Proposition 6 ([Classwise convergence of the normalization]). For r∈0,iπ/2 put σ(r)=38+2r²3π², A_m(r)=e^-Im² m^-σ(r) P_m(e^r). There are a fixed integer Q≥1 , a number δ>0 , and positive constants a_b(r) , 0≤b<Q , such that, as …

Use the transformed sum, whose exponent is. Choose a sufficiently large fixed g and put T=2a_0^ -1 m+g+iθ=T'+iθ, |θ|≤π/2. The stability estimate Lemma 5 ([Spatial stability at a real period]) and the [gas:grand-localization] localization bound continue to hold with this choice. The coercivity error is O(1/g) Q , absorbed by taking g large, and images and contour displacements cost only O(1) per particle in the linear term. Choose the period graph with a fixed cutoff profile in the real distance from either edge ±T'/2 . In coordinates relative to ±T/2 , its shape is then independent of m on the corresponding half interval. Multiply the lifted polynomial on this nome circle by e^ -2Am^2e^ -2T e^ -Im^2 m^ -σ(r) , and denote its value by F_m(θ,r) . The factor (-2Am^2e^ -2T ) is analytic in the nome and equals one at its center, so the circle mean is exactly A_m(r)=1π∫_ -π/2 ^ π/2 F_m(θ,r)\,dθ. We prove uniform exponentially fast convergence of F_m along each class; the same convergence then holds for this average. The scalar factor. Include (Tr^2/(4h_*)) from with the scalar prefactor. For the quadratic bulk term, retain both the residue at the pole -a_0i and its reflection in the sampling calculation--. After the displayed subtractions, the additional term is a constant multiple of m^2e^ -a_0T =e^ -a_0(g+iθ) , with error exponentially small in T' . For the logarithmic determinant in, use the complete all-mode sum of -(1-e(p)) and subtract its value at zero. The symbol is even and regular in a strip, decays exponentially, and continues to sectors of small slope about the real rays. Shifting the inverse Fourier contour and bending its tails therefore makes the nonzero period translates exponentially small. Its contribution is πT/(16L) plus a constant and an exponentially small error. The square-root factors have constant product. The powers of m from these terms are exactly m^ σ(r) , so the normalized scalar converges to a bounded function of θ,r , uniformly with exponentially small error. The edge integrals. Compare m with any larger m' in the same residue class, at the same θ . In both sums retain only particles whose real positions lie within εT' of an edge, where T' is the smaller period, retain at most T' particles, and set j equal to minus the total W -charge of the right edge. The omitted part has exponentially small integrated absolute weight. A particle away from the edges pays the localization factor f_0 ; large counts are suppressed by the factorials; and any other j leaves D+j≠0 throughout a central interval of length comparable to T' . Since g is fixed, the bound on the full integrated absolute weight is uniform in m . Parameterize the retained integrals by positions relative to their own period edges. For sufficiently small fixed ε , the two integrands agree up to an exponent error O(e^ -cT' ) . Here are the components of that comparison. The αWW^t/p^2 Schur energy combined with j -terms completes exactly to α∫( D +j)^2 ds in the quadratic form, where the staircase follows real order and integration is along the complex graph. Also the r term involving positions uses only ∑W_d s_d+jT , by the given formula for u_* . Both are unchanged in edge coordinates since D +j=0 between edges. The residual tilt terms (after extracting the bulk above) involve only counts, j , and kernel constants independent of positions and period. In the skew pole keep its sign part and the nearest translate in its smoothing correction, and in the regular symbol keep the nearest translate alone, with the same regularization on self terms. These prescriptions coincide for both sizes (sign of the original difference unchanged); all other images cost exponentially small times squared particle count by the sector decay already proved. In each chemical primitive the integral from 0 to the relevant edge ±T/2 equals ±1/2 before multiplying by i2πmσ_i (even periodized density of mass one, analytically continued). Between there and the particle, m times the density equals (√3 L)^ -1 e^ -a_0(g+iθ)/2 (e^ a_0 v +e^ -a_0 v ) up to uniform exponentially small additive error, v the coordinate of integration relative to the edge. This follows by expanding the two nearest images of the sinh quotient, all remaining ones exponentially suppressed; take ε small and include the fixed offset at the endpoint. Retain the discrete phase ε_m(ν) of each term in. It depends on the species counts and winding data, not on the positions or on T . The signs were independent of K_ count , so shifting its Poisson sum by the integer m introduces no new sign. The remaining m -dependent phases come from fixed logarithm branches raised to the power m and from the half-period chemical primitives. Every branch multiplier is a root of unity: the shifts and offsets are rational multiples of π , with fixed rational coefficients in the logarithmic factors. The sinh image products, the fixed powers of i from the trigonometric substitutions, and the integer branch jumps give only finitely many types of such coefficients. Write each such multiplier as e^ 2πiρ with ρ∈ℚ , and choose Q to be a common multiple of these denominators and of the parity modulus. Multiplying these coefficients by arbitrary integer particle counts or windings introduces no new denominator. Hence the entire discrete phase agrees for m,m' in the same class modulo Q . We do not require it to be independent of m across different classes. The image errors above are at most an exponentially small quantity times the square of the particle count. The retained count is at most T' , so they remain exponentially small after reducing the exponent c . The uniform integrated absolute bound now proves that F_m(θ,r) is Cauchy at exponential speed in T' along each residue class. Equation gives Proposition 6 ([Classwise convergence of the normalization]) with limits not yet known to be nonzero, since T' m . Saturation excludes a zero averaged limit. The individual saturation estimates Proposition 2 ([Saturation at the two physical parameters]) have exponents 3/8 at r=0 and 5/24 at r=iπ/2 . Thus A_m(r)=o( m) at both values. If the averaged limit on a residue class were zero, the convergence rate would give A_m(r)=O(m^ -δ ) on that class, a contradiction. Both center values are positive, so every nonzero class limit is positive. There are finitely many classes; taking their minimum and maximum gives bounded factors at both values. Finally σ(0)-σ(iπ/2)=1/6 , proving.