Proof. The calculation also holds on \(C_c^\infty(\mathbb R^{27};\mathcal F)\) before taking gauge invariants. Put \[y^a_{ij}=f_{abc}x_i^b x_j^c,\qquad A_\alpha=-i\gamma^i_{\alpha\beta}\theta_\beta^a\partial_{ia}, \qquad W_\alpha=\sum_{j<k,c}y^c_{jk}\gamma^{jk}_{\alpha\delta}\theta_\delta^c.\] The antisymmetry in \(j,k\) makes \(Q_\alpha=A_\alpha+W_\alpha\). The gamma relations give \[\operatorname{tr}((\gamma^i)^T\gamma^j)=16\delta_{ij},\qquad \operatorname{tr}((\gamma^{ij})^T\gamma^{kl}) =16\delta_{ik}\delta_{jl}\quad(i<j,\ k<l).\] Indeed, each indicated matrix has squared Hilbert–Schmidt norm \(16\), and a product of two or four distinct gamma matrices has trace zero: conjugation by one of its factors changes its sign. Clifford anticommutation, followed by these identities, gives on the smooth core \[\tfrac1{16}\sum_\alpha A_\alpha^2=-\tfrac12\Delta,\qquad \tfrac1{16}\sum_\alpha W_\alpha^2 =\tfrac12\sum_{a,j<k}(y^a_{jk})^2I.\] For the latter identity, the coefficients \(y^a_{jk}\) commute, so the two orders of each Clifford product combine into an anticommutator. Since \(f_{abc}=\sqrt2\,\varepsilon_{abc}\), the last scalar is \(\sum_{a<b}|x^a\wedge x^b|^2\).

In the cross term, the first-order part is \[-\tfrac i{16}\sum_{i,a,j<k} y^a_{jk}\operatorname{tr}((\gamma^i)^T\gamma^{jk})\partial_{ia}=0,\] because \(\gamma^i\) is symmetric and \(\gamma^{jk}\) is skew-symmetric. The remaining multiplication operator is \[-\tfrac i{16}\sum_{\substack{i,a,\beta,\delta\\j<k,c}} (\partial_{ia}y^c_{jk}) (\gamma^i\gamma^{jk})_{\beta\delta} \theta_\beta^a\theta_\delta^c.\] Here \[\partial_{ia}y^c_{jk} =f_{cab}\delta_{ij}x_k^b+f_{cba}x_j^b\delta_{ik}, \qquad \gamma^j\gamma^{jk}=\gamma^k,\quad \gamma^k\gamma^{jk}=-\gamma^j.\] Each spatial index occurs in eight pairs \(j<k\). Substitution gives \[-\tfrac i2 f_{cab}x_i^b\gamma^i_{\beta\delta} \theta_\beta^a\theta_\delta^c =\tfrac i2 f_{abc}x_i^a\gamma^i_{\beta\delta} \theta_\beta^b\theta_\delta^c,\] where the last equality uses antisymmetry of \(f\) and relabels the indices. Integration by parts now proves (2) on \(\mathcal C\).

Whenever \(f_{abc}\ne0\), the generators \(\theta_\beta^b\) and \(\theta_\delta^c\) anticommute. Their product is anti-Hermitian, so the displayed \(B\) is Hermitian. It is a linear function of \(x\) with coefficients in the finite-dimensional space \(\operatorname{End}(\mathcal F)\). This proves (4). ◻

Proof. The action \(S_g\) on each of the three color copies gives a homomorphism \(G\to\operatorname{SO}(48)\). Since \(G\) is simply connected, it lifts to \(\operatorname{Spin}(48)\). Acting on \(\mathcal F\) by the normalized Clifford generators \(\sqrt2\,\theta_\alpha^a\) gives \(U\) with the stated property.

The actions of \(G\) and \(\operatorname{SU}(2)\) on the generator labels commute. Their unitary commutator on \(\mathcal F\) therefore commutes with every Clifford generator and is scalar by irreducibility. For fixed \(g\in G\) this scalar, as a function of the gauge transformation, is a continuous character of \(\operatorname{SU}(2)\). Such a character is trivial. Thus the actions commute on \(\mathcal F\), and (6) preserves \(\mathcal H\).

The pullback in (6) sends the coordinate and derivative vectors to their \(R_g^{-1}\) transforms. The column of Clifford generators transforms by \(S_g^{-1}\). Applying (5), and its consequence for \(\gamma^{ij}\), to the two terms in (1) gives \[D_g Q_\alpha D_g^{-1} =(S_g^{-1})_{\alpha\beta}Q_\beta \quad\hbox{on }\mathcal C.\] The orthogonality of \(S_g\) proves \(q(D_g\Psi)=q(\Psi)\). Rotations preserve smoothness, and the rotated supports of a compact set lie in a common ball, so \(D_g\mathcal C=\mathcal C\).

The action is strongly continuous for the form norm \(\lVert \Psi\rVert_q^2=\lVert \Psi\rVert_2^2+q(\Psi)\). On \(\mathcal C\) this follows directly from smoothness, (1), and compact support; it extends to the closure because every \(D_g\) is a form-norm isometry. For normalized Haar measure and the irreducible character \(\chi_\lambda\), \[P_\lambda =(\dim V_\lambda)\int_G\overline{\chi_\lambda(g)}D_g\,dg.\] This integral is bounded in form norm. Differentiation under the integral and the common support bound show that it maps \(\mathcal C\) into \(\mathcal C\). If \(\Psi\in\operatorname{Dom}(q)\cap\mathcal H_\lambda\), project any core approximation to \(\Psi\) to obtain a form-norm approximation in \(P_\lambda\mathcal C\). This proves the core assertion.

Invariance of the closed form implies that its resolvent commutes with every \(D_g\): this follows, for example, from uniqueness of the solution of \[q(u,v)+\langle u,v\rangle=\langle f,v\rangle\qquad(v\in\operatorname{Dom}(q))\] defining \(u=(H+1)^{-1}f\), where \(q(u,v)\) denotes the polarization of \(q\). The resolvent therefore commutes with \(P_\lambda\). This proves reduction and identifies the restricted operator with the stated part of \(H\). ◻

Proof. This is the part of the classical orthogonal interlacing rule needed here; compare [3]. We give a character proof that includes the half-integral cases.

Use formal torus monomials \(Z^w\), allowing half-integral exponents, and write \(\Delta_G,\Delta_K\) for the Weyl denominators. The half-sums of positive roots are \[\rho_K=(3,2,1,0),\qquad \rho_G=\rho_K+(\tfrac12,\tfrac12,\tfrac12,\tfrac12).\] Thus \(\Delta_G=\Delta_K\prod_i(Z_i^{1/2}-Z_i^{-1/2})\). Set \(L_j=\lambda_j+4-j\). The Weyl character formula for type \(B_4\) gives \[ \chi_\lambda\Delta_K =\det\left( \frac{Z_i^{L_j+1/2}-Z_i^{-L_j-1/2}} {Z_i^{1/2}-Z_i^{-1/2}} \right)_{i,j=1}^4 =\det\left(\sum_{s=-L_j}^{L_j}Z_i^s\right)_{i,j=1}^4, \tag{10}\] where \(s\) increases in steps of one.

Let \(m_\eta\) be the multiplicity of the \(K\) type \(\eta\) in the restriction, and let \(W_K\) be the Weyl group of \(K\). The Weyl character formula for \(K\) gives the second expression \[\chi_\lambda\Delta_K =\sum_\eta m_\eta\sum_{w\in W_K}\det(w)\, Z^{w(\eta+\rho_K)}.\] Put \(t=\nu+\rho_K\). Its coordinates satisfy \(t_1>t_2>t_3>|t_4|\), so \(t\) is regular dominant for type \(D_4\), including when \(t_4=0\). The coefficient of \(Z^t\) in this expression is therefore \(m_\nu\): only \(\eta=\nu\) and the identity Weyl element contribute. In the determinant in (10), row \(i\) involves only \(Z_i\), so coefficient extraction acts row by row. If the coordinate lattices of \(\lambda\) and \(\nu\) differ, the coefficient is zero. Otherwise it is \[\det\bigl(\mathbf1_{\{|t_i|\leq L_j\}}\bigr)_{i,j=1}^4.\] The \(L_j\) strictly decrease, and so do the \(|t_i|\). Each row is an initial string of ones followed by zeros, and the string lengths are nondecreasing down the rows. A nonzero determinant has neither a zero row nor two equal rows, so these four lengths must be \(1,2,3,4\). In particular, \(t_1>L_2\), or \(\nu_1>\lambda_2-1\). The difference \(\nu_1-\lambda_2\) is integral because the lattices agree. Hence \(\nu_1\geq\lambda_2\). ◻

Proof. To restrict at fixed parameters, we first use the finite-dimensional span of the smooth translates of \(\Psi\). The isotypic representation has the form \(\mathcal H_\lambda\simeq V_\lambda\widehat\otimes\mathcal M_\lambda\), where \(\mathcal M_\lambda\) is a multiplicity Hilbert space. Writing \(d=\dim V_\lambda\), any individual vector has an expansion \(\sum_{j=1}^d e_j\otimes m_j\). Its span under \(G\) is contained in \(V_\lambda\otimes\operatorname{span}\{m_1,\ldots,m_d\}\) and is therefore finite dimensional. Apply this to \(\Psi\), and denote its translate span by \(W_\Psi\).

Every vector in \(W_\Psi\) has a smooth compactly supported representative, because it is a finite linear combination of translates of \(\Psi\). An equality between these representatives in \(L^2\) holds pointwise by continuity. Thus restriction at the fixed \(u,s_b\) is a well-defined linear map from \(W_\Psi\) to \(C_c^\infty((u^\perp)^2;\mathcal F)\). It intertwines \(K_u\), which fixes \(u\) and the longitudinal coordinates \(s_b\). Its image can have only types occurring in \(W_\Psi|_{K_u}\). Lemma 4 gives the claimed bound. The \(K_u\) action on the fiber \(L^2\) space is unitary, so distinct isotypic subspaces there are orthogonal. ◻

Proof. For \(r=|u|>0\), the oscillator \[h_r=-\tfrac14\Delta_z+r^2|z|^2\] has energies \(r(n+8)\), \(n=0,1,\ldots\). Indeed, the change of variables \(\xi=\sqrt{2r}\,z\) gives \(h_r=\frac r2(-\Delta_\xi+|\xi|^2)\) in sixteen dimensions. Each vector at level \(n\) is an invariant Gaussian times a polynomial of total degree at most \(n\). Tensor this level with \(\mathcal F\).

The transverse coordinates are two copies of the vector representation of \(K_u\), whose weights are \(\pm\epsilon_1,\ldots,\pm\epsilon_4\). Every weight in a polynomial of degree at most \(n\) has first coordinate at most \(n\), and every weight in the level tensored with \(\mathcal F\) consequently has first coordinate at most \(n+M\). In particular, the first coordinate of the highest weight of any constituent is at most \(n+M\).

Let \(\Pi_n\) be the projection onto level \(n\) tensored with \(\mathcal F\). It commutes with \(K_u\) and hence with every \(K_u\)-isotypic projection on the fiber \(L^2\) space. By Lemma 5, \(\Pi_n\Phi\) can therefore contain only types with \(\nu_1\geq\lambda_2\). Since it lies in level \(n\) tensored with \(\mathcal F\), its types also satisfy \(\nu_1\leq n+M\). For \(n\leq m\) these bounds are incompatible by (11), so \(\Pi_n\Phi=0\). Expanding in the complete Hermite basis, the least remaining energy is \(r(m+9)\), which proves (12). ◻

Proof. First take \(\Psi\in\mathcal C\cap\mathcal H_\lambda\) and fix a color \(a\). For fixed \(u\), the decomposition of the two other color variables into \(s_b,z_b\) is an orthogonal linear change of variables with unit Jacobian. The transverse gradient uses only derivatives in those two variables, so \(|\nabla_z\Phi|^2\leq|\nabla\Psi|^2\). No derivative of the decomposition with respect to \(u\) is taken. Fubini’s theorem, using local orthogonal frames for \(u^\perp\), therefore applies to (12); the set \(u=0\) has measure zero. Together with (8), it gives \[ \tfrac14\lVert \nabla\Psi\rVert_2^2+ \int V(x)\lVert \Psi(x)\rVert^2\,dx \ \geq\ (m+9)\int |x^a|\lVert \Psi(x)\rVert^2\,dx. \tag{14}\] Average (14) over the three colors; its left side is unchanged. Splitting the kinetic term in (2) into two quarters and using (4) then gives \[q(\Psi)\geq\tfrac14\lVert \nabla\Psi\rVert_2^2+ \int_{\mathbb R^{27}}\left(\tfrac{m+9}{3}\sum_a|x^a|-C|x|\right) \lVert \Psi(x)\rVert^2\,dx.\] Since \(\sum_a|x^a|\geq|x|\), this proves (13) on the core.

We now pass to the restricted form domain. By Lemma 3, any vector in that domain is a form-norm limit of vectors in \(\mathcal C\cap\mathcal H_\lambda\). Applied to differences of these vectors, (13) shows convergence of their gradients in \(L^2\) and of their products with \(|x|^{1/2}\) in \(L^2\). The gradient and multiplication by \(|x|^{1/2}\) are closed operators. Their limits are therefore the gradient and weighted product of the form limit. Passing to the limit proves (13) on the full restricted form domain.

Its form-norm unit ball is bounded in \(H^1(\mathbb R^{27};\mathcal F)\). Moreover, for \(R>0\), (13) gives the uniform tail estimate \[\int_{|x|>R}\lVert \Psi(x)\rVert^2\,dx \leq\frac{q(\Psi)}{c_m R}.\] Rellich compactness on bounded balls applies componentwise in the finite-dimensional space \(\mathcal F\). A diagonal subsequence and the tail estimate give compactness in \(L^2(\mathbb R^{27};\mathcal F)\). Hence the inclusion of the restricted form domain into \(\mathcal H_\lambda\) is compact. The resolvent \((H_\lambda+1)^{-1}\) maps the Hilbert space boundedly into that form domain, so it is compact.

If \(\Psi\in\ker H_\lambda\), then \(q(\Psi)=0\). The positive weighted term in (13) vanishes, forcing \(\Psi=0\) because \(|x|>0\) almost everywhere. This proves the kernel assertion. ◻

Proof. For \(1\leq j\leq8\) and \(1\leq a\leq3\), put \[c_j^a=\frac{\theta_{2j-1}^a+i\theta_{2j}^a}{\sqrt2}.\] These twenty-four operators and their adjoints satisfy the canonical anticommutation relations \(\{c_j^a,c_k^b\}=0\) and \(\{c_j^a,(c_k^b)^*\}=\delta_{jk}\delta_{ab}\). The occupation operators \(N_j^a=(c_j^a)^*c_j^a\) are commuting projections. Start with a nonzero joint eigenvector. Applying an occupied mode’s annihilator removes that occupation and preserves all others. The result is nonzero, since \(\lVert c_j^a v\rVert^2=\langle v,N_j^a v\rangle=\lVert v\rVert^2\). Repeating this gives a nonzero vector annihilated by every \(c_j^a\). For each subset of modes, apply its creation operators in a fixed order. The resulting vectors are nonzero and have distinct occupation patterns, hence are orthogonal. Their span is invariant under every creation and annihilation operator, so irreducibility makes it all of \(\mathcal F\). The common kernel of the annihilators is therefore exactly the original line.

Gauge transformations mix the color indices and preserve the span of the annihilators for each \(j\). They therefore preserve the common kernel line. Their action on that line is a continuous character of \(\operatorname{SU}(2)\) and is trivial. Thus \(\mathcal F^{\operatorname{SU}(2)}\ne0\). Its \(G\) invariance follows from the commutation in Lemma 3. ◻

Proof. Choose a nonzero highest vector \(v\) of some irreducible \(G\) type \(\mu\) in the finite-dimensional representation \(\mathcal F^{\operatorname{SU}(2)}\).

We construct a gauge-invariant polynomial that raises the first two highest-weight coordinates. In the complexified vector representation, set \[w_1=e_1+i e_2,\qquad w_2=e_3+i e_4,\] choosing the torus convention in which their weights are \(\epsilon_1,\epsilon_2\). Use the complex bilinear extension of the Euclidean dot product, and put \[\zeta_a=w_1\cdot x^a,\qquad \eta_a=w_2\cdot x^a,\qquad P(x)=\sum_{a<b}(\zeta_a\eta_b-\zeta_b\eta_a)^2.\] The identity \[P=(\zeta\cdot\zeta)(\eta\cdot\eta)-(\zeta\cdot\eta)^2\] uses bilinear dot products on the color space \(\mathbb C^3\). The real orthogonal adjoint action of \(\operatorname{SU}(2)\) preserves these products, so \(P\) is gauge invariant. At \(x=t(e_1,e_3,0)\) with \(t>0\), one has \(P(x)=t^4\) and \(|x|=\sqrt2\,t\). Thus \(P\) is nonzero on every nonempty radial shell.

For the action on spatial polynomials by pullback, \[(w\cdot R_g^{-1}x)=(R_gw)\cdot x.\] Thus \(w\mapsto(w\cdot x^a)\) intertwines the complexified vector representation with linear polynomials, with the same weight sign. The analogous map on exterior squares sends \(w_1\wedge w_2\) to \(\zeta_a\eta_b-\zeta_b\eta_a\). The bivector \(w_1\wedge w_2\) is highest of weight \(\epsilon_1+\epsilon_2\): a positive-root operator either kills both factors or sends \(w_2\) to a multiple of \(w_1\), and in the latter case its action on the wedge also vanishes. Hence each minor is highest of that weight. Root operators act as derivations on polynomials, so \(P\) is highest of weight \(2(\epsilon_1+\epsilon_2)\). Its nonzero power \(P^k\) is highest of weight \(2k(\epsilon_1+\epsilon_2)\).

Let \(\chi\in C_c^\infty((0,\infty))\) be nonzero. The function \[\Psi_\chi(x)=\chi(|x|)P(x)^k v\] is in \(\mathcal C\): it is smooth because its support avoids the origin, and its factors are gauge invariant. The preceding ray shows that \(\Psi_\chi\ne0\). Multiplication by the radial function is \(G\) equivariant. The translates of \(\Psi_\chi\) lie in the finite-dimensional representation obtained from homogeneous polynomials of degree \(4k\) tensored with \(\mathcal F\). The vector is highest of the dominant weight \[\lambda=2k(\epsilon_1+\epsilon_2)+\mu =(2k+\mu_1,\,2k+\mu_2,\,\mu_3,\,\mu_4).\] In a decomposition of that finite representation into irreducibles, each nonzero component of this vector is a weight-\(\lambda\) vector killed by all positive-root operators. It must be a highest vector, so its summand has type \(\lambda\). Therefore \(\Psi_\chi\in\mathcal H_\lambda\).

Choosing \(\chi\) with pairwise disjoint radial supports gives infinitely many linearly independent vectors in the same sector. Finally, \(\lambda_2=2k+\mu_2\) tends to infinity with \(k\). ◻

Proof of Theorem 1. Choose an integer \(m\geq0\) with \(c_m=(m+9)/3-C>0\). Lemma 9 supplies a type \(\lambda\) such that \(\lambda_2>m+M\) and \(\mathcal H_\lambda\) is infinite dimensional. By Proposition 7, \(H_\lambda\) has compact resolvent and no zero eigenvector.

Since \(H_\lambda\) is nonnegative with compact resolvent on an infinite-dimensional Hilbert space, the compact-resolvent spectral theorem gives an orthonormal eigenbasis \((\Psi_j)\) with eigenvalues \(E_j\to\infty\). Its trivial kernel makes every \(E_j\) positive. By Lemma 3, \(H_\lambda\) is the part of \(H\) on a reducing subspace. These vectors therefore belong to \(\operatorname{Dom}(H)\) and satisfy \(H\Psi_j=E_j\Psi_j\) in the full physical Hilbert space. ◻