Proof. The weight-plane relations give \[R(u)=\sigma(-2E_{12}+E_{45}+E_{67}+E_{89}).\] Off-torus coefficients vanish by orthogonality of distinct weight planes. On \(G\)-invariant vectors the generator of any coefficient vector with zero sum is zero. Thus, putting \(R_0=\sigma(E_{45}+E_{67}+E_{89})\), \(J(R(u))=J(R_0)/3\). Average \(R(gu)\) over \(g\in\mathop{\mathrm{Spin}}(3)\). The first block has average zero, and the last-six block stays fixed, so the average is \(R_0\). Proposition 2 now gives on the invariant core \[\int_{\mathop{\mathrm{Spin}}(3)}\left\lVert D_{gu}(h,m)\psi\right\rVert^2\,\mathrm dg =\mathcal H(h,m)[\psi]+m\left\langle \psi,J(R_0)\psi\right\rangle =3\left\lVert D_u(h,m)\psi\right\rVert^2-2\mathcal H(h,m)[\psi].\] Positivity proves the inequality. The domain argument below extends it to the closed graph domain of the restricted charge. ◻

Proof. Write \(D=D_0+V(x)\), where \(V\) is a smooth self-adjoint matrix and \(D_0\) has constant coefficients. Its symbol satisfies \[ \sigma_D(\xi)^2=\frac12\left\lvert \xi\right\rvert^2\mathrm{Id}, \qquad [D,\chi]=-\mathrm i\sum_{i,A}\theta^A(A_i(v))\partial_{i,A}\chi, \qquad \left\lVert [D,\chi](x)\right\rVert_{\mathrm{op}}=\frac{\left\lvert \nabla\chi(x)\right\rvert}{\sqrt2} \tag{12}\] for real scalar \(\chi\). If \(D\psi\in L^2\) distributionally, localizing in a compact set makes \(V\psi\) square integrable. The Fourier identity for \(D_0\), with a further compact cutoff, shows that \(\psi\in H^1_{\mathrm{loc}}\).

Take radial cutoffs \(\chi_R\), equal to one on \(\left\lvert x\right\rvert\leq R\) and zero on \(\left\lvert x\right\rvert\geq2R\), with \(\left\lvert \nabla\chi_R\right\rvert\leq C/R\). Then \[D(\chi_R\psi)=\chi_RD\psi+[D,\chi_R]\psi \longrightarrow D\psi\] in \(L^2\), and \(\chi_R\psi\to\psi\) in \(L^2\). Each compactly supported \(H^1\) vector can in turn be approximated in the graph norm by convolution and a fixed outer cutoff; on that compact set the potential is bounded and smooth. Thus the minimal domain equals the maximal domain. The adjoint of the symmetric core operator has precisely the distributional maximal domain, proving self-adjointness. Averaging these approximants over a commuting compact group preserves smooth compact support, and converges in graph norm because its action commutes with \(D\).

At \(m=0\), Proposition 2 makes the sixteen core graph norms, and that of any unit \(v\), identical. Their common completion is therefore both the closed single-charge graph domain and the form domain of \(\mathcal H(h,0)\). The associated nonnegative operator has zero form precisely when every charge vanishes. This proves the kernel statements. The same core approximation and Lemma 3 imply that the restricted massive graph domain is included in the averaged form domain and that (11) holds there. ◻

Proof. Orthogonality of gamma monomials gives \[\mathop{\mathrm{Tr}}(\gamma^jP_i(F))=F_{ji},\qquad \mathop{\mathrm{Tr}}(\gamma^{jk}P_i(F))=0,\qquad \sum_i\gamma^iP_i(F)^t=-\frac12S(F).\] For the last identity, the rank-one term contributes \(-S(F)/4\). For fixed distinct \(j,k\), exactly seven indices \(i\) remain in the rank-three term, and \(\gamma^i\gamma^{ijk}=\gamma^{jk}\); its transpose has a minus sign. Hence this term contributes another \(-S(F)/4\). The first trace identity yields the orbital term in the anticommutator. The second kills its bracket multiplier. The differentiated multiplier is \(-\mathrm i\theta^t(\sum_i\gamma^iP_i(F)^t)\theta\), which is exactly the spin term of \(J(F)\). ◻

Proof. Let \(\mathcal Z\) be the common massless kernel furnished by Lemma 4. Covariance preserves \(\mathcal Z\). Convolving its vectors with smooth approximate identities on the compact group \(\mathop{\mathrm{Spin}}(9)\) gives a dense invariant subspace \(\mathcal Z^\infty\) contained in the domains of all rotation generators. For \(\phi,\psi\in\mathcal Z^\infty\), use the radial cutoffs from Lemma 4 in (14). Local regularity implies that \(\chi_RM_v(F)\psi\) is in the charge domain. Pairing the anticommutator with \(\phi\), and using that both vectors are killed by every charge, gives \[0=\left\langle \phi,\chi_RJ(F)\psi\right\rangle +\sum_\nu\left\langle \phi,[D_{v_\nu}(h,0),\chi_R]M_{v_\nu}(F)\psi\right\rangle.\] The multiplier in the last term has operator norm bounded by \(C_F\left\lvert x\right\rvert\left\lvert \nabla\chi_R\right\rvert\), and is supported in \(R\leq\left\lvert x\right\rvert\leq2R\). Therefore that term is bounded in absolute value by \[C_F\left\lVert \mathbf 1_{\{\left\lvert x\right\rvert\geq R\}}\phi\right\rVert \left\lVert \mathbf 1_{\{\left\lvert x\right\rvert\geq R\}}\psi\right\rVert\longrightarrow0.\] Since \(J(F)\psi\in L^2\), the first term converges to \(\left\langle \phi,J(F)\psi\right\rangle\). The generator also preserves the closed invariant subspace \(\mathcal Z\), so the vanishing of these matrix elements for its dense smooth subspace implies \(J(F)\psi=0\). All infinitesimal generators consequently vanish on the smooth vectors of this unitary representation. Connectedness of \(\mathop{\mathrm{Spin}}(9)\), followed by density, makes the representation trivial on \(\mathcal Z\). The central minus one of \(\mathop{\mathrm{Spin}}(3)\) is exterior parity, as established above; thus every vector of \(\mathcal Z\) is even. The proof has used only escaping-annulus \(L^2\) tails of the uncut states. This is the rotational-singlet argument of Sethi–Stern (Sethi and Stern 2000), in the explicit anticommutator form of Hasler–Hoppe (Hasler and Hoppe 2002b). The underlying cutoff principle also appears in Hitchin’s invariant \(L^2\)-cohomology argument (Hitchin 2000, Theorem 3). ◻

Proof. The scaling identity follows term by term from (7). To prove the lower bound, suppose otherwise and choose \(x^{(n)}\) with \(r_n=\left\lvert x^{(n)}\right\rvert\to\infty\) and \(U_{1,1}(x^{(n)})/r_n^2\to0\). After passing to a subsequence, \(x_i^{(n)}/r_n\to a_i\). The last-six mass terms give \(a_d=0\), while the first-three squares give \[ [x_a^{(n)},x_b^{(n)}]-2x_c^{(n)}=o(r_n) \quad\text{for cyclic }(a,b,c). \tag{18}\] Dividing by \(r_n^2\) shows that the \(a_a\) commute. Their joint eigenspaces give an orthogonal decomposition \(\mathbb C^N=\bigoplus_\alpha E_\alpha\), with distinct joint eigenvalue triples \(\lambda_\alpha\in\mathbb R^3\). Choose \(t\in\mathbb R^3\) so that the numbers \(t\cdot\lambda_\alpha\) are distinct, and put \(y_n=\sum_a t_ax_a^{(n)}\).

The spectral subspaces of \(y_n\) in the corresponding clusters converge to the \(E_\alpha\). By unitaries tending to the identity we may identify these subspaces with fixed blocks, without changing the preceding limits. Their spectral gaps are at least \(c_0r_n\) for some \(c_0>0\). Equation (18) implies \(\left\lVert [y_n,x_a^{(n)}]\right\rVert=O(r_n)\). In bases diagonalizing the two relevant blocks of \(y_n\), an off-block matrix entry is multiplied in this commutator by a spectral difference of magnitude at least \(c_0r_n\). Hence \[ \left\lVert (x_a^{(n)})_{\alpha\beta}\right\rVert=O(1),\qquad\alpha\ne\beta. \tag{19}\] This is a finite-dimensional Hilbert–Schmidt estimate, uniform in the sequence.

The trace in a diagonal block of \([x_a^{(n)},x_b^{(n)}]\) has zero contribution from its internal commutator. Its remaining terms are products of the off-block entries in (19), so its block trace is \(O(1)\). Divide the block trace of (18) by \(r_n\). The result is \(2\dim(E_\alpha)(\lambda_\alpha)_c=0\) for every \(\alpha,c\). Thus every \(a_i\) vanishes. This contradicts \(\sum_i\left\lvert a_i\right\rvert^2=1\), proving (16). Grouping equal limiting joint eigenvalues into a single block was essential here; no gap was required inside such a block. ◻

Proof. The fermionic multiplier in (7), denoted below by \(V_F(h,m;x)\), acts on a fixed finite-dimensional space and satisfies \[ \left\lVert V_F(h,m;x)\right\rVert_{\mathrm{op}} \leq C_N(m+h\left\lvert x\right\rvert). \tag{22}\] Lemma 8 and scaling give, uniformly for \(h\in I\), \(U_{h,1}\geq c_I\left\lvert x\right\rvert^2-C_I\). Absorbing the linear term in (22) yields \[\mathcal H(h,1)[\psi]+C_I\left\lVert \psi\right\rVert^2 \geq \frac12\left\lVert \nabla\psi\right\rVert^2+c'_I\left\lVert \left\lvert x\right\rvert\psi\right\rVert^2.\] The identity for \(\mathcal H\) then also bounds \(\int U_{h,1}\left\lvert \psi\right\rvert^2\) by \(C_I(\mathcal H(h,1)[\psi]+\left\lVert \psi\right\rVert^2)\). Each bracket multiplier is controlled by this integral and \(\left\lVert \left\lvert x\right\rvert\psi\right\rVert\), because the first-three residuals are \(h[x_a,x_b]-2x_c\) and all the other brackets occur directly in the potential. On \(\mathscr H_G\), Lemma 3 consequently bounds (21) by \(C_I(\left\lVert D_h\psi\right\rVert+\left\lVert \psi\right\rVert)\). The reverse bound follows immediately from the formula for \(D_h\).

These statements first hold on the invariant core. Its graph completion equals the maximal domain by Lemma 4. Conversely a vector satisfying (20) has all terms of the charge in \(L^2\), so is in that maximal domain. This proves the common-domain assertion and norm equivalence. For a bounded set in this graph norm, Rellich compactness on balls gives local \(L^2\) compactness, whereas \[\left\lVert \mathbf 1_{\{\left\lvert x\right\rvert>R\}}\psi\right\rVert^2 \leq R^{-2}\left\lVert \left\lvert x\right\rvert\psi\right\rVert^2\] makes the tails uniformly small. The graph embedding is compact. Thus \(D_h\) has compact resolvent, finite-dimensional kernel, and closed range, and its odd part is Fredholm.

Writing \(D_h-D_{h_0}=(h-h_0)B\), the bracket bound shows that \(B:\mathcal D\to\mathscr H_G\) is bounded for the graph norm of \(D_{h_0}\). Thus \((D_h-D_{h_0})(D_{h_0}-\mathrm i)^{-1}\) tends to zero in operator norm. The resolvent identity and a Neumann series imply norm-resolvent continuity. For completeness, choose a symmetric interval \((-a,a)\) whose endpoints avoid the spectrum at \(h_0\). Its finite-rank spectral projection is norm-continuous for nearby \(h\), and preserves parity. The even and odd ranks of the projection are therefore locally constant. This is the standard stability of separated spectral projections; see (Kato 1995, IV, Section 3.4, Theorem 3.16). The graph estimates above and the parity argument here are specific to the present operators. On its nonzero spectral subspace, the odd operator \(D_h\) is an isomorphism between the even and odd parts, so their rank difference is exactly the graded kernel dimension. The index is locally constant, and hence constant on the connected interval \((0,\infty)\). ◻

Proof. Bosonic normal modes. Decompose \(\mathfrak k_\mathbb C\) into irreducible spin-\(l\) summands for \(\mathop{\mathrm{ad}}(j)\). Half-integral spins are allowed: maps between two different irreducible summands of the defining representation may carry such spins. The dimensions calculated in the complexification give the real dimensions of the original real operators after summing all their summands.

On the first three spatial components, the linearized residual in the potential is \[2\left(\sum_a\mathop{\mathrm{ad}}(j_a)L_a^{(1)}-\mathrm{Id}\right),\] where \(L_a^{(1)}\) generates cross product on \(\mathbb R^3\). Equation (24) gives the eigenvalues \(-l,1,l+1\) of \(\sum_a\mathop{\mathrm{ad}}(j_a)L_a^{(1)}\), on coupled spins \(l+1,l,l-1\), respectively, whenever the indicated representation exists. Taking the absolute values of the linearized residual gives normal frequencies \[ \begin{array}{c|c} \text{frequency}&\text{multiplicity}\\ \hline 2(l+1)&2l+3\\ 2l&\max(2l-1,0)\\ 2l+1&6(2l+1). \end{array} \tag{25}\] The last row comes from the last six components, whose squared frequency is \(1+4l(l+1)=(2l+1)^2\). For \(l>0\), the omitted zero-frequency spin-\(l\) summand is exactly the gauge tangent: \(w\mapsto([w,j_a])_a\) is an injective intertwiner from that color summand, its image is tangent to the zero orbit, and the dimensions agree. For \(l=0\), only the spin-one first-three summand exists and has frequency two; there is no gauge tangent. Thus all frequencies on \(S_q\) are positive. Their bosonic ground energy per color spin-\(l\) summand is \[ E_B(l)=\frac12\left((2l+2)(2l+3) +(2l)(2l-1)+6(2l+1)^2\right) =16l^2+16l+6. \tag{26}\] The middle term is omitted when its representation is absent; it is already zero for \(l=0\) and \(l=1/2\).

Fermionic energy and the ground line. The real skew spin-one-half matrices \(L_a^{(1/2)}=-T\gamma^a/2\) satisfy the same \(\mathfrak{su}(2)\) relations, and the fermion frequency matrix is \[M_q=\frac32T-2\mathop{\mathrm{ad}}(j_a)\gamma^a =T\left(\frac32-4\sum_a\mathop{\mathrm{ad}}(j_a)L_a^{(1/2)}\right).\] The coupled spins \(l+1/2\) and \(l-1/2\) have scalar factors \(2l+3/2\) and \(-2l-1/2\), respectively, multiplying \(T\). Since the complexified real sixteen-dimensional spinor has eight copies of spin one-half, the absolute frequencies and their complexified multiplicities are \[ \begin{array}{c|c} \text{absolute frequency}&\text{complexified multiplicity}\\ \hline 2l+3/2&8(2l+2)\\ 2l+1/2&8(2l). \end{array} \tag{27}\] There are no zero fermion frequencies. A real two-plane with skew frequency \(\omega\) contributes eigenvalues \(\pm\omega/2\) to \(\mathrm i\theta^tM_q\theta/2\), by the CAR normalization. The fermionic ground energy is consequently \(-\mathop{\mathrm{Tr}}\left\lvert M_q\right\rvert/4\). From (27) this is \(-E_B(l)\), with \(E_B(l)\) as in (26). Every bosonic oscillator and every fermionic two-plane has a unique ground vector, so their product is the unique zero-energy vector and has a positive gap above it.

Stabilizer action and parity. The unique ground line contributes to the invariant index only if it has trivial stabilizer character. We verify that condition in the fixed exterior realization, which also determines its parity. Decompose the defining representation as \[\mathbb C^N=\bigoplus_r(V_{j_r}\otimes M_r),\qquad H_q=\prod_r\mathop{\mathrm{U}}(M_r),\qquad m_r=\dim M_r,\] where the \(j_r\) are distinct and \(V_j\) denotes spin \(j\). The empty vector in the \(T\)-polarized Fock space is gauge invariant. The one-particle spin representation under \(\mathop{\mathrm{Spin}}(3)\times\mathop{\mathrm{Spin}}(6)\) is \(\mathbb C^2\otimes\mathbb C^4\). Relative to this polarization, the fermionic ground vector fills precisely the negative-factor space of \(M_q/T\). On a color spin-\(l\) summand this is \(V_{l-1/2}\otimes\mathbb C^4\), of complex dimension \[ q_l=4(2l)=8l\qquad(l>0). \tag{28}\] Notice the distinction from the doubled multiplicities of the complexified real frequency matrix in (27).

The color summand from maps between defining isotypic components is \[\operatorname{Hom}(V_{j_r},V_{j_s}) \otimes\operatorname{Hom}(M_r,M_s),\] and its spin list is \(l=\left\lvert j_r-j_s\right\rvert,\ldots,j_r+j_s\), once each. Thus the determinant character of its filled space is the product over that list, omitting \(l=0\), of \[ \left((\det g_s)^{m_r}(\det g_r)^{-m_s}\right)^{q_l}. \tag{29}\] The reverse pair \((s,r)\) has the identical spin list and the inverse character, so the factors cancel. For \(r=s\) the displayed character is already trivial. Removing the scalar color from \(\operatorname{End}(\mathbb C^N)\) removes only an \(l=0\) summand and does not affect this argument. Hence the determinant of the full filled space is the trivial character of \(H_q\). The bosonic ground Gaussian is also \(H_q\)-invariant. The entire ground line is therefore fixed by the stabilizer, including its permutation matrices exchanging identical summands. No ordering sign remains. Moreover every \(q_l\) is a multiple of four, including when \(l\) is half-integral. The occupied degree is even, proving the parity assertion.

Spatial rotations. First-three rotations of \(q\) can be compensated by the \(\mathop{\mathrm{SU}}(2)\) representation defined by \(j\). Their combined action, and the action of \(\mathop{\mathrm{Spin}}(6)\), preserve \(S_q\) and commute with \(H_q^{\mathrm{nor}}\). Each acts on its unique ground line by a continuous character of a connected semisimple group. Such a character is trivial: its derivative annihilates the commutator Lie algebra, which is the entire Lie algebra, and connectedness then annihilates the character. This proves all the claimed symmetries. ◻

Proof. The positive normal Hessians in Lemma 10, and the finitely many compact orbits, imply \(U_{1,1}\geq c\mathop{\mathrm{dist}}(\cdot,Z)^2\) in a fixed neighborhood of \(Z\). Indeed normal tubular coordinates have positive quadratic Taylor term and uniformly cubic Taylor remainder on these compact orbits. On a compact set outside that neighborhood the same inequality follows from the positive minimum of \(U_{1,1}/\mathop{\mathrm{dist}}(\cdot,Z)^2\). At infinity it follows from Lemma 8, since \(Z\) is bounded. Scaling then gives its first assertion for all \(x\).

Let \(M=\max_{q\in Z}\left\lvert q\right\rvert\). A nearest point of \(h^{-1}Z\) gives \(h\left\lvert x\right\rvert\leq hd_h(x)+M\). Equation (22) therefore implies \[U_{h,1}(x)+V_F(h,1;x) \geq c\,d_h(x)^2-C_N(1+M+hd_h(x)) \geq \frac c2d_h(x)^2-C.\] Adding the kinetic term proves (30). In particular the matrix-valued potential plus fermion term is nonnegative whenever \(d_h\geq R_0\), for a fixed \(R_0\), and the claimed tail bound follows by integration. ◻

Proof. Choose disjoint normal tubes of radius \(\rho\) about the compact orbits in \(Z\). The tubular neighborhood theorem here applies to the smooth compact embedded orbit \(\mathop{\mathrm{U}}(N)/H_q\) in the Euclidean configuration space, with its ordinary normal bundle. Scaling its normal exponential map gives the exact coordinates \[ \mathop{\mathrm{U}}(N)\times_{H_q}B_{S_q}(\rho/h) \longrightarrow\mathfrak k^9, \qquad [k,z]\longmapsto\operatorname{Ad}(k)(q/h+z). \tag{32}\] Within this tube, \(d_h=\left\lvert z\right\rvert\). Stabilizer equivariance is kept on the linear slice: if \(\rho_F\) denotes the gauge action on \(\mathcal F_N\), then restriction to the slice satisfies \[\psi(q/h+\operatorname{Ad}(a)z)=\rho_F(a)\psi(q/h+z), \qquad a\in H_q.\] We do not replace this equivariant linear slice by a singular quotient.

Let \(d_q=\dim\mathcal O\), and give \(\mathop{\mathrm{U}}(N)/H_q\) a fixed normalized invariant measure. Integration in (32) gives measure \(h^{-d_q}J_q(hz)\,\mathrm dz\) after orbit integration, with \(J_q\) smooth, positive, and \(H_q\)-invariant on \(B_{S_q}(\rho)\). This follows by differentiating the tube map: the \(d_q\) orbit columns are \(h^{-1}[w,q+hz]\), while the normal columns are fixed orthonormal vectors. Put \(j_{q,h}(z)=J_q(hz)/J_q(0)\). On every bounded slice set, \(j_{q,h}\to1\) smoothly and \(\nabla\log j_{q,h}=O(h)\). Define the flat-measure slice vector \[ f_{q,n}(z)= h_n^{-d_q/2}J_q(0)^{1/2}j_{q,h_n}(z)^{1/2} \psi_n(q/h_n+z),\qquad \left\lvert z\right\rvert<\rho/h_n. \tag{33}\] This identification is isometric on the tube. For general form-domain vectors we define it by \(L^2\) closure from smooth equivariant vectors; the derivative estimates below pass by form-core approximation. Distinct orbit dimensions cause no ambiguity: each tube has its own displayed constant normalization.

For any smooth equivariant vector, differentiation in an orthonormal basis of the fixed space \(S_q\) is ordinary ambient differentiation. Pointwise its squared slice derivative is therefore at most the full squared ambient gradient. After (33), the density derivative changes this bound only by \(O(h_n)\) on bounded sets. The uniform kinetic bound in Lemma 11 yields local \(H^1\) bounds for every \(f_{q,n}\). Rellich compactness, followed by a diagonal subsequence on expanding balls in the finitely many slices, gives strong local \(L^2\) convergence and weak local \(H^1\) convergence to \(f_q\). The distance bound gives both \(\left\lvert z\right\rvert f_q\in L^2\) and uniformly small squared norm tails \(O(R^{-2})\). For fixed \(R\), all the tubes \(\left\lvert z\right\rvert<R\) are disjoint when \(n\) is large, and their complement is contained in \(\{d_{h_n}\geq R\}\). Strong local convergence followed by \(R\to\infty\) proves the first identity in (31). The same estimates for a finite list, or the Cauchy–Schwarz inequality on its common tails, prove the polarized identity. All changes of coordinates and densities commute with parity, so parity passes to the profiles.

On a fixed bounded slice, Taylor expansion gives, uniformly there, \[ U_{h,1}(q/h+z)=U_q^{(2)}(z)+O_R(h),\qquad V_F(h,1;q/h+z)=\frac{\mathrm i}{2}\theta^tM_q\theta+O_R(h). \tag{34}\] For the first formula use \(U_{h,1}(q/h+z)=h^{-2}U_{1,1}(q+hz)\) and the vanishing of the constant and linear Taylor terms; for the second use the explicit linear fermion multiplier. Thus local potential energies converge under strong \(L^2\) convergence, and the slice-gradient inequality and weak lower semicontinuity give the kinetic liminf.

To justify the global energy statement, take real gauge-invariant cutoffs \(\chi_{q,h,R}\), equal to one in the \(R\)-tube about \(h^{-1}\mathcal O_q\) and supported in its \(2R\)-tube. For fixed \(R\) and small \(h\) the tubes are disjoint. Complete them to a squared partition of unity with an exterior cutoff, choosing the transition profiles so that the sum of squared gradients is at most \(C/R^2\). The first-order symbol identity (12) gives the localization formula \[ \sum_\alpha\mathcal H(h,1)[\chi_\alpha\psi] =\mathcal H(h,1)[\psi] +\frac12\int\sum_\alpha\left\lvert \nabla\chi_\alpha\right\rvert^2\left\lvert \psi\right\rvert^2. \tag{35}\] Every localized gauge-invariant form on the left is nonnegative, so the exterior term may be discarded. On each remaining tube the cutoff is a fixed function \(\chi_R(z)\). Applying the preceding local liminf yields \[\liminf_n\mathcal H(h_n,1)[\psi_n] \geq\sum_qH_q^{\mathrm{nor}}[\chi_Rf_q]-C/R^2.\] The uniform gradient and second-moment bounds already show that the limits belong to the confining oscillator form domains. Accordingly \(\chi_Rf_q\to f_q\) in these form domains as \(R\to\infty\). This proves the second assertion of (31) and completes the proof. ◻

Proof. Let \(\varphi_q\) be a normalized ground vector of \(H_q^{\mathrm{nor}}\), as in Lemma 10, and first fix a radial slice cutoff \(\chi_R\) supported in \(\left\lvert z\right\rvert<2R\). Use (33) inversely to put \(\chi_R\varphi_q\) in the tube, and extend by gauge equivariance and then by zero. The stabilizer character calculation makes this a well-defined vector, denoted \(\Phi_{q,h,R}\). Its support is inside the tube for all sufficiently small \(h\), and it is in the charge domain. The density and radial cutoff are preserved by the combined spatial and compensating gauge rotations. The invariance in Lemma 10 therefore makes \(\Phi_{q,h,R}\) invariant under the full deformation rotation group. It is even as well.

We verify energy convergence, including the derivative directions not contained in the fixed normal space. On a bounded slice the orbit columns in (32) are \([w,q/h+z]\), with \(w\) in a fixed complement of the stabilizer Lie algebra. At \(z=0\) they span the orthogonal complement of \(S_q\), with inverse of size \(O(h)\). On \(\left\lvert z\right\rvert\leq2R\) their inverse is still \(O_R(h)\). Gauge equivariance differentiates an orbit column to the fixed finite-dimensional operator \(\rho_{F,*}(w)\). Expressing a unit ambient tangent vector in these columns and the normal columns therefore bounds the additional derivatives by \[C_Rh\bigl(\left\lvert f\right\rvert+\left\lvert z\right\rvert\left\lvert \nabla f\right\rvert\bigr) \quad\text{for the corresponding fixed slice field }f.\] The density derivative is also \(O_R(h)\). It follows that the kinetic form of \(\Phi_{q,h,R}\) converges to the Euclidean slice kinetic form of \(\chi_R\varphi_q\). Together with (34), this gives \[\left\lVert \Phi_{q,h,R}\right\rVert\longrightarrow\left\lVert \chi_R\varphi_q\right\rVert,\qquad \mathcal H(h,1)[\Phi_{q,h,R}] \longrightarrow H_q^{\mathrm{nor}}[\chi_R\varphi_q].\] The latter is \(\frac12\left\lVert (\nabla\chi_R)\varphi_q\right\rVert^2\), since \(H_q^{\mathrm{nor}}\varphi_q=0\); it tends to zero as \(R\to\infty\). Rotation invariance and Proposition 2 identify this massive form energy with \(\left\lVert D_h\Phi_{q,h,R}\right\rVert^2\). Choose \(R\to\infty\) sufficiently slowly as \(h\downarrow0\) that all preceding fixed-\(R\) convergences hold. This is a diagonal choice: for each integer \(R\), first choose a positive upper bound on \(h\) making its finitely many errors smaller than \(1/R\). Supports for different orbits are disjoint for these choices. Normalizing the vectors proves (36). ◻

Proof of Proposition 7. The compact-domain and continuity assertions were proved in Lemma 9. We compute the constant index as \(h\downarrow0\).

There exist \(\delta>0\) and \(h_0>0\) such that \[ \left\lVert D_h\psi\right\rVert\geq\delta\left\lVert \psi\right\rVert, \qquad \psi\in\mathcal D\cap\mathscr H_G^-,\quad 0<h<h_0. \tag{37}\] Otherwise one could choose normalized odd \(\psi_n\), with \(h_n\downarrow0\) and \(\left\lVert D_{h_n}\psi_n\right\rVert\to0\). Lemma 3 would give \(\mathcal H(h_n,1)[\psi_n]\to0\). By Lemma 12, their profiles would have total squared norm one and would all belong to the zero-energy lines of the normal oscillators. Every such line is even by Lemma 10, whereas every extracted profile is odd. This contradiction proves (37).

Because \(D_h\) is odd and self-adjoint on the same invariant Hilbert space, it pairs positive even and odd eigenvectors of \(D_h^2\): for eigenvalue \(\lambda>0\), the map \(\lambda^{-1/2}D_h\) is an isometric parity-changing bijection. Compact resolvent makes this a statement about a complete discrete spectrum. Thus (37) excludes positive even eigenvalues below \(\delta^2\) as well, and the kernel is entirely even. If \(P_h\) denotes its orthogonal projection, we obtain \[ \left\lVert (1-P_h)\psi\right\rVert\leq\delta^{-1}\left\lVert D_h\psi\right\rVert, \qquad 0<h<h_0. \tag{38}\]

The kernel has dimension at most \(p(N)\) for all sufficiently small \(h\). If not, along a sequence \(h_n\downarrow0\) one could choose \(p(N)+1\) orthonormal null vectors. Simultaneous normal-slice extraction and the polarized norm identity in Lemma 12 would give an orthonormal list of that length in the direct sum of the \(p(N)\) one-dimensional normal ground spaces, a contradiction.

Conversely, Lemma 13 supplies \(p(N)\) asymptotically orthonormal even vectors with charge norm tending to zero. Equation (38) changes each by \(o(1)\) when it is projected onto \(\mathop{\mathrm{ker}}D_h\). Their projected Gram matrix tends to the identity, so these projected vectors are linearly independent for all sufficiently small \(h\). Thus the massive kernel has dimension exactly \(p(N)\) there, is even, and has the stated uniform gap. Its graded index is consequently \(p(N)\). Index constancy in Lemma 9 proves (15) for every \(h>0\). ◻

Proof. For fixed \(m\), its multiplication term is bounded on the support, so \(\|D_v(h,0)\psi_h\|=O(1)\). The undeformed square identity of Proposition 2, with the fermion term bounded below by \(-Ch|x|\), gives \[\|p\psi_h\|^2+h^2\sum_{i<j}\|[x_i,x_j]\psi_h\|^2 \leq C\bigl(\|D_v(h,0)\psi_h\|^2+h\|\psi_h\|^2\bigr).\] For (39), average the sixteen squared charges. The energy is \(o(m_h^2)\), and the fermion term on the support has norm at most \(C(h+m_h)\). In each shifted bosonic square use \(|a-b|^2\geq\frac12|a|^2-|b|^2\). The subtracted terms are bounded by \(Cm_h^2\|\psi_h\|^2\). Since \(m_h^2=O(h)\), the same conclusion follows. These form estimates extend from the invariant core by Lemma 4. Finally, on a compact set separated from the commuting locus, the continuous function \(\sum_{i<j}|[x_i,x_j]|^2\) has a positive minimum. The second estimate therefore makes the norm on that set tend to zero. ◻

Proof. The spectral projections of \(x_e\) onto the chosen windows identify its block subspaces uniquely up to \(K_0\). Smooth local choices of frames give the slice. The differential of conjugation in an off-block color \(w\in\mathfrak n\) has missing component \([w,b_e]\). All other differential components occur below this block in the Jacobian matrix, whose other diagonal blocks are the identity on the slice and the fixed homogeneous space measure. This proves (41) exactly. Invertibility of \(K_e\) and uniform bounds on its inverse follow from the spectral gaps. The same spectral-window parametrization allows arbitrary \(z\); small \(z\) is needed only to stay in the original compact neighborhood.

Here are coordinate formulas that also fix the content of the remainder. Let \((n_A)\) be an orthonormal basis of \(\mathfrak n\) and set \[w_A(b)=-K_e(b)^{-1}n_A, \qquad [w_A(b),b_e]=n_A.\] Write \(\theta_f\) and \(\theta_s\) for fast and slow color Clifford variables. Before density conjugation, gauge equivariance gives \[ \langle[w,b_e],p_e\rangle =-\sum_r\langle[w,b_r+z_r],p_r\rangle -\frac{\mathrm i}{2}\theta^t\mathop{\mathrm{ad}}(w)\theta. \tag{45}\] All coefficients in this identity multiply derivatives on their left. In particular, for \(t_e=0\), the exact fast operator is \[\begin{align*} E_v(b)={}&\sum_r\theta_f(A_r)\cdot p_{t_r} -\sum_{A,r}\theta_f^A(A_e) \langle[w_A,b_r],p_{t_r}\rangle\\ &+\sum_{i<j}\theta_f(B_{ij})\cdot \bigl([b_i,t_j]+[t_i,b_j]\bigr). \tag{46}\end{align*}\] If \(\Pi_0\) and \(\Pi_\perp\) denote the color projections onto \(\mathfrak k_0\) and \(\mathfrak n\), respectively, the remaining terms before the density correction are \[\begin{align*} R_v={}&\sum_{i<j}\theta(B_{ij})\cdot[t_i,t_j] +\frac m{\sqrt h}\sum_i\theta_f(L_i)\cdot t_i\\ &-\sum_{A,r}\theta_f^A(A_e) \langle\Pi_\perp[w_A,t_r],p_{t_r}\rangle\\ &-\frac1{\sqrt h}\sum_{A,r}\theta_f^A(A_e) \langle\Pi_0[w_A,t_r],p_{b_r}\rangle -\frac{\mathrm i}{2}\sum_A\theta_f^A(A_e) \theta^t\mathop{\mathrm{ad}}(w_A)\theta. \tag{47}\end{align*}\] The formula after density conjugation is obtained by replacing every \(p_{b_i}\) by \(p_{b_i}+\frac{\mathrm i}{2}\partial_{b_i}\log j\), and putting the added terms in \(R_v\). In this spatial frame \(j\) depends only on \(b_e\), so the added term is simply \[ \frac{\mathrm i}{2}\theta_s(A_e)\cdot\nabla_{b_e}\log j. \tag{48}\] These formulas include the Clifford signs of the actual full module; when it is written as a graded tensor product, the fast odd operator has the usual slow-parity factor. They show in particular that the only derivative terms in \(R_v\) have types \(t p_t\) and \(h^{-1/2}t p_b\). All coefficients and their derivatives are bounded on compact subcharts.

Tangential derivatives on the linear slice are restrictions of ambient derivatives. Gauge integration and the bounded positive density transfer the first raw estimate to these derivatives; density conjugation adds a bounded multiplier. The substitution \(z=t/\sqrt h\) then bounds \(p_t\) and \(p_b/\sqrt h\). Finally, \[[x_e,x_r]_{\mathfrak n}=[b_e,z_r]\] and the uniform inverse of \(K_e\) transfer the second raw estimate to \(\|t\psi_h\|\leq C\). ◻

Proof. We first compute the scalar-center oscillator and its gap. Perturbing that operator will give the exact fast line for nearby noncommuting blocks and its regularity. We then choose the phase needed for diagonal compression and for transport between cluster charts.

The oscillator at scalar centers. Change from the \(e\)-slice to the perpendicular slice for the linear conjugation directions \(([w,c_i])_i\), including the positive linear density factor. For one real two-dimensional color plane joining two distinct blocks, let \(\lambda\in\mathbb R^9\) be their signed center difference, \(r=|\lambda|\), and \(J\) the orthogonal complex structure on that plane, so that \(\mathop{\mathrm{ad}}(c_j)=\lambda_jJ\). There are sixteen real transverse bosonic coordinates: eight spatial directions times two color directions. The charge on them is \[\theta_f(A_i)\cdot p_i^q+ \sum_{i<j}\theta_f(B_{ij})\cdot J(\lambda_iq_j-\lambda_jq_i), \qquad q\perp\lambda,\quad p^q\perp\lambda.\] To check the square, rotate \(\lambda\) to \(r e_9\) and put \(s=v^t\gamma^9v\). The summand becomes \[\sum_{i=1}^8\bigl( \theta_f(\gamma^iv)\cdot p_i^q +r\theta_f(\gamma^i\gamma^9v)\cdot Jq_i\bigr),\] and the Clifford identity used in Proposition 2 gives \[ \frac12\bigl((p^q)^2+r^2q^2\bigr) +rs\,p^q\cdot Jq +\frac{\mathrm ir}{2}\theta_f^t \bigl[J\otimes(s\mathrm{Id}-\gamma^9)\bigr]\theta_f. \tag{49}\] There are no cross terms between disjoint Clifford summands. Summing their orbital terms before imposing neutrality gives the gauge generator for the single central color \(\sum_j(v^t\gamma^jv)c_j\). Equation (45) for this central generator cancels the scalar-spin terms in (49). Hence, on the neutral space, \[ E_v(c)^2=H_{\mathrm{osc}}(c) -\frac{\mathrm i}{2}\theta_f^t \bigl(\mathop{\mathrm{ad}}(c_j)|_{\mathfrak n}\otimes\gamma^j\bigr) \theta_f. \tag{50}\] In particular this square is independent of \(v\).

Each real color plane contributes bosonic ground energy \(8r\). Its thirty-two real fermions give sixteen Clifford oscillators of frequency \(r\), with unique spin ground energy \(-8r\). In the color-complex polarization the latter fills eight modes and leaves eight empty, so its central color charge is zero; the positive Gaussian is neutral as well. Their product is therefore in the neutral space. The nonnegative oscillator occupation numbers show that this product is the only zero vector up to scalar, and every other eigenvalue of the square is bounded below by the smallest center distance. Restriction to the neutral space cannot reduce that bound. This proves the kernel and gap assertions at centers.

Perturbation to noncommuting blocks. The subsequent first-order estimates use both differentiated projections and oscillator moments. At a reference center \(c_*\) define \(\mathcal W^s\) by the norm \(\|(1+\mathcal N)^{s/2}f\|\), where \(\mathcal N\) is its positive bosonic oscillator number operator; the fermion space is finite. These are restricted to neutral vectors. The creation and annihilation relations and (50) imply \[\|p_tf\|+\|tf\|\leq C(\|E_v(c_*)f\|+\|f\|).\] They also imply, on a circle enclosing zero alone, that \((E_v(c_*)-\zeta)^{-1}:\mathcal W^s\to\mathcal W^{s+1}\) is bounded for every fixed \(s\). For completeness, the square is an oscillator number operator plus a bounded spin matrix. Each application of \(t\) or \(p_t\) changes the number by one, and its coefficient is bounded by a constant times \((1+\mathcal N)^{1/2}\). The resolvent can be written \((E_v(c_*)+\zeta)(E_v(c_*)^2-\zeta^2)^{-1}\); its two factors respectively lose one and gain two such orders. This proves the claimed bound, including nonintegral \(s\) if needed.

The central torus fixes \(b\), and \(K_e(b)\), its inverse, and all \(\mathop{\mathrm{ad}}(b_r)|_{\mathfrak n}\) commute with its action. Thus \(E_v(b)\) preserves the same neutral space even when \(b\) is noncommuting. Its real linear coefficients make it symmetric on the neutral Schwartz core. The coefficient formula (46) gives, for every fixed finite range of \(s\), \[\|E_v(b)-E_v(c_*)\|_{\mathcal W^{s+1}\to\mathcal W^s} \leq C_s|b-c_*|.\] Choose the neighborhood so that the resolvent Neumann series converges on each of these finitely many spaces. On \(\mathcal W^0\) the same relative bound, made smaller than one, gives self-adjointness on the reference domain. The contour projection has constant rank one. It commutes with parity, and its range remains entirely even by continuity. Since \(E_v(b)\) preserves this range and reverses parity, its restriction to the range is zero. The complementary inverse can equivalently be formed from \((E_v(b)+P_v(b))^{-1}(1-P_v(b))\). Differentiating its resolvent formula and using \[\partial_b(E_v(b)-\zeta)^{-1} =-(E_v(b)-\zeta)^{-1}(\partial_bE_v(b)) (E_v(b)-\zeta)^{-1}\] proves all stated finite-order bounds after increasing the finite range of oscillator orders. None of this uses \([b_i,b_j]=0\).

A compatible phase. For a fixed oriented color pair, the spin ground line over \(\lambda/r\in S^8\) is homogeneous under \(\mathop{\mathrm{Spin}}(9)\). The underlying orthogonal action lifts to the Clifford module as an honest action because \(\mathop{\mathrm{Spin}}(9)\) is simply connected. Its stabilizer \(\mathop{\mathrm{Spin}}(8)\) acts on the line by a character. A connected semisimple compact group has no nontrivial continuous characters, so it fixes a chosen unit vector. Transport therefore defines a \(\mathop{\mathrm{Spin}}(9)\)-equivariant unit section. Its Berry one-form is invariant on \(S^8\), and hence zero: the tangent representation of \(\mathop{\mathrm{Spin}}(8)\) has no fixed covector. The normalized real positive Gaussian also has zero Berry form.

For directions in the last six axes, the unit skew ground-frequency matrix anticommutes with the complex structure \(T\) used for the exterior module. The projection between the corresponding complex annihilator spaces has squared norm \(1/2\) on unit vectors, and is invertible. The new annihilators are consequently a graph over the old ones. Their vacuum has a nonzero degree-zero coefficient: if the lowest nonzero exterior component had positive degree, the graph annihilation equations in one lower degree would make all ordinary contractions of that component zero, an impossibility. This graph vacuum is even. We fix its phase by positive overlap with the empty \(T\)-vacuum. The overlap convention is consistent as the last-six direction varies, by \(\mathop{\mathrm{Spin}}(6)\) covariance.

The block group and its block permutations preserve the empty \(T\)-vacuum by their natural exterior actions. Thus on last-six center configurations the product ground section is block invariant and has exactly the permutation transformation law of the underlying coordinates, without a scalar factor. Reordering its pair factors adds no sign, since each is even. These sections are parallel as functions of the labeled distinct centers. The quotient of two sections related by a permutation is therefore locally constant. The space of distinct labeled points in \(\mathbb R^9\) with trace-zero weighted mean is connected; removing the coincidence subspaces of codimension nine does not disconnect it. Agreement on last-six configurations proves agreement everywhere. The same construction gives covariance under \(\mathop{\mathrm{Spin}}(3)\times\mathop{\mathrm{Spin}}(6)\). These statements concern the fast line; the graded permutation on slow factors is still the ordinary one.

Let \(c=c(b)\) be the scalar block-center projection. If \(\xi(c)\) is the unit section just constructed, set \[ \chi_v(b)=\frac{P_v(b)\xi(c(b))} {\|P_v(b)\xi(c(b))\|}. \tag{51}\] The denominator is bounded away from zero on a smaller neighborhood. This prescription is \(K_0\)-equivariant. In internal directions at \(b=c\), \(\xi\) is constant and \(P(\partial P)P=0\) shows that \(\langle\chi_v,\partial_b\chi_v\rangle=0\). In center directions the same assertion is the zero Berry form already proved. Smooth transverse linear coordinate changes, implemented with the positive square root of the absolute Jacobian, do not change it: they send the normalized positive real Gaussian to a normalized positive real Gaussian, whose inner product with its derivative is zero. ◻

Proof. For each fixed \(h\), local ellipticity of the original charge puts a compactly supported graph-domain vector in \(H^1_{\mathrm{loc}}\). Gauge equivariance and the smooth orbit–slice coordinates transfer this regularity to the slice, by orbit integration, on compact subcharts. After a compact fast cutoff, \(E_v\psi\) and \(D_v^s\psi\) are therefore separately in \(L^2\): the slow multipliers are bounded on the compact slow support for that fixed \(h\). Local smoothing justifies the following squared-norm identities before any uniform bound on these separate operators is invoked.

Choose an even scalar cutoff \(\rho_M(t)\) equal to one for \(|t|\leq M\), supported in \(|t|\leq2M\), and with gradient bounded by \(C/M\). It is \(K_0\)-invariant. Equation (44) gives \(\|(1-\rho_M)\psi_h\|\leq C/M\). For fixed \(M\), the explicit remainder formula shows \[\|h^{-1/2}R_v\rho_M\psi_h\|\longrightarrow0.\] Indeed the respective bounds are constants depending on \(M\) times \(h^{-1/2}\) for \(t^2\), spin, and density terms; \(h^{-1/2}\|p_t\psi_h\|\) for \(tp_t\); \(h^{-1}\|p_b\psi_h\|\) for \(tp_b/\sqrt h\); and \(m/h\) for the mass term. The cutoff commutator with \(D_v/\sqrt h\) is bounded by \(C/M+o(1)\). Also \(\|D_v\psi_h\|/\sqrt h\to0\) in both regimes. Consequently \[ \limsup_{h\to\infty} \|(E_v+h^{-1/2}D_v^s)\rho_M\psi_h\|\leq C/M. \tag{53}\]

Every slow bracket and mass Clifford multiplier anticommutes with \(E_v\). In particular, the potentially large multiplier \(h[b,b]\) produces no anticommutator. Only slow differentiation of its coefficients remains: \[\{E_v,D_v^s\} =-\mathrm i\sum_{i,Q}\theta_s^Q(A_i)\,\partial_{b_i^Q}E_v.\] On \(\rho_M\psi_h\) its expectation is bounded uniformly for fixed \(M\), because \(\partial_bE_v\) is linear in \((p_t,t)\) and (44) bounds their norms. Squaring (53), dropping the nonnegative slow square, and using the fast gap thus gives \[\limsup_{h\to\infty}\|(1-P_v)\rho_M\psi_h\|\leq C'/M.\] The tail estimate and then \(M\to\infty\) prove the first assertion.

For the second assertion, it is enough to estimate the projected remainder. The rank-one projection, its oscillator moments, and integration by parts in \(t\) give the uniform bound \[ \|P_vR_v\psi_h\| \leq C\left((1+m/\sqrt h)\|\psi_h\| +\|p_b\psi_h/\sqrt h\|\right). \tag{54}\] To see each term, \(t^2\) moves onto the oscillator bra; in \(tp_t\) the derivative moves onto that bra and \(t\); spin and density terms are bounded; \(tp_b/\sqrt h\) uses the bounded bra \(t\chi_v\) and the last norm on the right; and the fast mass term uses \(t\chi_v\) and \(m/\sqrt h\). Thus no norm estimate for the unprojected \(R_v\psi_h\) is asserted or needed. Moreover \[[D_v^s,P_v]=-\mathrm i\sum_{i,Q}\theta_s^Q(A_i) \partial_{b_i^Q}P_v\] is bounded: the even projection commutes with all slow Clifford multipliers. Projecting (43) now gives \[D_v^sP_v\psi_h=P_vD_v\psi_h-P_vR_v\psi_h +[D_v^s,P_v]\psi_h.\] Equations (44) and (54) prove the claimed estimates. Finally, differentiating \(\chi_v\eta_h\) costs a bounded derivative of \(\chi_v\); the fixed mass multiplier is bounded on the chart. Removing the section therefore gives the asserted undeformed slow graph bound. Its equivariance follows from (51). ◻

Proof. The Berry term vanishes by Lemma 16. The outer fast fermion in each \(tp_t\) or \(tp_b\) term makes its diagonal matrix element zero, also when a slow derivative acts on the even section. The slow-color part of \([t_i,t_j]\) has zero Gaussian expectation: on each cross-block color space the covariance is a symmetric spatial matrix times the color identity, whose contraction with the antisymmetric color bracket is zero. Its fast-color part and the other purely fast odd terms have zero matrix element by parity.

We verify the remaining orbit-spin and density cancellation for every slow color, including internal traceless colors. For a slow orthonormal basis color \(q\in\mathfrak k_0\), put \[K_q=\mathop{\mathrm{ad}}(q)|_{\mathfrak n},\qquad K_e=\mathop{\mathrm{ad}}(c_e)|_{\mathfrak n}.\] There can be no term containing three slow fermions in the orbit-spin substitution: for \(w\in\mathfrak n\) the slow-to-slow block of \(\mathop{\mathrm{ad}}(w)\) is zero. Its fast-to-fast block, after the outer fast fermion is included, has odd fast degree and has zero expectation. Only the mixed slow-fast block contributes.

On a cross-block space joining blocks \(\alpha,\beta\), one has \(K_e=\delta_{\alpha\beta,e}J_{\alpha\beta}\) with \(\delta_{\alpha\beta,e}\ne0\). The operator \(K_q\) preserves this space and is real skew. The fast ground two-point matrix is \(\frac12\mathrm{Id}\) plus an imaginary skew matrix whose color factor is \(J_{\alpha\beta}\) and whose spin factor is the signed unit center-difference Clifford matrix. Its imaginary part in the mixed contraction has color trace \[ \mathop{\mathrm{Tr}}\bigl(J_{\alpha\beta}K_qK_e^{-1}\bigr) =\delta_{\alpha\beta,e}^{-1}\mathop{\mathrm{Tr}}K_q=0. \tag{55}\] The equality follows by cyclicity and \(J_{\alpha\beta}^2=-\mathrm{Id}\); it does not require \(q\) to be central. The spin coefficient can therefore be left arbitrary in this cancellation.

For clarity, the real scalar contraction can be computed without a symmetry argument. With the fast fermion ordered before the slow one, the mixed entries of \(\mathop{\mathrm{ad}}(w_A)\) are \([w_A,q]=-K_qw_A\), and their two orders combine to twice this coefficient. Multiplying by \(-\frac{\mathrm i}{2}\theta_f^A(A_e)\) from (47) and using \(\langle\theta_{f,\mu}^A\theta_{f,\nu}^B\rangle_{\mathrm{scalar}} =\frac12\delta_{AB}\delta_{\mu\nu}\) gives \[ -\frac{\mathrm i}{2}\mathop{\mathrm{Tr}}_{\mathfrak n}(K_e^{-1}K_q) \theta_s^q(A_e). \tag{56}\] On the other hand, differentiating the determinant in (41) gives \[\partial_{b_e^q}\log j\big|_{b=c} =\mathop{\mathrm{Tr}}_{\mathfrak n}(K_e^{-1}K_q).\] The density term (48) is exactly the opposite of (56). This proves the cancellation for every \(q\). There are no terms to check in other spatial directions: \(j\) depends only on \(b_e\), and only the \(e\)-momentum was eliminated. The finitely many remaining \(h^{-1/2}\) coefficients are explicitly visible in (47). Smoothness and the oscillator bounds prove the last assertion of the lemma. ◻

Proof. The assertion is immediate if the indicated interval is empty. Otherwise choose \(0\le\chi\le1\), equal to one on that interval, supported in \[Me^{-L}h^{-1/3}<r<e^L\varepsilon, \qquad |r\chi'(r)|\le C/L.\] One obtains such a function by multiplying two fixed smooth cutoffs in \(\log r\), each with transition length \(L\). Its support satisfies the hypothesis of (67). The Clifford normalization gives \(\left\lVert [D_u,\chi](x)\right\rVert_{\mathrm{op}}=|\nabla\chi(x)|/\sqrt2\). Thus \[c_n\left\lVert \chi F_h\right\rVert \le\left\lVert r\chi D_u(h,0)F_h\right\rVert+ \left\lVert r[D_u,\chi]F_h\right\rVert \le e^L\varepsilon K+CB/L.\] Approximation in the graph norm extends this computation beyond smooth vectors. The same estimate with spectators follows first for finite sums of spectator vectors, by squaring the exterior inequality and integrating, and then by approximation.

For a given \(\delta>0\), first choose \(L\) so that \(CB/(c_nL)<\delta/2\), then choose \(M_0>e^LR_n\), and finally choose \(\varepsilon_0\) so that \(e^L\varepsilon_0K/c_n<\delta/2\). This proves the stated order and uniformity of the limits. ◻

Proof. Local compactness will construct a kernel-valued profile. The intermediate-scale estimate is then needed to show that it accounts for all norm assigned to its stratum. We carry out these two steps before gluing the local profiles and treating inner products.

Construction of a local profile. On a compact set the fixed mass term is a bounded multiplier, uniformly in \(h_j\). The raw estimates of Lemma 14 therefore imply that any limiting norm measure is supported on commuting configurations. Indeed, on a compact subset where \(\sum_{i<j}|[x_i,x_j]|^2\ge a>0\), its mass is \(O(h_j^{-1}/a)\).

Fix a smaller compact set in a separated-center chart, and first multiply the states by a smooth cutoff supported in the chart and equal to one on a neighborhood of that set. Write the resulting states again as \(\psi_j\). All the estimates that follow are uniform on this fixed set. Lemma 17 gives \[\psi_j-P_u\psi_j\longrightarrow0, \qquad P_u\psi_j=\chi_u(b)\eta_j(b), \qquad \left\lVert D_u^s(h_j,m)P_u\psi_j\right\rVert\le C.\] The normalized fast section is even and has bounded derivatives as a vector in the fast Hilbert space. The slow Clifford multipliers act on the other factor. Consequently the Leibniz identity reads \[ D_u^s\bigl(\chi_u\eta_j\bigr) =\chi_uD_u^s\eta_j+ \sum_\ell a_\ell(\partial_{b_\ell}\chi_u)\eta_j, \tag{71}\] where the \(a_\ell\) are constant slow Clifford matrices. The last term has norm at most \(C\left\lVert \eta_j\right\rVert\). In particular, no factor of \(h_j\) is introduced when removing the fast section: the large internal bracket multiplier acts only on slow fermions. Subtracting the bounded mass multiplier gives \(\left\lVert D_u^s(h_j,0)\eta_j\right\rVert\le C\). Covariance of \(\chi_u\) means that \(\eta_j\) has the separate gauge symmetry in each internal \(\mathop{\mathrm{SU}}(n_\alpha)\) factor.

Put \(w_\alpha=b_\alpha-c_\alpha\mathrm{Id}_{n_\alpha}\). On the slow tensor product, with all graded signs included, \[D_u^s(h_j,0)=D_u^C(0)+\sum_{\alpha:n_\alpha>1} D_{u,\alpha}(h_j,0).\] These summands involve disjoint variables and odd Clifford factors, so their pairwise anticommutators vanish. For compactly supported smooth sections integration by parts gives \[ \left\lVert D_u^s(h_j,0)\eta_j\right\rVert^2 =\left\lVert D_u^C(0)\eta_j\right\rVert^2+ \sum_{\alpha:n_\alpha>1} \left\lVert D_{u,\alpha}(h_j,0)\eta_j\right\rVert^2. \tag{72}\] The identity extends by closure. One can see this directly here by approximating after the compact slow cutoff; every potential is bounded on its support for fixed \(h_j\), and ordinary local smoothing applies. In particular, each internal graph norm is bounded and \(\left\lVert \nabla_c\eta_j\right\rVert\le C\), since \(\left\lVert D_u^C(0)g\right\rVert^2=\tfrac12\left\lVert \nabla_cg\right\rVert^2\).

Rescale the internal variables unitarily by \(y_\alpha=h_j^{1/3}w_\alpha\), and denote the resulting function by \(\widetilde\eta_j(c,y)\). Then \[ \left\lVert D_{u,\alpha}(1,0)\widetilde\eta_j\right\rVert\le Ch_j^{-1/3}, \qquad \left\lVert \nabla_c\widetilde\eta_j\right\rVert\le C. \tag{73}\] For an internal cutoff \(\zeta\) of fixed compact support, write the constant-coefficient kinetic Dirac operator as \(D_{u,\alpha}(1,0)\) minus its bracket multiplier. That multiplier is bounded on \(\mathop{\mathrm{supp}}\zeta\). The kinetic norm identity and the bounded commutator with \(\zeta\) give \[\left\lVert \nabla_{y_\alpha}(\zeta\widetilde\eta_j)\right\rVert \le C_\zeta\bigl(\left\lVert D_{u,\alpha}(1,0)\widetilde\eta_j\right\rVert +\left\lVert \widetilde\eta_j\right\rVert\bigr).\] All spectators are integrated in these norms. Combining these inequalities for the finitely many blocks with the center derivative estimate gives a bounded ordinary \(H^1\) norm on every compact subset of the joint \((c,y)\) space. Its Clifford fiber is finite dimensional. The usual Rellich theorem therefore gives, by a diagonal subsequence, strong local \(L^2\) convergence to a function \(\eta_\infty(c,y)\). Compactness has been applied jointly in all these variables, rather than to functions with an uncontrolled infinite-dimensional spectator fiber.

Fatou’s lemma gives the global \(L^2\) bound for this local chart limit. Testing (73) against compactly supported smooth functions shows distributionally that \(D_{u,\alpha}(1,0)\eta_\infty=0\) for each \(\alpha\). Fubini’s theorem and a countable dense family of internal tests show that almost every slice in the remaining variables is in the corresponding distributional \(L^2\) kernel. Lemma 4 identifies this with \(\mathcal K_{n_\alpha}\). Equivalently, the orthogonal projection onto each internal kernel fixes \(\eta_\infty\); the product of these commuting projections identifies its coefficient space as \(\bigotimes_\alpha\mathcal K_{n_\alpha}\). This constructs the local profile.

Exact norm on the stratum. We now identify how much limiting norm is assigned to the stratum \(w=0\). The replacement of \(\psi_j\) by \(P_u\psi_j\) changes integrals against bounded multipliers by \(o(1)\). The physical fast coordinates tend to zero in norm measure, since \(z=t/\sqrt{h_j}\) and \(\left\lVert t\psi_j\right\rVert\le C\). On bounded \(y\) sets, smoothness also gives \(\chi_u(c+h_j^{-1/3}y)-\chi_u(c)\to0\) uniformly in fast Hilbert norm. It remains to decide whether any internal norm escapes all bounded \(y\) sets while still reaching \(w=0\).

Here is a precise measure comparison. Take a nonnegative continuous center test \(a(c)\) of compact support in a smaller chart. Let \(0\le\kappa_\varepsilon(w)\le1\) be a product of radial internal cutoffs, equal to one when every \(|w_\alpha|\le\varepsilon\) and zero when some \(|w_\alpha|\ge2\varepsilon\). Choose them decreasing as \(\varepsilon\downarrow0\). In testing the quotient measure, include a fixed physical fast cutoff equal to one near \(z=0\), so that the product extends continuously by zero inside the outer chart. Its removal changes the limit by zero, because \(\left\lVert t\psi_j\right\rVert\le C\) and \(z=t/\sqrt{h_j}\). The slice density has already been absorbed unitarily, and \(\int|\chi_u(b,t)|^2\,\mathrm dt=1\); thus, after removal of the fast cutoff, these tests of the projected states are exactly the corresponding slow integrals. For each fixed \(M\), strong local convergence identifies the integrals on \(\{|y_\alpha|\le M\text{ for all }\alpha\}\). The difference between such an integral and the integral with \(\kappa_\varepsilon\) is bounded above, apart from this local convergence error, by \[ \left\lVert a\right\rVert_\infty\sum_{\alpha:n_\alpha>1} \int_{Mh_j^{-1/3}<|w_\alpha|<2\varepsilon}|\eta_j|^2. \tag{74}\] Each summand satisfies Lemma 23, with all other variables as spectators, because of (72). For the converse comparison, the bounded \(y\) set lies where \(\kappa_\varepsilon=1\) for all sufficiently large \(j\). Thus its limiting integral is a lower bound for the one with \(\kappa_\varepsilon\).

To record the order of limits explicitly, put \[J_\varepsilon=\int a(c)\kappa_\varepsilon(w)\,\mathrm d\mu, \qquad A_M=\int_{\max_\alpha|y_\alpha|\le M} a(c)|\eta_\infty(c,y)|^2\,\mathrm dc\,\mathrm dy.\] For every fixed \(L\ge1\) and \(M>e^L\max_\alpha R_{n_\alpha}\), the preceding comparisons and (68), after \(j\to\infty\), give \[A_M\le J_\varepsilon\le A_M+ \left\lVert a\right\rVert_\infty\sum_{\alpha:n_\alpha>1} c_{n_\alpha}^{-2}\left(2e^L\varepsilon K_\alpha+CB/L\right)^2,\] where \(K_\alpha\) are the block graph bounds in (72). Decreasing \(\varepsilon\) restricts \(J_\varepsilon\) to \(w=0\), by continuity from above and concentration of the fast coordinates at zero. Next let \(M\to\infty\), keeping its lower threshold, and finally let \(L\to\infty\). The error vanishes and \(A_M\) exhausts the profile norm. Thus the restricted measure is exactly \(|F_C(c)|^2\,\mathrm dc\). If every block is scalar, the sums and internal cutoffs are absent and strong local center convergence gives the same conclusion directly. This argument also covers simultaneous fragmentation in several blocks, since their error sets are bounded by the finite sum in (74).

Compatibility and inner products. Local strong convergence and approximation by bounded \(y,t\) sets, followed by Proposition 20, give the local version of (70). It also identifies the profile independently of the outer chart cutoff. If two cutoffs \(\lambda,\widetilde\lambda\) equal one near the same smaller center support, then \(\| (\lambda-\widetilde\lambda)I_{h_j,u,m}f\|\to0\) for every test supported there: its leading lift concentrates in that neighborhood, and its correction tends to zero in norm. Pairing with the original bounded sequence proves that the two localizations have identical test overlaps. These identify the same profile coefficient, including its phase.

There are finitely many partition types and a countable exhaustion of each stratum by relatively compact charts. Repeating the extraction diagonally gives profiles on all of them. On overlapping charts, Lemma 21 identifies the leading tests and hence, by their overlaps, the local profiles. These consequently descend with the required permutation action. Exhaustion of the stratum gives the global norm bound, and the local overlaps give (70) for every compactly supported profile test.

For a finite list \(\psi_j^1,\ldots,\psi_j^q\), carry out the same extractions for the list and for the finitely many sums and imaginary sums needed for complex polarization. Equivalently, take weak limits of the matrix of measures with entries \(\pi_*(\left\langle \psi_j^a(x),\psi_j^b(x)\right\rangle\,\mathrm dx)\). Positivity of this matrix and Cauchy–Schwarz bound every entry by its diagonal measures. Linearity of the local limits then gives, on each stratum, the entry \(\left\langle F_C^a(c),F_C^b(c)\right\rangle\,\mathrm dc\). This proves the polarized assertion. The same leading identifications commute with the indicated symmetries, which proves the final assertion. ◻

Proof. Each collision subspace has codimension nine, also after imposing the trace-zero constraint. Let \(\rho_\varepsilon\) be the product of smooth cutoffs that vanish at distance at most \(\varepsilon\) from these finitely many subspaces and equal one at distance at least \(2\varepsilon\). On each compact set \(K\subset \mathcal Z_C\), tubular coordinates give \[ \left\lVert \nabla\rho_\varepsilon\right\rVert_{L^2(K)}^2 \le C_K\varepsilon^{-2}\varepsilon^9=C_K\varepsilon^7. \tag{75}\] Intersections of collision subspaces cause no difficulty, since the derivative of the finite product is bounded by the sum of the individual derivatives.

For a smooth compactly supported test \(\varphi\), apply the given distributional identity to \(\rho_\varepsilon\varphi\). The commutator error has absolute value at most \[2^{-1/2}\left\lVert F\right\rVert_{L^2(K)}\left\lVert \varphi\right\rVert_\infty \left\lVert \nabla\rho_\varepsilon\right\rVert_{L^2(K)},\] which tends to zero. All other terms converge by dominated convergence; the mass multiplier is bounded on \(K\). This proves the global distributional equation. The maximal and minimal graph domains coincide by the cutoff and local ellipticity argument of Lemma 4, which applies verbatim to this abelian operator. Tests with values in a finite-dimensional coefficient subspace suffice, followed by density if necessary. For permutation-equivariant vectors, averaging the tests over the finite permutation group gives the same conclusion. There is thus no additional collision boundary condition. ◻

Proof. Choose any finite family of profile tests \(g_C\in\mathcal T_C\) and put \(T_jg=\sum_C I_{h_j,u,m}g_C\). Proposition 20 and Lemma 21 give \[\left\lVert T_jg\right\rVert^2\longrightarrow\sum_C\left\lVert g_C\right\rVert^2, \qquad \left\langle D_u(h_j,m)\psi_j,T_jg\right\rangle \longrightarrow\sum_C\left\langle F_C,D_u^C(m)g_C\right\rangle.\] For the second identity, move the self-adjoint charge to the tests; its graph-output error pairs with the bounded norm of \(\psi_j\) and tends to zero. Cauchy–Schwarz therefore bounds the last functional by \[\left(\liminf_j\left\lVert D_u(h_j,m)\psi_j\right\rVert\right) \left(\sum_C\left\lVert g_C\right\rVert^2\right)^{1/2}.\] The tests are dense in the Hilbert direct sum of the profile spaces. Riesz representation identifies this bounded functional with the direct sum of the distributional derivatives, and proves (76) off the collisions. Lemma 25 extends each derivative across them. This argument uses convergence for each fixed finite test family; it requires no uniform estimate on the whole unit ball of test functions. ◻

Proof of Proposition 22 at size \(N\). Fix a compact annulus \(a\le r\le b\), where \(0<a<b<\infty\). We first claim that every sequence of unit vectors supported there, with \(h_j\to\infty\), satisfies \[ \liminf_j\left\lVert rD_u(h_j,0)F_j\right\rVert\ge\frac{5}{2\sqrt2}. \tag{77}\] It suffices to treat a subsequence for which the norms on the left are bounded. Because \(r\ge a\), its unweighted graph norms are bounded. Every limiting norm measure is supported on commuting configurations in this annulus. A traceless commuting configuration with only one joint center is zero. Thus only proper strata occur here, and Proposition 24, together with additivity of the measure over the finitely many strata, gives \[ \sum_{C\text{ proper}}\left\lVert F_C\right\rVert^2=1. \tag{78}\]

The profiles of \(rF_j\) are \(|c|F_C\). This follows either from the local rescalings or from (70), since \(r\) is bounded on the common support and restricts to \(|c|\) on the stratum. The commutator with \(r\) is bounded, so Proposition 26 applies to this sequence as well.

For a reference for the classical Euclidean Hardy inequality, see (Frank and Seiringer 2008, equation (1.1), with \(p=2\)). For completeness, completing a square in Euclidean dimension \(d>2\) gives, for smooth compactly supported \(g\) away from zero, \[0\le\left\|\nabla g+\frac{d-2}{2}\frac{c}{|c|^2}g\right\|^2 =\left\lVert \nabla g\right\rVert^2-\frac{(d-2)^2}{4}\left\lVert g/|c|\right\rVert^2.\] Cutoffs and closure extend this Hardy inequality to \(H^1\). Applied on \(\mathcal Z_C\), with its orthonormal trace coordinates and \(d_C=9(k-1)\ge9\), it yields \[\left\lVert D_u^C(0)(|c|F_C)\right\rVert =2^{-1/2}\left\lVert \nabla_c(|c|F_C)\right\rVert \ge\frac{d_C-2}{2\sqrt2}\left\lVert F_C\right\rVert \ge\frac7{2\sqrt2}\left\lVert F_C\right\rVert.\] The full-space graph statement needed here is supplied by Lemma 25. Hence (76) and (78) imply \(\liminf_j\left\lVert D_u(h_j,0)(rF_j)\right\rVert\ge7/(2\sqrt2)\). Finally, \(\left\lVert [D_u,r]F_j\right\rVert=1/\sqrt2\), proving (77).

Fix a logarithmic width \(A>0\). Since \(5/(2\sqrt2)>1\), a contradiction sequence in the fixed annulus \(1<r<e^A\) shows that there is \(H_A<\infty\) such that its annular inequality \(\left\lVert rD_u(h,0)F\right\rVert\ge\left\lVert F\right\rVert\) holds for every \(h\ge H_A\). Dilation \(x=s\xi\) replaces the coupling by \(hs^3\) and leaves the weighted operator \(rD_u\) unchanged. Therefore \[ \left\lVert rD_u(h,0)F\right\rVert\ge\left\lVert F\right\rVert \quad\text{if}\quad \mathop{\mathrm{supp}}F\subset\{s<r<e^As\},\quad h^{1/3}s\ge H_A^{1/3}. \tag{79}\]

We use the IMS localization principle (Simon 1983, Lemma 3.1), here in its weighted first-order form. Choose smooth functions \(\chi_j(\log r)\) forming a squared partition of unity, with supports in intervals of length \(A\), bounded overlap, and \(\sum_j|\chi_j'|^2\le C/A^2\). For example, normalize translates of a fixed bump by the square root of the sum of their squares, then rescale the real variable. The Clifford symbol and \(\sum_j\chi_j\nabla\chi_j=0\) give the exact localization identity \[ \sum_j\left\lVert rD_u(\chi_jF)\right\rVert^2 =\left\lVert rD_uF\right\rVert^2+ \frac12\int r^2\sum_j|\nabla\chi_j|^2|F|^2 \le\left\lVert rD_uF\right\rVert^2+\frac{C}{A^2}\left\lVert F\right\rVert^2. \tag{80}\] Fix \(A\) large enough that \(C/A^2\le1/2\), and only then set \(R_N>e^AH_A^{1/3}\). Every member of the partition meeting \(\mathop{\mathrm{supp}}F\subset\{h^{1/3}r>R_N\}\) is supported in an annulus satisfying (79): its lower radius is at least \(e^{-A}\) times a radius where it meets \(F\). Summing that bound and using (80) proves \(\left\lVert rD_uF\right\rVert^2\ge\tfrac12\left\lVert F\right\rVert^2\). Taking, for example, \(c_N=1/\sqrt2\) proves the assertion. The threshold can depend on \(N\). All identities initially use compact smooth vectors and extend to the stated domain by Lemma 4. ◻

Proof. Multiply by a fixed compact radial cutoff equal to one near zero. The resulting undeformed graph norms are bounded, since the mass term is bounded there. Unitary whole-configuration rescaling by \(h_j^{1/3}\) gives bounded \(L^2\) functions whose \(D_u(1,0)\) outputs tend to zero. Local ellipticity and Rellich compactness give a strong local limit. Fatou’s lemma makes that limit globally square integrable, and Lemma 4 puts it in \(\mathcal K_N\). Lemma 23, now available at size \(N\), identifies its norm with exactly the atom at zero by the same inner-ball and shrinking-physical-ball comparison used in (74). This construction is independent of the chosen cutoff near zero. It extracts one vector in \(\mathcal K_N\); it does not require compactness of its unit ball or prior finite dimensionality.

The commuting quotient is the disjoint union of zero and the proper strata. Tightness preserves total mass in the weak limit: approximate the constant function one by compact cutoffs and use the uniform tail bound. Adding the restricted measures gives (81). Applying this to the finite linear combinations used for polarization gives (82). Rotation and parity pass to \(F_0\) directly under radial cutoffs and dilation. ◻

Proof. For brevity write \(a_{v,j}\) for the state belonging to a spinor in the chosen basis, and put \(a_{0,j}=a_{v_1,j}\). The center mass multiplier is bounded by \(Cm_j\) on \(K\). Subtracting it from the charge and using the free center Dirac norm identity gives \[ \left\lVert p_Ca_{v,j}\right\rVert=\sqrt2\left\lVert D_v^C(0)a_{v,j}\right\rVert=O(m_j). \tag{85}\] This identity holds with spectators by integration, or first on finite spectator sums and then by closure.

Apply the abelian square formula of Proposition 2 and divide by \(m_j^2\). Dropping its nonnegative kinetic term yields \[ 0\ge \frac12\sum_{i=1}^9s_i^2\left\lVert c_i a_{v,j}\right\rVert^2 +\left\langle a_{v,j},A_{v,j}a_{v,j}\right\rangle+o(1), \qquad A_{v,j}=m_j^{-1}p_C\cdot R(v)c. \tag{86}\] The fermion multiplier and the spin part of the rotation generator contribute only \(O(m_j^{-1})\). The vector field \(R(v)c\) has zero divergence, so \(A_{v,j}\) is symmetric on these compactly supported \(H^1\) states.

Bounded center coordinates and (85) give \[\left\lVert A_{v,j}a_{v,j}\right\rVert+\left\lVert A_{v,j}a_{0,j}\right\rVert\le C.\] In the second term, the free Dirac identity for \(v_1\) bounds the full center gradient of \(a_{0,j}\), and hence bounds the angular operator for every \(v\). Symmetry therefore gives \[ \begin{split} &\left|\left\langle a_{v,j},A_{v,j}a_{v,j}\right\rangle -\left\langle a_{0,j},A_{v,j}a_{0,j}\right\rangle\right|\\ &\hspace{1cm}\le\left\lVert a_{v,j}-a_{0,j}\right\rVert \bigl(\left\lVert A_{v,j}a_{v,j}\right\rVert+\left\lVert A_{v,j}a_{0,j}\right\rVert\bigr)=o(1). \end{split} \tag{87}\] Only norm convergence of the difference is needed here. Bounded coordinate multiplication also permits replacement by \(a_{0,j}\) in the mass term with error \(o(1)\).

Average (86) over the spin basis. The identity \(\sum_\nu R(v_\nu)=0\) cancels the common angular expectation exactly. Since \(s_i^2\ge1\), \[0\ge\tfrac12\left\lVert |c|a_{0,j}\right\rVert^2+o(1) \ge\tfrac12b_0^2\left\lVert a_{0,j}\right\rVert^2+o(1).\] Thus \(a_{0,j}\to0\) in norm. The assumed norm differences give the same conclusion for every \(a_{v,j}\). ◻

Proof of Proposition 28. Suppose that the eventual uniform assertion (83) is false. Its quantified negation gives \(h_j\ge j\), \(R_j\ge j\), and unit vectors supported in \(R_j<|x|<4R_j\) such that \(\mathfrak q_{h_j,1}(\psi_j)/R_j^2\to0\). Dilation \(x=R_j\xi\), performed unitarily, and multiplication of the charge by \(R_j\) give parameters \[ h'_j=h_jR_j^3,\qquad m'_j=R_j^2,\qquad \frac{h'_j}{m'_j}=h_jR_j\longrightarrow\infty. \tag{88}\] Writing the dilated unit states as \(f_j\), we have \[\mathop{\mathrm{supp}}f_j\subset\{1<|\xi|<4\},\qquad \mathfrak q_{h'_j,m'_j}(f_j)=o((m'_j)^2).\] The sixteen nonnegative summands in this form imply \(\left\lVert D_v(h'_j,m'_j)f_j\right\rVert=o(m'_j)\) for each member of the spin basis.

First suppose, along a subsequence, that \(m'_j/\sqrt{h'_j}\to\infty\). The exact scaling of the bosonic potential is \[ U_{h',m'}(\xi)=\frac{(m')^4}{(h')^2} U_{1,1}\left(\frac{h'}{m'}\xi\right). \tag{89}\] Lemma 8 and (88) therefore imply \(U_{h'_j,m'_j}(\xi)\ge c(m'_j)^2|\xi|^2\) on the fixed annulus for all large \(j\). The fermionic multiplier in Proposition 2 has operator norm at most \(C(h'_j+m'_j)\) there, and \[\frac{h'_j+m'_j}{(m'_j)^2}\longrightarrow0.\] Dropping the nonnegative kinetic energy contradicts the assumed \(o((m'_j)^2)\) form bound.

Otherwise pass to a subsequence with \(m'_j\le C\sqrt{h'_j}\). This is the growing-mass regime of Lemmas 14 and 17. The norm measures concentrate on the commuting part of the compact annulus. Every such configuration has at least two joint centers. Cover this compact commuting set by finitely many separated-center charts, chosen small enough for the fast estimates for all sixteen spinors simultaneously. At each reference configuration its block-center vector is nonzero. Shrinking the corresponding chart gives \[ |c|\ge b_0>0 \tag{90}\] on it, with a possibly different positive constant in each of the finite charts. A fixed smooth gauge-invariant partition localizes to these charts and to a complementary region disjoint from the commuting set. Its charge commutators are \(O(1)=o(m'_j)\). The complementary norm tends to zero by the raw commutator estimate.

Consider one localized sequence, denoted by \(\psi_j\). In its full \((b,t)\) Hilbert space set \(a_{v,j}=P_v(b)\psi_j\). Lemma 17 gives \[ \left\lVert a_{v,j}-\psi_j\right\rVert=o(1),\qquad \left\lVert D_v^s(h'_j,m'_j)a_{v,j}\right\rVert=o(m'_j). \tag{91}\] We retain the fast Hilbert space as a spectator here. The slow center operator and the massive internal block operators anticommute, since they act on disjoint variables and odd Clifford factors. Their squared norm identity, first on a smooth compact slow core and then by closure, implies \[ \left\lVert D_v^C(m'_j)a_{v,j}\right\rVert=o(m'_j). \tag{92}\] This does not assert that \(P_v\) commutes with each block operator. Rather, it applies the identity for the slow operator to the vector \(a_{v,j}\), with its full dependence on \(b,t\). Compact slow support and local ellipticity ensure the individual slow domains, just as in (72).

The projections act only in the fast variables, so all the \(a_{v,j}\) are supported in the same compact center set as the localized states. By (90), this set is bounded away from zero. Regard the internal variables and the fast Hilbert space as spectators. Equations (91) and (92) give every hypothesis of Lemma 29, with mass \(m'_j\). The lemma therefore makes every \(a_{v,j}\), and hence \(\psi_j\), tend to zero in norm. Summing the finitely many charts and using the vanishing complementary norm contradicts \(\left\lVert f_j\right\rVert=1\). This proves (83) in both parameter regimes.

For the tail estimate, use a smooth squared radial partition consisting of an interior cutoff and annular functions \(\chi_j\), supported in \(R_j<r<4R_j\) with \(R_j\ge R\) and geometrically increasing radii. It can be chosen with bounded overlap, with the exterior functions exhausting the partition for \(r>4R\), and \(\sum_j|\nabla\chi_j|^2+|\nabla\chi_{\mathrm{in}}|^2\le C/R^2\). Averaging the unweighted Dirac localization identity over the spin basis gives \[\mathfrak q_{h,1}(\chi_{\mathrm{in}}\psi)+ \sum_j\mathfrak q_{h,1}(\chi_j\psi) \le\mathfrak q_{h,1}(\psi)+CR^{-2}\left\lVert \psi\right\rVert^2.\] The interior form is nonnegative, and the exterior forms are bounded below by \(aR^2\sum_j\left\lVert \chi_j\psi\right\rVert^2\). This proves (84). Graph approximation and monotone exhaustion of the partition justify the same argument for arbitrary form-domain vectors. Finally, Lemma 3 gives \(\mathfrak q_{h,1}(\psi)\le\tfrac32\left\lVert D_u(h,1)\psi\right\rVert^2\) on gauge and \(G\) invariants. The displayed tail bound proves their tightness. ◻

Proof. Let \(h_j\to\infty\), \(\left\lVert \psi_j\right\rVert=1\), and \(\left\lVert D_u(h_j,1)\psi_j\right\rVert\to0\), with the prescribed invariances. Proposition 28 and Corollary 27 give norm-exhausting profiles. By Proposition 26, every proper profile satisfies \(D_u^C(1)F_C=0\) in the full center graph domain.

Each internal kernel factor is a rotation singlet by Proposition 6. The phase and symmetry of the fast line in Lemma 16 and the compatibility in Lemma 21 therefore make \(G\) act on these profiles through its ordinary action on the center variables and center fermions. The abelian instance of Lemma 3, valid on the full graph domain by closure, gives \[0=\left\lVert D_u^C(1)F_C\right\rVert^2 \ge\tfrac23\left\langle F_C,\mathcal H_C(1)F_C\right\rangle.\] This is understood as a form inequality. The mass-only oscillator has a unique zero-energy line for each center color, as also follows from the zero-triple case of Lemma 10. In orthonormal center coordinates its bosonic factor is proportional to \[\exp\left(-\frac12\sum_{i=1}^9|s_i|\,|c_i|^2\right),\] and its fermionic factor is the empty \(T\)-polarized vacuum. Denote the normalized product by \(\Omega_C\). Its parity is even. Hence each proper profile belongs to \[ \left(\mathbb C\Omega_C\otimes\bigotimes_{\alpha=1}^k\mathcal K_{n_\alpha}\right) ^{\text{equal-size permutations}}. \tag{95}\] The full-collapse profile belongs to \(\mathcal K_N\) and is also even, by Proposition 6, regardless of the dimension of that space.

If every \(\psi_j\) were odd, symmetry transport would make every profile odd. The preceding identifications would force them all to vanish, contradicting (81). The negation of (93) would give exactly such a sequence, by choosing \(h_j\ge j\) and an odd unit vector with charge norm below \(1/j\). Thus the eventual uniform odd gap holds.

For each fixed positive \(h\), Proposition 7 gives self-adjointness and compact graph embedding on this invariant space. The square has a discrete spectrum with finite-dimensional eigenspaces. On the eigenspace of any positive eigenvalue \(\lambda\) of the square, \(D_u/\sqrt\lambda\) is an isometric bijection between its even and odd parts. The odd lower bound therefore excludes \(0<\lambda<\delta^2\) in both parities. Spectral expansion proves (94). There are no odd zero vectors; the index formula (15) consequently gives \(\dim\mathop{\mathrm{ker}}D_u(h,1)=p(N)\), all even, for \(h\ge h_1\). ◻

Proof. We first approximate \(\Phi_C\) in the center graph norm by admissible tests. Multiply its center factor by a smooth outer radial cutoff \(\zeta_R\), equal to one for \(|c|\le R\), zero for \(|c|\ge2R\), and with derivative bounded by \(C/R\). Multiply also by the collision cutoff \(\rho_\varepsilon\) of Lemma 25, chosen as a product over all unordered pairs. These cutoffs are rotation invariant and have the required permutation symmetry. The resulting vector \(f_{C,R,\varepsilon}\) belongs to \(\mathcal T_C\).

Since \(D_u^C(1)\Phi_C=0\), only cutoff commutators contribute to its charge. The outer error tends to zero by the Gaussian tail. For fixed \(R\), the smooth center factor is bounded on its support, so (75) bounds the squared collision error by \(C_R\varepsilon^7\). Its omitted \(L^2\) norm also tends to zero, since the collisions have measure zero. Consequently there is a sequence of such tests \(f_{C,\ell}\) with \[ \left\lVert f_{C,\ell}-\Phi_C\right\rVert\longrightarrow0, \qquad \left\lVert D_u^C(1)f_{C,\ell}\right\rVert\longrightarrow0. \tag{97}\] The order is explicit: for tolerance \(1/\ell\), choose the outer radius first and then choose the collision radius small enough for that fixed outer support.

For each fixed \(\ell\), Proposition 20 produces \(w_{C,h,\ell}=I_{h,u,1}f_{C,\ell}\), with the prescribed leading norm and charge output. We impose the remaining symmetries carefully, since the chartwise corrections need not have been chosen rotation invariant. Let \(A_G\) be Haar averaging over \(G\), and let \(E_+=(1+(-1)^F)/2\) be the even projection. For every fixed \(g\in G\), Lemma 21 and invariance of the profile give \[\left\langle w_{C,h,\ell},U_gw_{C,h,\ell}\right\rangle \longrightarrow\left\lVert f_{C,\ell}\right\rVert^2.\] These inner products are bounded uniformly in \(g,h\) for fixed \(\ell\). Haar dominated convergence and the projection identity \(\left\lVert A_Gw\right\rVert^2=\left\langle w,A_Gw\right\rangle\) show \(\left\lVert w_{C,h,\ell}-A_Gw_{C,h,\ell}\right\rVert\to0\). The identical argument for the two-element parity action gives \(\left\lVert w_{C,h,\ell}-E_+A_Gw_{C,h,\ell}\right\rVert\to0\). Moreover, \(D_u\) commutes with \(A_G\) and interchanges the parity projections, so \[\left\lVert D_u(h,1)E_+A_Gw_{C,h,\ell}\right\rVert \le\left\lVert D_u(h,1)w_{C,h,\ell}\right\rVert.\] Thus imposing these symmetries preserves the leading norms and does not increase the charge errors.

For fixed \(\ell\), the compact center supports on distinct partition strata are disjoint compact subsets of the gauge quotient. The leading identifications concentrate on those supports, so Lemma 21 makes the different lifts asymptotically orthogonal. Their projections have the same property by the norm approximation just proved.

There are only finitely many proper partitions. Choose an increasing sequence \(H_\ell\to\infty\) so that for every \(h\ge H_\ell\) the finitely many norm, overlap, symmetry, and graph-output errors at the fixed test level \(\ell\) are below \(1/\ell\). Taking that level for \(H_\ell\le h<H_{\ell+1}\), and normalizing the vectors, proves (96). Additional finitely many cross overlaps can be included when choosing \(H_\ell\); the required whole-block vectors and their concentration are constructed next. ◻

Proof. Put \(d=9(N^2-1)\), and let \[(S_h\phi)(x)=h^{d/6}\phi(h^{1/3}x).\] Then \(S_h\) is unitary and \(D_u(h,0)S_h=h^{1/3}S_hD_u(1,0)\). Choose a fixed compact radial cutoff \(\chi\) equal to one near zero, and set \(z_{a,h}=\chi S_h\phi_a\). The vectors \(S_h\phi_a\) concentrate at zero by their \(L^2\) integrability: for every \(\eta>0\), \[\int_{|x|>\eta}|S_h\phi_a|^2 =\int_{|y|>h^{1/3}\eta}|\phi_a(y)|^2\longrightarrow0.\] It follows that \(\left\lVert z_{a,h}-S_h\phi_a\right\rVert\to0\), proving the Gram matrix assertion. Lemma 4 allows the cutoff product in the graph domain, and its undeformed charge is \([D_u,\chi]S_h\phi_a\). The derivative of \(\chi\) is bounded and supported away from zero, so this norm tends to zero.

The massive multiplier has norm at most \(C|x|\). On the fixed support of \(\chi\), its norm on \(z_{a,h}\) tends to zero as well: if the support is contained in \(|x|\le B\), then \[\left\lVert |x|z_{a,h}\right\rVert^2 \le\eta^2\left\lVert \phi_a\right\rVert^2+ B^2\int_{|x|>\eta}|S_h\phi_a|^2.\] Take \(h\to\infty\) and then \(\eta\downarrow0\). This proves the charge estimate without assuming \(|y|\phi_a\in L^2\). Proposition 6 makes every \(\phi_a\) even and a rotation singlet; radial cutoff and dilation preserve these symmetries.

At any fixed level of the proper-partition construction, the compact center supports avoid collisions and hence avoid zero. Its lifts are therefore asymptotically orthogonal to all the \(z_{a,h}\). There are only finitely many such pairs. Include these overlaps in the threshold selection in Lemma 31 to obtain the simultaneous claim there. ◻

Proof of Theorem 1. Assume that \(\dim\mathcal K_n=1\) and Proposition 22 hold for \(2\le n<N\). Section 6 first proves the exterior estimate at size \(N\), using only proper profiles on compact annuli. It then constructs the full-collapse profile in \(\mathcal K_N\). Proposition 28 and Lemma 30 now show that, for all sufficiently large \(h\), the massive kernel on gauge and \(G\) invariants has dimension exactly \(p(N)\), with a uniform gap on its complement.

Choose any sequence \(h_j\to\infty\) in this range and an orthonormal \(p(N)\)-element basis of its massive kernel for each \(j\). Simultaneously extract profiles from this finite list. Equation (82) says that the resulting list is orthonormal in \[ \mathcal K_N\ \oplus\!\!\bigoplus_{C\text{ proper}}\mathbb C\Phi_C. \tag{99}\] The proper summand has dimension \(p(N)-1\). If \(\mathcal K_N\) were zero, this space could not contain the orthonormal list. Hence \(\mathcal K_N\ne\{0\}\).

For the opposite inequality, take an arbitrary finite orthonormal list \(\phi_1,\ldots,\phi_M\) in \(\mathcal K_N\). Lemmas 31 and 32 provide \(L=p(N)-1+M\) asymptotically orthonormal even gauge- and \(G\)-invariant vectors \(w_{1,h},\ldots,w_{L,h}\) with charge norms tending to zero. The gap (94) implies \[\left\lVert w_{a,h}-P_hw_{a,h}\right\rVert\longrightarrow0 \qquad(1\le a\le L).\] Their projected Gram matrix therefore tends to the \(L\)-by-\(L\) identity. For all sufficiently large \(h\) it is positive definite, so the \(L\) projected vectors are linearly independent in the \(p(N)\)-dimensional massive kernel. Thus \[p(N)-1+M\le p(N),\qquad M\le1.\] This applies to every finite orthonormal list, and hence proves \(\dim\mathcal K_N\le1\) without presupposing finite dimensionality. Together with existence it gives \(\dim\mathcal K_N=1\).

For \(N=2\), every proper internal block has size one. Its kernel factor is \(\mathbb C\), and there are no smaller nonabelian exterior estimates to assume. The same argument therefore starts the induction. At each subsequent size the order is: smaller-block kernel and exterior results, proper-stratum profiles, the current exterior estimate, the full-collapse profile, and finally the massive dimension comparison. This completes the simultaneous induction. Lemma 4 identifies \(\mathcal K_N\) with the kernel of the form Hamiltonian in (2), proving Theorem 1 on the entire configuration space. ◻