Proposition 23 ([Profiles on proper strata]). Assume the induction hypotheses at all sizes less than N . Fix m≥0 , let h_j→∞ , and suppose that ψ_j is a bounded sequence of gauge-invariant vectors with locally bounded D_u(h_j,m) graph …

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. 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 13 ([Raw local estimates]) therefore imply that any limiting norm measure is supported on commuting configurations. Indeed, on a compact subset where ∑_ i<j |[x_i,x_j]|^2≥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 ψ_j . All the estimates that follow are uniform on this fixed set. Lemma 16 ([Projection and the slow graph bound]) gives ψ_j-P_uψ_j⟶0, P_uψ_j=χ_u(b)η_j(b), D_u^s(h_j,m)P_uψ_j≤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(χ_uη_j) =χ_uD_u^sη_j+ ∑_ℓa_ℓ(∂_ b_ℓ χ_u)η_j, where the a_ℓ are constant slow Clifford matrices. The last term has norm at most Cη_j . 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 D_u^s(h_j,0)η_j≤C . Covariance of χ_u means that η_j has the separate gauge symmetry in each internal (n_α) factor. Put w_α=b_α-c_α_ n_α . On the slow tensor product, with all graded signs included, D_u^s(h_j,0)=D_u^C(0)+∑_ α:n_α>1 D_ u,α (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 D_u^s(h_j,0)η_j^2 =D_u^C(0)η_j^2+ ∑_ α:n_α>1 D_ u,α (h_j,0)η_j ^2. 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 ∇_cη_j≤C , since D_u^C(0)g^2=12∇_cg^2 . Rescale the internal variables unitarily by y_α=h_j^ 1/3 w_α , and denote the resulting function by η_j(c,y) . Then D_ u,α (1,0)η_j ≤Ch_j^ -1/3 , ∇_cη_j≤C. For an internal cutoff ζ of fixed compact support, write the constant-coefficient kinetic Dirac operator as D_ u,α (1,0) minus its bracket multiplier. That multiplier is bounded on ζ . The kinetic norm identity and the bounded commutator with ζ give ∇_ y_α (ζη_j) ≤C_ζ( D_ u,α (1,0)η_j +η_j). 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 η_∞(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 against compactly supported smooth functions shows distributionally that D_ u,α (1,0)η_∞=0 for each α . 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 3 ([Minimal and maximal domains]) identifies this with _ n_α . Equivalently, the orthogonal projection onto each internal kernel fixes η_∞ ; the product of these commuting projections identifies its coefficient space as _α_ n_α . This constructs the local profile. norm on the stratum. We now identify how much limiting norm is assigned to the stratum w=0 . The replacement of ψ_j by P_uψ_j changes integrals against bounded multipliers by o(1) . The physical fast coordinates tend to zero in norm measure, since z=t/√h_j and tψ_j≤C . On bounded y sets, smoothness also gives χ_u(c+h_j^ -1/3 y)-χ_u(c)→0 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≤κ_ε(w)≤1 be a product of radial internal cutoffs, equal to one when every |w_α|≤ε and zero when some |w_α|≥2ε . Choose them decreasing as ε0 . 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 tψ_j≤C and z=t/√h_j . The slice density has already been absorbed unitarily, and ∫|χ_u(b,t)|^2\, t=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_α|≤M for all α . The difference between such an integral and the integral with κ_ε is bounded above, apart from this local convergence error, by a_∞∑_ α:n_α>1 ∫_ Mh_j^ -1/3 <|w_α|<2ε |η_j|^2. Each summand satisfies Lemma 22 ([Quantitative exclusion of intermediate scales]), with all other variables as spectators, because of. For the converse comparison, the bounded y set lies where κ_ε=1 for all sufficiently large j . Thus its limiting integral is a lower bound for the one with κ_ε . To record the order of limits explicitly, put J_ε=∫a(c)κ_ε(w)\,μ, A_M=∫_ _α|y_α|≤M a(c)|η_∞(c,y)|^2\, c\, y. For every fixed L≥1 and M>e^L_αR_ n_α , the preceding comparisons and , after j→∞ , give A_M≤J_ε≤A_M+ a_∞∑_ α:n_α>1 c_ n_α ^ -2 (2e^LεK_α+CB/L)^2, where K_α are the block graph bounds in . Decreasing ε restricts J_ε to w=0 , by continuity from above and concentration of the fast coordinates at zero. Next let M→∞ , keeping its lower threshold, and finally let L→∞ . The error vanishes and A_M exhausts the profile norm. Thus the restricted measure is exactly |F_C(c)|^2\, c . 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 . and inner products. Local strong convergence and approximation by bounded y,t sets, followed by Proposition 19 ([Corrected test lifts]), give the local version of . It also identifies the profile independently of the outer chart cutoff. If two cutoffs λ,λ equal one near the same smaller center support, then \| (λ-λ)I_ h_j,u,m f\|→0 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 20 ([Compatibility of leading identifications]) 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 for every compactly supported profile test. For a finite list ψ_j^1,…,ψ_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 π_*(ψ_j^a(x) ψ_j^b(x) \, x) . 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 F_C^a(c) F_C^b(c) \, c . This proves the polarized assertion. The same leading identifications commute with the indicated symmetries, which proves the final assertion.