Proof. First consider one node \(n\). For \(u=s-n\) and real \(t_l\), direct integration gives the entire identities \[\begin{align*} \frac{e^{i\pi t_lu}-1}{u} &=i\pi\int_0^{t_l}e^{i\pi tu}\,\mathop{}\!\mathrm dt,\\ \frac{e^{i\pi t_lu}-1-i\pi t_lu}{u^2} &=-\pi^2\int_0^{t_l}(t_l-t)e^{i\pi tu}\,\mathop{}\!\mathrm dt. \end{align*}\] The integrals include the removable values at \(u=0\). Multiply these identities by \(P_le^{i\pi t_ln}\) and sum over \(l\). The subtracted constant and linear terms cancel because \(P(n)=P'(n)=0\). We therefore obtain spectral measures for the two entire node fractions in (12). Their negative-frequency parts are conjugate reflections of their positive-frequency parts. Folding doubles the latter parts and gives exactly (17). These node measures are absolutely continuous; the atoms come only from \(CP\), with the zero frequency counted once and every positive frequency counted twice.

The coefficient row in (8) gives \((|P_1|,\ldots,|P_6|)=(2,2,2,1,2,1)\). The constant column consequently has variation \(|P_0|+2\sum_{l=1}^6|P_l|=25\). Taking absolute values in the two integrals above gives, per unit \(c_n\) and \(d_n\), respectively, \[\pi^2\sum_{l=1}^6t_l^2|P_l|=\frac{65\pi^2}{18}<36, \qquad 2\pi\sum_{l=1}^6t_l|P_l|=\frac{32\pi}{3}<36.\] The phases involving \(n\) have modulus one at real frequencies, so these bounds are independent of the node. They apply to the full measure and to its folded form.

For an absolutely summable list, the node measures now converge in total variation, proving (15). On every compact subset of the complex \(s\)-plane, \(e^{i\pi ts}\) and each of its \(s\)-derivatives are uniformly bounded for \(|t|\le1\). Thus the measure integral and all its derivatives are locally uniform limits of those for finite lists. On a compact set avoiding the nodes, the reciprocal factors in the cardinal sum are uniformly bounded over all \(n\), so absolute summability also gives uniform convergence of that sum there. The integral thus defines its entire extension, with the stated removals.

For a finite list, (9) gives (13): the designated double and simple fractions give the two displayed terms, while every other term vanishes to order at least two at the node. Locally uniform convergence of the functions and their derivatives passes these identities to the full list. Finally, conjugate symmetry of the measure gives reality on the real axis and the folded formula. ◻

Proof. For a complex parameter \(\gamma\) with \(\mathop{\mathrm{Re}}\gamma>0\), the Gaussian transform in dimension two, with kernel \(e^{-2\pi i x\cdot\xi}\), is \[ \widehat{e^{-\pi\gamma|x|^2}}(\xi) =\gamma^{-1}e^{-\pi|\xi|^2/\gamma}. \tag{22}\] For real \(\gamma>0\), squaring the Gaussian integral and using polar coordinates gives the one-dimensional value \(\gamma^{-1/2}\) at frequency zero. Integration by parts shows that the one-dimensional transform obeys \(F'(\xi)=-(2\pi\xi/\gamma)F(\xi)\). Solving this differential equation and taking the product in two coordinates proves (22) for positive real \(\gamma\). Both sides are holomorphic on the right half-plane; differentiation under the integral is locally justified by a polynomial times a decaying Gaussian. The identity theorem gives the complex case.

After multiplication by \(e^{-\pi hs}\), the integrand in (14) is a Gaussian with parameter \(\gamma=b(h-it)=-ib(t+ih)\), whose real part is the fixed positive number \(bh\). Using \(B=1/b^2\), its transform is \[\frac1{b(h-it)} \exp\left(-\frac{\pi|\xi|^2}{b(h-it)}\right) =e^{-\pi hs}\lambda(t)e^{i\pi z(t)s},\qquad s=b|\xi|^2.\] Integrating this identity against \(\mu_a\) proves the forward Fourier formula. The exchanges with Fourier integration, derivatives, and polynomial weights are justified uniformly: the measure is finite, the frequency interval is compact, and every differentiated real-space Gaussian is bounded by a fixed polynomial times a decaying Gaussian. On the transformed side, the real part of the Gaussian parameter is \(h/[b(h^2+t^2)]\), with a positive lower bound on \([-1,1]\). The same domination applies there. Both functions are therefore Schwartz.

Fourier transformation twice reflects a Schwartz function. The functions here are radial, so reflection fixes them and proves the reverse transform. Linearity gives (21). ◻

Proof. Write \(q(t)=\pi\mathop{\mathrm{Im}}z(t)\) within this proof. Direct calculation gives \[\begin{align*} |\lambda(t)|&=\frac1{b\sqrt{t^2+h^2}},\\ |z(t)|^2&=h^2+\frac{B^2-2Bh^2}{t^2+h^2},\\ q(t)&=\pi h\left(\frac{B}{t^2+h^2}-1\right). \end{align*}\] Since \(B>2h^2\), all three expressions decrease for \(t\ge0\). At zero, \(|\lambda|=5/\sqrt3<3\) and \(|z|=44/15\). At the other relevant endpoints, \[q(1)=\frac{26\pi}{435}>.187,\qquad q(5/6)=\frac{862\pi}{3845}>.70,\] and \[|\lambda(5/6)|=\frac{60}{\sqrt{2307}}<\frac54, \qquad |z(5/6)|^2=\frac{33476}{19225}<\left(\frac{33}{25}\right)^2.\] Finally, \[-q'(t)=\frac{2\pi hBt}{(t^2+h^2)^2}.\] The function \(t/(t^2+h^2)^2\) decreases for \(t>h/\sqrt3\). Its value at the last endpoint therefore gives \(-q'(t)\ge2000\pi/2523>2.48\) throughout \([5/6,1]\). Integrating from \(t\) to 1 proves the last inequality in (26). All strict numerical comparisons here follow from \(157/50<\pi<22/7\) and squaring the positive rational bounds.

One unit of spectral variation at frequency \(t\) contributes at most \[ \frac{|\lambda(t)|}{Q_n} \bigl(1+|D_n|+\pi|z(t)|\bigr)e^{-q(t)n} \tag{28}\] to the sum of the two absolute jet coordinates at \(n\). By (11), its prefactor is less than 22 globally and less than 5.3 on the last interval, since \[\frac{3(1+2.12+3\pi)}{1.75}<22, \qquad \frac{1.25(1+2.12+1.32\pi)}{1.75}<5.3.\] The variation bound is at most \(36\|a\|_1\). Summing (28) over all positive integers, a superset of the positive nodes, therefore gives \[\|S\|\le\frac{22\cdot36}{e^{.187}-1}<4000.\] For the final comparison, \(e^{.187}-1>.187+(.187)^2/2>.198\).

For the sharper tail, split the folded measure into \([0,5/6]\) and \((5/6,1]\), assigning the atom at \(5/6\) to the first part. The first part has variation at most \(36w+25c\). Its contribution to the row pair at \(n\) is therefore at most \[22(36w+25c)e^{-.70n}\le800(w+c)e^{-.70n}.\] On the last open frequency interval, only \(l=6\) remains in (17). Here \(|P_6|=1\) and \(\pi(1-t)\le\pi/6<1\), so the density has absolute value at most \(2\pi w\). The only atom in \((5/6,1]\) is the one at 1, of mass \(2C\). This part contributes at most \[5.3e^{-.187n} \left(2\pi w\int_{5/6}^1e^{-2.48n(1-t)}\,\mathop{}\!\mathrm dt+2c\right) \le5.3e^{-.187n}\left(\frac{2.54w}{n}+2c\right),\] because \(2\pi/2.48<2.54\). Summing over the integers \(n\ge r\) and using \(1/n\le1/r\) proves (27). Only absolute-value sums were used, so the same argument applies to several lists with their summed inputs. ◻

Proof. We first bound each analytic integrand on a disk twice as wide as its real integration interval. We then use polynomial exactness and positivity to turn one Taylor remainder bound into a quadrature error bound. The estimates are uniform in the derivative order.

For the \(j\)th density piece in (17), put \(u=(2j-1)/12\) and consider \(|t-u|\le1/6\). On this disk, \(|t|\le13/12<1.1\). The pole \(-ih\) of \(z\) and \(\lambda\) lies outside, and \[|\lambda(t)|\le\frac1{b(h-1/6)}<5, \qquad |z(t)|\le\frac{B}{h-1/6}+h=\frac{214}{35}<6.2.\] Inversion sends the disk centered at \(u+ih\) with radius \(1/6\) to the disk centered at \((u-ih)/(u^2+h^2-1/36)\) with radius \((1/6)/(u^2+h^2-1/36)\). Thus \[ \mathop{\mathrm{Im}}z(t)\ge \frac{B(h-1/6)}{u^2+h^2-1/36}-h \ge-\frac{1402}{17505}>-.081. \tag{33}\] The middle expression decreases as \(u\) increases, so its minimum among the six disks occurs at \(u=11/12\).

For positive indices, the moduli of the coefficients in (8) are \((2,2,2,1,2,1)\). Hence \[\sum_{l=1}^6|P_l|=10,\qquad \sum_{l=1}^6t_l|P_l|=\frac{16}{3}.\] For a \(c_n\) basis column, the density on the disk has modulus at most \[2\pi^2e^{\pi n/6}\sum_{l=j}^6|P_l|(t_l-u+1/6) \le\frac{37\pi^2}{3}e^{\pi M/6}<220e^{\pi M/6}.\] The last sum is bounded by \(16/3+(1/12)10=37/6\); the first disk gives the largest possible added term. For a \(d_n\) column the smaller bound \(20\pi e^{\pi M/6}\) suffices. These arguments include \(n=0\). The constant column has only atoms, so it has no quadrature error.

The direct scaled kernel factor satisfies \[|e^{i\pi tm}|\frac{|i\pi ty|^a}{a!} \le e^{\pi M/6}e^{1.65\pi},\] and the transformed factor satisfies \[|\lambda(t)e^{i\pi z(t)m}|\frac{|i\pi z(t)y|^a}{a!} \le5e^{.081\pi M}e^{9.3\pi}.\] Here \(x^a/a!\le e^x\) for \(x\ge0\) removes all dependence on \(a\). The resulting direct and transformed integrand bounds are respectively \[220e^{\pi M/3+1.65\pi},\qquad 1100e^{\pi M/6+.081\pi M+9.3\pi}.\] For completeness, the elementary estimates used to bound these two numbers need no matrix or interval computation. Polynomial division gives \[0<\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,\mathop{}\!\mathrm dx=\frac{22}{7}-\pi.\] The positive exponential series, whose tail after degree six is bounded by a geometric series of ratio \(1/8\), gives \[e<\sum_{j=0}^6\frac1{j!}+\frac1{7!(1-1/8)} =\frac{95901}{35280}<\frac{2719}{1000}.\] With \(M=168\), the two exponents above are less than \(182\) and \(160\), respectively, after using \(\pi<22/7\). The exact rational comparisons \[220(2719/1000)^{182}<10^{83},\qquad 1100(2719/1000)^{160}<10^{83}\] therefore bound both analytic integrands by \(10^{83}\). This is the degree-six elementary route for the present allowance; it is separate from the scalar method developed in Appendix 9.

The actual interval is \(|t-u|\le1/12\), half the disk radius. Cauchy’s coefficient estimate bounds the Taylor tail after degree \(383\) there by \[10^{83}\sum_{r=384}^{\infty}2^{-r} =2\cdot10^{83}2^{-384}.\] The degree-\(383\) polynomial is integrated exactly. Both integration and quadrature are positive functionals of total mass one when the six intervals are combined, by (32). Applying each to the tail thus bounds their difference by \[4\cdot10^{83}2^{-384}<10^{-31};\] the final comparison is the integer inequality \(4\cdot10^{114}<2^{384}\). Taking real parts cannot enlarge the error, and no atom was changed. ◻

Proof. Write \(x=1/k\) and \(w_j=e^{-k(n_j-1)}\) for \(j=2,\ldots,5\). The physical vector lies in the box \[x\in[1/J,1/K],\qquad w_j\in[e^{-J(n_j-1)},e^{-K(n_j-1)}].\] On this larger box the expression \(a_1+xb_1+\sum_{j=2}^5(a_j+xb_j)w_j\) is affine in each variable while the others are fixed. Its minimum therefore occurs at a vertex. First minimizing each \(w_j\) and then the single shared \(x\) gives exactly (37). This is a containing-box bound, not sampling of the curve \(q(k)\). ◻

Finite verification. We specify the complete stronger exact-array comparisons in Tables 1–3 and the route by which each is checked. Their looser consequences are the proposition’s outputs, whose analytic use is separate. No precomputed coefficient list or approximate auxiliary function is an input: all arrays are reconstructed from (31)– (36).

The power sums use nine doubling steps. Start with \(U=D\) and \(V_+=V_-=I\). At step \(r=0,\ldots,8\), replace \(V_+\) by \(V_+(I+U)\) and \(V_-\) by \(V_-(I+\epsilon_rU)\), where \(\epsilon_0=-1\) and \(\epsilon_r=1\) for \(r>0\); square \(U\) before the next step. Before step \(r\), \(U=D^{2^r}\), and \[\prod_{r=0}^8(I+D^{2^r})=\sum_{j=0}^{511}D^j, \qquad (I-D)\prod_{r=1}^8(I+D^{2^r})=\sum_{j=0}^{511}(-D)^j.\] Thus the final sums are exactly those in (36). Table 1 gives strict upper bounds for \(U\) before each update and for both sums after it. Row 6 therefore bounds \(D^{64}\).

There are \(46\) centers: the seven complete residue blocks in \([0,83]\) contribute \(42\), and \(84,85,87,88\) contribute four more. Every center has a right half-gap and every center except zero has a left one, giving \(91\). Consequently the sign table consists of \(91\cdot2\cdot5\cdot41=37310\) separate real lower comparisons, and the tail column consists of ten. The power and column tests likewise compare every indicated norm, including each same-column paired norm.

The selected finite route encloses the exact mathematical quadrature arrays directly by Arb ball arithmetic, whose inclusion principle is described by Johansson (Johansson 2017). The local source implements these formulas at \(256\)-bit precision. For an upper comparison the required test is that the whole ball lies below its table cutoff; for a lower comparison it is that the whole ball lies above it. It performs these tests for all indices just counted. All comparisons in Tables 1–3 pass these strict inclusion tests. The containing-box lemma, Bernstein range property, and analytic remainders provide continuum coverage in \(s\) and \(k\).

Finally, the stronger cutoffs imply the displayed proposition bounds: \(.0031<.004\), \(32.12+1.85<36\), \(2.20<2.3\), \(5.40<5.6\), \(.00116>.001\), and \(.0022>.0021\). Appendix 11 gives a second, fully specified floor-rounded rational procedure and proves that a successful comparison with these same stronger cutoffs implies the looser proposition bounds after its roundoff allowances. ◻

Proof. At the origin, \(0\le s\le1/2\), and \[\frac{\partial T_k(s)}{\partial k} =T_k(s)\left(1-s-\frac1k\right)>0\] because \(k\ge2.36>2\). Replacing \(k\) by \(K\) therefore lowers the target. The Taylor polynomial of \(T_K(yx)\) through degree 40 has coefficients \((e^K/K)(-Ky)^r/r!\), exactly the first case in (39).

At the first positive node, write \(u=s-1\). The deleted quotient is \[\mathcal D_{1,2}T_k(1+u)=\frac{e^{-ku}-1+ku}{ku^2},\] with value \(k/2\) at \(u=0\). For \(u\ne0\), its derivative with respect to \(k\) is \[\frac{1-(1+ku)e^{-ku}}{k^2u^2}\ge0,\] by \(e^{ku}\ge1+ku\). The same monotonicity holds at the removable value. This works for both signed half-gaps, including negative \(u\). Replacing \(k\) by \(K\) again lowers the quotient, and its Taylor coefficients are \(K(-Ky)^r/(r+2)!\), the second case in (39).

In both cases the left endpoint satisfies \(K\le6\), even for the last, unbounded parameter interval. At the origin \(K|y|\le3\), and at node 1 \(K|y|\le6\). Also \(e^K/K\) increases for \(K>1\), so its value is at most \(e^6/6\). The absolute tails after degree 40 are bounded by \[\frac{e^6}{6}\frac{3^{41}}{41!(1-3/42)}<8\cdot10^{-29}<10^{-6}, \qquad \frac{6\cdot6^{41}}{43!(1-6/44)}<10^{-20}<10^{-6},\] using the scalar comparisons in Appendix 9. Thus no Taylor estimate with unbounded \(k\) is needed.

At every positive node, Taylor’s integral formula also gives the general nonnegative support \[ \mathcal D_{m,2}T_k(m+u) =k\int_0^1(1-v)e^{-k(m-1+vu)}\,\mathop{}\!\mathrm dv\ge0. \tag{63}\] This handles all cases in which (39) prescribes zero. It remains to justify the stronger constant at \(m=3\) for finite \(J\). Here \(-1\le u\le1/2\), so \(a=2+vu\ge1\). The function \(k e^{-ak}\) decreases for \(k\ge2.36\), and (63) is at least \[J\int_0^1(1-v)e^{-J(2+vu)}\,\mathop{}\!\mathrm dv.\] The probability density \(2(1-v)\) on \([0,1]\) has mean \(1/3\). Jensen’s inequality for the exponential therefore bounds this expression below by \[\frac J2e^{-J(2+u/3)}\ge\frac J2e^{-13J/6},\] which is the third case of (39). When \(J=\infty\), the prescribed support is zero and (63) applies. ◻

Proof. Choose a half-gap and a parameter interval containing \(k\). Combine the finite rows and the target support into \[\Phi_i(x;k)=\sum_{r=0}^{40} \bigl(l_{i,r}q(k)+\delta_i a_r^*\bigr)x^r.\] By the Bernstein conversion defining (38), the coefficient of the \(d\)-th degree-40 Bernstein basis function is \[L_{i,d}q(k)+\delta_i\sum_{r=0}^d\omega_{dr}a_r^*.\] The containing-box lower evaluation (37) applies to the physical vector \(q(k)\). Together with Proposition 9, it makes every coefficient above strictly greater than \(.001\). The Bernstein basis functions are nonnegative and sum to one on \([0,1]\), so \(\Phi_i(x;k)>.001\) on that entire interval.

Apply (60), (61), and Lemma 11 to (59). The result is the strict quotient margin \[\mathcal D_{m,\nu}F_i(m+yx) >.001-.00011-.000001-.000001=.000888>0.\] At the origin this is the desired inequality. At a positive center, multiplication by \((s-m)^2\) proves the inequality away from the center, and (58) supplies equality at the center. The half-gaps cover \([0,s_0]\), while the five parameter intervals cover every finite \(k\ge2.36\). ◻

Proof. The set \(\mathcal Z\) is discrete and periodic, so a nearest point exists. Between a zero \(m\) and either adjacent midpoint, the quotient \(P(s)/(s-m)^2\) is positive after its removable value is filled in. Its logarithm is concave. For the factor vanishing at \(m\), with \(u=s-m\), \[\frac{\mathop{}\!\mathrm d^2}{\mathop{}\!\mathrm du^2} \log\left(\frac{\sin^2(\pi u/12)}{u^2}\right) =\frac2{u^2}-2\left(\frac\pi{12}\right)^2 \csc^2\left(\frac{\pi u}{12}\right)\le0\] for \(u\ne0\), by \(|\sin x|\le|x|\), and the inequality extends through the removable value. Every other sine factor has nonpositive logarithmic second derivative on this half-gap. A concave function is bounded below by the smaller of its endpoint values, so the quotient is bounded below by its smaller value at the center and the midpoint.

At the center, the periodicity of \(P\) reduces the removable value to the corresponding node value \(Q_n>1.75\) in (11). At the midpoints, the exact values are \[\begin{array}{c|c} \text{gap modulo }12&P(s)/(s-m)^2\text{ at its midpoint}\\ \hline {[0,1],\ [3,4]}&12-8\sqrt2\\ {[1,3]}&1\\ {[4,7],\ [9,12]}&\dfrac89(3+\sqrt2+\sqrt3+\sqrt6)\\ {[7,9]}&9 \end{array}\] Here the denominator is the square of the half-gap length, independent of which endpoint is chosen at a tie. For clarity, the smallest entry can be derived directly. At \(s=1/2\), put \(\theta=\pi/24\) and pair the unsquared factors for residues \((1,7)\), \((3,9)\), and \((0,4)\). Product-to-sum gives \[\begin{align*} (2\sin(-\theta))(2\sin(-13\theta))&=2\sin(\pi/12),\\ (2\sin(-5\theta))(2\sin(-17\theta))&=2\sin(5\pi/12),\\ (2\sin\theta)(2\sin(-7\theta))&=1-\sqrt2. \end{align*}\] The product of the first two right sides is one. Thus \(P(1/2)/(1/2)^2=4(1-\sqrt2)^2=12-8\sqrt2\). The same product-to-sum calculation at the other midpoints gives the remaining entries. The identity \(P(4-s)=P(s)\), which follows from invariance of \(L\) under \(a\mapsto4-a\) modulo 12, accounts for the paired gap types in the table. The smallest entry satisfies \(12-8\sqrt2>.6862>.68\), as recorded in Appendix 9; every other entry is larger than \(.68\) as well. The half-gaps cover the real line, proving the claim, including the zero values themselves. ◻

Proof. We first bound the low-node rational contribution, then compare the remaining term on the same quadratic scale.

Proof of Theorem 2. For every real \(k\ge2.36\), Proposition 10 supplies real summable lists with the prescribed extras. Section 2 makes \(H_1,H_2\) entire and real on the real axis, and Lemma 5 gives the real radial Schwartz Fourier pair \[f(x)=e^{-\pi hb|x|^2}H_1(b|x|^2),\qquad \widehat f(\xi)=e^{-\pi hb|\xi|^2}H_2(b|\xi|^2).\] Lemmas 12 and 14 cover all \(s\ge0\), so \(H_1(s)\le T_k(s)\) and \(H_2(s)\ge0\). The exact jets are \[H_1(n)=T_k(n),\qquad H_1'(n)=T_k'(n),\qquad H_2(n)=H_2'(n)=0\qquad(n\in\mathcal N).\] This proves the theorem. ◻

Proof. For finite real signed measures \(\mu,\nu\) on \(\mathbb R^2\), define \[Q(\mu,\nu)=\iint f(x-y)\,\mathop{}\!\mathrm d\mu(x)\,\mathop{}\!\mathrm d\nu(y), \qquad \widehat\mu(\xi)=\int e^{-2\pi i x\cdot\xi}\,\mathop{}\!\mathrm d\mu(x).\] Fourier inversion and Fubini’s theorem give \[Q(\mu,\nu)=\int_{\mathbb R^2}\widehat f(\xi) \overline{\widehat\mu(\xi)}\widehat\nu(\xi)\,\mathop{}\!\mathrm d\xi.\] Indeed, \(\widehat f\) is integrable and the transforms of finite measures are bounded. Since \(\widehat f\ge0\), this is a positive-semidefinite form and therefore satisfies Cauchy–Schwarz.

Put \(a=\widehat f(0)=\int f\ge0\) and \(\mu_R=\sum_{x\in\mathcal C_R}\delta_x\). If \(a=0\), positivity gives \(Q(\mu_R,\mu_R)/N_R\ge a\) whenever \(N_R>0\). Suppose now that \(a>0\). Fix \(\varepsilon>0\) and let \[\nu_R=1_{B_{(1+\varepsilon)R}}\,\mathop{}\!\mathrm dx, \qquad V_R=\pi(1+\varepsilon)^2R^2.\] For every \(x\in B_R\), the disk \(B_{(1+\varepsilon)R}-x\) contains \(B_{\varepsilon R}\). Thus \[ \bigl|Q(\mu_R,\nu_R)-N_Ra\bigr| \le N_R\int_{|z|\ge\varepsilon R}|f(z)|\,\mathop{}\!\mathrm dz=o(N_R). \tag{70}\] The estimate is uniform in the locations of all points of \(\mathcal C_R\); it does not use a bound on their number in any smaller disk.

The area self-term has the limit \[\frac{Q(\nu_R,\nu_R)}{V_R} =\int_{\mathbb R^2}f(z) \frac{|B_{(1+\varepsilon)R}\cap(B_{(1+\varepsilon)R}-z)|}{V_R} \,\mathop{}\!\mathrm dz \longrightarrow a.\] For each fixed \(z\), the overlap fraction tends to one and is between zero and one. Dominated convergence therefore applies with dominator \(|f|\). In particular, \(Q(\nu_R,\nu_R)>0\) for all sufficiently large \(R\). Cauchy–Schwarz and (70) now yield \[\frac{Q(\mu_R,\mu_R)}{N_R} \ge\frac{|Q(\mu_R,\nu_R)|^2}{N_RQ(\nu_R,\nu_R)} =\frac{N_R}{V_R}\frac{(a+o(1))^2}{a+o(1)}.\] Since \(N_R/V_R\to(1+\varepsilon)^{-2}\), taking the lower limit in \(R\) and then letting \(\varepsilon\downarrow0\) proves \[\liminf_{R\to\infty}\frac{Q(\mu_R,\mu_R)}{N_R}\ge a.\] There are exactly \(N_R\) diagonal terms, each equal to \(f(0)\), so removing them proves (69). The assertion for \(\Phi\) follows by termwise comparison of each finite off-diagonal sum. ◻

Proof. For \(\alpha\ge1\), take \[k=\pi(\alpha/b-h)\ge\pi(2/\sqrt3-2/5)>2.36.\] The elementary comparison is proved in Appendix 9. Let \(f\) be the function supplied by Theorem 2 for this \(k\), and define \(f_\alpha=ke^{-k}f\). The multiplier is positive, so \(\widehat f_\alpha\ge0\). With \(s=b|x|^2\), the upper bound for \(f_\alpha\) becomes \[ke^{-k}e^{-\pi hs}T_k(s) =e^{-(\pi h+k)s}=e^{-\pi\alpha|x|^2}.\] The shell containment and \(A^*=JA\) from Section 2 place every nonzero direct and dual shell in \(\mathcal N\). The value contacts in (6) therefore give (71).

For \(0<\alpha<1\), put \(\beta=1/\alpha\) and use the function already constructed for \(\beta>1\). Define its nonnegative radial Schwartz slacks \[q_\beta=G_\beta-f_\beta,\qquad r_\beta=\widehat f_\beta.\] The Fourier transform applied twice is reflection, which is immaterial for these radial functions. Thus \(G_\beta=q_\beta+\widehat r_\beta\). The dimension-two Gaussian transform from (22) gives \[\beta^{-1}G_\alpha=\widehat G_\beta =\widehat q_\beta+r_\beta.\] Set \[ f_\alpha=\beta\widehat q_\beta. \tag{72}\] Then \(G_\alpha-f_\alpha=\beta r_\beta\ge0\) and \(\widehat f_\alpha=\beta q_\beta\ge0\). The known contacts say that \(q_\beta\) vanishes on \(A\setminus\{0\}\) and \(r_\beta\) vanishes on \(A^*\setminus\{0\}\). Since \(A^*\) is a rotation of \(A\) and both functions are radial, \(q_\beta\) also vanishes on the nonzero dual shells and \(r_\beta\) on the nonzero direct shells. The two displayed slack identities give the required contacts for \(f_\alpha\). ◻

Proof. Compactness of the measures. We first record an elementary compactness argument that preserves the limiting total mass without discarding possible endpoint atoms. Let \((\mu_n)\) be any sequence of positive measures on \([0,1]\) with uniformly bounded masses. By a diagonal subsequence, arrange that the masses converge to \(m\) and that \(\mu_n([0,q])\) converges to \(a(q)\) for every rational \(q\in(0,1)\). For \(0\le x<1\), set \[H(x)=\inf_{\substack{x<q<1\\q\text{ rational}}}a(q), \qquad H(1)=m.\] The function \(H\) is nondecreasing and right-continuous. Its associated positive Borel measure \(\mu\) has distribution \(\mu([0,x])=H(x)\), including mass \(H(0)\) at zero and mass \(m-H(1-)\) at one. At every continuity point \(x\in(0,1)\) of \(H\), rational points on either side of \(x\) bracket \(\mu_n([0,x])\) and give convergence to \(H(x)\). Step-function approximations on partitions with such internal endpoints, together with convergence of the total masses and uniform continuity, then give \[\int\varphi\,\mathop{}\!\mathrm d\mu_n\longrightarrow\int\varphi\,\mathop{}\!\mathrm d\mu \quad\text{for every continuous }\varphi:[0,1]\to\mathbb R.\] Thus the subsequence converges weakly. The endpoint atoms have not been discarded in this construction.

Discrete moments. Fix \(d>0\) and put \(m_j=F(jd)\) for integers \(j\ge0\). Writing \(\Delta m_j=m_{j+1}-m_j\), repeated integration gives, for \(r\ge1\), \[(-1)^r\Delta^r m_j =\int_{[0,d]^r}(-1)^r F^{(r)}(jd+u_1+\cdots+u_r) \,\mathop{}\!\mathrm du_1\cdots\mathop{}\!\mathrm du_r\ge0.\] Define a linear functional on polynomials by \(\Lambda(x^j)=m_j\). For every integer \(M\ge1\), place at \(p/M\), \(0\le p\le M\), the mass \[w_{M,p}=\Lambda\!\left(\binom Mp x^p(1-x)^{M-p}\right) =\binom Mp(-1)^{M-p}\Delta^{M-p}m_p\ge0.\] Let \(\mu_{M,d}=\sum_{p=0}^M w_{M,p}\delta_{p/M}\). Its total mass is \(m_0\), because the Bernstein basis sums to one. For fixed \(0\le j\le M\), the identity \(\binom Mp\binom pj=\binom Mj\binom{M-j}{p-j}\) gives \[\sum_{p=0}^M\frac{\binom pj}{\binom Mj} \binom Mp x^p(1-x)^{M-p}=x^j.\] Applying \(\Lambda\) shows that \(\sum_p w_{M,p}\binom pj/\binom Mj=m_j\). For fixed \(j\) and \(M\ge\max\{1,j\}\), put \[P_{M,j}(x)=\prod_{r=0}^{j-1}\frac{Mx-r}{M-r}.\] These polynomials converge uniformly to \(x^j\) on \([0,1]\) as \(M\to\infty\): each of their finitely many factors converges uniformly to \(x\). At \(x=p/M\) their values are exactly \(\binom pj/\binom Mj\), so the identity above says \(\int P_{M,j}\,\mathop{}\!\mathrm d\mu_{M,d}=m_j\). Since \(\mu_{M,d}\) is positive with mass \(m_0\), \[\left|\int x^j\,\mathop{}\!\mathrm d\mu_{M,d}-m_j\right| \le m_0\|x^j-P_{M,j}\|_\infty\longrightarrow0.\] The compactness argument supplies one weakly convergent subsequence of \((\mu_{M,d})_{M\ge1}\). Since the moment convergence holds for every fixed \(j\) along the full sequence, its limit \(\eta_d\) satisfies all the moment identities \[F(jd)=\int_{[0,1]}x^j\,\mathop{}\!\mathrm d\eta_d(x) \qquad(j=0,1,2,\ldots).\] In particular, \(\eta_d\) has mass \(F(0)\).

Positive real exponents. For each positive integer \(l\), push \(\eta_{1/l}\) forward by \(x\mapsto x^l\) and call the resulting measure \(\rho_l\). It has mass \(F(0)\) and \[\int_{[0,1]}v^{j/l}\,\mathop{}\!\mathrm d\rho_l(v)=F(j/l) \qquad(j=1,2,\ldots).\] Choose one weakly convergent subsequence \(\rho_{l_n}\to\rho\); necessarily \(l_n\to\infty\), and the limit still has mass \(F(0)\). Now fix any \(t>0\) and set \(t_n=\lceil l_nt\rceil/l_n\). Since \(0\le t_n-t<1/l_n\) and \(\sup_{0<v\le1}v^t|\log v|=1/(et)\), the mean value theorem in the exponent gives the following bound; at \(v=0\) both powers are zero because their exponents are positive: \[\sup_{0\le v\le1}|v^{t_n}-v^t| \le\frac{t_n-t}{et}\longrightarrow0.\] Continuity of \(F\) and weak convergence against the continuous function \(v^t\) now imply \[F(t)=\lim_nF(t_n) =\lim_n\int v^{t_n}\,\mathop{}\!\mathrm d\rho_{l_n} =\int v^t\,\mathop{}\!\mathrm d\rho.\] The subsequence was chosen before \(t\), so the same \(\rho\) works for every \(t>0\). Finally, as \(t\downarrow0\), the functions \(v^t\) increase to \(1_{(0,1]}(v)\). Monotone convergence and continuity of \(F\) at zero give \(\rho((0,1])=F(0)=\rho([0,1])\), proving \(\rho(\{0\})=0\). ◻

Proof. Fix \(\varepsilon>0\) and set \(F_\varepsilon(t)=g(\varepsilon+t)\) for \(t\ge0\). It satisfies Lemma 17, with a finite representing measure \(\rho_\varepsilon\) of mass \(g(\varepsilon)\). Choose any real sequence \(R_n\to\infty\), discarding finitely many terms so that \(N_{R_n}>0\), and define \[A_n(v)=\frac1{N_{R_n}} \sum_{\substack{x,y\in\mathcal C_{R_n}\\x\ne y}}v^{|x-y|^2} \qquad(0\le v\le1).\] Every exponent is positive, so \(A_n\) is continuous and nonnegative on \([0,1]\), with \(0^t=0\) for \(t>0\). Define the nonnegative extended-valued function \[L(v)=\sum_{a\in A\setminus\{0\}}v^{|a|^2}\qquad(0\le v\le1).\] For \(0<v<1\), the Gaussian hypothesis with \(\alpha=-\log(v)/\pi\) gives \[\liminf_n A_n(v)\ge L(v).\] A full-radius lower-limit inequality holds along every sequence tending to infinity. The same comparison holds at the endpoints: \(A_n(0)=L(0)=0\), while \(A_n(1)=N_{R_n}-1\to\infty=L(1)\).

The finite-sum representation, Fatou’s lemma, and Tonelli’s theorem now give \[\begin{align*} &\liminf_n\frac1{N_{R_n}} \sum_{\substack{x,y\in\mathcal C_{R_n}\\x\ne y}} F_\varepsilon(|x-y|^2)\\ &\qquad=\liminf_n\int A_n(v)\,\mathop{}\!\mathrm d\rho_\varepsilon(v)\\ &\qquad\ge\int\liminf_n A_n(v)\,\mathop{}\!\mathrm d\rho_\varepsilon(v) \ge\int L(v)\,\mathop{}\!\mathrm d\rho_\varepsilon(v)\\ &\qquad=\sum_{a\in A\setminus\{0\}}F_\varepsilon(|a|^2). \end{align*}\] Every integrand and summand is nonnegative, so these steps remain valid when the last sum is infinite. Since the sequence \(R_n\) was arbitrary, this is the full-radius lower bound for \(F_\varepsilon\).

Complete monotonicity gives \(g'\le0\), hence \(g(t)\ge F_\varepsilon(t)\) for \(t>0\). Termwise comparison therefore yields \[E_g(\mathcal C)\ge \sum_{a\in A\setminus\{0\}}g(\varepsilon+|a|^2) \qquad(\varepsilon>0).\] As \(\varepsilon\downarrow0\), the summands increase to \(g(|a|^2)\). Monotone convergence in the lattice sum proves (76). No limit in \(\varepsilon\) has been interchanged with the configuration lower limit, and no unshifted representing measure has been used. ◻

Completion of the proof of Theorem 1. Equation (74) verifies the proposition’s hypothesis for every locally finite configuration of centered disk density one. This proves the inequality in Theorem 1. We finish by checking attainment, including the case of an infinite lattice sum.

Choose a bounded half-open fundamental cell of \(A\) with vertex zero and area one, and let \(d\) bound the distance of its points from its lattice vertex. Up to boundary sets of area zero, for \(R>d\) the union of cells whose vertices lie in \(A\cap B_R\) contains \(B_{R-d}\) and is contained in \(B_{R+d}\). Its area is \(\#(A\cap B_R)\), so \(A\) has centered disk density one. For each \(x\in A\cap B_R\), the differences \(y-x\) with \(y\in A\cap B_R\setminus\{x\}\) form a subset of \(A\setminus\{0\}\). Nonnegativity gives \[\sum_{\substack{y\in A\cap B_R\\y\ne x}}g(|x-y|^2) \le\sum_{a\in A\setminus\{0\}}g(|a|^2).\] If the lattice sum is finite, averaging gives the upper bound for \(E_g(A)\), and (76) applied to \(A\) gives the reverse bound. If the lattice sum is infinite, the same lower bound gives \(E_g(A)=\infty\). Thus the equality in Theorem 1 holds in every case. ◻

Proof. Put \(\alpha=1-s/2\). We first identify the scalar Green function whose heat representation has the field normalization used here. On the base plane, \((-\Delta)^\alpha g_s(\cdot,0)=q_s\delta_0\), with the extra coordinate omitted when \(s=0\). On \(T=\mathbb R^2/L\), the spectral operator \((-\Delta)^\alpha\) multiplies the Fourier mode \(e^{2\pi i\xi\cdot x}\), \(\xi\in L^*\), by \(\lambda_\xi^\alpha\), where \(\lambda_\xi=(2\pi|\xi|)^2\). Its mean-zero unit-source Green function \(G_L\) therefore has coefficient \(N^{-1}\lambda_\xi^{-\alpha}\) for each \(\xi\ne0\). To compare it with the full-space extension, suppose first that \(0<s<2\). For \(\lambda>0\), the even profile \[u_\lambda(y)=\frac1{\Gamma(\alpha)}\int_0^\infty t^{\alpha-1}e^{-\lambda t-y^2/(4t)}\,\mathop{}\!\mathrm dt\] decays at infinity and satisfies \[u_\lambda(0)=\lambda^{-\alpha},\qquad u_\lambda''+\frac{1-2\alpha}{y}u_\lambda'-\lambda u_\lambda=0 \quad(y>0).\] Differentiation and the substitution \(v=y^2/(4t)\) give its full, two-sided weighted flux: \[-2\lim_{y\downarrow0}y^{1-2\alpha}u_\lambda'(y) =2^{2-2\alpha}\frac{\Gamma(1-\alpha)}{\Gamma(\alpha)} =\frac{\kappa_s}{q_s}.\] Thus the unit-source extension mode is \((q_s/(N\kappa_s))u_{\lambda_\xi}\), whose trace is \((q_s/\kappa_s)N^{-1}\lambda_\xi^{-\alpha}\). The zero mode has no source: finite tail energy forces it to be constant, and the mean-zero trace sets that constant to zero. For \(s=0\) there is no extension coordinate; both Green functions solve \(-\Delta G=\delta_0-N^{-1}\) on \(T\) with zero mean, and \(q_0/\kappa_0=1\). Hence, throughout \(0\le s<2\), \(\mathcal G_L(x,0)=(q_s/\kappa_s)G_L(x)\), with the trace notation omitted at \(s=0\). The periodic formula above consequently becomes \[W_{s,L}=\kappa_sq_s\left[ \frac1N\sum_{i\ne j}G_L(a_i-a_j) +\lim_{x\to0}\left(G_L(x)-\frac{g_s(x)}{q_s}\right)\right].\]

Set \[B_{\mathcal C}(t,x)=\frac1N\sum_{i,j}P_{t,L}(a_i-a_j+x),\] so \(B_{\mathcal C}(t,0)=H_{\mathcal C}(t)+\Psi_t(0)\). The heat representation (Petrache and Serfaty 2020, Lemma 1) is \[G_L(x)=\frac1{\Gamma(\alpha)}\int_0^\infty \left(P_{t,L}(x)-\frac1N\right)t^{\alpha-1}\,\mathop{}\!\mathrm dt.\] Its spectral proof applies to any flat torus. To compare two periodic configurations \(\mathcal C\) and \(\mathcal D\), set \(S_{\mathcal C}(x)=N^{-1}\sum_{i,j}G_L(a_i-a_j+x)\), and define \(S_{\mathcal D}\) using its own period lattice and point count. Each \(S\) has singular part \(g_s(x)/q_s\). In their heat-integrand difference the background terms are both \(1\) and the diagonal terms are both \(\Psi_t(x)\), so these terms cancel. Uniformly for \(x\) near zero, the remaining difference is bounded by \(Ct^{-1}e^{-c/t}\) as \(t\downarrow0\), because the other image distances are positive, and by \(Ce^{-\lambda t}\) as \(t\to\infty\), by the two torus spectral gaps. Dominated convergence as \(x\to0\) consequently gives \[V_s(\mathcal C)-V_s(\mathcal D) =\frac{\kappa_sq_s}{\Gamma(1-s/2)}\int_0^\infty \big(H_{\mathcal C}(t)-H_{\mathcal D}(t)\big)t^{-s/2}\,\mathop{}\!\mathrm dt.\] This calculation also covers \(s=0\): the logarithmic singularity cancels before either endpoint is integrated. ◻

Proof of Corollary 19. Periodic configurations. The subtraction in \(H_{\mathcal C}\) removes only the coincident point; all other images, including images of the same point, remain. Periodic counting gives \[H_{\mathcal C}(t)=\frac1{4\pi t}E_{g_t}(\mathcal C), \qquad g_t(u)=e^{-u/(4t)}.\] Indeed, every fixed image displacement has its periodic frequency in large disks, and the Gaussian tail is summable uniformly over the finitely many point types. The Gaussian estimate (74), with Gaussian parameter \(1/(4\pi t)\), multiplied by the positive factor \((4\pi t)^{-1}\), therefore gives \(H_{\mathcal C}(t)\ge H_A(t)\) for every \(t>0\).

Applying Lemma 20 with \(\mathcal D=A\) therefore proves \(V_s(\mathcal C)\ge V_s(A)\) for every periodic \(\mathcal C\). Repetition on \(nA\) preserves \(H_A\) and hence \(V_s(A)\), which proves the stated torus inequality.

The infinite-system field infimum.

Put \(m_s=\inf_E\mathcal W_s(E)\), where the infimum runs over all compatible gradients in the fixed-\(p\) class with simple locally finite configurations and unit background. No centered-density assumption is added to this infimum. Proposition 22 supplies canonical periodic gradients whose values tend to \(m_s\). Each value is at least \(V_s(A)\) by the preceding inequality, so \(m_s\ge V_s(A)\). Conversely the canonical field of \(A\) is admissible, so \(m_s\le V_s(A)\). For the configuration infimum we now have \[m_s\le W_s(A)\le V_s(A)=m_s,\] and every configuration has \(W_s(\mathcal C)\ge m_s\). This lower bound uses inclusion in the field class, not a conversion between disk and square density. The cubic sequence suffices because the heat comparison applies to every period lattice.

The scalar jellium value. Let \(G_L\) again denote the mean-zero unit-source Green function of \((-\Delta)^{1-s/2}\) on \(\mathbb R^2/L\), as in the heat-comparison lemma. To identify the jellium value, use the physical periodic kernel \(v_L=q_sG_L\) and its regular part \(r_L=\lim_{x\to0}(v_L(x)-g_s(x))\). The unordered scalar periodic energy, including its self term, is \[\mathcal E_{\mathrm{per},L} =\sum_{i<j}v_L(a_i-a_j)+\frac N2r_L, \qquad \frac{2\mathcal E_{\mathrm{per},L}}N =\frac{W_{s,L}}{\kappa_s}.\] The scalar thermodynamic input, in this notation, is \[ \lim_{N\to\infty}\frac1{2N} \min_{\mathbf a\in(\mathbb R^2)^N}J_{s,\sqrt N}(\mathbf a) = \liminf_{N\to\infty}\frac1N \min_{\mathbf a\in(\mathbb R^2/(\sqrt N\mathbb Z^2))^N} \mathcal E_{\mathrm{per},\sqrt N\mathbb Z^2}(\mathbf a) =:e_{\mathrm{per}}^{(s)} . \tag{78}\] For \(0<s<2\), this is Lewin et al. (Lewin et al. 2019, secs. II, (3)–(4), and Section VI, (25), (27)–(31), Theorem 2); for \(s=0\), it is Lauritsen (Lauritsen 2021, sec. II and Theorem II.1), who also proves existence of the full periodic limit. Both inputs use \(N\) particles against a background of density one and area \(N\), with unrestricted particle positions on the left. Their scalar energies count each pair once. Their periodic kernels have zero mean, and their self contribution is \(Nr_L/2\), exactly as in \(\mathcal E_{\mathrm{per},L}\) above.

The approximating periods in Proposition 22 are cubic. Every canonical periodic value is at least \(m_s\), and the values along that sequence tend to \(m_s\). The finite identity \(2\mathcal E_{\mathrm{per},L}/N=W_{s,L}/\kappa_s\) therefore identifies the lower limit in (78) as \(e_{\mathrm{per}}^{(s)}=m_s/(2\kappa_s)\).

Confinement to the background square. Finite minimization is unchanged by restricting the particles to \(K_R\). Fix the other \(N-1\) particles. For \(0<s<2\), their one-particle potential extends to \(X=(x,z)\in\mathbb R^3\) as \[U(X)=\sum_{j\ne i}|X-(a_j,0)|^{-s} -\int_{K_R}|X-(y,0)|^{-s}\,\mathop{}\!\mathrm dy .\] The background potential is continuous because \(s<2\), the point poles tend to \(+\infty\), and \(U(X)=-|X|^{-s}+O(|X|^{-s-1})\) at infinity. Hence \(U\) attains a negative global minimum away from its point poles.

We use the weighted strong minimum principle: a continuous weak solution \(u\in H^1_{\mathrm{loc}}(D,|z|^\gamma\,\mathop{}\!\mathrm dX)\) of \(\operatorname{div}(|z|^\gamma\nabla u)=0\) on a connected open set \(D\), with \(-1<\gamma<1\), cannot attain an interior minimum unless it is constant (Fabes et al. 1982, Corollary 2.3.10, p. 98). This follows from their Harnack inequality for \(A_2\) weights, Theorem 2.3.8; here \(|z|^\gamma\in A_2\).

Apply this principle on \[D=\mathbb R^3\setminus\bigl((K_R\times\{0\}) \cup\{(a_j,0):j\ne i\}\bigr), \qquad \gamma=s-1 .\] The function \(U\) is smooth away from these sources and belongs to \(H^1_{\mathrm{loc}}(D,|z|^{s-1}\,\mathop{}\!\mathrm dX)\). It is weighted harmonic also across the source-free part of \(z=0\): there \(\partial_zU=O(|z|)\), so \(|z|^{s-1}\partial_zU=O(|z|^s)\to0\) from both sides. The set \(D\) is connected. Constancy there is incompatible with the negative minimum and \(U\to0\) at infinity, so a minimum lies on \(K_R\times\{0\}\). For \(s=0\), ordinary harmonicity outside the square and the point poles, together with \(U(x)=\log|x|+o(1)\) at infinity, gives the same conclusion by the ordinary strong minimum principle. Moving each exterior particle to such a minimum does not increase the energy or create a collision. On \(K_R^N\), the extended energy is lower semicontinuous, finite at a distinct tuple, and diverges at collisions, so its infimum is attained. The confined and unrestricted minima therefore agree, and \[e_{\mathrm{Jel}}^{(s)} =2e_{\mathrm{per}}^{(s)} =\frac{m_s}{\kappa_s}.\] ◻

Proof. Put \[A(x)=\sum_{j=0}^{65}\frac{(-1)^jx^{2j+1}}{2j+1}.\] For \(0<x<1\), integrate the finite geometric expansion of \(1/(1+x^2)\). Its remainder has the sign and magnitude giving \(A(x)<\arctan x<A(x)+x^{133}/133\). The tangent addition formula gives \(\tan(4\arctan(1/5))=120/119\); subtracting an angle with tangent \(1/239\) gives tangent one, and the resulting angle is in \((0,\pi/2)\). Therefore \(\pi/4=4\arctan(1/5)-\arctan(1/239)\). The two required rational comparisons are \[\begin{split} p_*&<16A(1/5)-4\bigl(A(1/239)+239^{-133}/133\bigr),\\ 16\bigl(A(1/5)+5^{-133}/133\bigr)-4A(1/239)&<p_*+\epsilon. \end{split}\] They follow by clearing positive denominators. Likewise, the second bracket is exactly the pair of integer-square comparisons \(b_*^2<3/4<(b_*+\epsilon)^2\). ◻

Proof. Testing the smeared field equation against a nonnegative smooth function equal to one near all spheres centered in a fixed unit cell gives \[ N_B^2\le C_{B,\eta}\left(1+ \int_{B'\times[-h,h]^k}w|E_\eta|^2\right). \tag{85}\] Here \(B'\) and \(h\) are fixed enlargements. These enlargements have bounded overlap over a unit-cell tiling. Also \[\int w|\nabla(f_\alpha-f_\eta)|^2 =\kappa[g(\alpha)-g(\eta)],\qquad 0<\alpha<\eta.\] Consequently, if \(Q_\alpha(E,R)=O(R^2)\) at one fixed cutoff, then \(Q_\eta(E,R)=O(R^2)\) at every fixed cutoff: apply (85) to the finite-range difference \(\sum_pN_p\nabla(f_\alpha-f_\eta)(\cdot-p)\).

The smeared Gauss equation also gives \(\nu(K_R)/R^2\to1\) whenever one cutoff has volume-order energy. For completeness, choose a lateral boundary at distance at most one from \(\partial K_R\) with squared flux energy \(O(R^2)\). For \(k=1\), choose height \(L\in[1,\sqrt R]\) with top and bottom squared energy \(O(R^{3/2})\). Weighted Cauchy–Schwarz bounds the total flux by \[C L^{(\gamma+1)/2}R^{3/2} +C L^{\gamma/2}R^{7/4}=o(R^2),\] since \(-1<\gamma<1\) (for negative \(\gamma\), use \(L^{\gamma/2}\le1\)). The smeared count lies between the counts in squares whose sidelengths differ from \(R\) by a fixed constant. Applying these inequalities at the correspondingly shifted radii proves full-radius density convergence. In the logarithmic case only the lateral term is needed. This is the argument of Petrache and Serfaty (Petrache and Serfaty 2017, Lemma 2.1), with the actual flux in the weak equation.

Put \(S=\sum_pN_p(f_\alpha-f_\eta)(\cdot-p)\le0\). For a horizontal cutoff \(\chi\), integration by parts gives \[\begin{align*} \int\chi w(|E_\eta|^2-|E_\alpha|^2) &=\kappa\int\chi S(\nu_\eta+\nu_\alpha-2\delta_{\mathbb R^2})\\ &\quad-\int S w(E_\alpha+E_\eta)\cdot\nabla\chi. \end{align*}\] Distinct-charge terms are nonpositive. An interior self term is \(-\kappa[g(\alpha)-g(\eta)]N_p^2\), bounded above by the same expression with \(N_p\) in place of \(N_p^2\). The background contributes at most \(2\kappa F_\eta\) per particle. The commutator and transition terms are bounded by the sum of the two energies and the counts in a fixed-width shell, using (85).

To remove that shell, choose \(r_j\) realizing the \(\eta\)-limsup. Averaging both fixed-cutoff shell energies over \(t\in[r_j+b_0,r_j+\sqrt{r_j}]\), where \(b_0\) exceeds twice the shell width, gives a radius \(t_j\) at which their sum is \(O(r_j^{3/2})=o(r_j^2)\). Since \(t_j/r_j\to1\) and \(Q_\eta\) is nonnegative and increasing, \[\frac{Q_\eta(E,t_j)}{t_j^2} \ge\frac{r_j^2}{t_j^2}\frac{Q_\eta(E,r_j)}{r_j^2}.\] Thus these radii still realize the \(\eta\)-limsup. Only an upper limsup bound is needed for \(Q_\alpha(E,t_j)/t_j^2\). The count shell is \(o(t_j^2)\) by the density conclusion. Hence \[ \mathcal W_\eta(E)\le\mathcal W_\alpha(E)+2\kappa F_\eta, \qquad 0<\alpha<\eta<\eta_0. \tag{86}\] These selected radii suffice for the upper-limit comparison. Comparing with one fixed cutoff proves a uniform lower bound; then (86) proves existence of the full cutoff limit and \[ -C_s\le\mathcal W_\eta(E)\le\mathcal W(E)+2\kappa F_\eta. \tag{87}\] If every cutoff has infinite energy, the conclusion is immediate. These arguments also hold with integer multiplicities. A canonical unit-lattice field has finite renormalized value, so \(-\infty<m<\infty\). ◻

Proof. Choose simple fields \(E^j\) with \(\mathcal W(E^j)\le m+1/j\). For each \(j\) choose one radius \(S_j\), with \(S_j/j\to\infty\), large enough that the upper limsup estimates for \(E^j\) hold simultaneously at \(\eta_1,\ldots,\eta_j\) on \(K_{S_j+2j}\). This enlargement handles all observation windows contained in the centered exhaustion square \(K_{2j}\). This is possible because there are only finitely many estimates at each stage. Let \(P_j\) be the law of \((E^j(\cdot+(x,0)),\nu_j(\cdot+x))\) when \(x\) is uniform in \(K_{S_j}\). For the unit square \(Q\) fixed above, Tonelli’s theorem and the enlargement by \(2j\) give the limiting bounds \[ \limsup_j\int\Gamma_{\eta_\ell}\,dP_j \le m+2\kappa F_{\eta_\ell}+\kappa g(\eta_\ell), \qquad \Gamma_\eta(E)=\int_{Q\times\mathbb R^k}w|E_\eta|^2. \tag{88}\] The analogous bounds hold on every fixed base square.

Here are the compactness details. Use \(H^{-r}_{\rm loc}\) for \(E\), with \(r>(2+k)/2+1\), together with the vague topology for the positive charge measure. Bounded local \(L^p\) sets are relatively compact in the first space. Equation (85) bounds second moments of local counts. Weighted Hölder bounds \(E_\eta\) in local unweighted \(L^p\) when \(p<\min(2,2/s)\); the singular correction belongs to local \(L^p\) when \(p<3/(s+1)\). For the logarithm both requirements reduce to \(p<2\). Thus Markov’s inequality and a countable exhaustion give tightness. Allow multiplicities in the resulting integer point measure.

On sets with bounded local counts, vague convergence identifies every fixed singular correction distributionally (indeed in the permitted local \(L^p\) topology). On bounded-energy subsequences, the truncated fields have weak \(L^2(w)\) limits, identified by that correction. It is this weighted convergence that passes the smeared divergence equation, since test gradients belong to \(L^2(w)\). Adding back the singular correction recovers the unsmeared equation; weighted Cauchy–Schwarz and the locally integrable singular flux give \(wE\in L^1_{\rm loc}\). Distributional curl remains zero and the local \(L^p\) bounds persist, so the limit is a gradient in the prescribed class. These statements hold on compact sets of arbitrarily large probability, by a summable allocation of the Markov bounds. They therefore apply to the limiting law, not merely to a deterministic subsequence of configurations.

Weighted quadratic energy is lower semicontinuous in this topology. On a bounded cylinder this follows also from \[\int w|F|^2 =\sup_{\psi\in C_c^\infty} \left(2\langle F,\psi\rangle-\int w^{-1}|\psi|^2\right).\] Take the increasing supremum over cylinder heights for the full extension integral. Consequently a subsequential law \(P\) satisfies (88) with \(P\) in place of \(P_j\). At this point the limit law is supported on compatible gradients and has the required energy bounds; it may still have multiple charges. It is horizontally stationary: for a fixed translation \(z\), the total variation discrepancy of an orbit law and its translate is at most \(|K_{S_j}\mathbin\triangle(K_{S_j}+z)|/S_j^2\to0\).

The \(L^1\) pointwise and mean ergodic theorems along tempered Følner sequences (Lindenstrauss 1999, sec. 1), applied to the centered square sequence in \(\mathbb Z^2\), apply to each integrable observable \(\Gamma_{\eta_\ell}\). The tiled sums of this observable are exactly the raw energies on integer squares; positivity and nested squares extend the limits to every real radius. Each limit is the conditional expectation of its observable on the \(\mathbb Z^2\)-invariant sigma-algebra. It may depend on the realization; no ergodicity of \(P\) is assumed. Its expectation equals the mean of the observable. Thus, on one event of full probability for all selected cutoffs, \[a_{\eta_\ell}:=\mathbb E_P\mathcal W_{\eta_\ell} =\mathbb E_P\Gamma_{\eta_\ell}-\kappa g(\eta_\ell) \le m+2\kappa F_{\eta_\ell}.\] Fatou and the uniform lower bound give \(\mathbb E_P\mathcal W\le m\). This does not yet use minimality, because the limit could have multiplicities. It shows that \(\mathcal W\) is finite and integrable. Equation (87) then gives dominated convergence \(a_{\eta_\ell}\to\mathbb E_P\mathcal W\).

Unit intensity and simplicity. The local count bound gives finite point intensity \(\rho\). In the logarithmic case the expected smeared PDE immediately gives \(\rho=1\) by horizontal stationarity. In the extension case, let \(b_\eta(y)=\mathbb E_P(E_\eta)_y\) and let \(\mu_{\eta,y}\) be the vertical probability marginal of one smeared charge. Jensen gives \(\int |y|^\gamma|b_\eta|^2<\infty\), and the mean PDE is \[-\partial_y(|y|^\gamma b_\eta) =\kappa(\rho\mu_{\eta,y}-\delta_0).\] Outside \([-\eta,\eta]\) the weighted flux is constant on each tail. They must vanish, since \(\int_1^\infty y^{-\gamma}\,dy=\infty\). Integrating the equation proves \(\rho=1\).

For cutoffs \(0<\alpha<\eta\) from the selected dyadic sequence, put \(d=\sum_p(f_\eta-f_\alpha)(\cdot-p)\), counting labeled particles. Then \[ w(|E_\alpha|^2-|E_\eta|^2) =\operatorname{div}[wd(E_\alpha+E_\eta)] +\kappa d(\nu_\alpha+\nu_\eta-2\delta_{\mathbb R^2}). \tag{89}\] At positive cutoffs the kernels in \(d\) are bounded and have finite range. The second count moment and Cauchy–Schwarz justify all local expected products. Test with a smooth horizontal function of integral one and, if needed, a vertical cutoff equal to one on \(|y|\le\eta\). Stationarity annihilates the expected horizontal divergence; the vertical cutoff varies outside the support of \(d\). Separating self, distinct-labeled-pair, and background terms gives exactly \[ D_{\alpha,\eta}=a_\alpha-a_\eta+2\kappa(F_\eta-F_\alpha), \qquad D_{\alpha,\eta}\ge0. \tag{90}\] Here \(D_{\alpha,\eta}\) is the ordered-pair intensity of the kernel \[\kappa\int(f_\eta-f_\alpha)(X-(p,0)) (d\delta_q^{(\alpha)}+d\delta_q^{(\eta)})(X), \qquad p\ne q\text{ as labeled particles}.\] A coincident pair contributes \(\kappa[g(\alpha)-g(\eta)]\). If \(c_{\rm coll}=\mathbb E_P\sum_{p\in Q}N_p(N_p-1)\), then \[0\le\kappa[g(\alpha)-g(\eta)]c_{\rm coll} \le D_{\alpha,\eta}.\] The right side stays bounded as \(\alpha\downarrow0\), whereas \(g(\alpha)\to\infty\). Hence \(c_{\rm coll}=0\); stationarity and a countable tiling show simplicity almost surely. Only now do we use \(\mathcal W\ge m\), proving \(\mathcal W=m\) almost surely and \(a_{\eta_\ell}\to m\).

We have obtained simple minimizing fields. The remaining task is to select one whose close pairs and vertical tail remain controlled during screening.

Vanishing defect and selection of a realization. Fatou in (90), with only \(\alpha\downarrow0\), gives \[ 0\le D_\eta:=\kappa\mathbb E_P \sum_{p\in Q}\sum_{q\ne p}q_\eta(p-q) \le m-a_\eta+2\kappa F_\eta\longrightarrow0. \tag{91}\] The measure \(\mu_\eta\) thus has finite mean intensity \(D_\eta\). The ergodic theorem therefore gives its spatial density \[d_\eta(E)=\lim_{R\to\infty}\frac{\mu_\eta(K_R)}{R^2}, \qquad \mathbb E_Pd_\eta=D_\eta\] almost surely. For \(k=1\), apply the same ergodic theorem to the raw energy over each fixed integer height \(|y|>z>1\). These integrable observables decrease to zero, so their spatial densities decrease to zero almost surely. Intersect all the energy, tail, and selected defect events, a countable family of full-measure events. For each selected \(\eta\), choose a realization \((E,\mathcal C)\) in this intersection with \(d_\eta\le2D_\eta\); if \(D_\eta=0\), choose \(d_\eta=0\). This realization may depend on \(\eta\). For every fixed \(\varepsilon>0\), define \[ e_{\varepsilon,R}:=\frac1{\varepsilon^4R^2} \int_{K_R\times\{|y|>\frac14\varepsilon^2R\}}w|E|^2 \longrightarrow0. \tag{92}\] The coefficient \(\frac14\) in this threshold is fixed independently of the cutoffs and the realization. Indeed, compare the moving height with each fixed integer height and then let that fixed height tend to infinity. No uniform rate in \(\varepsilon\) is needed. ◻

Proof. Keep the field and both cutoffs fixed. We first choose the inner cylinder where its truncated field is retained, then construct the shell and caps. Coarea supplies an inner rectangle \(K'_R\) whose corresponding faces are at distances between \(\varepsilon R\) and \(2\varepsilon R\) from the outer boundary, with \[\int_{\partial K'_R\times\mathbb R^k}w|E_\eta|^2 \le C M_{R,\eta}\varepsilon^{-1}R, \quad \int_{(\partial K'_R)_b\times\mathbb R^k}w|E_{\eta'}|^2 \le C M_{R,\eta'}\varepsilon^{-1}R,\] where \(b\) is a fixed sufficiently large shell width and \(M_{R,\rho}=R^{-2}Q_\rho(E,R)\). For \(k=1\), choose \(\ell\in[\frac12\varepsilon^2R,\varepsilon^2R]\). Such a height with the required trace bound exists because \[\begin{align*} \int_{\frac12\varepsilon^2R}^{\varepsilon^2R} \left(\int_{K'_R\times\{\pm t\}}w|E|^2\right)\,dt &\le \int_{K_R\times\{|y|>\frac14\varepsilon^2R\}}w|E|^2\\ &=\varepsilon^4R^2e_{\varepsilon,R}. \end{align*}\] Dividing by the interval length \(\frac12\varepsilon^2R\) gives \[\int_{K'_R\times\{\pm\ell\}}w|E|^2 \le2\varepsilon^2R e_{\varepsilon,R}.\] For \(k=0\), retain the choice \(\ell=\varepsilon^2R\). In both cases \(R\) is then taken large enough that \(\ell>2\sqrt2\). The fixed-cutoff energy limits bound both \(M\)’s as \(R\to\infty\).

Let \(D_0=K'_R\times[-\ell,\ell]^k\), \(A_{\rm sh}=|K_R|-|K'_R|\), and, when \(k=1\), let \(T_\pm\) be the outward weighted flux of the old field through the top and bottom of \(D_0\). The cap constants must be \[ C^\pm=-\frac{T_\pm}{A_{\rm sh}\ell^\gamma}. \tag{93}\] They impose separate compatibility in the disconnected upper and lower caps \(K_R\times(\ell,R)\) and \(K_R\times(-R,-\ell)\). For \(k=0\) all \(T_\pm,C^\pm\) and cap terms are omitted. Let \(\Lambda_0=\{p\in\mathcal C:\operatorname{dist}(p,\partial K'_R)\le\eta\}\) be the retained crossing centers and \(\rho_0=\sum_{p\in\Lambda_0}N_p\delta_p^{(\eta)}\) their full smeared measure. The retained old centers are \(\Lambda_{\rm old}=(\mathcal C\cap K'_R)\cup\Lambda_0\); \(N_{\rm old}\) counts all of them, including multiplicity if present. All pieces assembled below are finite-energy components of the truncated field. In particular the new-charge component uses smeared sources; the point singularities are restored only after the pieces have been assembled. On a shell cell \(H_i\), denote the old side flux by \(B_i=\int_{(\partial K'_R\cap\partial H_i)\times[-\ell,\ell]^k}wE_\eta\cdot\nu_0\), where \(\nu_0\) points outward from \(D_0\). The truncated shell field is the sum of five components, constructed below:

For \(k=0\), \(E_{\rm vert}=0\) and there are no caps. We use two independent fixed partitions of the shell, so that corner completion and integer charge counts do not require moving its faces.

For the large cells, include the coordinates of every inner and outer face as breakpoints in each coordinate axis. Subdivide each resulting interval into lengths in \([\ell,2\ell]\), and remove the product cells inside \(K'_R\). The remaining rectangles \(H_i\) form a connected annular grid. Adjacent rectangles share full edges of length at least \(\ell\), and there are \(O_\varepsilon(1)\) rectangles at fixed \(\varepsilon\). All the one-dimensional intervals exceed \(\ell\) for sufficiently large \(R\) and the chosen small \(\varepsilon\).

We spell out the field estimates needed in this construction. First consider a fixed cell \(\Omega=S\times[-1,1]^k\), where \(S\) is a rectangle with sidelengths in \([1,2]\); when \(k=0\), take \(\Omega=S\). Suppose the cell carries a restriction \(\nu\) of smeared charges, of total mass \(Q\). A constant outward normal component \(b_F\) on a chosen lateral face \(F\) is compatible with \(Lh=\kappa\nu\) precisely when \[b_F\int_Fw=-\kappa Q.\] Use the representative \(u\) of weighted cell mean zero; compatibility allows this choice because the Neumann functional annihilates constants. The weighted Poincaré and trace estimates on these cells can be seen directly. Apply ordinary Poincaré and trace in the horizontal variables, then integrate against \(|y|^\gamma\). In the vertical variable use \[|v(y)-v(z)|^2 \le\left(\int_{-1}^1|t|^\gamma|v'(t)|^2\,dt\right) \left(\int_{-1}^1|t|^{-\gamma}\,dt\right).\] Both \(w\) and \(w^{-1}\) are integrable on this interval because \(-1<\gamma<1\). Averaging this inequality proves weighted Poincaré and the trace bound at \(y=0\); horizontal trace gives the lateral-face bound. The constants are uniform over the stated sidelengths.

For the extension, reflect across the coordinate faces of the base rectangle \(S\), which preserves \(w\). If \(k=1\), extend in \(y\) only to \([-3/2,3/2]\), reflecting each added collar into \(1/2<|y|<1\), where the original and reflected weights are comparable. Multiply by a smooth cutoff equal to one on \(\Omega\) and supported in this fixed enlargement. Poincaré controls the term \(u\nabla\chi\) from that multiplication, giving an extension \(Tu\) with \[\|\nabla Tu\|_{L^2(w)} \le C_\gamma\|\nabla u\|_{L^2(w;\Omega)}.\] When \(k=0\), the same construction uses only the horizontal reflections.

Put \(g_\eta=g-f_\eta\), the fundamental solution capped at \(g(\eta)\). The retained centers contributing to this cell lie in a fixed enlargement of it. On the support of \(Tu\), radial integration of the fundamental solution’s flux gives \[\int_{\operatorname{supp}(Tu)}w |\nabla g_\eta(X-(p,0))|^2\,dX \le C_s(1+g(\eta)).\] For \(s>0\) the full-space integral is \(\kappa g(\eta)\); for the logarithm the integral over a ball of fixed radius is \(\kappa\log(1/\eta)+O(1)\). Only this local integral is used. Positivity of the smearing measure then gives \[\left|\int_\Omega u\,d\delta_p^{(\eta)}\right| \le\int |Tu|\,d\delta_p^{(\eta)} =\kappa^{-1}\left|\int w\nabla g_\eta(\cdot-p) \cdot\nabla|Tu|\right| \le C\sqrt{1+g(\eta)}\,\|\nabla u\|_{L^2(w)}.\] The Neumann functional annihilates constants. Lax–Milgram therefore gives energy at most \(C N^2(1+g(\eta))\) when at most \(N\) labeled charges contribute and the sum of the absolute prescribed face fluxes is at most \(C\kappa N\). The same proof allows several constant face data whose total is \(-\kappa Q\): apply the trace bound to each face. This proves the local estimate needed from Petrache and Serfaty (Petrache and Serfaty 2017, Lemma 6.6), with the outward flux fixed by \(Lh=\kappa\nu\).

For crossing-charge completion, form a separate tensor grid with sidelengths in \([1,2]\), again containing all inner and outer face coordinates. Restrict \(\rho_0\) to its exterior cells times \([-1,1]^k\). Its exterior support lies within distance \(2\eta\) of \(\partial K'_R\). With \(2\eta<1/2\), every such cell either lies in the first layer outside exactly one inner coordinate face, and shares a full face with \(K'_R\), or is one of the four diagonal corner cells. Route each corner cell to an adjacent first-layer cell through their full shared face. These are trees of uniformly bounded size; cells not receiving a corner contribution are single-cell trees.

All face data in the following construction are constant normal derivatives: a prescribed integrated weighted flux \(F_0\) means the value \(F_0/\int_Fw\) on that face. On a leaf, prescribe outward weighted flux \(-\kappa\) times its restricted source mass through the face leading to the root. At a root, prescribe the opposite incoming flux through each child face, and export \(-\kappa\) times the total tree mass through its full inner face. All other face data are zero. More generally the outgoing flux of a tree vertex is \(-\kappa\) times the mass in its subtree. The sum of its outward fluxes is exactly \(-\kappa\) times its own source mass, so the cell problems are compatible. Internal normal fluxes cancel. This gives one global completion field \(E_{\rm cross}\) with source \(\kappa\rho_0\) on the exterior part, and a nonnegative outlet density \[j=-E_{\rm cross}\cdot\nu_{\rm shell} =\sum_{F\ {\rm outlet}}\mathbf1_F \frac{\kappa Q_F}{\int_Fw} \quad\hbox{on the inner shell boundary}.\] Here \(Q_F\) is the total tree mass and \(\nu_{\rm shell}=-\nu_0\). Figure 3 shows the local corner relay.

The completion field is not cut at the boundaries of the large cells \(H_i\); thus no trace on an arbitrary such cut is required. The local count estimate and bounded overlap of the unit trees give \[\int w|E_{\rm cross}|^2 \le C(1+g(\eta))\sum_{\rm trees}N_{\rm tree}^2, \qquad \int_{\partial K'_R\times\mathbb R^k}w j^2 \le C\sum_{\rm trees}N_{\rm tree}^2 \le C(1+M_{R,\eta'})\varepsilon^{-1}R.\] The additive \(1\) in the count-square bound is required.

The crossing field now completes every retained smeared charge and provides its actual outlet flux on the inner boundary. We next balance that flux against the background, then choose an integer number of new particles.

For each large cell set \(Q_i=\int_{(\partial K'_R\cap\partial H_i)\times[-\ell,\ell]^k}w j\), its actual inner-face outlet flux. This need not equal the restricted source mass \(\kappa\rho_0(H_i\times[-\ell,\ell]^k)\), because the completion field may transport that source across a large-cell boundary. Define the ideal density by \[ \kappa(m_i-1)|H_i| =B_i-\ell^\gamma(C^++C^-)|H_i|-Q_i. \tag{94}\] Since \(\sum_iQ_i=\kappa\rho_0(D_0^c)\), Gauss’ equation and (93) give \(\sum_i m_i|H_i|=R^2-N_{\rm old}=:N_{\rm new}\in\mathbb N\). At fixed parameters the lateral-flux, cap-flux and outlet bounds make \(\max_i|m_i-1|\to0\) as \(R\to\infty\); choose this maximum below \(1/4\). For example the outlet term divided by \(|H_i|\) is bounded by \(C(1+M_{R,\eta'})\varepsilon^{-5}R^{-1}\).

Set \(r_i=m_i|H_i|\). Round the \(r_i\) down, and increase the required number of entries by one, obtaining integers \(n_i\) with \(\sum_i n_i=N_{\rm new}\) and \(|n_i-r_i|\le1\). Put \(\widetilde m_i=n_i/|H_i|\). For large \(R\) these densities lie in \([1/2,3/2]\). To compensate this rounding, write \(\delta_i=n_i-r_i\), whose sum is zero, and choose a rooted spanning tree of the large-cell adjacency graph. On the face from a child cell \(i\) to its parent prescribe outward weighted flux \(-\kappa\sum_{a\text{ in the subtree of }i}\delta_a\), with opposite flux on the adjacent cell. The total outward flux of cell \(i\) is \(-\kappa\delta_i\). There is therefore a compatible Neumann field \(E_{{\rm round},i}\) with source \(\kappa\delta_i|H_i|^{-1}\delta_{\mathbb R^2}\), these face fluxes, and zero flux on all other faces, including the caps. The paired fluxes make the union \(E_{\rm round}\) a global field. It has no flux through the inner or outer shell boundary. After dilation by \(\ell\), weighted trace and Poincaré on rectangles of bounded aspect ratio give \[\int w|E_{\rm round}|^2\le \begin{cases}C_{s,\varepsilon}\ell^{-s},&s>0,\\ C_\varepsilon,&s=0. \end{cases}\] Indeed there are \(O_\varepsilon(1)\) cells and each tree flux is \(O_\varepsilon(1)\); the unit-scale variational problem has bounded energy, and dilation scales it by \(\ell^{-s}\) for Riesz kernels and by one for the logarithm. This is \(o(R^2)\) at fixed parameters. For sufficiently large \(R\), every sidelength of \(H_i\) is at least \(\ell>2\sqrt2\ge2\widetilde m_i^{-1/2}\), and \(\widetilde m_i|H_i|=n_i\in\mathbb N\). Thus Petrache and Serfaty (Petrache and Serfaty 2017, Lemma 6.3) partitions \(H_i\) into \(n_i\) rectangles of area \(1/\widetilde m_i\), with every sidelength in \[\left[\frac14\widetilde m_i^{-1/2}, 4\widetilde m_i^{-1/2}\right] \subset[a_0,a_1].\] In particular, the lower bound \(a_0\) fixed at the start applies to every new-charge rectangle, independently of all later choices.

The vertical correction on each shell cell is the gradient of \[h_2(y)=\frac{\ell^\gamma}{1-\gamma} \begin{cases}C^+y^{1-\gamma},&y\ge0,\\ C^-|y|^{1-\gamma},&y\le0. \end{cases}\] It has source \(-\ell^\gamma(C^++C^-)\delta_{\mathbb R^2}\). The side-flux correction has the opposite cap source plus \(\kappa(m_i-1)\delta_{\mathbb R^2}\) and datum \(-E_\eta\cdot\nu_0+j\) on the inner side, with zero data elsewhere. Equation (94) is exactly its compatibility.

The large-cell estimate used for this correction is as follows. Let \(H\) have sidelengths in \([\ell,2\ell]\), set \(\Omega=H\times[-\ell,\ell]\) and \(\Sigma=\partial H\times[-\ell,\ell]\), and let \(-1<\gamma<1\). For a constant \(a\) and \(b\in L^2(\Sigma,|y|^\gamma)\) satisfying \[a|H|=-\int_\Sigma |y|^\gamma b,\] the weak Neumann problem \[-\operatorname{div}(|y|^\gamma\nabla h) =a\delta_{\mathbb R^2}\quad\hbox{in }\Omega, \qquad \partial_\nu h=b\quad\hbox{on }\Sigma, \qquad \partial_\nu h=0\quad\hbox{on }H\times\{\pm\ell\}\] has a solution modulo constants and obeys \[ \int_\Omega |y|^\gamma|\nabla h|^2 \le C_\gamma\left(a^2\ell^{3-\gamma} +\ell\int_\Sigma |y|^\gamma|b|^2\right) \le C_\gamma\ell\int_\Sigma |y|^\gamma|b|^2. \tag{95}\] The constants depend only on \(\gamma\) and the stated aspect-ratio bounds. To see the scaling, choose the weighted-mean-zero representative. Weighted Poincaré and the one-dimensional trace inequalities, after dilation by \(\ell\), give \[\int_H|h(x,0)|^2\le C_\gamma\ell^{1-\gamma} \int_\Omega |y|^\gamma|\nabla h|^2, \qquad \int_\Sigma |y|^\gamma|h|^2 \le C_\gamma\ell\int_\Omega |y|^\gamma|\nabla h|^2.\] Testing the weak equation gives the first inequality in (95). Compatibility, \(|H|\asymp\ell^2\), and \(\int_\Sigma|y|^\gamma\asymp\ell^{\gamma+2}\) give \(a^2\ell^{3-\gamma}\le C_\gamma\ell\int_\Sigma|y|^\gamma|b|^2\), which proves the second. This is the constant-source case of Petrache and Serfaty (Petrache and Serfaty 2017, Lemma 6.4), with the source coefficient written explicitly. For the logarithm use \(\Omega=H\), \(-\Delta h=a\), and \(\partial_\nu h=b\) on \(\partial H\); ordinary Poincaré, trace, and compatibility give the same final bound \(C\ell\int_{\partial H}|b|^2\).

For the side field on \(H_i\), take \[a_i=\kappa(m_i-1)+\ell^\gamma(C^++C^-),\qquad b_i=-E_\eta\cdot\nu_0+j \quad\hbox{on its inner face,}\] with \(b_i=0\) on its other lateral faces. The cap term is omitted when \(k=0\). The density equation gives \(a_i|H_i|=B_i-Q_i=-\int_{\Sigma_i}w b_i\) exactly, so the cap contribution to the constant source is included in the compatibility estimate. The inner faces partition the old lateral boundary. Consequently the old-field coarea bound and the outlet bound yield \[\begin{align*} \int w|E_{\rm side}|^2 &\le C_s\ell\sum_i\int_{\Sigma_i}w|b_i|^2\\ &\le C_s\ell\left( \int_{\partial K'_R\times[-\ell,\ell]^k}w|E_\eta|^2 +\int_{\partial K'_R\times\mathbb R^k}wj^2\right)\\ &\le C_{s,\eta'}(\varepsilon^2R)(\varepsilon^{-1}R) (M_{R,\eta}+1+M_{R,\eta'})\\ &=C_{s,\eta'}\varepsilon R^2(M_{R,\eta}+1+M_{R,\eta'}). \end{align*}\] Here \(\Sigma_i=\partial H_i\times[-\ell,\ell]^k\). This calculation supplies the length factor in the side-field row of the shell-energy estimate. Its constant may depend on the single fixed auxiliary cutoff \(\eta'\), but not on \(\eta\), \(\varepsilon\), \(R\), or the realization.

Finally, let \(S\subset H_i\) be one of the area-\(1/\widetilde m_i\) rectangles, with barycenter \(p\), and work on the fixed cylinder \(\Omega_S=S\times(-1,1)^k\) (or \(\Omega_S=S\) when \(k=0\)). Take the \(\eta\)-truncation of the neutral Neumann field with source \(\kappa(\delta_{(p,0)}-\widetilde m_i\mathbf1_S\delta_{\mathbb R^2})\) and zero normal component on its entire boundary. The truncated source is \(\kappa(\delta_p^{(\eta)}-\widetilde m_i\mathbf1_S\delta_{\mathbb R^2})\). Extend this truncated field by zero into \(H_i\times[-\ell,\ell]^k\) and the shell. The sidelength bounds \([a_0,a_1]\), fixed height, and \(\eta<a_0/16\) keep the charge uniformly away from the cell boundary. To verify its energy, write the point-source potential as \(g(\cdot-(p,0))+v\). The correction solves \[Lv=-\kappa\widetilde m_i\mathbf1_S\delta_{\mathbb R^2},\qquad \partial_\nu v=-\partial_\nu g(\cdot-(p,0)) \quad\hbox{on }\partial\Omega_S.\] The weighted trace and Poincaré estimates give a uniform bound on \(\int_{\Omega_S}w|\nabla v|^2\); no smoothness across \(y=0\) is needed. Test this compatible weak equation against \(g_\eta(\cdot-(p,0))\). The resulting plane integral is uniformly bounded because \(s<2\), and the boundary terms are uniformly bounded by the clearance from the pole. Hence \(\int_{\Omega_S}w\nabla v\cdot\nabla g_\eta=O_s(1)\). The capped fundamental solution itself has energy \(\kappa g(\eta)+O_s(1)\) in \(\Omega_S\), so the truncated cell field has the same energy, uniformly over the allowed rectangles. In addition to the preceding \(o(R^2)\) bound for \(E_{\rm round}\), the shell fields satisfy \[\begin{align*} \int w|E_{\rm cross}|^2 &\le C(1+g(\eta))(1+M_{R,\eta'})\varepsilon^{-1}R,\\ \int w|E_{\rm vert}|^2 &\le C e_{\varepsilon,R}\varepsilon^3R^2,\\ \int w|E_{\rm side}|^2 &\le C_{s,\eta'}\varepsilon(M_{R,\eta}+1+M_{R,\eta'})R^2,\\ \int w|E_{\rm new}|^2 &\le C(1+g(\eta))N_{\rm new}, \end{align*}\] where \(N_{\rm new}=O(\varepsilon R^2)+o(R^2)\) at fixed parameters. The first bound uses the sum of squared tree counts; the third uses the weighted rectangle trace estimate with height \(\ell\). Write \(\nu_{{\rm new},\eta}=\sum_{p\ {\rm new}}\delta_p^{(\eta)}\). The five shell fields have combined source \[\kappa\rho_0 -\ell^\gamma(C^++C^-)\delta_{\mathbb R^2} +\bigl[\kappa(m_i-1)+\ell^\gamma(C^++C^-)\bigr]\delta_{\mathbb R^2} +\kappa(\widetilde m_i-m_i)\delta_{\mathbb R^2} +\kappa(\nu_{{\rm new},\eta}-\widetilde m_i\delta_{\mathbb R^2}),\] which is \(\kappa(\rho_0+\nu_{{\rm new},\eta}-\delta_{\mathbb R^2})\). On the inner shell boundary the crossing and side data sum to \(-j+(-E_\eta\cdot\nu_0+j)=-E_\eta\cdot\nu_0\). The rounding field has zero inner data and paired internal data; the completion field remains global across the large-cell boundaries. Thus all interface fluxes cancel. We sum squared energies only in the shell, where a fixed factor of five is harmless. The old field inside \(D_0\) is retained exactly, with coefficient one in the energy bound; no global factor multiplying its energy is introduced.

The shell now has the required source and lateral flux. It remains to close the two vertical ends without changing these sources.

The upper and lower weighted harmonic caps are solved separately. On the inner horizontal face of the upper or lower cap, prescribe the outward normal derivative \(-E_\eta(x,\pm\ell)\cdot(\pm e_y)\) for \(x\in K'_R\), and \(-C^\pm\) for \(x\in K_R\setminus K'_R\); here \(e_y\) is the positive vertical unit vector. Prescribe zero normal derivative on every other face. By (93), the weighted integral of these data is \(-T_\pm-A_{\rm sh}\ell^\gamma C^\pm=0\), so each cap problem is compatible. If \(u\) is adjusted by its ordinary mean on one cap, ordinary trace and Poincaré, followed by weight comparison, give \[\int_{K_R\times\{\ell\}}w|u|^2 \le CR\max\{1,(\ell/R)^\gamma\} \int_{\rm cap}w|\nabla u|^2.\] Testing the compatible cap problem yields total cap energy at most \[C e_{\varepsilon,R} \varepsilon^{1+\min(2\gamma,0)}R^2.\] The extra factor for \(\gamma<0\) is retained. It vanishes at fixed \(\varepsilon\) by (92). There are no cap terms for \(k=0\).

Retain \(E_\eta\) in \(D_0\), use the sum of the five finite-energy components in the shell, and use the two harmonic cap fields above and below. Extend this assembled field \(\widehat F_\eta\) by zero past the outer square and the cap heights. It has zero exterior normal flux and source \(\kappa(\sum_{p\in\widehat\Lambda}\delta_p^{(\eta)} -\mathbf1_{K_R}\delta_{\mathbb R^2})\), where \(\widehat\Lambda\) consists of the retained old and new barycenter charges, exactly \(R^2\) points. Now define \[\widehat E=\widehat F_\eta+ \sum_{p\in\widehat\Lambda}\nabla f_\eta(\cdot-(p,0)).\] The singular corrections lie inside the outer square by its clearance. Thus \(\widehat E_\eta=\widehat F_\eta\) exactly, and the fundamental-solution identity proves the required unsmeared divergence equation, with zero exterior normal flux. At fixed cutoffs the preceding bounds give \[\begin{align*} R^{-2}\int_{K_R\times\mathbb R^k}w|\widehat E_\eta|^2 &\le M_{R,\eta} +C_{s,\eta'}\varepsilon(1+g(\eta)) (1+M_{R,\eta}+M_{R,\eta'})+o_R(1)\\ &\quad+C_s e_{\varepsilon,R} \varepsilon^{1+\min(2\gamma,0)}. \tag{96}\end{align*}\] Here \(R\) is chosen after \(\eta,\eta',\varepsilon\). For example, the count estimate is controlled once \(R\gg(1+M_{R,\eta'})\varepsilon^{-5}\).

Every new barycenter is at least \(a_0/2\) from each face of its rectangle. Retained old points lie inside \(K'_R\) or within \(\eta\) of its boundary. Since \(\eta<\eta_0\le a_0/16\) was fixed before the realization was selected, every new–old distance is at least \(a_0/2-\eta>2\eta\). Choose the integer radius large enough that \(\varepsilon R-\eta\ge a_0/2\); this gives the same clearance at the outer boundary and hence at every reflected seam. New–new distances and all reflection-seam distances exceed \(2\eta\). Thus every close pair after screening was already an old close pair. ◻

Proof of Proposition 22. Lemma 23 proves existence and the uniform lower bound of the cutoff limit, as well as finiteness of \(m\). To approximate \(m\), choose a selected dyadic cutoff \(\eta\) and a realization supplied by Lemma 24. Apply Lemma 25 after choosing \(\varepsilon\), and then choose its integer radius \(R\). We now convert that screened field into a canonical periodic gradient and estimate its energy after removing the cutoff.

Reflect the screened square in both coordinate directions and repeat with period \(2R\mathbb Z^2\). The reflected vector components have matching zero normal flux, so there are no interface charges. The period cell has \(4R^2\) points and area \(4R^2\). Reflection multiplies both energy and defect by four. The nonnegative defect of the resulting configuration is at most the old anchored defect in \(K_R\), divided by \(R^2\), and hence tends to at most \(d_\eta(E)\le2D_\eta\).

Write \(E^{\rm per}\) for the reflected field, which need not be curl free, and let \(E^{\rm can}\) be the canonical periodic gradient with exactly its charges and background. At fixed cutoff, \(D=E^{\rm per}-E^{\rm can}=E^{\rm per}_\eta-E^{\rm can}_\eta\) belongs to \(L^2(w)\) on the full period cylinder and has zero weighted divergence. Test against the canonical truncated potential, with vertical cutoffs tending to infinity; its nonconstant Fourier modes decay vertically and its zero mode has compactly supported derivative. The cross term vanishes, so \[\int_{T\times\mathbb R^k}w|E^{\rm per}_\eta|^2 =\int_{T\times\mathbb R^k}w|E^{\rm can}_\eta|^2 +\int_{T\times\mathbb R^k}w|D|^2.\] Canonical projection therefore decreases the cutoff energy and leaves the configuration defect unchanged. It also returns to the prescribed admissible class. For \(s>0\), finite truncated \(L^2(w)\) energy gives local unweighted \(L^p\) by weighted Hölder when \(p<\min(2,2/s)\), and each of the finitely many local singular corrections belongs to \(L^p\) when \(p<3/(s+1)\). For the logarithm both bounds reduce to \(p<2\). Weighted Cauchy–Schwarz and the locally integrable singular flux give \(wE^{\rm can}\in L^1_{\rm loc}\). Thus the canonical gradient is a member of \(\mathcal A_1\) with the exponent fixed at the beginning, even though the intermediate screened field was not required to be a gradient.

For a finite simple periodic configuration, integrate (89) over its period cell and let the auxiliary cutoff tend to zero. This gives \[ V(\mathcal C)-\mathcal W_\eta(E^{\rm can}_{\mathcal C}) =\frac{\kappa}{|T|}\sum_{p\in T}\sum_{q\ne p}q_\eta(p-q) -2\kappa F_\eta. \tag{97}\] The second sum includes periodic images; the self image at zero is the only omitted particle.

Keep the auxiliary cutoff \(\eta'\) fixed at its initial value as \(\eta\) tends to zero. For each selected \(\eta\), take the realization chosen above, and choose \(\varepsilon\) so the middle term of (96) is as small as desired; then choose an arbitrarily large integer \(R\) so that all energy and defect limits, the moving-tail estimate, the size conditions, and the seam clearance hold simultaneously. Subtract \(\kappa g(\eta)\), project canonically, and apply (97). Equations (87) and (91) give \[V(\mathcal C_{\eta,R}) \le m+2\kappa F_\eta+o(1)+2D_\eta-2\kappa F_\eta =m+2D_\eta+o(1).\] Let the chosen cutoffs tend to zero, taking the screening thickness and integer radius in that order for each cutoff. The canonical periodic gradients belong to \(\mathcal A_1\), so their values are also at least \(m\). This proves the asserted cubic minimizing sequence. Every use of the box limit preceded removal of the cutoff; no interchange of the two limits has been made. ◻

Proof. We control elementary functions first, then a single moment, then the matrix products, and finally the scalar lower evaluations. This order keeps the error due to the coefficient lists separate from the error in the primitive moments.