: \[ \begin{array}{ccc} k[A_0,A_1,\ldots]&\xrightarrow{\ A_r\mapsto2q_r\ }&k[q_0,q_1,\ldots]\\ {\scriptstyle D_l}\downarrow&&\downarrow{\scriptstyle\delta_l}\\ k[A_0,A_1,\ldots]&\xrightarrow{\ A_r\mapsto2q_r\ }&k[q_0,q_1,\ldots]. \end{array} \tag{RC.C8} \] *The left vertical map is the current-coordinate derivation (RC.C3); the right map is the inverse-coordinate action on projective-connection functions (RC.C5). Their scalar terms agree precisely with the factor two in (RC.C6). The exponential and nilpotent-translation argument extends the square to every ordinary coordinate substitution.* Combining (RC.P5), (RC.P6) and (RC.C6) proves the coordinate-independent rank-one vacuum-center isomorphism \[ \operatorname{End}_{\widehat{\mathfrak{sl}}_{2,R}}(V_{-2,R}) \simeq R[\operatorname{Op}_{PGL_2}(D_R)] \tag{RC.C9} \] for every ordinary parameter algebra, with the exact current cocycle and scalar-oper convention of this lesson. The proof supplies all generators, their independence, exhaustion and coordinate action. Sections 3.3–3.7 supply the type A basic lifts, polynomial and completed centers, intrinsic critical Poisson structures and full scalar-oper Poisson comparison. Basic lifting and the full center/oper-Poisson comparison in other types, derived/global Hamiltonian reduction, chiral/factorization, Satake and derived-family compatibility remain the additional scopes of §4. ### 3.3. General current invariants and the quantum-lifting problem #### 3.3.1. Invariants of split semisimple current jets Let \(k\) be a characteristic-zero field and \(\mathfrak g\) a split semisimple Lie algebra of rank \(\ell\). Fix homogeneous polynomial generators \(P_1,\ldots,P_\ell\) of its adjoint infinitesimal invariant ring. Their existence and the polynomial principal section are the inputs proved in §1.2 of [Opers, critical level and the Beilinson–Drinfeld construction](opers-critical-level-and-the-beilinson-drinfeld-construction.md), at its explicitly retained root/pinning and invariant-polynomial foundations. We explain the extension from an algebraically closed field below. For \(N\ge1\), put \(\mathfrak g_N=\mathfrak g\otimes_k k[t]/(t^N)\), write \(A(t)=\sum_{s=0}^{N-1}A_st^s\), and define the coefficient polynomials by \[ P_i(A(t))=\sum_{r=0}^{N-1}p_{i,r}(A)t^r \pmod{t^N}. \] We will prove, including algebraic independence, \[ \boxed{ k[\mathfrak g_N]^{\mathfrak g_N} =k[p_{i,r}:1\le i\le\ell,\ 0\le rE\). For finite \(N\), put \(q=\min(E,N-1)\). The invariant part of a bidegree space is therefore \[ \ker L_{d,E},\qquad L_{d,E}:B_{d,E}\longrightarrow \bigoplus_a\bigoplus_{r=0}^{q}B_{d,E-r}, \qquad F\longmapsto(\delta_{y_a,r}F)_{a,r}. \tag{GJ.11} \] Negative-energy spaces are zero. All spaces and maps in (GJ.11) are finite over \(k\). Bases of the kernel and image, extended to bases of the domain and target, show that for every ordinary \(k\)-algebra \(R\), \(\ker(R\otimes L_{d,E})=R\otimes\ker L_{d,E}\). The maps after tensoring are exactly the relative current derivations over \(R\). Every polynomial is a finite sum of bidegree components. For a fixed \(r\), different bidegrees go to different bidegrees, so invariance holds componentwise. Taking their direct sum, and then using (GJ.1), proves the natural algebra isomorphism \[ R[\mathfrak g_N\otimes_kR]^{\,\mathfrak g_N\otimes_kR} =R\otimes_k k[\mathfrak g_N]^{\mathfrak g_N} =R[p_{i,r}:1\le i\le\ell,\ 0\le rm,\\0,&j\leq m. \end{cases} \tag{GJ.P2} \] The form \(\kappa\) has disappeared from this symbol action, because its terms decrease the number of letters. Identify a symbol \(y_{-j}\) with the function \(A(t)\mapsto B(y,A_{j-1})\), where \(A(t)=\sum_{a\geq0}A_at^a\). The pairing is nondegenerate, so this is an isomorphism of polynomial algebras. Invariance of \(B\) gives \[ B([x,y],A_{j-m-1})=B(y,[A_{j-m-1},x]). \] Thus (GJ.P2) is exactly the current derivation \(A_a\mapsto[A_{a-m},x]\) used in §3.3.1. In particular, the leading symbol of every vacuum invariant belongs to the polynomial algebra proved there. There is an injective map of graded algebras \[ \boxed{\operatorname{gr}_F Z_\kappa \hookrightarrow \bigl(\operatorname{gr}V_\kappa\bigr)^{\mathfrak g[[t]]} =k[p_{i,r}:1\leq i\leq\ell,\ r\geq0].} \tag{GJ.P3} \] We justify both “injective” and “algebras.” The kernel of the map from an invariant of degree at most \(d\) to its degree-\(d\) symbol consists exactly of invariants of degree at most \(d-1\), giving injectivity. If \(w=P_wv\), with \(P_w\) a finite polynomial in negative modes, the endomorphism corresponding to \(w\) sends \(P_zv\) to \(P_zP_wv\). This product has degree at most the sum of the two degrees, and its top symbol is the product of their symbols in the commutative algebra (GJ.P2). The filtration is therefore multiplicative and the symbol map is an algebra map. In particular the associated graded algebra is commutative. Equation (GJ.P3) alone does not prove that the unfiltered algebra is commutative, or that this inclusion is surjective. Give \(y_{-j}\) energy \(j\). Each energy-\(E\) space of the vacuum is finite-dimensional: it involves only the finitely many modes with \(j\leq E\), and only finitely many ordered words whose positive indices add to \(E\). The mode \(x_m\) has energy shift \(-m\). This follows directly from the affine bracket: a negative replacement subtracts \(m\) from the energy, and a central replacement is possible only when the removed indices add to \(m\). Hence \(m>E\) kills that energy space. For any ordinary \(k\)-algebra \(R\), each invariant energy space after extension is the kernel of the finite map \[ (V_\kappa)_E\otimes_k R\longrightarrow \bigoplus_{x\in\mathcal B}\ \bigoplus_{0\leq m\leq E} (V_\kappa)_{E-m}\otimes_k R, \qquad w\longmapsto(x_mw)_{x,m}, \tag{GJ.P4} \] where \(\mathcal B\) is a basis of \(\mathfrak g\). All vector spaces in this formula are finite-dimensional. Tensor over \(k\) is exact on its kernel. Invariance is energy graded, since for a fixed \(m\) the outputs of distinct input energies have distinct energies. Every vector has finite energy support. The ordered negative-mode basis remains free over \(R\), so the induced-module endomorphism argument also holds over \(R\). Summing the finite-kernel equalities gives, as algebras, \[ Z_{\kappa,R}=R\otimes_k Z_\kappa. \tag{GJ.P5} \] The same finite map restricted to the letter-length filtration proves \(F_dZ_{\kappa,R}=R\otimes_k F_dZ_\kappa\). Consequently the associated graded inclusion in (GJ.P3) also extends to every ordinary \(R\), including nonreduced rings. Formal nonnegative series impose precisely the same equations: on an energy-bounded vector their tails vanish. No completed Hom, derived invariant or pointwise criterion was substituted for this argument. **Translation.** On the affine algebra the rule \[ T(x_m)=-m x_{m-1},\qquad T(K)=0 \tag{GJ.P6} \] is a derivation. The central difference in its bracket identity is \(-m(m+n-1)\kappa(x,y)\delta_{m+n-1,0}K=0\). It preserves the vacuum relations: \(T(x_0)=0\), and for \(m\geq1\) the index \(m-1\) is still nonnegative. Thus \(Tv=0\) defines an operator on the vacuum satisfying \([T,x_m]=-m x_{m-1}\). If \(w\) is invariant, then \(x_mTw=T x_mw+m x_{m-1}w=0\) for every \(m\geq0\); the exceptional index at \(m=0\) has zero coefficient. Translation preserves invariants and their length filtration and raises energy by one. On symbols, \(TA_a=(a+1)A_{a+1}\). Its formal exponential therefore sends \[ \exp(sT)A_0=\sum_{a\geq0}A_as^a=A(s). \] The Leibniz rule makes this exponential multiplicative, coefficient by coefficient. Applied to \(P_i(A_0)\), it proves \[ p_{i,r}=\frac{T^r}{r!}p_{i,0},\qquad Tp_{i,r}=(r+1)p_{i,r+1}. \tag{GJ.P7} \] All identities are identities of individual polynomial coefficients. The letter length of \(p_{i,r}\) is \(D_i\), and its energy is \(D_i+r\). These two gradings should not be identified. **The coordinate action on symbols.** Let \(\phi\) be a continuous coordinate substitution over an ordinary \(R\), including a nilpotent constant coefficient, and write \(\psi=\phi^{-1}\). The inverse and coefficientwise finite substitutions were proved in §1.1.8. Substitution sends \(xF(t)\) to \(xF(\phi(t))\). It preserves the affine cocycle: the proof of (RC.C7) uses only residues of Laurent monomials and therefore applies to any \(\kappa\). It also preserves \(\mathfrak g_R[[t]]\). The formula \(U_\phi(uv)=\sigma_\phi(u)v\) consequently gives an invertible map of the vacuum: the defining induction relations are respected, and every formal current appearing in a substituted finite word acts through a finite truncation. Its inverse is \(U_\psi\). These maps preserve the letter filtration. A substituted generator is linear in modes; positive modes that occur in a substituted negative word can be commuted to the right, where each extra bracket or central term decreases its length. The inverse has the same property. On the associated graded, only the negative part of a substituted generator remains. Its pairing with \(A(t)\,dt\) is \[ \begin{split} \operatorname{Res}_t B(y,A(t))\phi(t)^{-j}\,dt &=\operatorname{Res}_u B\bigl(y,\psi'(u)A(\psi(u))\bigr)u^{-j}\,du. \end{split} \tag{GJ.P8} \] This is the residue-change identity, with the inverse substitution specified. Thus current substitution on polynomial functions corresponds to inverse pullback of the Lie-algebra-valued one-form. Homogeneity of \(P_i\) gives the full transformation rule \[ P_i(A(t))\longmapsto \psi'(t)^{D_i}P_i(A(\psi(t))). \tag{GJ.P9} \] Its coefficient formulas are finite on each polynomial, also for a nilpotent translation: only finitely many additional Taylor coefficients can occur before a power of the nilpotent constant vanishes. The invariant algebra and the inclusion (GJ.P3) are therefore stable under every such coordinate change. At \(\phi(t)=t+\epsilon t^{l+1}\), \(\epsilon^2=0\), the inverse variation gives \[ \delta_l p_{i,r} =-\mathbf1_{r\geq l} \bigl(r+D_i+(D_i-1)l\bigr)p_{i,r-l}, \qquad l\geq-1. \tag{GJ.P10} \] Indeed the variation of the series is \(-t^{l+1}(P_i(A))'-D_i(l+1)t^lP_i(A)\); reading its coefficient proves the formula. At \(l=-1\) it is \(-(r+1)p_{i,r+1}\), agreeing with translation. At \(l=0\) it is the negative energy weight \(-(r+D_i)\). For \(D_i=2\), it is the leading quadratic part of (RC.C3). The scalar Schwarzian term there has lower letter degree and is consequently absent from the symbol formula. The maps and the ordinary-family boundary are displayed in the following square: \[ \begin{array}{ccc} \operatorname{gr}_F Z_{\kappa,R}&\hookrightarrow&R[p_{i,r}]\\ {\scriptstyle\operatorname{gr}U_\phi}\downarrow&& \downarrow{\scriptstyle P_i(A)\mapsto\psi'^{D_i}P_i(A\circ\psi)}\\ \operatorname{gr}_F Z_{\kappa,R}&\hookrightarrow&R[p_{i,r}]. \end{array} \tag{GJ.P11} \] *The horizontal arrows are the proved symbol inclusion (GJ.P3), extended by the finite equations (GJ.P4)–(GJ.P5). The right vertical arrow is inverse pullback of degree-\(D_i\) differential coefficients, proved by the residue pairing (GJ.P8). The square does not assert surjectivity of the horizontal arrows or the full coordinate law of quantum generators.* #### 3.3.3. Algebraic reconstruction and commutativity of the general vacuum center Let \(k\) be a field of characteristic zero, let \(\mathfrak g\) be finite-dimensional, and let \(B\) be an invariant symmetric bilinear form. Fix a scalar level \(\ell\). We use the affine bracket and vacuum of §3: \[ [x_m,y_n]=[x,y]_{m+n}+m\ell B(x,y)\delta_{m+n,0}\operatorname{Id}, \qquad \mathfrak g[t]v=0. \tag{VC.1} \] The ordered-word proof identifies \(V_\ell\) with \(U(t^{-1}\mathfrak g[t^{-1}])v\). Every state is a finite linear combination of finite negative-mode words; the sum of their positive mode indices gives an annihilation bound for sufficiently high currents. Thus all states have finite energy in this algebraic sense. The full formal-current action and the bijection \[ \operatorname{End}_{\widehat{\mathfrak g}}(V_\ell) \longrightarrow V_\ell^{\mathfrak g[[t]]},\qquad E\longmapsto Ev, \tag{VC.2} \] are the proved induction and smoothness statements of §3. The translation constructed in §3.2.2 works for this general bracket as well: \[ Tv=0,\qquad [T,x_m]=-m x_{m-1}. \tag{VC.3} \] Indeed \(D(x_m)=-m x_{m-1}\), \(D(K)=0\), preserves the affine bracket. Its possible central defect is \(-m(m+n-1)B(x,y)\delta_{m+n-1,0}K=0\). It preserves the defining vacuum ideal, so descends to \(T\). Taking B=κ and ℓ=1 applies this construction to every form in §3.3.2. No simplicity or critical-level hypothesis is needed below. **Theorem.** The algebra \(\operatorname{End}_{\widehat{\mathfrak g}}(V_\ell)\) is commutative. Every invariant state has a field reconstructed from normally ordered derivative currents; all its modes commute with all currents and with the modes of every other invariant field. Its negative modes are exactly the endomorphisms corresponding to its divided translates. **Fields and finite sums.** A field on \(V_\ell\) is a series \(F(z)=\sum_{n\in\mathbb Z}F_{(n)}z^{-n-1}\) such that \(F_{(n)}u=0\) for all sufficiently large \(n\), for each fixed state \(u\). Equivalently, \(F(z)u\in V_\ell((z))\). The current \(x(z)=\sum_mx_mz^{-m-1}\) is a field by smoothness. Write \[ x_-(z)=\sum_{m<0}x_mz^{-m-1},\qquad x_+(z)=\sum_{m\ge0}x_mz^{-m-1}, \qquad x^{[n]}(z)=\frac{\partial_z^{n-1}x(z)}{(n-1)!}\quad(n\ge1). \tag{VC.4} \] The derivative's minus and plus parts are the derivatives of the indicated parts. The finitely many derivatives of \(z^0,\ldots,z^{n-2}\) vanish, so its minus part still has only nonnegative powers of \(z\). Define the left normal-multiplication operator on fields by \[ L_{x,n}F=x^{[n]}_-(z)F(z)+F(z)x^{[n]}_+(z). \tag{VC.5} \] Both terms are fields. On \(u\), the second uses finitely many nonnegative current modes. In the first, any fixed coefficient uses finitely many nonnegative powers from \(x^{[n]}_-\), because \(F(z)u\) has a lower bound on its powers. These observations also justify a finite iteration. More explicitly, expand a fixed iterated normal product into its finitely many choices of minus or plus parts. Each term has all negative-mode factors to the left of all nonnegative-mode factors. The latter act first on \(u\); smoothness leaves a finite tree of intermediate states and finite mode cutoffs. The remaining minus-part factors have nonnegative powers, so only finitely many of their terms can contribute to a fixed coefficient. This proves coefficientwise finiteness, including after replacing \(u\) by any finite list of states. **Reconstruction respects the state relations.** These operators obey \[ [L_{x,n},L_{y,r}]=L_{[x,y],n+r}. \tag{VC.6} \] Here is the calculation. Put \(X=x^{[n]}\), \(Y=y^{[r]}\). Left and right multiplication commute, so the cross terms cancel and the left side on \(F\) is \([X_-,Y_-]F-F[X_+,Y_+]\). There is no central term in either bracket: two strictly negative indices cannot sum to zero, and two nonnegative indices can sum to zero only at zero, where the cocycle coefficient vanishes. The explicit expansions are \[ x^{[n]}_-= \sum_{s\ge0}\binom{n+s-1}{n-1}x_{-(n+s)}z^s, \quad x^{[n]}_+=(-1)^{n-1}\sum_{p\ge0} \binom{n+p-1}{n-1}x_pz^{-n-p}. \tag{VC.7} \] The convolution identity \[ \sum_{i+j=s}\binom{n+i-1}{n-1}\binom{r+j-1}{r-1} =\binom{n+r+s-1}{n+r-1} \] follows by multiplying the formal series \((1-u)^{-n}\) and \((1-u)^{-r}\); their displayed coefficients follow by differentiating the geometric series. It gives \([X_-,Y_-]=[x,y]^{[n+r]}_-\) and \([X_+,Y_+]=-[x,y]^{[n+r]}_+\), proving (VC.6). Every coefficient comparison is finite by the preceding bounds. Thus \(x_{-n}\mapsto L_{x,n}\) is a representation of the negative current Lie algebra on fields. The enveloping-algebra universal property makes the following linear map well defined on the actual vacuum space: \[ Y(v,z)=\operatorname{Id},\qquad Y(x_{-n}u,z)=L_{x,n}Y(u,z). \tag{VC.8} \] In particular \(Y(x_{-1}v,z)=x(z)\). Equation (VC.6) proves independence from all exchanges of negative-mode letters; the construction is not a prescription that ignores their brackets. **Creation and translation.** The reconstructed field has \[ Y(u,z)v=e^{zT}u=\sum_{a\ge0}\frac{z^aT^au}{a!}. \tag{VC.9} \] For the vacuum this is immediate. For \(x_{-n}u\), the plus part kills \(v\), while repeated use of (VC.3) gives \[ e^{zT}x_{-n}e^{-zT} =\sum_{a\ge0}\binom{n+a-1}{a}x_{-(n+a)}z^a =x^{[n]}_-(z). \] This identity follows coefficientwise from \((\operatorname{ad}T)^ax_{-n}=n(n+1)\cdots(n+a-1)x_{-(n+a)}\), and the finite binomial expansion for conjugation. Multiplying the induction hypothesis by its left side proves (VC.9). The exponential is a formal power series, with a finite algebraic state in each coefficient. We also have \[ Y(Tu,z)=\partial_zY(u,z),\qquad [T,Y(u,z)]=\partial_zY(u,z). \tag{VC.10} \] For the first identity, use \(T(x_{-n}u)=n x_{-(n+1)}u+x_{-n}Tu\) and \(\partial_z(L_{x,n}F)=nL_{x,n+1}F+L_{x,n}\partial_zF\). For the second, (VC.3) gives \([T,x^{[n]}_\pm]=\partial_zx^{[n]}_\pm\); the only splitting boundary would have mode index zero and coefficient zero. Apply the product rule in (VC.5) and induction from the identity field. The same finite coefficient bounds justify both calculations. **Locality, proved algebraically.** Two fields are called local when \((s-z)^M[A(s),B(z)]=0\) for some nonnegative integer \(M\), as a coefficient identity of operator-valued distributions. This definition does not assume an operator-product theorem. Define the formal delta distribution \[ \delta(s,z)=\sum_{m\in\mathbb Z}s^{-m-1}z^m. \] The affine bracket directly gives \[ [x(s),y(z)]=x,y\delta(s,z) +\ell B(x,y)\partial_z\delta(s,z). \tag{VC.11} \] Indeed its \(s^{-m-1}\) coefficient is \(z^mx,y+m\ell B(x,y)z^{m-1}\). Shifting indices gives \((s-z)\delta=0\); differentiation gives \((s-z)\partial_z\delta=\delta\). Thus currents are local with order at most two. Differentiating a locality relation and multiplying by one additional factor \(s-z\) proves locality of a derivative; iteration covers all derivative currents. We need closure under products, and prove it here. For an integer \(n\), set \[ (A_{(n)}B)(z)=\operatorname{Res}_s \left(\iota_{s,z}(s-z)^n A(s)B(z) -\iota_{z,s}(s-z)^n B(z)A(s)\right). \tag{VC.12} \] The notation \(\iota_{s,z}\) means the binomial expansion in nonnegative powers of \(z/s\); \(\iota_{z,s}\) means the expansion in nonnegative powers of \(s/z\). For \(n\ge0\) both are the same finite polynomial. The result is then the finite sum \(\sum_{i=0}^n\binom ni(-z)^{n-i}[A_{(i)},B(z)]\), hence a field. For \(n=-r<0\), expansion and residue extraction give \[ A_{(-r)}B= \left(\frac{\partial_z^{r-1}A}{(r-1)!}\right)_-B +B\left(\frac{\partial_z^{r-1}A}{(r-1)!}\right)_+. \tag{VC.13} \] For example \(r=1\) uses \(\iota_{s,z}(s-z)^{-1}=\sum_{a\ge0}s^{-a-1}z^a\) and \(\iota_{z,s}(s-z)^{-1}=-\sum_{a\ge0}s^az^{-a-1}\). The higher formula follows by differentiating these expansions \(r-1\) times with respect to \(z\) and dividing by \((r-1)!\). It is a field by precisely the finite normal-product argument given above, which applies to any fields with the truncation property. Suppose \(A,B,C\) are pairwise local, with respective orders \(a,b,c\) for \((A,B),(A,C),(B,C)\). Then \(A_{(n)}B\) is local with \(C\). To check this without importing a closure theorem, write \(p=s-z\), \(q=z-u\), and \(s-u=p+q\), and choose \(M=a+b+c+|n|+1\). Multiply the commutator with \(C(u)\) in (VC.12) by \(q^M\). The factor \(q^c\) allows \(C\) to pass through \(B\), leaving, inside the residue, the two orders \[ \iota_{s,z}p^n[C(u),A(s)]B(z),\qquad \iota_{z,s}p^nB(z)[C(u),A(s)]. \] Expand the remaining polynomial as \[ q^{M-c}=((s-u)-p)^{M-c} =\sum_{j=0}^{M-c}\binom{M-c}{j}(s-u)^j(-p)^{M-c-j}. \] Terms with \(j\ge b\) vanish by locality of \(A,C\). For \(jk\) already give zero by energy. Then (TC.A5), with \(D_0w_k=-kw_k\) and \(D_{-1}w_k=-Tw_k\), proves \[ D_lW_k(t)=-t^{l+1}W_k'(t)-k(l+1)t^lW_k(t) +\sum_{i=0}^{k-1}C_{ki} \bigl(t^{l+1}\bigr)^{(k-i+1)}W_i(t). \tag{TC.A7} \] For the term with \(j=k-i\ge1\), its coefficient in (TC.A5) is \(\binom{l+1}{j+1}t^{l-j}(j+1)!\), exactly the derivative of \(t^{l+1}\) of order \(j+1\). The two remaining terms are the displayed derivative and weight terms. Derivatives of order greater than \(l+1\) vanish. This proves every boundary case as well. By (TC.A4), the identical formula holds for the corresponding endomorphism series. The finite basic-vector identity (TC.A6) is proved by the quantum calculations (SL.X6), (SL.X10) and their positive-Witt induction (SL.X13). It is stronger than homogeneity and the classical symbol law (GJ.P10); it retains every lower-filtration term required by the oper comparison. #### 3.4.2. Full scalar-oper coordinate law ##### Density transport on the full ordinary coordinate group Let \(k\) be a characteristic-zero field, \(R\) any ordinary commutative \(k\)-algebra, and \(n\geq2\). All derivatives below are relative to \(R\). The finite-jet construction in §§2.2–2.6 of [Opers, critical level and the Beilinson–Drinfeld construction](opers-critical-level-and-the-beilinson-drinfeld-construction.md), especially (O4.2)–(O4.6), gives the scalar presentation of an ordinary \(PGL_n\)-oper. In a density coordinate its operator is \[ D_t=\partial_t^n+\sum_{i=2}^n s_i(t)\partial_t^{n-i}, \qquad h=\frac{1-n}{2}, \qquad D:F_h\longrightarrow F_{h+n}. \tag{SC.O1} \] Here \(F_h\) denotes the source density frame and \(F_{h+n}\) the target density frame. In the theta presentation these are \(\theta^{1-n}\) and \(\theta^{1+n}\). This notation specifies the two local weights; it does not choose a global trivialization of either line. The coefficient functor in this coordinate is \[ R\longmapsto\prod_{i=2}^n R[[t]] =\operatorname{Hom}_{k\text{-alg}} \bigl(k[s_{i,r}:2\leq i\leq n,\ r\geq0],R\bigr), \qquad s_i(t)=\sum_{r\geq0}s_{i,r}t^r. \tag{SC.O2} \] The finite product permits every power-series coefficient sequence. It uses the completed ring \(R[[t]]\). First consider a passive coordinate change \(t=\eta(u)\). Write \(\alpha=\eta'(u)\), so \(\alpha\in R[[u]]^\times\). The full continuous coordinate group also permits a nilpotent constant \(\eta(0)\); it is larger than the subgroup fixing the origin. Section 1.1.8 constructs these substitutions and their inverses over every ordinary \(R\). The finiteness assertion can be seen directly. If \(\eta(0)^\nu=0\), a term contributing to the coefficient of \(u^r\) in \(\eta(u)^m\) uses at least \(m-r\) constant factors. Thus \[ [u^r]\eta(u)^m=0\quad(m\geq r+\nu), \qquad [u^r]F(\eta(u)) =\sum_{m=0}^{r+\nu-1}[t^m]F(t)\,[u^r]\eta(u)^m. \tag{SC.O3} \] No reduction of the ring is involved. For completeness, the inverse construction also retains nilpotents. Put \(I=(\eta(0))\). Starting with \(a=0\), the iteration \(a\mapsto a-\eta(a)/\eta'(a)\) takes place in \(I\). Evaluation there is finite, and \(\eta'(a)\) is a unit modulo the nilpotent ideal \(I\), hence a unit. Taylor expansion shows that an error in \(I^d\) becomes an error in \(I^{2d}\). Finitely many steps give a root \(a\in I\). The difference \(\eta(a)-\eta(a')\) is \((a-a')\) times a unit for \(a,a'\in I\), so this root is unique. The pointed series \(\eta(u+a)\) has an inverse by coefficient recursion, using its unit linear coefficient. Adding back \(a\) gives the two-sided continuous inverse of \(\eta\). These formulas prove that the coordinate changes used below are actual automorphisms, including over nonreduced bases. A density coefficient transforms by \[ P_w(\eta)f(u)=\alpha(u)^w f(\eta(u)). \] For integral \(w\) this is an ordinary formula. Half-integral weights will be justified below. Transporting both the source and target of the operator gives \[ \begin{aligned} \mathcal T_\eta(D) &=P_{h+n}(\eta)\,D\,P_h(\eta)^{-1}\\ &=\alpha^{h+n} \left(\sum_{i=0}^n s_i(\eta(u)) (\alpha^{-1}\partial_u)^{n-i}\right)\alpha^{-h}, \qquad s_0=1,\quad s_1=0. \end{aligned} \tag{SC.O4} \] The order of the factors matters: the derivative operators act on the factor to their right. Formula (SC.O4) follows by applying the operator to a section, expressing its input in the old density frame, and expressing its output in the new density frame. It is also the transition formula of the jet construction. The chain and product rules defining \(J^{n-1}F_h\) give the following comparison: \[ \begin{array}{ccc} D_t:F_h(t)\longrightarrow F_{h+n}(t) &\xrightarrow{\ (O4.2)\text{--}(O4.6)\ }& (J^{n-1}F_h,\nabla_D,F_\bullet)_{\mathrm{proj}}\\ \big\downarrow{\mathcal T_\eta}&& \big\downarrow{\,t=\eta(u)\,}\\ D_u:F_h(u)\longrightarrow F_{h+n}(u) &\xrightarrow{\ (O4.2)\text{--}(O4.6)\ }& (J^{n-1}F_h,\nabla_{D_u},F_\bullet)_{\mathrm{proj}} . \end{array} \tag{SC.O5} \] Every arrow on the right is induced by the finite change-of-jet matrix, whose diagonal entries are units. The horizontal Taylor recursion in (O4.3) commutes with that matrix. Hence the square compares intrinsic connections and flags in ordinary families. There is no square-root choice in the resulting operator action. If \(n\) is odd, the weights are integral. If \(n\) is even, adjoin a square root of the unit \(\alpha(0)\). The extension \(R[z]/(z^2-\alpha(0))\) is free of rank two and faithfully flat; \(z\) and \(2z\) are units. Recursively solving \(q(u)^2=\alpha(u)\), with \(q(0)=z\), gives a unique series because each new equation has coefficient \(2z\). In this extension the factors in (SC.O4) are \(q^{n+1}\) and \(q^{n-1}\). If \(\widehat q\) is another root, \(a=\widehat q/q\) satisfies \(a^2=1\). Differentiation gives \(a'=0\), since \(2a\) is a unit. Thus \(a\) commutes with the derivative operators, and replacing \(q\) by \(\widehat q\) multiplies the full expression by \(a^{2n}=1\). This argument allows different signs on different idempotent components. In the jet presentation the source frame changes by the single relatively constant scalar \(a^{1-n}\), and every jet component changes by that same scalar. Its projectivization is unchanged. Section 2.6 supplies the corresponding flat-line comparison between different global theta presentations; the present coordinate calculation requires no global theta frame. The square-root-free formula below proves descent directly. It also proves the composition law \[ \mathcal T_\zeta\bigl(\mathcal T_\eta(D)\bigr) =\mathcal T_{\eta\circ\zeta}(D). \tag{SC.O6} \] Indeed \((\eta\circ\zeta)'=(\eta'\circ\zeta)\zeta'\), and the density pullbacks compose with exactly these two factors. The signs cancel as just proved. Coefficient-finite substitution makes the same identity valid for nilpotent coordinate constants. ##### A finite formula for every transported coefficient Put \(b=\alpha'/\alpha\). Conjugating one derivative in (SC.O4) gives \[ \alpha^h(\alpha^{-1}\partial_u)\alpha^{-h} =\alpha^{-1}(\partial_u-hb)=L. \] Consequently \[ \boxed{\quad \mathcal T_\eta(D)= \alpha^n\sum_{i=0}^n s_i(\eta(u))L^{n-i}, \qquad L=\alpha^{-1}(\partial_u-hb). \quad} \tag{SC.O7} \] This expression uses integer powers of \(\alpha\), its inverse, relative derivatives and rational constants. It belongs to the original coefficient ring before any square root is adjoined. Here is an explicit finite coefficient recursion. Define differential operators \[ B_a=\alpha^a L^a =\sum_{d=0}^a C_{a,d}(b)\partial_u^{a-d}, \qquad B_0=1 . \] Since \[ B_{a+1}=(\partial_u-(h+a)b)B_a, \] their coefficients satisfy \[ \begin{gathered} C_{0,0}=1,\qquad C_{a,d}=0\quad(d<0\text{ or }d>a),\\ C_{a+1,d} =C_{a,d}+\partial_u C_{a,d-1} -(h+a)b\,C_{a,d-1}. \end{gathered} \tag{SC.O8} \] All \(C_{a,d}\) are finite differential polynomials with rational coefficients. This recursion specifies every one of them, using only \(a\) steps. It also gives \[ C_{a,0}=1,\qquad C_{a,1}=-\left(ah+\frac{a(a-1)}2\right)b. \tag{SC.O9} \] For the second identity, each successive factor contributes \(-(h+a)b\) to the next-to-leading coefficient. Their sum is the displayed coefficient. Write \(\mathcal T_\eta(D)=\sum_{i=0}^n\widetilde s_i(u)\partial_u^{n-i}\). Extracting each derivative order from (SC.O7) gives the complete formula \[ \boxed{\quad \widetilde s_i(u) =\sum_{j=0}^i \alpha(u)^j s_j(\eta(u))C_{n-j,i-j}(b) \quad(0\leq i\leq n). \quad} \tag{SC.O10} \] In particular \(\widetilde s_0=1\). For \(i=1\), the only possible contributions are \(C_{n,1}\) and \(\alpha s_1(\eta)\). They both vanish, because \(h=(1-n)/2\) and \(s_1=0\). This proves monicity and vanishing of the subprincipal coefficient for the full coordinate action. For \(i\geq2\), the formula has the following triangular structure: | Contribution to \(\widetilde s_i\) | Exact expression | |---|---| | The coefficient of the same order | \(\alpha^i s_i(\eta)\) | | Lower-index coefficients | \(\sum_{j=2}^{i-1}\alpha^j s_j(\eta)C_{n-j,i-j}(b)\) | | The coordinate correction of the leading derivative | \(C_{n,i}(b)\) | The table displays (SC.O10): a scalar coefficient has its weight-\(i\) term, mixing from lower-index coefficients, and a universal correction from the monic leading derivative. It describes the raw scalar coordinates rather than a chosen splitting of the global affine oper torsor. The quadratic correction has a useful closed form for every \(n\). Put \[ K_n=\frac{n(n^2-1)}{12}, \qquad \mathcal S(\eta)=b'-\frac12b^2. \] For fixed \(h=(1-n)/2\), set \(m_a=ah+a(a-1)/2=a(a-n)/2\). Summing the recurrence for \(C_{a+1,2}\) gives \[ C_{n,2}=-\sum_{a=0}^{n-1}m_a\,b' +\sum_{a=0}^{n-1}(h+a)m_a\,b^2. \] The first sum is \(-K_n\), by the sums of consecutive integers and their squares. For the second, pair \(a\) with \(n-a\): \(m_{n-a}=m_a\) and \((h+a)+(h+n-a)=1\). The zero endpoint terms allow the pairing over \(0\leq a\leq n\), so the second sum is one half of the first, namely \(-K_n/2\). Therefore \[ C_{n,2}=K_n\mathcal S(\eta), \qquad \widetilde s_2=\alpha^2s_2(\eta)+K_n\mathcal S(\eta). \tag{SC.O10a} \] This proves the scalar Schwarzian coefficient in every order. The divided Taylor coefficients are \[ s_{i,r}=[t^r]s_i(t)=\frac{1}{r!}s_i^{(r)}(0). \] Set \(s_{0,m}=\delta_{m,0}\) and \(s_{1,m}=0\). For a fixed nilpotence bound \(\eta(0)^\nu=0\), coefficient extraction from (SC.O10) is the finite formula \[ \boxed{\quad \widetilde s_{i,r} =\sum_{j=0}^i \sum_{\substack{a+c+d=r\\a,c,d\geq0}} [u^a]\alpha^j\, [u^d]C_{n-j,i-j}(b)\, \sum_{m=0}^{c+\nu-1}s_{j,m}[u^c]\eta(u)^m . \quad} \tag{SC.O11} \] Thus a fixed transformed coefficient is a polynomial in finitely many original coefficients, coordinate coefficients and the inverse of the linear coordinate coefficient. The rational constants belong to \(k\). More explicitly, the original coefficients used have indices at most \(r+\nu-1\); the coordinate coefficients used have indices at most \(r+n+1\). The latter bound follows because \(C_{a,d}\) uses derivatives of \(b\) of order at most \(d-1\), and \(b=\eta''/\eta'\). Indeed (SC.O8) raises this maximum derivative order by at most one when \(d\) increases by one; its initial coefficient has no derivative. Inverse-series coefficients are determined by finite recursion from the unit \(\eta'(0)\). Every formula therefore commutes with \(R\to R'\), without flatness or reducedness assumptions. The nilpotence bound may depend on the individual coordinate; an algebra homomorphism preserves any bound already satisfied. On the coordinate chart with \(\eta(0)^\nu=0\), (SC.O11) is an ordinary polynomial formula with the linear coefficient inverted. These charts cover the full continuous coordinate functor. Together with (SC.O6), they give the full ordinary coordinate action, not only its first-order approximation. The same differential-operator formula is valid for finite-pole Laurent coefficients: the finite expansion of \((\eta(0)+u\,a(u))^{-1}\) from §1.1.8 supplies the required Laurent substitution. ##### Inverse variation and all divided coefficient modes To specify the action on functions, let \(\phi\) be a coordinate automorphism and put \(\psi=\phi^{-1}\). The inverse-pullback action is \[ (\phi\cdot f)(D)=f(\mathcal T_\psi(D)). \tag{SC.O12} \] This is the convention already used for the projective-connection coefficients in (RC.C4)–(RC.C5). In (SC.O7)–(SC.O11), take \(\eta=\psi\) to obtain its full finite substitution law. Let \(\phi(t)=t+\epsilon v(t)\), where \(\epsilon^2=0\) and \(v(t)\in R[[t]]\). The constant coefficient \(\epsilon v(0)\) is nilpotent, so this is a coordinate in the full group. Its inverse is \(t-\epsilon v(t)\). For any density weight \(w\), \[ P_w(\phi)=1+\epsilon A_w(v), \qquad A_w(v)=v\partial_t+w v'. \] Expanding the source and target transports in (SC.O4) therefore gives \[ \boxed{\quad \delta_vD =-(v\partial_t+(h+n)v')D +D(v\partial_t+h v'). \quad} \tag{SC.O13} \] This identity is an equality of finite-order differential operators over \(R[\epsilon]/(\epsilon^2)\). We calculate every coefficient. The product rule, proved by induction on \(q\), is \[ \partial_t^q f =\sum_{a=0}^q\binom qa f^{(a)}\partial_t^{q-a}. \tag{SC.O14} \] The induction differentiates each coefficient and shifts each derivative order; the two contributions combine by Pascal's identity. In (SC.O13), the terms \(s_jv\partial_t^{n-j+1}\) from \(Dv\partial_t\) and \(-v\partial_tD\) cancel. The latter also contributes \(-v s_j'\partial_t^{n-j}\). At derivative order \(n-i\), the remaining coefficient is \[ -v s_i'-(h+n)v's_i +\sum_{j=0}^i s_j \left(\binom{n-j}{i-j+1} +h\binom{n-j}{i-j}\right)v^{(i-j+1)} . \] For \(j=i\), the term in parentheses is \(n-i+h\); combining it with the preceding \(v's_i\) term gives \(-i v's_i\). For \(j=0\), set \[ \gamma_{n,i} =\binom n{i+1}+h\binom ni =-\frac{(i-1)(n+1)}{2(i+1)}\binom ni. \tag{SC.O15} \] The second equality uses \(\binom n{i+1}=\binom ni(n-i)/(i+1)\) and \(h=(1-n)/2\). For \(2\leq j0\) vanish, and the full transport is \(s_i(u+a)\). The coefficient formula becomes \[ \widetilde s_{i,r} =\sum_{q=0}^{\nu-1}\binom{r+q}{r}a^q s_{i,r+q}. \tag{SC.O24} \] For the inverse action replace \(a\) by \(-a\). Thus nilpotent translations are finite on each coefficient even though they can involve coefficients beyond a fixed pointed jet truncation. For \(n=1\), the normalized operator is \(\partial_t:F_0\to F_1\); (SC.O7) gives \(\mathcal T_\eta(\partial_t)=\partial_u\). There are no scalar coefficients and the \(PGL_1\) coefficient functor is a point. These are complete algebraic coordinate formulas for the ordinary scalar-oper coefficient functor. Their identification with intrinsic opers uses the finite-jet, vector-lift and ordinary bundle-descent inputs specified in §§2.2–2.6. The recursive global curve/Picard foundations in §2.1 and the bundle-algebraization foundation in §2.8 remain unproved in their stated global uses. No such global theorem is needed for the coefficient recursion or its ordinary base change. The coefficients here are those of the scalar differential operator (SC.O1); Section 3.4.3 compares these coefficients with the quantum determinant generators. Sections 3.6–3.7 supply the smooth completed center in type A, the intrinsic critical Poisson structures and the full type A scalar-oper Poisson comparison. Completed centers and oper-Poisson comparison in other types, derived/global Hamiltonian reduction, chiral/factorization and derived parameter spaces retain their separate mathematical requirements. #### 3.4.3. Quantum coordinate law of the critical determinant Let \(k\) have characteristic zero. We use the critical affine matrix bracket (SL.L1)–(SL.L3), the right-ordered determinant coefficients (SL.L4)–(SL.L7), and the proved exterior relations (SL.L8)–(SL.L12). The determinant construction is credited there to Chervov–Molev, [*On higher order Sugawara operators*, arXiv:0808.1947v2](https://arxiv.org/abs/0808.1947v2). The raw coefficients already have the law of an order-\(n\) scalar density operator from weight \(h=(1-n)/2\) to weight \(h+n\). No triangular derivative correction is required for this law. The calculation below includes the lower-filtration terms, the scalar anomaly, every divided translate, and ordinary coordinate substitutions with nilpotent constant coefficient. ##### SL.X1. The operators and pinned minors Initially work in \(\mathfrak{gl}_n\), with \(\kappa(X,Y)=-n\operatorname{tr}(XY)+\operatorname{tr}(X)\operatorname{tr}(Y)\), \(K=1\), and \(T=\tau\) on the vacuum. Introduce the central parameter \(u\), and write \[ M_{aj}(u)=\delta_{aj}(T+u)+E_{aj}[-1],\quad \theta_j=\sum_a\psi_aM_{aj}(u),\quad P=\theta_1\cdots\theta_n=\Psi D(u). \] \[ \Psi=\psi_1\cdots\psi_n,\qquad F(u)=D(u)v=\sum_{k=0}^n w_k u^{n-k},\qquad w_0=v,\quad w_k=S_kv. \tag{SL.X1} \] Each \(\psi_a\) anticommutes with every exterior-odd expression and commutes with every ordinary operator coefficient. The \(\theta_j\) anticommute by the already proved explicit affine identity (SL.L8), not by an additional determinant theorem. For a set \(J\subseteq\{1,\ldots,n\}\), let \(P_J\) be the ordered product with \(\theta_j\) replaced by \(\psi_j\) at precisely its positions in \(J\), and let \(D_J(u)\) delete those same rows and columns from \(M(u)\). Expansion gives \[ P_J=\Psi D_J(u),\qquad \partial_u^a P=a!\sum_{|J|=a}P_J. \tag{SL.X2} \] Here the pinned sign is positive. A surviving permutation fixes every index in \(J\). Moving those fixed rows to the front and moving those fixed columns to the front use the same sequence of index positions, so their signs multiply to \(+1\); the remaining sign is the permutation sign on the ordered complement. No operator coefficient is reordered in this argument. The derivative formula follows from the finite product rule: a factor can be differentiated only once, with derivative \(\psi_j\), and each chosen set of \(a\) factors occurs in its \(a!\) possible differentiation orders. We abbreviate \(P_{\{i,j\}}=P_{ij}\), and similarly for minors and triples. For \(l\geq-1\), current substitution has infinitesimal action \[ [D_l,E_{ab}[r]]=rE_{ab}[r+l],\qquad D_lv=0. \tag{SL.X3} \] This is the ordinary affine derivation proved in (RC.C1). Directly, its possible central defect is \(r(r+s+l)\kappa(X,Y)\delta_{r+s+l,0}=0\). Nonnegative currents remain nonnegative; for \(l=-1,r=0\) the coefficient is zero. Thus it acts on the induced vacuum. Comparing on currents and on the cyclic vacuum proves \[ [D_l,D_m]=(m-l)D_{l+m},\qquad [D_l,T]=(l+1)D_{l-1}\ (l\geq0),\qquad D_{-1}=-T. \tag{SL.X4} \] In particular \([D_0,\theta_j]=-\theta_j+u\psi_j\). The product rule already gives \(D_0F=(u\partial_u-n)F\), so \(D_0w_k=-kw_k\). ##### SL.X2. The complete D1 calculation Put \[ H_i=\sum_a\psi_a E_{ai}[0],\qquad Z_i=2\psi_iD_0-H_i. \] Then \([D_1,\theta_i]=Z_i\) and \(Z_iv=0\). Formula (SL.L12), summed against \(\psi_a\), proves \[ \{H_i,\theta_j\}=-\psi_i\theta_j-\psi_j\theta_i, \qquad \{Z_i,\theta_j\}=-\psi_i\theta_j+\psi_j\theta_i+2u\psi_i\psi_j. \tag{SL.X5} \] For the first identity, the sum is \(\sum_a\psi_a(\psi_iM_{aj}-\delta_{aj}\theta_i)\), which is exactly its displayed right side. For the second, use \(2\psi_i[D_0,\theta_j]-\{H_i,\theta_j\}\). In the product-rule term inserting \(Z_i\) in column \(i\), move it through each later column until it reaches \(v\), where it is zero. Its interaction with a later column \(j\) has a sign \((-1)^m\), \(m=j-i-1\), from the intermediate odd factors. The three terms in (SL.X5) give \[ -P_i-P_j+2uP_{ij}. \] The signs are explicit: moving \(\psi_i\) left past the \(m\) intermediate factors supplies \((-1)^m\), canceling the interaction sign in the first and third terms. In the second, first \(\psi_j\theta_i=-\theta_i\psi_j\), and then moving \(\theta_i\) left past those \(m\) factors supplies another \((-1)^m\). Its result is therefore \(-P_j\). Summing all pairs, each single pinned position belongs to \(n-1\) pairs. By (SL.X2), \[ \boxed{D_1F(u)=uF''(u)-(n-1)F'(u).} \tag{SL.X6} \] This is an equality of actual vacuum vectors, before passage to symbols. ##### SL.X3. The complete D2 calculation and its central correction Set \[ R_i=\sum_a\psi_aE_{ai}[1],\qquad B_i=3\psi_iD_1-R_i. \] Then \([D_2,\theta_i]=B_i\) and \(B_iv=0\). Use the full entry bracket (SL.L14), including its trace and critical central terms. Summing it gives \[ \{R_i,\theta_j\}=-2\psi_jH_i-\psi_iH_j+(n+1)\psi_i\psi_j. \tag{SL.X7} \] Indeed the four types of terms in that sum are \(\sum_a\psi_a\psi_jE_{ai}[0]=-\psi_jH_i\), \(\sum_a\psi_a\psi_iE_{aj}[0]=-\psi_iH_j\), \(-\psi_jH_i\), and \(\psi_i\psi_j-n\psi_j\psi_i=(n+1)\psi_i\psi_j\). Consequently \[ \{B_i,\theta_j\}=K_{ij}-(n+1)\psi_i\psi_j, \quad K_{ij}=6\psi_i\psi_jD_0-2\psi_iH_j+2\psi_jH_i. \tag{SL.X8} \] The operator \(K_{ij}\) is even and kills \(v\). Its full commutator with a later column is \[ [K_{ij},\theta_k] =-2\psi_i\psi_j\theta_k+2\psi_i\psi_k\theta_j -2\psi_j\psi_k\theta_i+6u\psi_i\psi_j\psi_k. \tag{SL.X9} \] To verify every coefficient, use \([D_0,\theta_k]=-\theta_k+u\psi_k\) and \([\psi_iH_j,\theta_k]=\psi_i\{H_j,\theta_k\} =-\psi_i\psi_j\theta_k-\psi_i\psi_k\theta_j\). The corresponding formula with \(i,j\) swapped gives the remaining terms of (SL.X9). The coefficients of \(\psi_i\psi_j\theta_k\) are \(-6+2+2=-2\). There is no uncomputed operator remainder. Move \(B_i\) through the later columns. At each \(ik).} \tag{SL.X13} \] Together with \(D_0w_k=-kw_k\) and \(D_{-1}w_k=-Tw_k\), this is the full basic-vector action. In particular, \[ D_1w_k=-(k-1)(n-k+1)w_{k-1}, \] \[ D_2w_k=-\frac{(n-k+2)(n-k+1)(n+2k-3)}2w_{k-2}. \tag{SL.X14} \] These are exact quantum identities; their right sides give all lower-length corrections for the basic vectors. ##### SL.X5. The traceless quotient does not remove the trace correction The ideal generated by \(I[r]=\sum_aE_{aa}[r]\) is stable under \(D_l\), since \(D_lI[r]=rI[r+l]\). Consequently all preceding identities pass to the traceless vacuum, with \(w_k=\overline S_kv\), \(w_0=v\), and \(w_1=0\). The central term \(+\delta_{ab}\delta_{ij}\) of (SL.L14) must remain in the traceless matrix notation. Indeed for \(F_{ab}=E_{ab}-\delta_{ab}I/n\), \[ -n\operatorname{tr}(F_{ab}F_{ij}) =-n\delta_{bi}\delta_{aj}+\delta_{ab}\delta_{ij}. \tag{SL.X15} \] Thus taking the quotient agrees with the same \(n+1\) in (SL.X7); dropping it would give a different and incorrect coordinate anomaly. Also \(C_{10}=\binom n2+hn=0\), so the resulting subprincipal coefficient remains zero under the whole law. ##### SL.X6. Every divided translate and the complete coefficient series Let \(A_{k,r}\) be the endomorphism with vacuum image \(T^rw_k/r!\), and put \(A_{0,0}=\operatorname{Id}\), \(A_{0,r}=0\) for \(r>0\), \(A_{1,r}=0\), and \(A_{k,r}=0\) for \(r<0\). These conventions refer to the traceless quotient. The proved all-integer reconstruction (VC.17)–(VC.20) identifies its invariant field with the regular series \[ s_k(t)=Y(w_k,t)=\sum_{r\geq0}A_{k,r}t^r, \qquad s_0=1,\quad s_1=0. \tag{SL.X16} \] All nonnegative field modes vanish as actual operators by (VC.18) and cyclicity. Thus (SL.X16) describes the entire reconstructed field, not a truncation imposed by hand. The conjugation action on an endomorphism satisfies \([D_l,E_w]=E_{D_lw}\). Its commutator with a current is zero by Jacobi and (SL.X3), and its vacuum image is \(D_lw\); the induction correspondence then proves the assertion. Repeated use of (SL.X4) gives, for \(l\geq0\), \[ D_l\frac{T^rw_k}{r!} =\sum_{a=0}^{\min(r,l+1)} \frac{(l+1)_{\underline a}}{a!(r-a)!}\, T^{r-a}D_{l-a}w_k. \tag{SL.X17} \] Here \((b)_{\underline a}=b(b-1)\cdots(b-a+1)\). The formula follows by induction on \(r\): moving \(D_l\) across the last \(T\) supplies \((l+1)D_{l-1}\), and the two adjacent binomial terms combine. Equivalently, its iterated right commutators with \(T\) are \((l+1)_{\underline a}D_{l-a}\) through \(a=l+1\), after which the commutator of \(-T\) with \(T\) is zero. This proves both the finite sum and its last boundary term. Inserting (SL.X13) into (SL.X17) yields the following fully explicit action on the endomorphisms, for \(l\geq0\): \[ \begin{split} [D_l,A_{k,r}]={}& -\mathbf1_{r\geq l}\bigl(r-l+k(l+1)\bigr)A_{k,r-l}\\ &+\sum_{i=0}^{k-1} C_{ki}(l+1)_{\underline{k-i+1}} A_{i,r-l+k-i}. \end{split} \tag{SL.X18} \] A term with a negative coefficient index is zero. A falling factorial is zero if its order exceeds \(l+1\). To see the matching coefficients explicitly, in the first part of (SL.X17) put \(i=k-l+a\). Its factor becomes \((l+1)!/(l-k+i)!\), exactly the falling factorial in (SL.X18). The remaining terms \(a=l\) and \(a=l+1\) contribute \(-k(l+1)\) and \(-(r-l)\) when present. For \(l=0\) this gives \(-(k+r)A_{k,r}\). For \(l=-1\), directly \([D_{-1},A_{k,r}]=-(r+1)A_{k,r+1}\). Therefore, for \(v(t)=t^{l+1}\), \(l\geq-1\), summing the coefficients gives \[ \boxed{ \delta_v s_k=-v s_k'-kv's_k +\sum_{i=0}^{k-1}C_{ki}v^{(k-i+1)}s_i. } \tag{SL.X19} \] For a general formal vector field the same formula is extended linearly. Each fixed generator has finite energy \(k+r\), so \(D_l\) vanishes on it for \(l>k+r\); thus this formal action uses actual finite sums on each generator. The formula lists every lower-filtration term. No additional term can be hidden by passage to symbols: (SL.X6), (SL.X10), positive-Witt induction and (SL.X17) were all exact operator/vector identities. ##### SL.X7. Equality with the scalar density law Set \[ \mathcal L=\partial_t^n+s_2(t)\partial_t^{n-2}+\cdots+s_n(t) =\sum_{i=0}^n s_i(t)\partial_t^{n-i}. \] As an operator from density weight \(h\) to weight \(h+n\), inverse coordinate pullback has variation \[ \delta_v\mathcal L =-(v\partial_t+(h+n)v')\mathcal L +\mathcal L(v\partial_t+hv'). \tag{SL.X20} \] We can compare it without assuming any quantum–oper identification. The finite Leibniz rule \(\partial^m f=\sum_{a=0}^m\binom ma f^{(a)}\partial^{m-a}\) follows inductively by the ordinary product rule. Apply it with \(m=n-i\). At coefficient \(\partial^{n-k}\), the \(i=k\) terms are exactly \(-v s_k'-kv's_k\); for \(ik+r\) acts trivially on a generator. Equality of the exact derivations (SL.X18) therefore implies equality of their finite exponentials and their coefficientwise successive products. The translation and scaling factors complete the proof for every ordinary substitution, including nonreduced parameters. Finally, (SL.C1) gives the whole ordinary type-A polynomial vacuum algebra, and the scalar-oper construction (O3)–(O5) gives the polynomial ring on the coefficients of \(\mathcal L\). Write \(s_{k,r}\) for the independent coefficient coordinates in that scalar-oper ring. The map \[ A_{k,r}\longmapsto s_{k,r}\qquad(2\leq k\leq n,\ r\geq0) \tag{SL.X26} \] is therefore an algebra isomorphism, with the specified determinant normalization. Equations (SL.X18)–(SL.X24) and the integration argument prove its coordinate equivariance over every ordinary parameter algebra. Products of type-A factors are handled componentwise. This establishes the type-A ordinary formal-disc coordinate comparison at the previously stated scalar-oper and invariant-theory foundations. Sections 3.6–3.7 prove the type A punctured-disc completed center and its full scalar-oper Poisson comparison; the regular comparison is a Poisson vertex comparison. Ordinary principal Hamiltonian reduction in other types and derived/global reduction, chiral/factorization, Satake and derived-family compatibility, and basic lifting and full coordinate/oper-Poisson identification in other types retain their separate proof obligations. #### 3.4.4. General quadratic critical states and their coordinate action Let \(k\) be a field of characteristic zero and let \(\mathfrak g\) be a finite-dimensional split semisimple Lie algebra. Fix a nondegenerate symmetric invariant form \(B\), with dual bases \(a_\alpha,a^\alpha\). The form \(B\) identifies the current coordinates with their duals; it is separate from the affine form \(\kappa\). Write \[ [x_m,y_j]=[x,y]_{m+j}+m\kappa(x,y)\delta_{m+j,0}\,1. \tag{GQ.1} \] The vacuum \(v\) is annihilated by \(\mathfrak g[[t]]\). The PBW construction, its smooth finite-energy bound, the induced-module universal property, and the ordinary-base-change result of §3.3.2 are the foundations used below. The endomorphism commutativity proved in §3.3.3 is available once a state is shown to be invariant. No assertion about the higher-degree generators of the centre is needed here. For an invariant form \(F\), define \(K_F\in\operatorname{End}_k(\mathfrak g)\) by \[ B(K_Fx,y)=F(x,y),\qquad C_B=K_{\mathrm{Kil}}, \quad \mathrm{Kil}(x,y)=\operatorname{tr}(\operatorname{ad}x\operatorname{ad}y). \tag{GQ.2} \] Define the half-Casimir state and its normally ordered modes by \[ Q=\frac12\sum_\alpha(a_\alpha)_{-1}(a^\alpha)_{-1}v, \qquad S_n=\frac12\sum_{\alpha,j\in\mathbb Z}:(a_\alpha)_j(a^\alpha)_{n-j}:. \tag{GQ.3} \] Our ordering convention places the first mode to the left when its index is negative and to the right when its index is nonnegative. The sum defining \(S_n\) is an operator on the smooth vacuum module: for each input vector, the positive mode at the right kills that vector once its index exceeds a fixed bound. All sums below have this meaning. **The inverse-form tensor and the Killing correction.** The tensor \(\sum a_\alpha\otimes a^\alpha\) is symmetric and invariant. Symmetry follows from the symmetric inverse Gram matrix. To prove invariance, pair its simultaneous adjoint variation with \(y\otimes z\), using \(B\) in each factor. The result is \[ -B([x,y],z)-B(y,[x,z])=0. \] Consequently \[ \sum_\alpha[a_\alpha,a^\alpha]=0, \qquad \sum_\alpha[[x,a_\alpha],a^\alpha]=C_Bx. \tag{GQ.4} \] For the second identity, pairing with \(y\) gives \[ \begin{aligned} B\left(\sum_\alpha[[x,a_\alpha],a^\alpha],y\right) &=-\sum_\alpha B([x,a_\alpha],[y,a^\alpha])\\ &=\sum_\alpha B(\operatorname{ad}y\operatorname{ad}x(a_\alpha),a^\alpha) =\mathrm{Kil}(x,y). \end{aligned} \] This proves the required arbitrary-type Casimir identity directly from the invariant pairing. Two further contractions are \[ \sum_\alpha\kappa([x,a_\alpha],a^\alpha)=0, \qquad \sum_\alpha\kappa(a_\alpha,a^\alpha)=\operatorname{tr}(K_\kappa). \tag{GQ.5} \] Indeed, interchange the two factors of the symmetric tensor in the first sum and use invariance of \(\kappa\): the sum becomes its own negative. The second equality is the dual-basis trace formula. **The all-integer current commutator.** Put \(\theta(j)=1\) for \(j\geq0\) and \(0\) otherwise. Moving the first factor across the ordering boundary gives \[ \begin{aligned} [x_m,:a_jb_{n-j}:] &=:[x_m,a_j]b_{n-j}:+:a_j[x_m,b_{n-j}]:\\ &\quad +(\theta(j+m)-\theta(j))[[x,a]_{j+m},b_{n-j}]. \end{aligned} \tag{GQ.6} \] This identity follows by writing the two possible orders before and after the commutator. Their difference is precisely the bracket of the two factors when the first index crosses zero. Central terms in the inner current bracket cause no additional ordering difference. In the sum (GQ.3), reindexing the ordinary Lie terms in (GQ.6) makes them the simultaneous adjoint variation of the inverse-form tensor, hence zero. The two affine central terms contribute \[ \frac m2\sum_\alpha \bigl(\kappa(x,a_\alpha)(a^\alpha)_{m+n} +\kappa(x,a^\alpha)(a_\alpha)_{m+n}\bigr) =m(K_\kappa x)_{m+n}. \] The boundary double bracket is \((C_Bx)_{m+n}\), by (GQ.4). Its possible scalar part is \((j+m)\delta_{m+n,0}\sum_\alpha\kappa([x,a_\alpha],a^\alpha)\), which vanishes by (GQ.5). Finally \[ \sum_{j\in\mathbb Z}(\theta(j+m)-\theta(j))=m \] for every integer \(m\): there are \(m\) positive crossings if \(m>0\), and \(-m\) negative crossings if \(m<0\). Thus \[ \boxed{[x_m,S_n] =m\bigl((K_\kappa+\tfrac12C_B)x\bigr)_{m+n}} \qquad(m,n\in\mathbb Z). \tag{GQ.7} \] This is a finite normal-order calculation, not a manipulation of unconvergent sums. On a given input, choose a common cutoff for that input, its image under \(x_m\), both reindexed normally ordered families, the two scalar positions, and the finite crossing interval. Formula (GQ.6) can then be summed over a finite interval. Increasing its endpoints adds zero on both sides. This also proves (GQ.7) on every smooth vector. At the critical form \[ \kappa_{\mathrm{crit}}=-\frac12\mathrm{Kil}, \tag{GQ.8} \] we have \(K_\kappa=-C_B/2\), so every \(S_n\) commutes with every current. On the vacuum, only two negative indices survive in (GQ.3). Therefore \[ S_nv=0\ (n\geq-1),\qquad S_{-r-2}v=Q_r\ (r\geq0), \qquad Q_r=\frac12\sum_{\alpha,\ i,j\geq1,\ i+j=r+2} (a_\alpha)_{-i}(a^\alpha)_{-j}v. \tag{GQ.9} \] In particular, \(Q\) and all \(Q_r\) are critical vacuum invariants. By cyclicity, \(S_n=0\) as a critical vacuum operator when \(n\geq-1\). The negative \(S_n\) are the invariant endomorphisms corresponding to the states in (GQ.9). They commute with one another by the full invariant-endomorphism theorem of §3.3.3. The translation operator satisfies \(Tv=0\) and \([T,x_m]=-m x_{m-1}\). Expanding a product with this derivation gives \[ Q_r=\frac{T^rQ}{r!}. \tag{GQ.10} \] For example, distributing \(r\) derivatives between the two factors of \(Q\) cancels the two factorials against the binomial coefficient and \(r!\); the resulting indices are exactly \(i+j=r+2\). This proof works at every affine form. Under the PBW pairing \(y_{-j}\leftrightarrow B(y,A_{j-1})\), the leading symbols of \(Q_r\) are the coefficients of \(\frac12B(A(t),A(t))\). Each \(Q_r\) has PBW degree two and energy \(r+2\). **The complete regular-coordinate action.** For \(l\geq-1\), the derivation \[ D_lx_m=m x_{m+l},\qquad D_lv=0 \tag{GQ.11} \] exists on the affine algebra and induced vacuum. Its central-bracket defect is a multiple of \(m(m+n+l)\kappa(x,y)\delta_{m+n+l,0}\), hence zero. It preserves the vacuum subalgebra, including for \(l=-1\), since the possible index \(-1\) comes only from \(m=0\) and has zero coefficient. Notice \(D_{-1}=-T\). Negative \(l<-1\) would not preserve this vacuum and are outside this statement. There is again an explicit ordering boundary: \[ \begin{aligned} [D_l,:a_jb_{n-j}:] &=j:a_{j+l}b_{n-j}:+(n-j):a_jb_{n-j+l}:\\ &\quad+j(\theta(j+l)-\theta(j))[a_{j+l},b_{n-j}]. \end{aligned} \] Reindexing the first ordinary sum changes its coefficient to \(j-l\); adding the second coefficient \(n-j\) leaves \(n-l\). The boundary Lie term contracts to zero by (GQ.4). Its scalar contribution is \[ \frac12\operatorname{tr}(K_\kappa)\delta_{n+l,0} \sum_j j(j+l)(\theta(j+l)-\theta(j)). \] For \(l\geq1\), write \(j=-s\), \(1\leq s\leq l\), and evaluate \[ \sum_{s=1}^{l}(s^2-ls) =\frac{l(l+1)(2l+1)}6-\frac{l^2(l+1)}2 =-\frac{l^3-l}{6}. \] For \(l=0\) the interval is empty; for \(l=-1\) its only possible index is \(j=0\), with zero coefficient. Define \[ \gamma_\kappa=-\frac{\operatorname{tr}(K_\kappa)}{12}. \] We have proved the complete mode identity \[ \boxed{[D_l,S_n]=(n-l)S_{n+l} +\gamma_\kappa(l^3-l)\delta_{n+l,0}\,1} \qquad(n\in\mathbb Z,\ l\geq-1). \tag{GQ.12} \] As for (GQ.7), a common cutoff for the input, its image under \(D_l\), the reindexed sums, and this finite crossing interval makes the proof finite. Applying (GQ.12) to \(v\) gives, with a missing-index term interpreted as zero, \[ \boxed{D_lQ_r=-\mathbf1_{r\geq l}(r+l+2)Q_{r-l} +\gamma_\kappa(l^3-l)\delta_{r,l-2}v.} \tag{GQ.13} \] Here \(r\geq0\) and \(l\geq-1\), so the first term for \(l=-1\) is always present. In particular \[ D_0Q=-2Q,\quad D_1Q=0,\quad D_2Q=-\frac12\operatorname{tr}(K_\kappa)v, \quad D_lQ=0\ (l\geq3). \] At critical level put \[ \gamma_B=\frac{\operatorname{tr}(C_B)}{24}. \tag{GQ.14} \] Then \(D_2Q=\operatorname{tr}(C_B)v/4=6\gamma_Bv\). The current anomaly in (GQ.7) has disappeared, while the coordinate anomaly (GQ.13) generally remains. Let \(\mathcal Q(t)=\sum_{r\geq0}Q_rt^r\) and let \(f(t)\partial_t\) be a regular formal vector field, with \(f(t)=\sum_{l\geq-1}f_lt^{l+1}\). Summing (GQ.13) gives \[ \boxed{D_f\mathcal Q=-f\mathcal Q'-2f'\mathcal Q +\gamma_\kappa f'''v.} \tag{GQ.15} \] Each coefficient has only finitely many contributing \(l\): in its ordinary part \(l\leq r\), and its scalar part has \(l=r+2\). Thus this identity is defined over every ordinary coefficient ring. For completeness, this infinitesimal law has the following full formal-coordinate form. Let \(R\) be any ordinary \(k\)-algebra. Let \(\phi(t)\in R[[t]]\) define a continuous formal-disc automorphism, allowing a nilpotent constant term, and put \(\psi=\phi^{-1}\). On currents use substitution \(xF(t)\mapsto xF(\phi(t))\); its induced vacuum action is \(U_\phi\). Then \[ \boxed{U_\phi\mathcal Q(t) =\psi'(t)^2\mathcal Q(\psi(t)) -\gamma_\kappa\operatorname{Sch}(\psi,t)v,} \qquad \operatorname{Sch}(\psi,t)=\frac{\psi'''}{\psi'} -\frac32\left(\frac{\psi''}{\psi'}\right)^2. \tag{GQ.16} \] The inverse coordinate is essential: \(U_\phi\) acts on the coefficient functions. For \(\phi=t+\varepsilon f\), one has \(\psi=t-\varepsilon f\) and \(\operatorname{Sch}(\psi)=-\varepsilon f'''\), recovering (GQ.15). Here is an algebraic integration proof, including nonreduced \(R\). Substitution preserves the affine cocycle because it preserves residue. For a Laurent monomial of exponent other than \(-1\), the substituted differential is an exact derivative and has residue zero. In the remaining case factor \(\phi=(t-b)u(t)\), where \(b=\psi(0)\) is nilpotent and \(u\) is a unit power series. Then \(\phi'/\phi=1/(t-b)+u'/u\) has residue one. These facts prove the residue identity coefficientwise; nilpotence makes every negative-power substitution finite in the necessary direction. The induced vacuum action therefore exists. The candidate right side of (GQ.16) composes under substitution. Indeed, writing \(L_F=F''/F'\) gives \(L_{F\circ G}=G'(L_F\circ G)+L_G\); substituting into \(\operatorname{Sch}(F)=L_F'-L_F^2/2\) gives the Schwarzian chain rule. For a scaling \(\phi=at\), both actions send \(Q_r\) to \(a^{-r-2}Q_r\). For a nilpotent translation \(\phi=t+b\), both actions are the finite exponential \(\exp(bD_{-1})\) on each coefficient. Finally, a pointed coordinate with linear term one is a successive composition of \(\exp(c_nt^{n+1}\partial_t)\), \(n\geq1\): choose \(c_n\) to remove the first remaining coefficient, then proceed to the next degree. Every fixed coordinate coefficient is settled after finitely many steps. On each finite-energy state, \(D_n\) lowers energy by \(n\), so the operator exponentials and the necessary composition are also finite. Their infinitesimal actions agree by (GQ.15); termwise differentiation of the finite exponential proves equality. Scaling, translation, and these pointed coordinates give every \(\phi\), proving (GQ.16) over \(R\), without an inference from field-valued points. **Projective connections and the separate simple factors.** Our projective-connection coefficient is \(q\) in the scalar operator \(\partial_t^2-q(t)\). Its direct coordinate law and its induced law on coefficient functions are respectively \[ P_\phi(q)=\phi'^2q\circ\phi-\tfrac12\operatorname{Sch}(\phi), \qquad \delta_fq=-fq'-2f'q+\tfrac12f'''. \tag{GQ.17} \] Thus, at critical level, the exact quadratic-sector assignment is \[ Q_r\longmapsto 2\gamma_Bq_r. \tag{GQ.18} \] Whenever \(\gamma_B\ne0\), the normalized endomorphisms \(\widehat q_r=Q_r/(2\gamma_B)\) have precisely the projective-connection law. This assertion identifies the normalization and equivariance of the quadratic coefficient; it does not identify the full oper algebra or prove exhaustion of the affine centre. To state this safely for an arbitrary invariant \(B\), decompose \(\mathfrak g=\bigoplus_s\mathfrak g_s\) into split simple ideals. Cross terms of every invariant form vanish: each simple ideal is perfect, and \(B([x,y],z)=B(x,[y,z])=0\) when \(z\) lies in another ideal. The forms therefore restrict to each factor, and \(Q=\sum_sQ^{(s)}\). The map \(C_B\) commutes with the adjoint action. On an absolutely simple factor it is scalar: over an algebraic closure an eigenspace kernel is a nonzero adjoint-stable ideal, hence the whole factor; comparison of matrix entries descends the scalar to \(k\). Nondegeneracy of the Killing form on a semisimple characteristic-zero algebra, and the split-simple decomposition used here, are the retained structural Lie-algebra foundations. With \(C_B|_{\mathfrak g_s}=c_s\operatorname{id}\), its scalar \(c_s\) is nonzero, and \[ \gamma_s=\frac{c_s\dim\mathfrak g_s}{24}\ne0, \qquad Q_r^{(s)}\longmapsto2\gamma_sq_r^{(s)}. \tag{GQ.19} \] Each of these scalars is a unit after extension to any ordinary \(k\)-algebra. There is no such guarantee for \(\gamma_B=\sum_s\gamma_s\): taking two equal simple factors with opposite Killing forms makes it zero. In that case the aggregate \(\mathcal Q\) transforms as a quadratic differential; one must retain and normalize the separate factors. The canonical choice \(B=\mathrm{Kil}\) gives \(C_B=\operatorname{id}\) and \(\gamma_B=\dim\mathfrak g/24\) for a nonzero semisimple algebra. The following square displays the mechanism with the exact scalar retained: \[ \begin{array}{ccc} \mathcal Q^{(s)}(t)&\longmapsto&2\gamma_sq^{(s)}(t)\\[2pt] D_f\downarrow&&\downarrow\,\delta_f\\[2pt] -f(\mathcal Q^{(s)})'-2f'\mathcal Q^{(s)}+\gamma_sf'''v &\longmapsto& 2\gamma_s\bigl(-f(q^{(s)})'-2f'q^{(s)}+\tfrac12f'''\bigr). \end{array} \] The top arrow is (GQ.19), the left arrow is (GQ.15), and the right arrow differentiates the displayed multiple using (GQ.17). Current invariance comes from the cancellation in (GQ.7), whereas the displayed coordinate scalar remains. Formula (GQ.16) is the full inverse-coordinate version of this square. **The trace normalization for \(\mathfrak{sl}_n\).** Take \(n\geq2\) and \(B(X,Y)=\operatorname{tr}(XY)\). The trace pairing is nondegenerate on \(\mathfrak{sl}_n\), since \(M_n=kI\oplus\mathfrak{sl}_n\) is an orthogonal decomposition and \(n\) is a unit. Write \(\operatorname{ad}X=L_X-R_X\) on \(M_n\). A matrix-unit calculation gives \[ \operatorname{tr}(L_XL_Y)=\operatorname{tr}(R_XR_Y)=n\operatorname{tr}(XY), \qquad \operatorname{tr}(L_XR_Y)=\operatorname{tr}(X)\operatorname{tr}(Y). \] For the last equality, the diagonal coefficient on \(E_{ij}\) is \(X_{ii}Y_{jj}\); summing proves it. The scalar line has zero adjoint action. On traceless matrices this yields \[ \mathrm{Kil}(X,Y)=2n\operatorname{tr}(XY),\quad C_B=2n\operatorname{id},\quad \kappa_{\mathrm{crit}}=-n\operatorname{tr},\quad \gamma_B=\frac{n(n^2-1)}{12}. \tag{GQ.20} \] In particular \(D_2Q=n(n^2-1)v/2\). We also derive the sign relating this \(Q\) to the raw coefficient of the column determinant. This deduction uses only the determinant ordering of SL.L2, not an unproved determinant-invariance theorem. Let \(a_{ij}=E_{ij}[-1]\), impose the trace-free quotient, and use \[ \operatorname{cdet}(\delta_{ij}\tau+a_{ij}) =\tau^n+\mathbb S_1\tau^{n-1}+\mathbb S_2\tau^{n-2}+\cdots, \qquad [\tau,E_{ij}[-1]]=E_{ij}[-2], \] where column factors are multiplied in increasing column order and \(\tau\) is moved to the right. Put \(d_j=E_{jj}[-1]\) and \(d_j'=E_{jj}[-2]\). The coefficient state is \[ \mathbb S_2v= \left(\sum_{i