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Choose two separating axes valid on the overlapping compact center support. To put b+z into the second slice, solve its off-block slice equation by a gauge rotation with off-block generator w(b,z) . The differential of that equation at z=0 is the invertible map w↦[w,b_ e' ] on n . The implicit function theorem with the fixed spectral gap gives w(b,z)=O(|z|) with uniformly bounded smooth coefficients. Since [ n, k_0]⊂ n , its first action on b is off-block. Thus the block-diagonal part of the transformed configuration satisfies b'=b+O(|z|^2), z'=L_bz+O(|z|^2), where L_c is the linear change of slice obtained by projecting along ([w,c_i])_i . A fixed change of block frames or labels can be applied before this calculation. On bounded rescaled sets |y|,|t|≤M , h^ 1/3 |b'-b|=O_M(h^ -2/3 ), t'=L_ct+o(1), c'=c+O_M(h^ -1 ). The spin action of the gauge rotation tends uniformly to the identity there. The density factors in , together with the Jacobian of L_c , are exactly those of the unitary change between the two flat-measure transverse slices. This can also be seen by mapping each slice to the same perpendicular slice for ([w,c_i])_i : both start from the same ambient Euclidean volume and the same homogeneous space orbit measure. All these density square roots are positive. The transformed Gaussian is therefore the same normalized oscillator Gaussian, and its spin factor agrees by the phase convention of Lemma 15 ([The neutral fast line]). For bounded smooth internal profiles these statements prove norm convergence by change of variables and dominated convergence on bounded (y,t) sets. They extend to an arbitrary fixed internal L^2 kernel tensor by approximation: truncate and approximate that tensor on a bounded y set, use the unitary coordinate changes and bounded physical cutoffs to bound the approximation error, and then remove the truncation. The fast tails are uniformly small by the oscillator moments. No estimate of an uncut y moment is involved. The same argument applies to D_v^C(m)f , which has the same finite collection of internal kernel factors and a smooth compact center coefficient. For block relabelings the only additional transformation is the ordinary graded transport of the slow factors: the fast line has no extra character by Lemma 15 ([The neutral fast line]). Rotating a chart by a fixed element g of (3)×(6) gives the same construction and the natural profile action. Its charge output transforms with v replaced by gv , as follows from charge covariance. In particular for rotations fixing v it transforms the same charge. These are statements for each fixed g ; a uniform assertion over all group elements is not needed. To finish the global construction, take smooth functions λ_ν(c) subordinate to finitely many labeled center charts and summing to one on the test support. Sum the local lifts of λ_νf , making the finite block permutation average when passing to the quotient. The overlap calculation identifies their leading sum with J_hf , and their outputs with J_h∑_νD_v^C(m)(λ_νf) = J_hD_v^C(m)f. The product-rule terms cancel because ∑_νλ_ν=0 on the support. Thus the partition introduces exactly the center Dirac product rule already present in the required output. Finally, compact center supports away from collisions for distinct unordered partitions are disjoint compact subsets of the commuting gauge quotient. The norm measures of their leading products concentrate on those supports. Disjoint neighborhoods and Cauchy--Schwarz therefore make their mutual inner products tend to zero. Charge outputs have the same limiting supports, so this also proves orthogonality of the indicated outputs and the mixed inner products. Within one partition, the unitary rescalings, normalized fast vector, and fixed orbit-volume convention give exactly the profile inner products; polarization proves the finite-family assertion.
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