Historical context

Classical work on the lattice-restricted problem proceeds through the binary Epstein zeta function. Rankin and Cassels studied its minimization, with corrections and refinements by Ennola and Diananda [20, 3, 4, 12, 10]. This line of work establishes the triangular minimum for convergent inverse-power lattice sums. Montgomery’s theta theorem gives the Gaussian minimum among planar lattices of fixed covolume for every positive parameter [16]. These lattice results do not by themselves compare the triangular lattice with a nonperiodic set.

The Fourier bounds of Cohn and Elkies for sphere packing [6] provided a framework in which pointwise inequalities for a function and its transform control a geometric optimization problem. Cohn and Kumar developed the corresponding energy bounds and formulated the Euclidean universal-optimality program [7]. In dimensions eight and twenty-four, Cohn, Kumar, Miller, Radchenko, and Viazovska proved the \(E_8\) and Leech lattice cases using sharp Gaussian auxiliaries and simultaneous Fourier interpolation formulas [8].

Other planar comparisons retain restrictions on the competitors or the allowed perturbations. Faulhuber, Shafkulovska, and Zlotnikov compare the triangular lattice with the honeycomb and further periodic classes [13]. Hardin and Tenpas treat four-point and six-point configurations on a fixed triangular torus and a fixed rectangular torus, respectively, with the extension from periodized Gaussians to completely monotone interactions subject to their rapid-decay hypothesis [14]. For interactions of finite triangular-lattice energy, Leblé proves local optimality under sufficiently small bounded displacements, including nonperiodic displacements, with the allowed size depending on the interaction [15]. Theorem 1 imposes no assumption that the competitor is a bounded perturbation of a lattice.

A separate companion develops a modulo-\(12\) interpolation construction for the same energy endpoint [18]. Its finite part uses \(168\) positive-node coordinates in each block, frequency quadrature, and Gaussian-parameter intervals [18]. The construction here uses modulo-\(36\) nodes and twenty positive-node coordinates in each block. The undamped finite columns are trigonometric polynomials in squared radius, so their spectral measures are finite sums of atoms, and one containing polytope controls the Gaussian data. The auxiliary statements of the two methods are not identical, even though their energy consequences agree. The smaller finite system does not enlarge the competitor class, and it does not eliminate the infinite correction or the estimates between nodes.

The interpolation mechanism has several precedents. Cardinal formulas built from squared sine or cosine factors and value-and-derivative data occur in Cohn and Kumar’s one-dimensional energy bound [7] and in the Gaussian extremal approximations of Carneiro, Littmann, and Vaaler [2]. Radchenko and Viazovska developed simultaneous interpolation of an even real Schwartz function and its Fourier transform on the line [19]. Sardari’s planar construction uses periodic supersets of triangular quadratic-form values and shows that the values and first radial derivatives of a function and its Fourier transform at the triangular shells do not determine the function uniquely in the full radial Schwartz class [21]. The uniqueness established below is only for the specified summable coefficient problem. The sharp packing companion develops Gaussian cardinal functions, Fourier kernels, finite-block and infinite-tail inversion, and Bernstein sign certification that we adapt here [17]. All parameter-dependent identities and estimates required for the present Gaussian functions are proved in this paper.

For the transfer to general configurations, Cohn and de Courcy-Ireland broaden the competitor class beyond periodic configurations to the present density-based class, using a real continuous, positive-definite, integrable auxiliary satisfying their minorant condition [5]; see also [8]. We prove the Schwartz case needed here locally, using a Fourier cutoff near zero. The reciprocal-parameter construction is the Gaussian instance of the unnumbered duality calculation immediately following Conjecture 6.1 in [9]. We also prove locally the Bernstein–Widder positive Laplace representation used to mix the Gaussian kernels [1, 23]. Cohn–Kumar Conjecture 9.4 asks in addition for an auxiliary satisfying their Proposition 9.3 and proving sharpness for each sufficiently decaying completely monotone potential; it does not require that auxiliary to be Schwartz. Mixing the Gaussian energy inequalities does not by itself construct such an auxiliary for a general potential; the results here are the universal energy comparison and the sharp Gaussian functions.

Proof overview

Section 2 first explains why the Gaussian functions have exactly the properties needed for energy. For \(\mathcal C_R=\mathcal C\cap B_R\), define \(M_R(\xi)=\sum_{x\in\mathcal C_R}e^{2\pi i x\cdot\xi}\). Fourier inversion gives the finite identity \[\sum_{x,y\in\mathcal C_R}f_\alpha(x-y) =\int_{\mathbb R^2}\widehat f_\alpha(\xi)|M_R(\xi)|^2\,d\xi.\] A smooth cutoff of the disk and the density assumption show that, for every fixed \(\varepsilon>0\), \[\liminf_{R\to\infty}\frac1{N_R} \int_{|\xi|\le\varepsilon}|M_R(\xi)|^2\,d\xi\ge1.\] Nonnegativity and continuity of \(\widehat f_\alpha\) therefore give the off-diagonal lower bound \(\widehat f_\alpha(0)-f_\alpha(0)\), after removing the \(N_R\) diagonal terms. This step uses no count in a translated smaller disk.

The contact conditions identify that lower bound. The dual lattice is a quarter turn of \(A\), and Poisson summation with unit covolume gives \[\widehat f_\alpha(0)-f_\alpha(0) =\sum_{a\in A\setminus\{0\}}k_\alpha(a).\] The same duality turns a minorant for \(k_{1/\alpha}\) into one for \(k_\alpha\) when \(0<\alpha<1\), so only the construction for \(\alpha\ge1\) remains. Finally a positive Laplace representation expresses \(g\) as a mixture of Gaussian kernels. Fatou’s lemma is applied to their nonnegative energy sums along an arbitrary sequence of radii; this permits both an atom at the constant kernel and infinite total representing mass. These arguments prove the energy theorem from the later local construction, without requiring a sharp auxiliary for the mixed potential.

For \(\alpha\ge1\), Section 3 works in the scalar coordinate \(s=b|x|^2\). The interpolation nodes are the positive integers whose residues modulo \(36\) are represented by \(m^2+mn+n^2\) for integers \(m,n\); they contain every coordinate \(b|a|^2\) for \(a\in A\setminus\{0\}\) and may include additional nodes. A product of squared sine factors, one for each residue, has a double zero at every node. Multiplying it by suitable double- and simple-pole factors produces columns that prescribe a value and a derivative. Periodic pole columns handle ten positive nodes, together with fixed data at zero; rational columns with one pole location handle the remaining positive nodes.

The periodic columns use stronger Gaussian damping. Before damping they are trigonometric polynomials, so their spectral measures consist of finitely many atoms. They also contribute value-and-derivative data at later nodes in their residue classes modulo \(36\); the stronger damping makes these repeated contributions summably small. The rational columns have absolutely continuous spectral measures with total variation bounded uniformly in their node. These representations prove that arbitrary absolutely summable tail coefficients give radial Schwartz functions, with all needed differentiations justified. Two real input functions \(U_1,U_2\) are formed from the damped columns, together with opposite fixed multiples of the damped sine product. Define \[F_1=U_1+\widehat U_2,\qquad F_2=U_2+\widehat U_1.\] Fourier inversion and radiality give \(\widehat F_1=F_2\).

Section 4 constructs a finite approximation to the simultaneous value-and-slope equations. There are twenty unknown coordinates on each side; taking sums and differences separates the forty coupled coordinates into two blocks of order twenty. A single containing polytope encloses the Gaussian data. Outward-rational interval arithmetic certifies the fixed inverse bounds and coefficient bounds on that polytope. For the polynomial inequalities, every Bernstein coefficient is bounded on the whole polytope, with no mesh of parameter or spatial values. Appendix 7 gives the exact evaluation recipes and proves that their interval arithmetic encloses the required quantities.

Section 5 then proves uniform bounds for the maps between the finite lists and the full unweighted \(\ell^1\) tail space. These analytic bounds include the repeated jets of the periodic columns. Together with the finite inverse bounds, they permit a Schur elimination and a Neumann series on the full tail space. The resulting functions satisfy the interpolation equations exactly at every node, including those beyond the finite block. The same argument bounds their difference from the finite approximation. Thus finite arithmetic supplies quantitative inputs, while the full-space argument supplies the exact infinite interpolation.

Section 6 proves the remaining pointwise inequalities on the entire half-line. Near a node, subtract the exact value and linear term before dividing by the squared displacement. The finite polynomial tests, correction bounds, and Gaussian comparison then control the exact functions on whole half-cells; a separate cubic argument handles the first right half-cell. Farther out, signed sine-product terms dominate the remaining curvature on every gap. Positive normalization gives the \(\alpha\ge1\) minorant and closes the input to the energy reduction.

Duality and the lattice normalization

The following lattice calculation is independent of the input (1). Let \(M\) have columns \(b^{-1/2}(1,0)\) and \(b^{-1/2}(1/2,b)\), so that \(A=M\mathbb Z^2\) and \(\det M=1\). If \(J=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\), the two-dimensional identity \(M^{-\mathsf T}=JMJ^{-1}\) gives \[A^*=M^{-\mathsf T}\mathbb Z^2=JA,\] because \(J^{-1}\mathbb Z^2=\mathbb Z^2\). Thus \(A\) and \(A^*\) have identical sets of radii, and the covolume in Poisson summation is exactly one.

A Fourier comparison using only centered density

Let \(\mathcal C\subset\mathbb R^2\) be locally finite, put \(\mathcal C_R=\mathcal C\cap B_R\) and \(N_R=\#\mathcal C_R\), and assume \(N_R/(\pi R^2)\to1\). In particular, \(N_R>0\) for all sufficiently large \(R\).

Under the input (1), Lemma 3 and Proposition 5 give \[ \liminf_{R\to\infty}\frac1{N_R} \sum_{\substack{x,y\in\mathcal C_R\\x\ne y}}e^{-\pi\alpha|x-y|^2} \ge \sum_{a\in A\setminus\{0\}}e^{-\pi\alpha|a|^2} \qquad(\alpha>0). \tag{5}\] Only nonnegative kernels will be mixed in the remaining argument; there is no need to integrate the auxiliary functions themselves.

Positive mixtures without a boundedness assumption at zero

The next lemma is the classical Bernstein representation of completely monotone functions; see [23]. For a related construction using Taylor coefficients and step functions, see [11]. We give the details here on the open half-line for a locally finite measure that may have infinite total mass and an atom at zero.

Lower limits, singular interactions, and attainment

We now prove the energy theorem under the single input (1). Once Sections 3–6 establish that input, Lemma 3 also completes the all-parameter Gaussian theorem.

Spectral measures for the interpolation columns

We establish exact spectral formulas for the columns and prove that summable coefficient lists define Schwartz functions. These facts justify the infinite interpolation system and termwise differentiation.

A finite complex measure \(\mu\) on \([-5/6,5/6]\) will be called conjugate symmetric if reflection takes it to its complex conjugate. Then \(\int e^{i\pi ts}\,d\mu(t)\) is real for real \(s\). Its folded measure is \[\mu_+=\mu(\{0\})\delta_0+2\mu|_{(0,5/6]}, \qquad \int e^{i\pi ts}\,d\mu(t) =\operatorname{Re}\int_{[0,5/6]}e^{i\pi ts}\,d\mu_+(t).\] The atom at zero is not doubled. The total variations of these two measures agree, because the atom at zero is real and the positive and negative parts have equal total variation.

For \(n\in L\) consider the entire, real-valued columns obtained by removing their apparent poles: \[A_n(s)=\kappa^2 P(s)\csc^2\kappa(s-n),\qquad B_n(s)=\kappa P(s)\cot\kappa(s-n).\] Set \(b_j=2-\mathbf1_{j=0}\). Their folded spectral measures are atomic at \(t_j\), \(0\le j\le15\), with respective masses \[\begin{align*} M_{j;c,n} &=-4\kappa^2b_j \sum_{l=j+1}^{15}(l-j)P_l e^{i\pi(t_l-t_j)n}, \tag{11}\\ M_{j;d,n} &=i\kappa b_j\left(P_j+ 2\sum_{l=j+1}^{15}P_l e^{i\pi(t_l-t_j)n}\right). \tag{12}\end{align*}\] The folded masses for the column \(P(s)\) itself are \(V_j=b_jP_j\).

To prove these identities, put \(u=s-n\), \(Y=e^{2i\kappa u}\), and \(a_l=P_l e^{i\pi t_ln}\). The Laurent polynomial \(P(n+u)=\sum_l a_lY^l\) has a factor \((Y-1)^2\). Moreover, \[\csc^2\kappa u=-\frac{4Y}{(Y-1)^2},\qquad \cot\kappa u=i\left(1+\frac2{Y-1}\right).\] The products are therefore Laurent polynomials. Expanding at infinity, \[\frac{Y}{(Y-1)^2}=\sum_{r\ge1}rY^{-r},\qquad \frac1{Y-1}=\sum_{r\ge1}Y^{-r},\] shows that the coefficients of \(Y^j\), \(j\ge0\), in these products are \(-4\kappa^2\sum_{l>j}(l-j)a_l\) and \(i\kappa(a_j+2\sum_{l>j}a_l)\). Changing back from \(Y^j\) to \(e^{i\pi t_js}\) gives (11)–(12) after folding. The negative coefficients are conjugates of the positive coefficients. For completeness, \(P(n)=P'(n)=0\) give \[P_0+2\operatorname{Re}\sum_{l>0}a_l=0, \qquad \operatorname{Im}\sum_{l>0}l a_l=0.\] Thus both displayed zero-frequency masses are real, as folding requires.

For \(n\in T\) the other two undamped columns are \[C_n(s)=\frac{P(s)}{(s-n)^2},\qquad D_n^{\mathrm{col}}(s)=\frac{P(s)}{s-n},\] again with removable values. Their folded spectral measures are absolutely continuous. On \(t\in(t_{j-1},t_j)\), \(1\le j\le15\), their densities are \[\begin{align*} \rho_{c,n}(t) &=-2\pi^2\sum_{l=j}^{15}(t_l-t)P_l e^{i\pi(t_l-t)n},\tag{13}\\ \rho_{d,n}(t) &=2i\pi\sum_{l=j}^{15}P_l e^{i\pi(t_l-t)n}. \tag{14}\end{align*}\] Indeed, for real \(a\) the oriented integral identities \[\begin{align*} \frac{e^{i\pi au}-1}{u} &=i\pi\int_0^a e^{i\pi tu}\,dt,\\ \frac{e^{i\pi au}-1-i\pi au}{u^2} &=-\pi^2\int_0^a(a-t)e^{i\pi tu}\,dt \end{align*}\] hold first for \(u\ne0\) and then by continuity for \(u=0\). They are valid also when \(a<0\), with the usual orientation of the integral. Since \(\sum_l a_l=\sum_l t_la_l=0\), summing these identities with \(a=t_l\) and coefficient \(a_l\) gives the asserted columns. Replacing \(u\) by \(s-n\) yields the phase in (13)–(14). The negative-frequency densities are the conjugates of their reflected positive-frequency densities. The measures are absolutely continuous even at zero; endpoint values of their densities are immaterial. In particular, no zero-frequency atom is to be inserted when folding.

The phases depending on \(n\) have modulus one. Consequently the variation of either folded measure on \([t_{j-1},t_j]\) is bounded by the corresponding entry of \[ \begin{gathered} W_{j,*}=\left( 2(\pi/18)^2\sum_{l=j}^{15}(l-j+\tfrac12)|P_l|, (2\pi/18)\sum_{l=j}^{15}|P_l|\right), \\ W_{0,*}=(0,0). \end{gathered} \tag{15}\] For the first entry, integrate \(t_l-t\) on this interval to obtain \((l-j+1/2)/18^2\); for the second, the interval length is \(1/18\). These bounds hold uniformly for every positive tail node \(n\), however large it is.

Schwartz convergence and the Fourier waves

For \(k>0\) and \(|t|\le5/6\) put \[\Phi_{k,t}(x)=e^{-\pi b(k-it)|x|^2},\qquad x\in\mathbb R^2.\] For each pair of multi-indices \(\beta,\gamma\) there is a finite constant \(C_{\beta,\gamma,k}\), independent of \(t\), such that \[ \sup_{x\in\mathbb R^2} |x^\beta\partial_x^\gamma\Phi_{k,t}(x)| \le C_{\beta,\gamma,k}. \tag{16}\] To see this directly, differentiating the Gaussian gives a polynomial in \(x\) with coefficients polynomial in \(b(k-it)\), times the same Gaussian. Those coefficients are uniformly bounded on the compact frequency interval, and the absolute value of the Gaussian is \(e^{-\pi bk|x|^2}\). Every polynomial times this fixed decaying Gaussian is bounded.

Thus integration of \(\Phi_{k,t}\) against a finite measure defines a Schwartz function, and its \((\beta,\gamma)\) seminorm is at most \(C_{\beta,\gamma,k}\) times the measure’s total variation. Differentiation under the integral follows from the same bounds. In particular, define \[w_c=\sum_{j=1}^{15}W_{j,c},\qquad w_d=\sum_{j=1}^{15}W_{j,d}.\] For any real lists satisfying \(\sum_{n\in T}(|c_n|+|d_n|)<\infty\), the series of tail measures converges in total variation and has total variation at most \[w_c\sum_{n\in T}|c_n|+w_d\sum_{n\in T}|d_n|.\] The corresponding series of functions therefore converges in every Schwartz seminorm. This proves the claimed convergence for arbitrary summable coefficients; no moment condition such as \(\sum n^r|c_n|<\infty\) is needed. It also justifies taking removable values and derivatives at every node. Equivalently, before damping, the \(r\)th \(s\)-derivative of each integral is bounded on the real axis by \((5\pi/6)^r\) times its total variation, uniformly in \(n\). The finitely many atomic columns cause no further convergence issue.

With Fourier kernel \(e^{-2\pi ix\cdot\xi}\) the Gaussian integral in dimension two is \[ \widehat\Phi_{k,t}(\xi) =\frac1{b(k-it)} \exp\left(-\frac{\pi|\xi|^2}{b(k-it)}\right). \tag{17}\] For a positive real parameter \(a\), the one-dimensional identity \(\widehat{e^{-\pi ax^2}}(\xi)=a^{-1/2}e^{-\pi\xi^2/a}\) follows by differentiating the Fourier integral in \(\xi\), integrating by parts, and using \(\int e^{-\pi ax^2}\,dx=a^{-1/2}\) at \(\xi=0\). Taking the product of two such integrals gives the two-dimensional formula. Both sides of that formula are holomorphic in \(a\) on \(\operatorname{Re}a>0\): compact subsets have a common integrable Gaussian bound for every parameter derivative. The identity theorem extends the formula to that half-plane and gives (17) with \(a=b(k-it)\). This argument fixes both the Fourier normalization and the complex prefactor without any square-root branch ambiguity in dimension two.

Write \(s=b|\xi|^2\), \(B=b^{-2}\), and set \[ X_k(t)=\frac{B}{k^2+t^2},\qquad \lambda_k(t)=\frac{k+it}{b(k^2+t^2)},\qquad \zeta_{k,v}(t)=-tX_k(t)+i\bigl(kX_k(t)-v\bigr). \tag{18}\] After division by \(e^{-\pi vs}\), the right side of (17) is exactly \(\lambda_k(t)e^{i\pi\zeta_{k,v}(t)s}\). The Fourier transform commutes with the above measure integrals by Fubini, since \(\int|\Phi_{k,t}(x)|\,dx=1/(bk)\) uniformly in \(t\). It also commutes with summation of the tail columns, either by the same domination or by convergence in the Schwartz space. At opposite frequencies the transformed waves are conjugates, so the same folded measures may be used, followed by taking the real part.

For \(k,v\in\{h,H\}\) the additional decay exponent is positive. Indeed \(kX_k(t)\) decreases as \(t^2\) increases; its smallest difference from \(v\) is attained at \(|t|=5/6\). With \(h=17/50\) and \(H=27/50\), the four endpoint differences, for \((k,v)=(h,h),(h,H),(H,h),(H,H)\), are respectively \[\frac{100079}{455650},\quad \frac{8949}{455650},\quad \frac{216419}{554650},\quad \frac{105489}{554650}.\] In particular each is positive. The undivided transformed waves also have uniform Schwartz bounds: their Gaussian parameters \(1/(b(k-it))\) form a compact subset of the open right half-plane. All input functions and their transforms are consequently real radial Schwartz functions. If \(U_1,U_2\) denote the two complete input sums, then \[F_1=U_1+\widehat U_2,\qquad F_2=U_2+\widehat U_1 \quad\Longrightarrow\quad \widehat F_1=F_2,\] because Fourier inversion gives \(\widehat{\widehat U_2}(x)=U_2(-x)\) and \(U_2\) is radial.

Jets and repeated nodes

At a positive node \(n\) let \(v=H\) on \(f\) and \(v=h\) on \(T\). After division of the input centered at \(n\) by \(e^{-\pi vs}\), its value and first derivative at \(n\) are \[ Q_n\bigl(c_n,d_n+D_nc_n\bigr). \tag{19}\] For the rational columns this follows immediately from (8). For the trigonometric columns use \[\kappa^2\csc^2\kappa u=u^{-2}+O(1),\qquad \kappa\cot\kappa u=u^{-1}+O(u).\] Every other direct column has zero value and first derivative at \(n\), except for a trigonometric column with the same residue modulo \(36\). The direct constant column \(P\) always has a double zero there.

For any additional divided contribution \(Z(s)\), cancelling its jets requires the coefficient correction \[Q_n^{-1}\bigl(-Z(n),D_nZ(n)-Z'(n)\bigr).\] Applying this to one Fourier wave from (18) yields exactly \[ Q_n^{-1}\lambda_k(t)e^{i\pi\zeta_{k,v}(t)n} \bigl(-1,D_n-i\pi\zeta_{k,v}(t)\bigr). \tag{20}\] Integration against the folded measures, followed by taking real parts, gives the full transformed contribution to the coefficient equations.

There is one additional direct contribution on the tail. For \(n\in L\) its positive repeated poles are precisely \(m=n+36l\), \(l\ge1\). All belong to \(T\) because \(0\le n\le21\). At such \(m\), divide by \(e^{-\pi hs}\). The original damping \(H\) leaves \(e^{-\pi\eta s}\), so the jets of this repeated column are \[Q_m e^{-\pi\eta m} \bigl(c_n,d_n+(D_m-\pi\eta)c_n\bigr).\] In the tail coefficient coordinates of (19), this is \(e^{-\pi\eta m}(c_n,d_n-\pi\eta c_n)\). It must therefore be subtracted from the desired coefficients at \(m\). This includes the column centered at \(0\); no division by an uncontrolled quotient \(Q_m/Q_n\) appears. There are no other direct cross-terms: distinct residues have no pole at \(m\), and a rational column has only its own possible pole. The induced map from the finite lists to the summable tail lists has norm at most \[ (1+\pi\eta)\sum_{l\ge1}e^{-36\pi\eta l} =\frac{1+\pi\eta}{e^{36\pi\eta}-1}, \tag{21}\] since \(e^{-\pi\eta n}\le1\) for \(n\in L\).

Finally let \(G(s)=Q_1z e^{(1-s)/z}\), with \(z>0\) as defined for the Gaussian parameter, so the desired divided target at a node using damping \(v\) is \(e^{\pi(v-h)s}G(s)\) on side \(1\) and zero on side \(2\). Applying the inverse of (19) to that target gives \[ \frac{Q_1}{Q_n}e^{\pi(v-h)n+(1-n)/z} \bigl(z,-1+(\pi(v-h)-D_n)z\bigr). \tag{22}\] Equations (20) and (22), together with subtraction of the repeated column contribution, therefore impose exactly the desired value and first derivative at every positive node. All operations used to derive these equations are justified for arbitrary summable tail coefficients by the spectral variation and Schwartz bounds above.

Atoms, jets, and the finite system

Use the folded matrices \(M,V\) of (11)–(12), with columns \(c_n,d_n\) for \(n\in L\) and the separate constant column \(V\). The Fourier waves \(X_k,\lambda_k,\zeta_{k,v}\) are those in (18). For a folded atom column \(S\), its opposite-side contribution to the change of the two coefficients at \(n\) is \[ \mathcal R_{k,v}(n)S =\frac1{Q_n}\operatorname{Re} \sum_{j=0}^{15} \lambda_k(t_j)e^{i\pi\zeta_{k,v}(t_j)n} \begin{pmatrix}-1\\D_n-i\pi\zeta_{k,v}(t_j)\end{pmatrix}S_j. \tag{23}\] Indeed the direct coefficients supply jets \(Q_n(c_n,d_n+D_nc_n)\) after removing their own damping. Subtracting an opposite-side value \(u\) and derivative \(u'\) changes the coefficients by \(Q_n^{-1}(-u,D_nu-u')\), which is exactly (23).

For the finite equations use \(v=H\), \(k=H\) on \(M\), and \(k=h\) on \(V\). After putting the fixed columns \(c_0,d_0,C\) first, denote the resulting twenty-row matrix by \((R_0\ R)\), where \(R\) is \(20\) by \(20\). The fixed coefficients are \(c_{1,0}=c_{2,0}=0.44\), \(d_{1,0}=d_{2,0}=0\), and \(C_1=-0.006\), \(C_2=0.006\).

The finite columns use damping \(H\), whereas the prescribed functions \(J_i\) use damping \(h\). To obtain a finite coefficient pair, we therefore first multiply the target \(G_z(s)\) by \(e^{\pi\eta s}\) and then invert the direct jet map (19). Formula (22) gives the result. Its dependence on the Gaussian parameter enters through the two quantities \(ze^{(1-n)/z}\) and \(e^{(1-n)/z}\) at each node \(n\).

For the finite approximation we retain the targets at \(n=1,3,4,7\). Their exponential factors are \(1,w,u,p\), respectively, where \[w=e^{-2/z},\qquad u=e^{-3/z},\qquad p=e^{-6/z}.\] Thus the retained target pairs depend linearly on the eight coordinates \[e=(z,1,zw,w,zu,u,zp,p)^{\mathsf T}.\] At the six remaining finite nodes \(9,12,13,16,19,21\), this approximation replaces the target pairs by zero. The exact interpolation conditions are not changed: the omitted target pairs are a small residual, bounded in Lemma 7 and restored by the infinite correction in Section 5.

We enclose these Gaussian data in a single polytope. Put \[Z=0.391,\quad W_*=0.0061,\quad \Delta_*=0.00018, \quad U_*=0.00048,\quad P_*=0.0000003,\] and define \[ \begin{split} \mathcal E=\{&(z,1,Zw-\Delta,w,a,u,r,p)^{\mathsf T}:\quad 0\leq z\leq Z,\quad0\leq w\leq W_*,\quad 0\leq\Delta\leq\Delta_*,\\[-2pt] &0\leq u\leq U_*,\quad0\leq a\leq Zu,\quad 0\leq r,p\leq P_*\}. \end{split} \tag{24}\] For \(\alpha\geq1\) and \(z=1/(\pi(\alpha/b-h))\), the actual vector \(e\) belongs to \(\mathcal E\). The finite checks below give \(1/(\pi(1/b-h))<Z\) and \((e^{-2/Z},e^{-3/Z},e^{-6/Z})<(W_*,U_*,P_*)\) coordinatewise. The remaining coordinate is \(\Delta=(Z-z)e^{-2/z}\); its maximum occurs at \(z=\sqrt{1+2Z}-1\) and is less than \(0.00015<\Delta_*\). Also \(zp\leq p\leq P_*\) because \(Z<1\).

Let \(\gamma_i e=(0.44,0,C_i)^{\mathsf T}e_2\). Define the \(20\)-by-\(8\) target matrix \(Y_f\) by zero row pairs except at the first four nodes \(n\in f\). At the node in position \(j\in\{1,2,3,4\}\) its row pair is \[(Y_fe)_n=\frac{Q_1e^{\pi\eta n}}{Q_n} \begin{pmatrix}e_{2j-1}\\-e_{2j}+(\pi\eta-D_n)e_{2j-1}\end{pmatrix}.\] Let \(A_i\) have twenty-three rows, ordered as \(c_n,d_n\) on \(L\), followed by \(C\). Fix its three prescribed rows by \(\gamma_i\) and solve \[ (A_1)_f=R(A_2)_f+R_0\gamma_2+Y_f, \qquad (A_2)_f=R(A_1)_f+R_0\gamma_1. \tag{25}\] The inverse bounds below prove unique solvability. Explicitly, put \[U_\pm=(I\mp R)^{-1} \bigl(Y_f+R_0(\gamma_2\pm\gamma_1)\bigr).\] Then \((A_1)_f=(U_++U_-)/2\) and \((A_2)_f=(U_+-U_-)/2\). Recall that \(J_i=e^{\pi hs}F_i\). The finite function \(\widetilde J_i\) is formed by the same rule, using coefficients \(A_ie\) and no tail columns.

The finite inequalities

The coefficient bounds will control the infinite correction. For the sign bounds, it is important to compare curvature after removing the prescribed value and derivative at each node. For a twice continuously differentiable function \(\varphi\) on an interval containing \(m\) and \(m+v\), define the second-order divided remainder \[ U_m[\varphi](v)= \begin{cases} \displaystyle\frac{\varphi(m+v)-\varphi(m)-v\varphi'(m)}{v^2},&v\ne0,\\[5pt] \varphi''(m)/2,&v=0. \end{cases} \qquad U_m[\varphi](v)=\int_0^1(1-t)\varphi''(m+tv)\,dt. \tag{26}\] The integral identity follows by integrating twice and holds for either sign of \(v\). It shows that a bound for \(|\varphi''|/2\) on the joining segment also bounds \(|U_m[\varphi](v)|\). If two functions have matching value and derivative at \(m\), their difference at \(m+v\) is \(v^2\) times the difference of their divided remainders. We will use this identity for the exact interpolants; here we prepare polynomial estimates for the divided remainders of the finite approximations.

For \(m\in\{1,3,4,7,9,12,13\}\), let \(y\) be each displacement from \(m\) to the endpoints of its nearest-positive-node interval on \([0,\infty)\). At \(m=1\) the two values are \(-1\) and \(1\); elsewhere they are half the signed gaps to the neighboring positive nodes. The degree-twenty Taylor polynomial of \(U_m[\widetilde J_i](y\tau)\) is \[ T_i(\tau;e)=\sum_{\ell=0}^{20} \frac{\widetilde J_i^{(\ell+2)}(m)y^\ell}{(\ell+2)!}\tau^\ell. \tag{27}\] Its coefficients are linear in \(e\). In the exceptional case \(m=y=1\), write \(A(\tau)\) and \(D(\tau)\) for \(-T_1\) at \(z=0\) and \(z=Z\), fixing \(w,\Delta,r,p\) and a chosen allowed pair \((a,u)\) in (24). Only \(z\) varies along this segment. These scalar polynomial names are distinct from the coefficient matrices \(A_i\).

The matrix \(W\) in (15) has columns ordered \((c,d)\). Its row \(j\) bounds the integrated folded variation on \([t_{j-1},t_j]\) of one tail column, uniformly in its node. Set \(T_0=\{n\in T:25\le n<61\}\).

We use three kinds of envelopes. Jet envelopes bound the coefficient maps of the interpolation system. Curvature envelopes, after weighting by the relevant coefficient bounds, bound half the absolute second derivative of the pole-column and transformed-constant contributions to \(J_i\). Taylor envelopes bound the degree-twenty truncation error in the divided Taylor expansion of the finite approximation \(\widetilde J_i\). For input damping \(k\in\{h,H\}\), divide the direct wave by \(e^{-\pi hs}\) and its Fourier transform by \(e^{-\pi vs}\), where \(v\in\{h,H\}\). At frequency \(t_j\), let \(\ell=|\lambda_k(t_j)|\) be the transformed-wave amplitude. Let \(d_a,d_b\) be the absolute derivative multipliers of the divided direct and transformed waves, and \(E_a,E_b\) their decay rates. Explicitly, put \[\begin{gathered} \ell=\sqrt{X_k},\quad d_a=\pi\sqrt{t_j^2+(k-h)^2},\quad E_a=\pi(k-h),\\ d_b=\pi\sqrt{v^2+(B-2kv)X_k},\qquad E_b=\pi(kX_k-v). \end{gathered}\] The row for the jet envelope at \(n\) is \[\psi_j(n)=\frac{(1+|D_n|+d_b)\ell e^{-E_bn}}{Q_n}, \qquad \Psi_{f,k,H}(S)=\left[\sum_{n\in f}\psi(n),S\right].\] For \(\Psi_{T,k,h}(S)\), sum over \(T_0\) and divide its \(j\)th entry by \(1-e^{-36E_b(t_j)}\). Periodicity makes this the exact geometric sum of the node envelope over \(T\). The missing-target envelopes are \[Q_1\sum_{n\in f\cap[9,21]} \frac{1+Z+Z|\pi\eta-D_n|}{Q_n} e^{\pi\eta n+(1-n)/Z}\] and the sum over \(T_0\) of \(Q_1(1+Z+Z|D_n|)e^{(1-n)/Z}/Q_n\), divided by \(1-e^{-36/Z}\).

The raw \(L\) curvature expression is \[\tfrac12[d_a^2e^{-sE_a},M] +\tfrac12[d_b^2\ell e^{-sE_b},M] \qquad(k=H,\ v=h).\] For the transformed constant it is \(\tfrac12[d_b^2\ell e^{-sE_b},V]\), with \(k=v=h\). For \(T\), use \[60\bigl(d_a(5/6)^2+ d_b(t_*)^2\ell(t_*)e^{-sE_b(t_*)}\bigr), \qquad k=v=h,\] where \(t_*=0\) at \(s=0\) and \(t_*=5/6\) at the other listed values. The factor \(60\) bounds one half of \(\|W\|\). The frequency-extremum justification and the passage to uniform curvature bounds are given in Lemma 9.

For the Taylor remainder, define \[g_{m,y}(d,E)= \frac{d^2(|y|d)^{21}}{23!} \frac{e^{-mE}}{1-|y|d/24},\qquad \mathcal E_{m,y}(k,S) =[g_{m,y}(d_a,E_a),S]+[\ell g_{m,y}(d_b,E_b),S],\] using \(v=h\). Every denominator is positive. The tested envelope is \(3\mathcal E_{m,y}(H,M)+0.006\mathcal E_{m,y}(h,V)\). The geometric majorant of the exponential Taylor series explains precisely this remainder: after the degree-twenty divided polynomial, the first term has derivative order \(23\), and subsequent absolute-term ratios are at most \(|y|d/24<1\).

The lemma is the finite computational input, not the conclusion of the construction. The next section proves the uniform frequency and all-node operator bounds and solves the infinite system. Section 6 then transfers the finite polynomial inequalities to the exact functions and proves their signs at every real radius.

Bounds for the infinite coefficient maps

We use the sum of absolute values on every coefficient list. For a matrix between such lists, the operator norm is the supremum of its absolute column sums. The same characterization holds for infinite matrices acting on \(\ell^1\): a uniform finite bound on all column sums gives absolute convergence of all matrix sums by Tonelli’s theorem, and a bounded operator follows first on finite lists and then by density.

We explain why the finite wave estimates control every tail node. For \(k,v\in\{h,H\}\) and \(0\le t\le5/6\), let \[X=X_k(t),\quad \ell=\sqrt X, \quad d_b=\pi\sqrt{v^2+(B-2kv)X}, \quad E_b=\pi(kX-v).\] Equation (18) gives \(|\lambda_k|=\ell\) and \(|i\pi\zeta_{k,v}|=d_b\). The sum of the absolute values of the two entries in the jet correction (20) is therefore at most \[ \psi_{k,v,n}(t) =\frac{(1+|D_n|+d_b)\ell}{Q_n}e^{-E_bn}. \tag{31}\] Its use with atomic columns requires only summation over their atoms. For a tail column its value at the right endpoint \(t_j\) bounds the whole frequency interval \([t_{j-1},t_j]\). Here is the monotonicity needed for that assertion. Since \(B-2kv>0\), the logarithmic derivative in \(X\) of the increasing factor \((1+|D_n|+d_b)\sqrt X\) is at most \(1/X\). The logarithmic derivative of the exponential contributes \(-\pi kn\). For \(n\ge1\), \[\frac1X\le\frac{k^2+25/36}{B}<3k<\pi kn \qquad(k=h,H).\] The middle inequalities are direct rational inequalities at \(k=17/50,27/50\). Thus (31) decreases in \(X\) and increases in \(t\) on the nonnegative frequency interval. Multiplication by the interval variation bounds \(W_{j,*}\) of the finite certificate therefore gives a valid bound without quadrature or sampling.

The real numbers \(Q_n,D_n\) are periodic with period \(36\). If \(T_0=\mathcal N\cap[25,61)\), then every element of \(T\) is uniquely \(n+36l\) with \(n\in T_0\) and \(l\ge0\). Consequently for each frequency entry the full tail sum is exactly bounded by the finite-period sum using \[ \sum_{m\in T}\psi_{k,h,m}(t) =\frac{\sum_{n\in T_0}\psi_{k,h,n}(t)}{1-e^{-36E_b}}. \tag{32}\] The denominator is positive by the decay established above. Together with the repeated-node norm (21), these formulas are the analytic justification of the finite-to-tail, tail-to-finite and tail-to-tail bounds in Lemma 7.

To express these estimates as operator bounds, measure coefficient corrections after removing the local damping and inverting the direct jet map (19). Finite lists lie in \(\mathbb R^{20}\), indexed by the coefficient pairs on \(f\), and tail lists lie in \(\ell^1(T;\mathbb R^2)\). The map \(\mathcal B\) takes opposite-side tail lists to finite corrections; \(\mathcal C\) takes opposite-side finite trigonometric lists to tail corrections; and \(\mathcal D\) takes opposite-side tail lists to tail corrections. The map \(\mathcal A\) takes own-side finite trigonometric lists to direct repeated-node contributions on the tail. The equations below subtract these direct contributions. Finally, \(\mathcal K_{T,V}\) takes a scalar constant-column coefficient to its transformed tail correction.

Integrating the row of (31) against the entrywise absolute atomic matrix \(|M|\), the constant column \(|V|\), or the tail variation matrix \(W\), and adding the direct repetition bound, gives \[\begin{align*} \|\mathcal B\|&<340, &\|\mathcal D\|&<3\cdot10^{-8},\\ \|\mathcal C\|+\|\mathcal A\|&<\beta:=3\cdot10^{-10}, &\|\mathcal K_{T,V}\|&<1.3\cdot10^{-7}. \tag{33}\end{align*}\] The stated bound for \(\mathcal C\) and \(\mathcal A\) remains valid when the fixed trigonometric columns at zero are included. It follows from the stricter bounds \(7\cdot10^{-15}\) for the transformed columns and \(2.5\cdot10^{-10}\) for the repetitions. There is no direct repetition map on the tail: each rational column has only one possible pole.

Solving every interpolation equation

We adapt the finite-block and summable-tail elimination in [17] and [18], now including the own-side repetition map of the periodic finite columns.

With the coefficient lists \(u_i,w_i\) of Proposition 8 and the finite opposite-side coefficient-correction matrix \(R\), the finite equations are \[u_1=Ru_2+\mathcal B w_2+r_1,\qquad u_2=Ru_1+\mathcal B w_1+r_2.\] Here \(r_i\) collect the target data in finite coefficient coordinates and the coefficient corrections from the opposite-side transformed fixed columns. The tail equations have the form \[w_1=\mathcal C u_2+\mathcal D w_2-\mathcal A u_1+s_1, \qquad w_2=\mathcal C u_1+\mathcal D w_1-\mathcal A u_2+s_2.\] Each \(s_i\) is the sum of the target data in tail coefficient coordinates and the corrections from transformed fixed columns, minus the repeated-node contribution from the fixed trigonometric columns at zero. It is summable: the fixed-column contributions are summable by (33), and the targets decay geometrically by (22) with \(v=h\) on \(T\) and \(z>0\).

For \(\varepsilon\in\{1,-1\}\) define \(u_\varepsilon=u_1+\varepsilon u_2\) and \(w_\varepsilon=w_1+\varepsilon w_2\), and define \(r_\varepsilon,s_\varepsilon\) in the same manner. The two sides then separate exactly as \[\begin{align*} (I-\varepsilon R)u_\varepsilon -\varepsilon\mathcal B w_\varepsilon&=r_\varepsilon, \\ (I-\varepsilon\mathcal D)w_\varepsilon -(\varepsilon\mathcal C-\mathcal A)u_\varepsilon&=s_\varepsilon. \tag{34}\end{align*}\] The own-side repeated-node map has the same minus sign for both \(\varepsilon\); it does not exchange sides when the lists are subtracted.

Lemma 7 establishes \[\|(I-R)^{-1}\|<N_+:=17,\qquad \|(I+R)^{-1}\|<N_-:=5.\] Write \(L_\varepsilon=(I-\varepsilon R)^{-1}\). Eliminating the finite variables from (34) yields the tail equation \[\begin{align*} &[I-\varepsilon\mathcal D -(\varepsilon\mathcal C-\mathcal A)L_\varepsilon \varepsilon\mathcal B] w_\varepsilon\\ &\hspace{2em}=s_\varepsilon+ (\varepsilon\mathcal C-\mathcal A)L_\varepsilon r_\varepsilon. \tag{35}\end{align*}\] The operator subtracted from the identity has norm less than \[\delta+340\beta N_\varepsilon, \qquad \delta=3\cdot10^{-8},\] which is less than one for both values of \(\varepsilon\). The Neumann series therefore inverts this operator on the full Banach space \(\ell^1(T;\mathbb R^2)\) with norm at most \[ (1-\delta-340\beta N_\varepsilon)^{-1}. \tag{36}\] Recover \(u_\varepsilon=L_\varepsilon(r_\varepsilon+ \varepsilon\mathcal B w_\varepsilon)\) and then take half-sums and half-differences. These reversible steps give the unique finite and summable tail lists. The spectral convergence in Section 3 makes their inputs and Fourier pair Schwartz functions, and (20)–(22) identify the solved equations with all positive-node jets. It remains to bound the correction to the finite interpolant.

Error relative to the finite interpolant

Let \(A_i e\) be the finite coefficient lists from the arithmetic construction, with no tail columns. They contain the same fixed coefficients as the exact lists, satisfy the finite equations with the truncated target \(Y_f e\), and obey \(\|A_i e\|_1<3\) for actual parameter data. Their omitted target residual has norm less than \[d=8.2\cdot10^{-8}\quad\hbox{on }f, \qquad 10^{-26}\quad\hbox{on }T.\] Because the target appears only on side one, the same bound \(d\) applies to each of the sum and difference residuals. The sum of the norms of the two finite trigonometric lists is less than six, so their total tail effect is less than \(6\beta\). The constant columns cancel in the sum because \(C_1+C_2=0\); in the difference their absolute coefficient is \(|C_1-C_2|=0.012\). Thus valid tail residual bounds are \[d'_+=10^{-26}+6\beta, \qquad d'_-=10^{-26}+6\beta+0.012(1.3\cdot10^{-7}).\] Apply the same Schur elimination to the difference between the exact solution and this finite interpolant. From (35)–(36), its sum and difference errors are bounded on the tail by \[ e'_\varepsilon= \frac{d'_\varepsilon+\beta N_\varepsilon d} {1-\delta-340\beta N_\varepsilon}, \tag{37}\] and on the finite unknowns by \[ e_\varepsilon=N_\varepsilon(d+340e'_\varepsilon). \tag{38}\] All constants here are rational. Direct rational evaluation gives \[\begin{array}{c|cc} & e'_\varepsilon & e_\varepsilon\\\hline \varepsilon=1 & <1.800004\cdot10^{-9} & <1.179803\cdot10^{-5}\\ \varepsilon=-1 & <3.360002\cdot10^{-9} & <6.122004\cdot10^{-6}. \end{array}\] Taking half the sum of the two bounds gives the individual estimates (30), completing the proof of Proposition 8. It is not necessary that either sum/difference finite bound be below \(10^{-5}\) separately; the individual side is controlled by their average. The same observation applies to the difference tail bound. These are the uniform error bounds used by the global sign argument.

Curvature and Taylor remainders

For a direct wave after division by \(e^{-\pi hs}\), write its exponent as \(w_a=i\pi t-\pi(k-h)\). For a transformed wave write \(w_b=i\pi\zeta_{k,h}\), with amplitude \(\lambda_k\). Set \[\begin{gathered} d_a=|w_a|,\qquad E_a=-\operatorname{Re}w_a=\pi(k-h),\\ d_b=|w_b|,\qquad E_b=-\operatorname{Re}w_b=\pi(kX_k-h),\\ \ell=|\lambda_k|=\sqrt{X_k},\qquad X_k=\frac{B}{k^2+t^2}. \end{gathered}\] Thus \(d_a=\pi\sqrt{t^2+(k-h)^2}\) and \(d_b=\pi\sqrt{h^2+(B-2kh)X_k}\). The decay rates \(E_b\) are strictly positive, and \(E_a\ge0\). For a nonnegative row \(g\) and a matrix \(S\) of folded spectral coefficients, abbreviate \([g,S]=\max_c\sum_j g_j|S_{jc}|\).

We first apply these curvature bounds to the difference between the exact and finite deleted jets. The cells of the first seven positive interpolation nodes, and their signed half-gap lengths, are \[ \begin{array}{c|c|c} m&\text{cell}&\text{two values of }y\\\hline 1 &[0,2]&-1,\ 1\\ 3 &[2,3.5]&-1,\ .5\\ 4 &[3.5,5.5]&-.5,\ 1.5\\ 7 &[5.5,8]&-1.5,\ 1\\ 9 &[8,10.5]&-1,\ 1.5\\ 12&[10.5,12.5]&-1.5,\ .5\\ 13&[12.5,14.5]&-.5,\ 1.5 \end{array} \tag{42}\] Each half-cell is \(s=m+y\tau\), \(0\le\tau\le1\). The first cell includes \(s=0\): its definition uses positive interpolation nodes, although \(P\) itself also vanishes at zero.

The Gaussian comparison near the first nodes

The sine product and the unbounded range

We next turn the finite midpoint bounds into a lower bound for \(P\) between every pair of consecutive zeros.

Normalization and completion of the theorems

Finally \(Q_1ze^{1/z}>0\) and, because \(s=b|x|^2\) and \(z^{-1}=\pi(\alpha/b-h)\), \[\frac{e^{-\pi hs}G(s)}{Q_1ze^{1/z}} =e^{-\pi\alpha|x|^2}.\] Thus \(f_\alpha=F_1/(Q_1ze^{1/z})\) is the claimed Gaussian minorant for \(\alpha\ge1\). Its Fourier transform is \(F_2/(Q_1ze^{1/z})\ge0\), and its contact and Fourier-zero conditions follow from the exact positive-node jets. Every nonzero squared-radius coordinate of \(A\) or \(A^*\) belongs to \(\mathcal N\); the exact positive-node jets therefore supply every contact required by (1). This proves that input for every \(\alpha\ge1\). Lemma 3 completes Theorem 2 for all \(\alpha>0\), and the deduction in Section 2 completes Theorem 1.