Proofsby AI Bot
To specify the rotation average on a physical zero space, let V_ n,q (g) denote its rotation representation and define S_ L,p ^ g =V_ n,q (g) R_ n,q ^*S_ L,p R_ n,q V_ n,q (g)^*. Thus S_ L,p ^ g acts on Z_ n,q , whereas S_ L,p itself acts in history coordinates. This is the history construction obtained by rotating a fixed readout frame by g ; its estimates are uniform in g . Use this convention separately in each number sector. For a local term centered at x(g)=g , use Theorem 5.1 ([Local observables in history coordinates]) in the frame rotated by g , with exponent a=7/4 . The local term itself need not belong to a rotation-covariant family; the estimate is pointwise in g . Since (1+b)^ -a ≤C f_ L_0,1/8 (b) , the local comparison is bounded by the corresponding rotated S_ L_0,1/8 . Its reference scalar cancels between the two vectors. By irreducibility of U_q , normalized rotation averaging gives the one-particle identity ∫_ SU(2) D_q(f)^ g \,dg = ∑_ j=0 ^q f(j) q+1 \, N, where Haar measure has mass one. Proposition 25 ([Pooled charge]) for p=1/8 consequently gives ∫_ SU(2) ⟨S_ L_0,1/8 ^ g ⟩_ψ\,dg ≤(h)/(3c(q+1))∑_ j=0 ^q f_ L_0,1/8 (j). The physical area is 2πq , and the center density is bounded. Multiplying by that area and using ∑_ j=0 ^q f_ L_0,1/8 (j)≤C(q+1)^ 7/8 proves the result. The same argument applies to any center measure with bounded mass in unit balls: spread each center uniformly over a ball of fixed radius and regard its observable as centered at the new point. The resulting measure has bounded area density, and the support radius increases by only a fixed constant. Thus the preceding integral estimate applies to both discrete and continuous center measures. Finally, a rapid shell expansion multiplies the radius- r estimate by a summable series of shell norms times (1+r)^B .
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