Fourier certificates and interpolation

Cohn and Elkies formulated their sharpness conjecture in dimensions \(2\), \(8\), and \(24\) (Cohn and Elkies 2003, Conjecture 7.3). Their finite Laguerre–Gaussian approximations already combine prescribed double roots, numerical searches for the remaining parameters, and rigorous sign checks by rational arithmetic and Sturm root counts (Cohn and Elkies 2003, sec. 7, pp. 704–705). Exact attainment of the conjectured sharp bound remained unresolved in that approach. Viazovska constructed the required function in dimension \(8\) (Viazovska 2017, Theorems 3 and 4), and Cohn, Kumar, Miller, Radchenko, and Viazovska constructed its dimension-\(24\) counterpart (Cohn et al. 2017, sec. 4). Their constructions use modular forms to control a function and its Fourier transform simultaneously. Cohn and Elkies report that Gorbachev independently obtained essentially the same linear programming bound (Cohn and Elkies 2003, 694).

Interpolation gives a broader setting for these constructions. Radchenko and Viazovska recovered even Schwartz functions on the real line from their values and Fourier values at the square roots of nonnegative integers (Radchenko and Viazovska 2019, Theorem 1). Cohn, Kumar, Miller, Radchenko, and Viazovska developed interpolation using values and first derivatives in dimensions \(8\) and \(24\) and used it to prove universal optimality of the \(E_8\) and Leech lattices (Cohn et al. 2022, Theorems 1.4, 1.7, and 1.9).

The planar shell data have a different feature. Cohn, Kumar, Miller, Radchenko, and Viazovska conjectured that values and first derivatives of a radial Schwartz function and its Fourier transform on the nonzero shells of the triangular lattice normalized to covolume one do not determine the function uniquely (Cohn et al. 2022, Conjecture 7.5). Sardari proved this nonuniqueness by constructing infinitely many linearly independent radial Schwartz functions with double zeros, before and after Fourier transformation, on those shells (Talebizadeh Sardari 2021, Theorem 1.7). Thus prescribing the shell data alone does not single out a planar certificate. We solve a specified interpolation problem within an explicit family of Gaussian-damped cardinal functions and establish the signs separately.

A 2026 bachelor’s thesis abstract by Zhitniaia (Zhitniaia 2026) announces an integral representation of a magic function for the hexagonal lattice using weakly holomorphic modular forms and theta functions for \(\Gamma_0(12)\). Our comparison is limited to the institutional abstract, which does not specify the complete sign, normalization, and regularity statement. The precise relation of that claim to Theorem 1 remains to be determined from the full thesis.

The construction

For a sharp Fourier bound on a lattice, Poisson summation forces the auxiliary function and its Fourier transform to vanish on the nonzero shells of the lattice and its dual, respectively (Cohn and Elkies 2003, sec. 5). After normalizing the triangular lattice to covolume one, it is isometric to its dual, so the two sets of shell radii coincide. This normalization makes the coupled Fourier equations symmetric. At the end we dilate the resulting function to put the first physical-space shell at the exclusion radius one in Theorem 1. We cover their squared radii, in a fixed scaled coordinate, by a periodic set of interpolation nodes. Periodic supersets of the values of the triangular quadratic form also occur in Sardari’s construction (Talebizadeh Sardari 2021, sec. 6.1); here a fixed periodic set permits an explicit trigonometric cardinal family.

An interior zero of a differentiable function with a fixed sign must also have zero first derivative. We therefore impose zeros of order at least two on the Fourier side and at every physical-space shell beyond the first. The first physical-space shell is the boundary of the required sign region, so a simple zero is possible there; its prescribed negative derivative will fix the normalization.

A trigonometric polynomial vanishes doubly on the chosen set. Dividing it by linear and quadratic factors produces functions that prescribe values and first derivatives at individual nodes. This Hermite-cardinal pattern has a precedent in the Gaussian Beurling–Selberg constructions of Carneiro, Littmann, and Vaaler (Carneiro et al. 2013, formulas (1.2)–(1.3)); the Fourier coupling and sign estimates below are proved for the present family. Multiplication by a fixed Gaussian gives radial Schwartz functions. Their Fourier transforms admit integral representations with additional Gaussian decay. We use that decay to solve the coupled interpolation equations: a finite matrix handles the first nodes, and explicit estimates control the remaining infinite system. The inversion acts on the full unweighted \(\ell^1\) coefficient space. Its uniqueness concerns the prescribed coefficient lists within this chosen family.

The resulting functions have exact zeros, but the coefficient lists used for computation are finite approximations. An explicit per-list \(\ell^1\) error bound connects these approximations to the exact solution. Near a zero, a raw error divided by its vanishing power would be singular. We instead subtract the lower Taylor terms of the error before division; positive Bernstein coefficients then certify the signs on complete half-gaps. A separate sine-product barrier and second-derivative estimate control the entire unbounded region. Poisson summation first equates the two origin values. Differentiating its dilation identity then determines their common positive value from the prescribed first-shell derivative, so the final normalization is exact.

Section 2 develops the cardinal functions and their Fourier transforms. Section 3 solves the interpolation equations, and Section 4 proves the global signs and identifies their zeros. Section 5 establishes the exact normalization, performs the final dilation, and derives the packing consequences. Appendix 6 gives the finite data and the primary interval certificate, including the analytic integration error that connects its finite sums to exact moments. Appendix 7 gathers the rational alternatives and independent checks, with their precise mathematical and computational scopes.

The cardinal family

The values and first derivatives will be encoded by double-pole and simple-pole terms multiplied by a common vanishing factor. This is the same cardinal pattern that occurs in Gaussian Beurling–Selberg interpolation (Carneiro et al. 2013, formulas (1.2)–(1.3)). The nodes and simultaneous Fourier constraints here are different.

For \(-6\le j\le6\), put \(t_j=j/6\), and define \[ P(s)=\prod_{a\in A}\left(2\sin\frac{\pi(s-a)}{12}\right)^2 =\sum_{j=-6}^{6}P_j e^{i\pi t_js}. \tag{6}\] Here \(P_{-j}=\overline{P_j}\), and \[ (P_0,P_1,\ldots,P_6) =\left(5,-1+2ib,1+2ib,-2,-\frac12+ib,-1-2ib,1\right). \tag{7}\] For example, with \(w=e^{i\pi s/6}\) and \(\rho=e^{i\pi/6}\), each factor is \[\left(2\sin\frac{\pi(s-a)}{12}\right)^2 =-\rho^{-a}w^{-1}(w-\rho^a)^2.\] The six minus signs cancel, and \(\rho^{-\sum_{a\in A}a}=\rho^{-24}=1\). Thus the product in (6) equals \[w^{-6}\prod_{a\in A}(w-\rho^a)^2.\] Expanding this Laurent polynomial gives (7).

Every zero \(n\in12\mathbb Z+A\) of \(P\) is double. Define \(Q_n,D_n\) by \[ P(n+u)=Q_nu^2\bigl(1+D_nu+O(u^2)\bigr). \tag{8}\] If \(a\in A\) is the residue of \(n\) modulo \(12\), then \[ \begin{split} Q_n&=\left(\frac\pi6\right)^2 \prod_{\substack{a'\in A\\a'\ne a}} \left(2\sin\frac{\pi(a-a')}{12}\right)^2,\\ D_n&=\frac\pi6\sum_{\substack{a'\in A\\a'\ne a}} \cot\frac{\pi(a-a')}{12}. \end{split} \tag{9}\] Thus \(Q_n,D_n\) are periodic in \(n\), and their exact values are \[\begin{array}{c|c|c} a&Q_n&D_n\\\hline 0&\pi^2/3&-7\pi\sqrt3/18\\ 1&(2\pi^2/3)(2-\sqrt3)&\pi(3+\sqrt3)/18\\ 3&(2\pi^2/3)(2-\sqrt3)&-\pi(3+\sqrt3)/18\\ 4&\pi^2/3&7\pi\sqrt3/18\\ 7&(2\pi^2/3)(2+\sqrt3)&-\pi(3-\sqrt3)/18\\ 9&(2\pi^2/3)(2+\sqrt3)&\pi(3-\sqrt3)/18 \end{array}\] The expansion follows by isolating the vanishing sine factor; logarithmic differentiation of the remaining factors gives \(D_n\). The table then uses the elementary trigonometric values at multiples of \(\pi/12\). In particular, \[ Q_n>\frac74=1.75,\qquad \left\lvert D_n\right\rvert<\frac{53}{25}=2.12. \tag{10}\] To check these inequalities with rational bounds, use \(157/50<\pi<22/7\) and \(\sqrt3<26/15\). The minimum of \(Q_n\) is greater than \(197192/112500>7/4\), and the maximum of \(\left\lvert D_n\right\rvert\) is less than \(286/135<53/25\).

Let \(C\in\mathbb R\), and let \((c_n,d_n)_{n\in\{0\}\cup\mathcal N}\) be a real coefficient list with \[ \left\lVert(c,d)\right\rVert_1 :=\sum_{n\in\{0\}\cup\mathcal N}(\left\lvert c_n\right\rvert+\left\lvert d_n\right\rvert)<\infty. \tag{11}\] Away from the nodes, define \[ p(s)=P(s)R(s),\qquad R(s)=C+\sum_{n\in\{0\}\cup\mathcal N} \left(\frac{c_n}{(s-n)^2}+\frac{d_n}{s-n}\right). \tag{12}\] We next show that the apparent singularities are removable, with uniform control of the resulting entire function.

In particular, for every integer \(k\ge0\), \[ \left\lvert p^{(k)}(s)\right\rvert\le\pi^k\left\lVert\mu\right\rVert_{\mathrm{TV}} \qquad(s\in\mathbb R). \tag{17}\] The finite cardinal sums and their derivatives converge locally uniformly: the assertion for derivatives follows either directly from the measure bound or from Cauchy’s integral formula on a slightly larger compact set. Passing their node values and derivatives to the limit in (8) gives the cardinal identities \[ p(n)=Q_nc_n,\qquad p'(n)=Q_n(d_n+D_nc_n) \qquad(n\in\{0\}\cup\mathcal N). \tag{18}\] The other summands in (12) vanish to second order at \(n\), which explains why these identities involve only its own pair of coefficients.

Fourier transformation

For a cardinal function \(p\), let \[q_p(x)=e^{-\pi hs}p(s),\qquad s=b\left\lvert x\right\rvert^2.\] Define, for real \(-1\le t\le1\), \[ \lambda(t)=\frac{i}{b(t+ih)},\qquad z(t)=-\frac{B}{t+ih}-ih, \tag{19}\] and, using the measure from Lemma 2, put \[ K_p(s)=\int_{-1}^{1}\lambda(t)e^{i\pi z(t)s}\,\mathop{}\!\mathrm d\mu(t). \tag{20}\]

No cardinal representation of \(K_p\) is assumed. Only its entireness, its values and derivatives, and the estimates obtained from (20) will be used in the interpolation equations. In particular, the measure in that formula is supported on the real interval \([-1,1]\); later complex-disk estimates concern specified finite integrands, not an analytic continuation in the integration variable of an arbitrary \(\ell^1\) density.

The coupled interpolation conditions

We use two cardinal functions \(p_1,p_2\), with coefficients \(c_{i,n},d_{i,n},C_i\) for function index \(i\in\{1,2\}\). Their coefficients at positive nodes will be determined later; fix \[ (c_{1,0},d_{1,0},C_1)=(1,0.44,-0.013),\qquad (c_{2,0},d_{2,0},C_2)=(0,-0.368,0.017). \tag{23}\] All terminating decimals in the construction denote exact rational numbers. Write \(K_i=K_{p_i}\) and \(H_i=p_i+K_{3-i}\). The associated pair is \[ F(x)=e^{-\pi hs}H_1(s),\qquad \widehat F(\xi)=e^{-\pi hs}H_2(s), \quad s=b\left\lvert x\right\rvert^2\ \text{or}\ b\left\lvert\xi\right\rvert^2, \tag{24}\] respectively. Indeed, \(F=q_{p_1}+\widehat{q_{p_2}}\), and Fourier inversion gives \(\widehat F=\widehat{q_{p_1}}+q_{p_2}\), since the functions are even. Thus both functions in (24) are real, radial, and Schwartz for every pair of admissible coefficient lists.

The required interpolation conditions are \[ \begin{split} H_1(n)&=H_2(n)=0,\\ H_2'(n)&=0,\qquad H_1'(n)=-Q_n\mathbf1_{\{n=1\}} \qquad(n\in\mathcal N). \end{split} \tag{25}\] Here primes denote derivatives with respect to \(s\). They give zeros of order at least two for both functions at every positive node except that \(H_1\) has a prescribed simple zero at \(1\).

Uniform envelopes for the Fourier terms

We use the following rational upper and lower bounds. As throughout the construction, every finite decimal denotes its exact rational value.

At each endpoint \(t_j=j/6\), \[\left\lvert\lambda(t_j)\right\rvert\le b_j^*,\qquad \left\lvert z(t_j)\right\rvert\le a_j^*,\qquad \operatorname{Im}z(t_j)\ge g_j.\] On the interval \(t_{j-1}\le t\le t_j\), the corresponding bounds are \[ \left\lvert\lambda(t)\right\rvert\le b_{j-1}^*,\quad \left\lvert z(t)\right\rvert\le a_{j-1}^*,\quad \operatorname{Im}z(t)\ge g_j+\beta_j(t_j-t),\quad \left\lvert u_n(t)\right\rvert,\left\lvert v_n(t)\right\rvert\le M_j. \tag{29}\] The density bounds hold uniformly for \(n\in\{0\}\cup\mathcal N\).

Here are direct ways to verify these estimates. The identities \[\left\lvert\lambda(t)\right\rvert=\frac{1}{b\sqrt{t^2+h^2}},\qquad \left\lvert z(t)\right\rvert^2=h^2+\frac{B^2-2Bh^2}{t^2+h^2},\qquad \operatorname{Im}z(t)=h\left(\frac{B}{t^2+h^2}-1\right)\] show that the first two quantities decrease on \([0,1]\). The nonnegative quantity \(2hBt/(t^2+h^2)^2\), which is minus the derivative of \(\operatorname{Im}z\), has its minimum on each of the six intervals at an endpoint. Its only interior critical point on the positive axis is a maximum. These observations give the first three bounds in (29) by endpoint substitution. For the densities, \(e^{i\pi t_l n}\) depends only on \(n\) modulo \(12\), whereas \(\left\lvert e^{-i\pi tn}\right\rvert=1\) on the real interval. The norm of an affine function of \(t\) is convex. Thus an upper bound for both folded densities on the \(j\)th interval is \[ 2\max_{a\in A}\max\left\{ \pi\left\lvert\sum_{l=j}^6P_le^{i\pi t_la}\right\rvert,\quad \pi^2\max_{u\in\{t_{j-1},t_j\}} \left\lvert\sum_{l=j}^6(t_l-u)P_le^{i\pi t_la}\right\rvert \right\}. \tag{30}\] Substitution gives a value less than the displayed \(M_j\) in every case. For the constant input \(C=1\), the absolute folded atom masses are \[(m_0^*,\ldots,m_6^*)=(5,4,4,4,2,4,2).\]

For \(s\ge0\), set \[ \begin{aligned} L_j(s)&=b_{j-1}^*e^{-\pi g_js} \min\left\{\frac16,\frac{1}{\pi\beta_js}\right\} &&(1\le j\le6),\\ L_j^P(s)&=m_j^*b_j^*e^{-\pi g_js} &&(0\le j\le6). \end{aligned} \tag{31}\] When \(\beta_js=0\), the minimum in the first line is understood to be \(1/6\). Integrating the exponential envelope in (29) over a piece of length \(1/6\) gives its factor \(L_j(s)\). Differentiating the kernel contributes \((i\pi z(t))^\ell\). Consequently, for every integer \(\ell\ge0\) and every input \(p\) of the form (12), \[ \left\lvert K_p^{(\ell)}(s)\right\rvert \le w\sum_{j=1}^6 M_jL_j(s)(\pi a_{j-1}^*)^\ell +\left\lvert C\right\rvert\sum_{j=0}^6L_j^P(s)(\pi a_j^*)^\ell, \qquad s\ge0, \tag{32}\] where \(w=\sum_{n\in\{0\}\cup\mathcal N}(\left\lvert c_n\right\rvert+\left\lvert d_n\right\rvert)\). The argument applies first to finite lists and then to all such lists by the finite-measure representation and absolute summability.

Both \(Q_n\) and \(D_n\) are \(12\)-periodic on the nodes. Moreover, \[L_j(n+12k)\le e^{-12k\pi g_j}L_j(n)\qquad(k\ge0),\] because the minimum factor in (31) is nonincreasing. For every real cutoff \(x\ge0\), the value and derivative terms in (27), together with (32), therefore imply \[ \left\lVert S_{>x,*}\right\rVert \le \sum_{n\in\mathcal N\cap(x,x+12]} \sum_{j=1}^6 \frac{1+\left\lvert D_n\right\rvert+\pi a_{j-1}^*}{Q_n}\, \frac{M_jL_j(n)}{1-e^{-12\pi g_j}}. \tag{33}\] Here \(>x\) means a strict inequality for the row node. This is a bound uniform over all input columns. It applies as well to a single input coefficient \(c_0\) or \(d_0\).

For completeness, this estimate also justifies the operator definition on the whole space. Apply the bound first to one positive-node basis column, then sum with the absolute coefficients of a finitely supported list. The case \(x=0\) bounds the resulting output norm by a constant times the input norm. Finite lists are dense in \(X\), so this map has a unique bounded extension to \(X\). The basis measures converge in total variation by Lemma 2; hence \(K_{p_x}(n)\) and \(K_{p_x}'(n)\) converge at each fixed node as the lists are truncated. The extended map therefore has exactly the coordinates in (27), not merely the coordinates of an auxiliary matrix limit.

For \(x=0,4,16,100\), respectively, the right side of (33) is less than \[27.7,\qquad .272,\qquad .0145,\qquad 4.7\cdot10^{-10}.\] We shall use the slightly weaker bounds \[ \left\lVert S_{>0,*}\right\rVert<28,\qquad \left\lVert S_{>4,*}\right\rVert<.28,\qquad \left\lVert S_{>16,*}\right\rVert<.015,\qquad \left\lVert S_{>100,*}\right\rVert<5\cdot10^{-10}. \tag{34}\] For the input \(C=1\), the analogue of (33) replaces \(M_jL_j\) by \(L_j^P\), uses \(a_j^*\) in place of \(a_{j-1}^*\), and sums over \(0\le j\le6\). For rows greater than \(100\), it gives the bound \[ \sum_{n\in\mathcal N,\ n>100} \left(\frac{\left\lvert K_P(n)\right\rvert}{Q_n} +\frac{\left\lvert D_nK_P(n)-K_P'(n)\right\rvert}{Q_n}\right) <3.2\cdot10^{-8}. \tag{35}\] The scalar substitutions in these envelope estimates use only the explicit constants and the node data. Section 6.6 records the finite scalar bounds and the precise coverage of their supplied verifier.

The finite certificates

There are \(51\) nodes in \(I_f\) and hence \(102\) scalar coordinates. The approximate lists \(\widetilde x_1,\widetilde x_2\) are given by Table 2, with the indicated factor \(10^{-10}\), and are zero at positive nodes beyond \(100\). Append the fixed data (23) to obtain \(\widetilde p_i\), and write \(\widetilde K_i=K_{\widetilde p_i}\).

Inversion on the whole coefficient space

We first prove an inverse bound that keeps finite and tail residuals separate.

The uniqueness here is within the specified coefficient space, with the fixed node-zero data and constants. Triangular-shell values and first derivatives do not determine an arbitrary radial Schwartz function and its Fourier transform uniquely (Talebizadeh Sardari 2021, Theorem 1.7).

Taylor quotients on half-gaps

Set \(\mathcal N_0=\{0\}\cup\mathcal N\). For each \(m\in\mathcal N_0\cap[0,40]\), let \(m^+\) be its successor in \(\mathcal N_0\) and include \[y=\frac{m^+-m}{2}.\] If \(m>0\), also include \(y=(m^--m)/2\), where \(m^-\) is its predecessor. Thus \(y\) is signed, and the associated half-gap is \[I_{m,y}=\{m+yv:0\le v\le1\}.\] For function index \(2\) we use every such interval, with order \(\nu=0\) at \(m=0\) and \(\nu=2\) otherwise. For function index \(1\) we use only those intervals contained in \([1,\infty)\), with \(\nu=1\) at \(m=1\) and \(\nu=2\) otherwise. In particular, the only odd order occurs on the rightward half-gap from \(1\). There are \(22\) centers from \(0\) through \(40\). They give \(43\) half-gaps for index \(2\) and, after removing the two intervals not contained in \([1,\infty)\), \(41\) for index \(1\), for a total of \(84\) certified intervals.

Fix one of the specified triples \((i,m,y)\). For \(s=m+u\) in its interval, we will prove positivity of \[\frac{\sigma_i H_i(m+u)}{u^\nu}.\] Here \(u^0=1\), and for \(\nu>0\) the quotient at \(u=0\) is interpreted by continuity. The interpolation conditions make the derivatives of \(H_i\) of orders below \(\nu\) vanish at \(m\), so this extension exists. Positivity of the quotient gives the required sign: every order is even except for the rightward half-gap from \(1\), where \(u\ge0\). Figure 1 shows the node-to-midpoint intervals on which these quotients are studied.

The tabulated function \(\widetilde H_i\) approximates \(H_i\) but need not reproduce its exact zero and slope at a node. We therefore compare the exact quotient with the Taylor series of \(\widetilde H_i\) after deleting the terms of orders below \(\nu\). Put \(\Delta_i=H_i-\widetilde H_i\). Since \(\widetilde H_i\) is entire and the lower derivatives of \(H_i\) vanish, this Taylor series gives \[\begin{align*} &\frac{\sigma_i H_i(m+u)}{u^\nu} -\sigma_i\sum_{r=0}^{\infty} \frac{\widetilde H_i^{(r+\nu)}(m)}{(r+\nu)!}u^r \\ &\hspace{1cm}= \frac{\sigma_i}{u^\nu} \left(\Delta_i(m+u)- \sum_{j=0}^{\nu-1}\frac{\Delta_i^{(j)}(m)}{j!}u^j\right). \tag{42}\end{align*}\] For \(\nu=0\), the sum on the second line is empty and the right-hand side is simply \(\sigma_i\Delta_i(m+u)\). For \(\nu>0\), Taylor’s integral remainder gives the exact identity \[\frac{\Delta_i(m+u)-\sum_{j=0}^{\nu-1} \Delta_i^{(j)}(m)u^j/j!}{u^\nu} =\frac{1}{(\nu-1)!}\int_0^1(1-t)^{\nu-1} \Delta_i^{(\nu)}(m+tu)\,\mathop{}\!\mathrm dt.\] It holds for either sign of \(u\) and extends continuously to \(u=0\). Since \(m+tu\) stays in the same half-gap, the absolute value of the right-hand side of [signs:deleted-jets] is at most \[ \frac{1}{\nu!}\sup_{s\in I_{m,y}} \left\lvert\Delta_i^{(\nu)}(s)\right\rvert. \tag{43}\] The same bound applies at \(u=0\) by continuity and for either orientation of the half-gap. Deleting the lower Taylor terms is essential here: dividing the raw table error by \(u^\nu\) could be singular at the node.

After writing \(u=yv\), the first \(29\) terms of the tabulated series in [signs:deleted-jets] form a polynomial of degree at most \(28\) in \(v\). We bound it on \([0,1]\) using the standard range enclosure by Bernstein coefficients (Titi and Garloff 2019, sec. 3), retaining its elementary proof below.

The relevance of this finite statement is the identity \[ \sum_{r=0}^{28}T_r v^r =\sum_{k=0}^{28} B_k\binom{28}{k}v^k(1-v)^{28-k}. \tag{46}\] Indeed, for \(0\le r\le k\le d=28\), the binomial identity \[\frac{\binom{k}{r}}{\binom{d}{r}}\binom{d}{k} =\binom{d-r}{k-r}\] gives \[\sum_{k=r}^{d}\frac{\binom{k}{r}}{\binom{d}{r}} \binom{d}{k}v^k(1-v)^{d-k} =v^r\sum_{l=0}^{d-r}\binom{d-r}{l}v^l(1-v)^{d-r-l}=v^r.\] Multiply by \(T_r\), sum over \(r\), and interchange the two finite sums to obtain (46). For \(0\le v\le1\), the basis functions on the right are nonnegative and sum to one. Hence the lower bounds in (45) hold on the whole interval for the polynomial on the left. To obtain a lower bound for the exact quotient, it remains to control the coefficient error in (43) and the terms beyond degree \(28\) in the tabulated series.

Bounds for the two errors

The density bounds and (32), together with the separate coefficient error bounds \(\delta\) for the two functions, give, for \(s\ge s_*\ge0\), \[ \left\lvert\Delta_i^{(\nu)}(s)\right\rvert \le \delta\sum_{j=1}^6 M_j \left(\frac{(\pi t_j)^\nu}{6} +L_j(s_*)(\pi a_{j-1}^*)^\nu\right). \tag{47}\] There are no atom contributions because the constants \(C_i\) have not changed. Substitution of the envelopes proves the following uniform bounds for (43): \[ \frac{1}{\nu!}\sup_{I_{m,y}}\left\lvert\Delta_i^{(\nu)}\right\rvert <\begin{cases} 0.004,&m=0,\\ 0.011,&m=1,\\ 0.002,&m\ge3. \end{cases} \tag{48}\] Here one uses \(s_*=0\) for \(m=0\), \(s_*=1/2\) for \(m=1\) and both possible orders, and \(s_*=2\) for \(m\ge3\). For example the first bound is \(\delta\sum_{j=1}^6(M_j/6+M_jL_j(0))<0.004\). All these are bounds on intervals, obtained from the derivative envelopes.

It remains to truncate the infinite series in [signs:deleted-jets]. Put \(Y=\left\lvert y\right\rvert\) and \(q=29+\nu\). The total variation of the direct spectral measure of either tabulated function is less than \(80\), since \[2.29\sum_{j=1}^6\frac{M_j}{6} +0.017\sum_{j=0}^6 m_j^*<80.\] For the transformed part, use coefficient norm less than \(2.3\) and \(\left\lvert C_i\right\rvert\le0.017\). Expanding each spectral exponential at \(m\) gives the following bound for the omitted quotient terms: \[\begin{align*} \mathcal E(m,Y,\nu) := {}&\frac{Y^{-\nu}}{q!} \frac{1}{1-\pi a_0^*Y/(q+1)} \Bigg(80(\pi Y)^q \\[-2pt] &\qquad+2.3\sum_{j=1}^6 M_jL_j(m)(\pi a_{j-1}^*Y)^q +0.017\sum_{j=0}^6 L_j^P(m)(\pi a_j^*Y)^q\Bigg). \tag{49}\end{align*}\] Indeed the first omitted exponential term has degree \(q\) and subsequent absolute-value ratios are at most \(\pi a_0^*Y/(q+1)<1\). After division by \(u^\nu\), all powers still have degree at least \(29\), so their maximum for \(\left\lvert u\right\rvert\le Y\) is bounded by evaluating these absolute-value majorants at \(Y\). This proves [signs:taylor-bound] also when \(y<0\).

The five representative substitutions are \[ (m,Y,\nu)=(0,\tfrac12,0),\ (1,1,1),\ (1,1,2),\ (3,1,2),\ (4,\tfrac32,2). \tag{50}\] In each case [signs:taylor-bound] is less than \(0.001\). Increasing \(m\) or decreasing \(Y\) only decreases the bound. The first three cases handle centers \(0\) and \(1\), the fourth handles \(m=3\), and the last handles every center \(m\ge4\). Thus the truncation error is less than \(0.001\) on every required half-gap.

Combining (46), [signs:deleted-jets], (48), and [signs:taylor-bound], we obtain \[ \frac{\sigma_i H_i(s)}{(s-m)^\nu} >\begin{cases} 0.735,&m=0,\\ 0.168,&m=1,\\ 0.017,&m=3,4,\\ 0.006,&4<m\le40, \end{cases} \qquad s\in I_{m,y}, \tag{51}\] with endpoint values interpreted by continuity. Since the only odd order has \(s-m\ge0\), these inequalities imply the desired signs on all of these intervals. Their union is \([0,41.5]\) for index \(2\) and \([1,41.5]\) for index \(1\): adjacent half-gaps meet at their common midpoint, and the last interval runs from \(40\) halfway to \(43\).

The unbounded region

For \(s\ge s_0:=41.5\), split the exact direct part as \[ \begin{aligned} p_i(s)&=P(s)R_{i,0}(s)+p_{i,1}(s),\\ R_{i,0}(s)&=C_i+ \sum_{\substack{n\in\mathcal N_0\\n\le40}} \left(\frac{c_{i,n}}{(s-n)^2}+\frac{d_{i,n}}{s-n}\right),\\ \widetilde R_{i,0}(s)&=C_i+ \sum_{\substack{n\in\mathcal N_0\\n\le40}} \left(\frac{\widetilde c_{i,n}}{(s-n)^2} +\frac{\widetilde d_{i,n}}{s-n}\right). \end{aligned} \tag{52}\] Here \(p_{i,1}\) is the cardinal contribution from exact coefficients at nodes greater than \(40\). The function \(R_{i,0}\) uses the exact coefficients, whereas \(\widetilde R_{i,0}\) uses the finite table; their constants and node-zero coefficients agree. All denominators in these two rational functions are at least \(1.5\). Write the corresponding decomposition of the exact function as \[E_i:=p_{i,1}+K_{p_{3-i}},\qquad H_i=P R_{i,0}+E_i.\] For each \(s\ge s_0\), the final comparison will use a nearest node \(m\ge43\). At such a node, the exact interpolation conditions and the double zero of \(P\) make \(E_i\) have zero value and slope; we verify this at the comparison step below. The useful scale for both terms is therefore \((s-m)^2\). We will bound \(\sigma_iR_{i,0}\) below, bound \(P(s)\) below by a multiple of \((s-m)^2\), and bound \(|E_i(s)|\) above at the same scale through a uniform second-derivative estimate.

We first show \[ \sigma_i R_{i,0}(s)>0.011\qquad(s\ge s_0,\ i=1,2). \tag{53}\] For the tabulated coefficients put \(u_n^*=\sigma_i\widetilde c_{i,n}\), \(v_n^*=\sigma_i\widetilde d_{i,n}\), and \([w]_-:=\min\{w,0\}\). Using \[\frac{1}{s-n}=\frac1s+\frac{n}{s(s-n)}\] and discarding positive terms gives the following lower bound for \(\sigma_i\widetilde R_{i,0}(s)\), uniformly for \(s\ge s_0\): \[ \sigma_i C_i+ \frac{[\sum_{n\le40}v_n^*]_-}{s_0} +\sum_{n\le40} \left(\frac{[u_n^*]_-}{(s_0-n)^2} +\frac{[v_n^*]_-n}{s_0(s_0-n)}\right), \tag{54}\] where both sums include \(n=0\). Each negative reciprocal term is smallest at \(s=s_0\). The exact rational sums in (54) exceed \(0.012\) for index \(1\) and \(0.016\) for index \(2\). To pass to the exact rational part, use \(s-n\ge1.5>1\) and the coefficient enclosure to obtain \[\begin{split} \left\lvert R_{i,0}(s)-\widetilde R_{i,0}(s)\right\rvert &\le\sum_{n\le40}\left( \frac{\left\lvert c_{i,n}-\widetilde c_{i,n}\right\rvert}{(s-n)^2} +\frac{\left\lvert d_{i,n}-\widetilde d_{i,n}\right\rvert}{s-n}\right)\\ &<\frac{\delta}{s_0-40}=0.00002. \end{split}\] The finite lower sums therefore imply the asserted exact bound (53).

The coefficient norm of the tabulated part with \(n>40\) is less than \(0.00002\) for either function. Hence the exact coefficient norm for \(p_{i,1}\) is less than \(0.00005\). The derivative envelopes imply \[\begin{align*} \sup_{s\ge s_0}\left\lvert E_i''(s)\right\rvert \le {}&0.00005\cdot77 +2.3\sum_{j=1}^6 M_jL_j(s_0)(\pi a_{j-1}^*)^2 \\ &+0.017\sum_{j=0}^6 L_j^P(s_0)(\pi a_j^*)^2 <0.006. \tag{55}\end{align*}\] For the direct part we used \(\sum_{j=1}^6 M_j(\pi t_j)^2/6<77\); for the transformed part we used the all-node coefficient norm less than \(2.3\) from Proposition 9.

It remains to give the quadratic lower bound for \(P\).

For any \(s\ge41.5\), choose a nearest node \(m\ge43\); at \(s=41.5\), choose \(43\) rather than the tied node \(40\). Both \(H_i\) and \(P R_{i,0}\) vanish to order at least two at \(m\), by (25) and the absence of a pole of \(R_{i,0}\) there. Thus \(E_i=H_i-P R_{i,0}\) vanishes to order at least two at \(m\). The segment between \(s\) and \(m\) lies in \([41.5,\infty)\), and [signs:tail-second-derivative] gives \[\left\lvert E_i(s)\right\rvert\le\frac{0.006}{2}(s-m)^2.\] Using (53) and Lemma 11, we obtain on the entire unbounded region \[ \sigma_iH_i(s) \ge\left(0.011\cdot0.68-\frac{0.006}{2}\right)(s-m)^2 =0.00448(s-m)^2\ge0. \tag{56}\]

The dilation identity

Use the lattice \(\Lambda\) of covolume one and the shell coordinate \(s_a\) from (3)–(4). Every positive \(s_a\) lies in \(\mathcal N\). Exactly six lattice vectors have \(s_a=1\): from \(4(j^2+jk+k^2)=(2j+k)^2+3k^2=4\), one obtains \[(j,k)=(1,0),\ (-1,0),\ (0,1),\ (0,-1),\ (1,-1),\ (-1,1).\]

As shown in Section 2, \(\Lambda^*=\mathcal R\Lambda\). Apply Poisson summation on \(\Lambda\) to \(F_t(x)=F(\sqrt t\,x)\), whose Fourier transform is \(t^{-1}\widehat F(\xi/\sqrt t)\). Radiality makes the sums over \(\Lambda^*\) and \(\Lambda\) equal, and gives, for every \(t>0\), \[ \sum_{a\in\Lambda}G_1(ts_a) =\frac1t\sum_{a\in\Lambda}G_2(s_a/t). \tag{59}\]

We will differentiate this identity twice. Here is the needed convergence justification. On a compact subinterval of \((0,\infty)\), each of the first two \(t\) derivatives of \(F(\sqrt t\,x)\) is a finite sum of spatial derivatives of \(F\) multiplied by polynomials in \(x\) and bounded functions of \(t\). The same statement holds for \(t^{-1}\widehat F(x/\sqrt t)\). The Schwartz bounds therefore imply, for every \(M\), a common bound \(C_M(1+|x|)^{-M}\) for these functions and their first two \(t\) derivatives. The number of points of a fixed planar lattice in a radius-\(R\) disk is \(O(R^2)\), so \(\sum_{a\in\Lambda}(1+|a|)^{-M}\) converges for \(M>2\). Uniform convergence on compact \(t\) intervals now permits both termwise differentiations of (59).

The final dilation

The scaling also records the prescribed contact data in the final radial coordinates. Write \(f_{\mathrm{rad}}(r)=f(r,0)\) and \(\widehat f_{\mathrm{rad}}(\rho)=\widehat f(\rho,0)\) for \(r,\rho\ge0\). Then \[f_{\mathrm{rad}}(r)=\frac{G_1(r^2)}{bF(0)},\qquad \widehat f_{\mathrm{rad}}(\rho)=\frac{G_2(b^2\rho^2)}{F(0)}.\] Proposition 12 therefore gives the exact zero sets in the respective sign domains: \[ \begin{aligned} \{r^2:r\ge1,\ f_{\mathrm{rad}}(r)=0\}&=\mathcal N,\\ \{\rho^2:\rho\ge0,\ \widehat f_{\mathrm{rad}}(\rho)=0\} &=b^{-2}\mathcal N=\frac43\mathcal N. \end{aligned} \tag{62}\] The physical zero at \(r=1\) is simple, every other listed physical zero is exactly double, and every listed Fourier zero is exactly double. Indeed, the Gaussian and scalar factors do not vanish, while the changes of variable \(s=r^2\) and \(s=b^2\rho^2\) have nonzero derivatives at these positive radii, so they preserve the orders established for \(H_1,H_2\). At the exclusion radius, the radial derivative is \[ f_{\mathrm{rad}}'(1) =\frac{2G_1'(1)}{bF(0)}=-\frac{1}{3b}=-\frac{2}{3\sqrt3}. \tag{63}\]

The packing consequence

The relation between the signs and packing density is particularly visible for a periodic packing. Suppose the center set is \(\bigcup_{j=1}^N(L+t_j)\), where \(L\) has covolume \(V\), the \(t_j\) are distinct modulo \(L\), and distinct centers have distance at least one. Absolute convergence follows from the Schwartz bounds. Every summand below except the \(N\) diagonal terms at the origin is nonpositive, so \[\mathcal S:=\sum_{j,k=1}^N\sum_{v\in L}f(v+t_j-t_k)\le Nf(0).\] Poisson summation on each translated lattice gives \[\mathcal S=\frac1V\sum_{w\in L^*}\widehat f(w) \left|\sum_{j=1}^N e^{2\pi i w\cdot t_j}\right|^2 \ge\frac{N^2}{V}\widehat f(0),\] because every Fourier summand is nonnegative. It follows that \(N/V\le f(0)/\widehat f(0)=2/\sqrt3\). The disks have area \(\pi/4\), hence their covered-area density is at most \(\pi/(2\sqrt3)\).

For an arbitrary, not necessarily periodic, packing, the same implication is Theorem 3.1 of Cohn and Elkies (Cohn and Elkies 2003). Its admissibility hypotheses hold because both \(f\) and \(\widehat f\) are Schwartz. For disks of radius \(1/2\), the Cohn–Elkies center-density normalization is one quarter of the center-number density in their upper-density convention. Multiplying it by the unit-disk area \(\pi\) therefore gives the corresponding covered-area density, equivalently \(\pi/4\) times that center-number density. Their upper density takes a supremum over observation centers, so it bounds the fixed-origin limsup used in the introduction. For the fixed-origin convention and separated centers, the covered area inside \(B_R\) lies between \((\pi/4)\) times the number of centers in \(B_{R-1/2}\) and \((\pi/4)\) times the number in \(B_{R+1/2}\); division by \(\pi R^2\) and passage to the limsup matches the convention in the introduction. The lattice generated by \((1,0)\) and \((1/2,b)\) has one center per area \(b\) and attains this value. This is the classical packing-density theorem recovered from the new Fourier certificate, not a new density claim.

The restriction of the physical zero set to integer squared radii also recovers the classical periodic equality case. We use the even integral lattice argument of Cohn and Elkies (Cohn and Elkies 2003, sec. 8).

A second-derivative consistency identity

The same dilation identity contains an additional relation among the exact curvatures. It is not needed for the construction or its sign proof. For \(n\in\mathcal N\), let \(r_\Lambda(n)=\#\{a\in\Lambda:s_a=n\}\), allowing the value zero for nodes that are not lattice radii; in particular, \(r_\Lambda(1)=6\).

Exact input lists and matrices

Table 2 gives the coefficients at every node of \(I_f=\mathcal N\cap[1,100]\). Its four coefficient columns are multiplied by \(10^{10}\): an integer \(a\) in the table means the coefficient \(a\,10^{-10}\). At all nodes \(n>100\) both approximate lists are zero. The data at zero and the constant terms are, exactly, \[ (\widetilde c_{1,0},\widetilde d_{1,0},C_1) =(1,.44,-.013),\qquad (\widetilde c_{2,0},\widetilde d_{2,0},C_2) =(0,-.368,.017). \tag{66}\] The zero-node data are included in all evaluations of the tabulated functions. They are excluded when forming a column of the operator \(S\). Direct integer addition gives \[ \begin{aligned} \sum_{n\ge0}(|\widetilde c_{1,n}|+|\widetilde d_{1,n}|) &=2.2516666247<2.29,\\ \sum_{n\ge0}(|\widetilde c_{2,n}|+|\widetilde d_{2,n}|) &=.8927477314<2.29,\\ \sum_{n>40}(|\widetilde c_{1,n}|+|\widetilde d_{1,n}|) &=.0000197471<.00002,\\ \sum_{n>40}(|\widetilde c_{2,n}|+|\widetilde d_{2,n}|) &=.0000161623<.00002. \end{aligned} \tag{67}\] Here and below an index \(n\ge0\) in a coefficient sum ranges over \(\{0\}\cup\mathcal N\).

Write \(J=\{1,3,4\}\) and \(E=I_f\setminus J\). Matrix rows and columns on a node set always list all \(c\) entries first and all \(d\) entries second, with nodes increasing within each group. In particular the order on \(J\) is \((c_1,c_3,c_4,d_1,d_3,d_4)\). The two rational matrices used to certify the small inverses are \[ W_+=10^{-4}\begin{pmatrix} \input{data/W_plus.tex} \end{pmatrix}, \tag{68}\] \[ W_-=10^{-4}\begin{pmatrix} \input{data/W_minus.tex} \end{pmatrix}. \tag{69}\] These matrices are exact inputs. The \(W\)-based inverse argument below uses their products and column sums; it does not require a computed matrix inverse. We describe the supplied program’s additional inverse checks in Section 6.3.

Moments and the integration rule

For a finite input \(p=PR\) supported on nodes at most \(100\), put \[ U_{p,l}(m,y)=\frac{y^l p^{(l)}(m)}{l!},\qquad X_{p,l}(m,y)=\frac{y^l K_p^{(l)}(m)}{l!}. \tag{70}\] The domain for the quadrature error estimate is stated explicitly in Lemma 16.

We use the first Fejér quadrature rule, whose cosine-sum weights are described in (Waldvogel 2006, sec. 2, formula (2.3)). Our weights are half the mass-two weights there, normalized to average over an interval; for even \(N=256\) the terminal cosine term in that formula vanishes. We prove the exactness, positivity and error bound needed here. Set \(N=256\), and for \(0\le v<N\) define \[ \begin{aligned} \alpha_v&=\frac{v+1/2}{N},\qquad o_v=\frac1N\left(1-2\sum_{a=1}^{127} \frac{\cos(2a\pi\alpha_v)}{4a^2-1}\right),\\ \tau_{jv}&=\frac{2j-1+\cos(\pi\alpha_v)}{12} \qquad(1\le j\le6). \end{aligned} \tag{71}\] Integration over \((t_{j-1},t_j)\) is replaced by evaluation at \(\tau_{jv}\) with weights \(o_v/6\). To check exactness, expand a polynomial in the Chebyshev basis \(T_k(\cos\theta)=\cos(k\theta)\). Discrete cosine orthogonality at \(\theta_v=\pi\alpha_v\) recovers its coefficients through degree \(255\). The integral averages of this basis on \([-1,1]\) are zero at odd \(k\) and \(1/(1-k^2)\) at even \(k\). These are exactly the coefficients in (71), so the rule is exact through degree \(255\). Moreover, \[o_v\ge\frac1{256}\left(1-\sum_{a=1}^{127}\frac2{4a^2-1}\right) =\frac1{256\cdot255}>0,\qquad \sum_v o_v=1.\] Thus the six pieces together have total weight one.

For clarity, the complete density on piece \(j\) used in this appendix is \[ f_j(t)=2\pi\sum_n e^{-i\pi tn} \left\{-\pi c_n\sum_{l=j}^6(t_l-t)P_l e^{i\pi t_ln} +i d_n\sum_{l=j}^6 P_l e^{i\pi t_ln}\right\}. \tag{72}\] It includes node \(0\) when that coefficient is part of the input. The folded atoms have masses \(e_jCP_j\) at \(t_j\), where \(e_0=1\) and \(e_j=2\) for \(j>0\). The factors multiplying this density and these masses for \(U\) and \(X\), respectively, are \[ e^{i\pi tm}\frac{(i\pi ty)^l}{l!},\qquad \lambda(t)e^{i\pi z(t)m}\frac{(i\pi z(t)y)^l}{l!}. \tag{73}\] The real part is taken after summation.

This domain contains all basis inputs and the two tabulated inputs used below.

Interval evaluation

For a finite input \(p\) in the real finite-input domain of Lemma 16 and a node row \(m\in I_f\), form the two real numbers \[ \mathcal S_m(p)= \left(-\frac{X_{p,0}(m,1)}{Q_m},\; \frac{D_mX_{p,0}(m,1)-X_{p,1}(m,1)}{Q_m}\right). \tag{76}\] With input \(c_n=1\) or \(d_n=1\) and all other data zero, these are the entries of the corresponding column of \(S\) at row node \(m\).

Let \(D=S_{J,J}\) and \(U=S_{J,E}\).

For the Bernstein certificates, use exactly the triples \((i,m,y)\) and orders \(\nu\) specified in Lemma 10; in particular \(y\) is signed and no leftward interval at \(m=1\) is used for index \(1\). Express its power coefficients in terms of the moments as \[ T_r=\sigma_i y^{-\nu} \left(U_{\widetilde p_i,r+\nu}(m,y) +X_{\widetilde p_{3-i},r+\nu}(m,y)\right). \tag{77}\] The signed scaling is exact. Definition (70) gives \(U_{p,l}(m,y)=y^lU_{p,l}(m,1)\) and the same identity for \(X\). Therefore, for positive or negative \(y\), \[T_r=\sigma_i y^r\left(U_{\widetilde p_i,r+\nu}(m,1) +X_{\widetilde p_{3-i},r+\nu}(m,1)\right).\] This is the identity used when the implementation caches unscaled moments at \(y=1\); it does not discard the orientation of a half-gap.

Equations (71), (72), (73), (76), and (77) already specify all sums in the finite certificates below. The following pseudocode records their algebraic organization. A measure is a list of pairs \((t,g)\) in the order used by the supplied interval implementation: first the \(6\cdot256\) quadrature pairs \((\tau_{jv},o_v f_j(\tau_{jv})/6)\), with \(j=1,\ldots,6\) outermost and \(v=0,\ldots,255\) innermost, and then the seven atom pairs \((t_j,e_jCP_j)\) for \(j=0,\ldots,6\). The atom masses are zero for a basis input. The pseudocode is not a byte-for-byte execution trace. The supplied source fixes the detailed arithmetic and summation order for an actual interval run, including the factoring and caching described below.

Moments(measure, m, y, L):
    U[0..L] = 0; X[0..L] = 0
    for (t,g) in measure:
        lam = i/(b*(t+i*h))
        z = -B/(t+i*h)-i*h
        u = g*exp(i*pi*t*m)
        v = g*lam*exp(i*pi*z*m)
        for r = 0..L:
            U[r] += real(u); X[r] += real(v)
            u *= (i*pi*t*y)/(r+1)
            v *= (i*pi*z*y)/(r+1)
    return U,X

Pair(measure,m):
    U,X = Moments(measure,m,1,1)
    return -X[0]/Q[m], (D[m]*X[0]-X[1])/Q[m]

for basis column c[n] or d[n], n in I_f:
    measure = Measure(basis column; all other data zero)
    column = stack Pair(measure,m), m in J
    if n in J:
        also form rows m in E intersect [7,16]
    test column sums of W[eta] and the defect/product matrices

for i in {1,2}, m in I_f:
    c,d = Pair(Measure(full table input 3-i),m)
    test abs(table_c[i,m]-c)
    test abs(table_d[i,m]+indicator(i==1 and m==1)-d)

for (i,m,y,nu) in the prescribed half-gap cases:
    U,X = Moments for the two full table inputs, through L=30
    T[r] = sigma[i]*(U[i,r+nu]+X[3-i,r+nu])/y**nu
    for k = 0..28:
        Bk = T[0]; a = 1
        for r = 1..k:
            a *= (k+1-r)/(29-r)
            Bk += a*T[r]
        test the appropriate lower bound for Bk

Here stack means all first entries followed by all second entries. The recurrence for \(a\) equals \(\binom{k}{r}/\binom{28}{r}\). Measure denotes this finite list and introduces no additional approximation. The fixed data at zero are used only for the two full table inputs. Reusing a measure means reusing its whole finite list.

For the accompanying certified evaluation, every real or complex quantity in these sums is enclosed using interval arithmetic. The pair calculation forms the moments at \(y=1\). For the Bernstein calculation, the implementation also forms and caches the unscaled moments at \(y=1\), then uses the signed \(y^r\) identity following (77). Each unscaled real moment enclosure is enlarged by \([-10^{-22},10^{-22}]\), as justified by Lemma 16. Consequently the analytic-remainder contribution to the enclosure radius of a Bernstein coefficient is bounded by \[\begin{aligned} 2\cdot10^{-22}\sum_{r=0}^k \frac{\binom{k}{r}}{\binom{28}{r}}|y|^r &\le2\cdot10^{-22}\sum_{r=0}^{28}(3/2)^r\\ &=4\cdot10^{-22}\bigl((3/2)^{29}-1\bigr)<5.2\cdot10^{-17}. \end{aligned}\] The last inequality is an exact rational comparison. It is not correct to regard \(10^{-22}\) as an unamplified per-Bernstein error. Arb rounding and dependency effects can further widen the balls; all of them are propagated by the subsequent products, absolute values, column sums, and Bernstein combinations. An upper bound is accepted only when the complete enclosure lies strictly below the asserted rational bound, and a lower bound only when it lies strictly above. This procedure uses enclosures of the exact integrals, not a quadrature error inferred from observed numerical agreement.

The ball-arithmetic inclusion methodology is described in (Johansson 2017). The supplied implementation uses Arb through python-flint at \(192\) bits of precision; the citation does not by itself certify this local environment, its call domains, or a run. The entry point runs two verifiers in the same directory, and . Their output records, and , and the aggregate record are placed in its subdirectory. The first verifier checks the exact bounds in (78), (79), (80), and (81), as well as all 2436 individual Bernstein inequalities in Lemma 10. It also asserts four finite brackets: the two defect brackets, the exterior bracket, and the bracket for \((2,16,-1/2,28)\). All its comparisons and extrema use complete enclosures.

The mathematical \(W\) argument in the next subsection uses only products and column sums. The supplied program also calls a certified inverse for \(I-\eta D\) and tests its inverse and cross-product norms as additional checks; these calls are not needed for the \(W\) deduction.

Matrix and residual certificates

The following are strict upper bounds for the indicated column norms. The last three columns use only columns whose node lies in the stated range. They refer to the exact operator entries in (76). \[ \begin{array}{c|cc|ccc} \eta& \|W_\eta\|& \|I-W_\eta(I-\eta D)\|& \|W_\eta U_{7:9}\|& \|W_\eta U_{12:16}\|& \|W_\eta U_{19:100}\|\\ \hline +1&30.84&.001&3.54&.44&.25\\ -1&1.83&.001&.73&.32&.17 \end{array} \tag{78}\] The unmultiplied exterior block also satisfies \[ \|S_{E\cap[7,16],J}\|<.088. \tag{79}\] The certified enclosure calculation described in Section 6.3 proves these inequalities. For example, the two defect norms are enclosed respectively in \[\begin{gathered} [.00025422624910652544,\ .00025422624910652546],\\ [.00042288165102553355,\ .00042288165102553357], \end{gathered}\] and the exterior block norm is enclosed in \([.08688748182289641591,\ .08688748182289641592]\). These intervals are stated with rational endpoints rounded outwards.

The three printed product ranges partition \(E\), so \(\|W_\eta U\|\) is the maximum of the three grouped product norms. The exact bounds in (78) and (79) give the primary matrix conclusion directly. Put \(A_\eta=I-\eta D\). Since \(\|I-W_\eta A_\eta\|<.001\), the finite square matrix \(W_\eta A_\eta\) is invertible by a geometric series, hence so is \(A_\eta\), and \[A_\eta^{-1}=(W_\eta A_\eta)^{-1}W_\eta.\] It follows from (78) that \[\|A_\eta^{-1}\|<\frac{30.84}{1-.001}<32,\qquad \|A_\eta^{-1}U\|<\frac{3.54}{1-.001}<3.7.\] Equation (79) gives the remaining bound \(<.09\) in Lemma 7.

For the two full tabulated inputs, including (66), the residual certificate is \[ \begin{aligned} \left|\widetilde c_{i,m}+\frac{X_{\widetilde p_{3-i},0}(m,1)}{Q_m}\right| &<3\cdot10^{-10},\\ \left|\widetilde d_{i,m}+\mathbf1_{\{i=1,\ m=1\}} +\frac{X_{\widetilde p_{3-i},1}(m,1) -D_mX_{\widetilde p_{3-i},0}(m,1)}{Q_m}\right| &<3\cdot10^{-10} \qquad(i=1,2,\ m\in I_f). \end{aligned} \tag{80}\] The primary evaluation in Section 6.3 proves (80) for these exact residuals. Since \(3\cdot10^{-10}<10^{-9}\), these inequalities, the preceding primary matrix conclusion, and (67) prove every finite assertion of Lemma 7.

Bernstein coefficient certificates

These bounds concern the exact derivatives of the tabulated functions, not derivatives rounded to satisfy interpolation identities. For example, the coefficient at \((i,m,y,k)=(2,16,-1/2,28)\) is enclosed in \[[.00951875955680,\ .00951875955684].\] The primary evaluation in Section 6.3 proves the grouped bounds (81) for the exact derivatives of the tabulated functions. Each printed lower bound is stronger than its corresponding threshold in (45). Therefore these exact grouped bounds imply Lemma 10.

Scalar envelope and sign bounds

The remaining finite substitutions involve elementary real expressions. For the endpoint and density envelopes in (29), it suffices to test the finitely many endpoints and residues specified there. The endpoint formulas for \(|\lambda|\), \(|z|\), and \(\operatorname{Im}z\) use rational numbers and square roots. The density expressions additionally use sine and cosine at rational multiples of \(\pi\). The endpoint tests are certified by the same elementary enclosures used above; their monotonicity and convexity extensions to whole intervals were proved in Section 3.

For the row envelope in (33), certified outward bounds are \[ \begin{array}{c|c} \text{row cut }x&\text{upper bound for the finite envelope sum}\\ \hline 0&27.618723\\ 4&.269094\\ 16&.014122\\ 100&4.617010\cdot10^{-10} \end{array} \tag{82}\] The atom-column analogue at cut \(100\) is less than \(3.131186\cdot10^{-8}\). These bounds imply all the row-tail constants used in Section 3.

For the sign proof, the following outward bounds record the relevant substitutions. The expressions themselves, and the reason they bound whole intervals or tails, are given at the cited equations.

In the fourth row the expression means the explicit sum with \(.00005\cdot77\), not the derivative supremum to its left. For the five cases of (50), the corresponding bounds for \(\mathcal E(m,Y,\nu)\) in [signs:taylor-bound] are \[ \begin{array}{c|c} (m,Y,\nu)&\text{upper bound}\\ \hline (0,1/2,0)&1.578\cdot10^{-10}\\ (1,1,1)&2.007\cdot10^{-5}\\ (1,1,2)&5.735\cdot10^{-6}\\ (3,1,2)&9.581\cdot10^{-11}\\ (4,3/2,2)&6.133\cdot10^{-7} \end{array} \tag{83}\] The first two rational-part bounds can be obtained by exact fraction addition alone. All the other displayed scalar bounds follow by substitution and outward enclosure of the explicit exponential, trigonometric and rational expressions. The scalar verifier records 22 named strict comparisons: the four row substitutions, the atom tail, the nine sign-table rows, the five Taylor representatives, and the three disk-integrand and quadrature comparisons. It also checks the endpoint and residue envelopes, the coarser row-tail comparisons, the finite-inverse substitution giving a bound below \(90\), and the distance substitution using \(91\) and \(2540\). The derivation of these whole-space coefficients from the Schur estimates remains the analytic argument in Section 3. Its disk assertions use the coarser bounds \(5\) and \(6.2\) for \(|\lambda|\) and \(|z|\); the finer \(4.774\) and \(5.912\) bounds were proved analytically above and are not literal scalar-program assertions. At the six gap midpoints the program evaluates the product defining \(P\) and checks that each quotient is greater than \(.68\); it does not assert the symbolic radical identities for those values. No sampling of an unknown function between interpolation nodes is involved. The logarithmic concavity argument and exact midpoint values in Lemma 11 supply the global lower bound for \(P\).

Rational exponential primitives

The fixed rational procedures below use rational arithmetic alone, with a complex number represented by a pair of rationals. Put \[\begin{aligned} p_*&=3.141592653589793238462643383279502884,\\ b_*&=.866025403784438646763723170752936183, \end{aligned}\] and retain \(B=4/3\) and \(h=2/5\) exactly. Both \(|p_*-\pi|\) and \(|b_*-b|\) are less than \(10^{-36}\). For example, the identity \(\pi/4=4\arctan(1/5)-\arctan(1/239)\) and alternating series through power \(57\) certify the first assertion; comparing \((b_*\pm10^{-36})^2\) with \(3/4\) certifies the second. Define \[[w]_{40}=10^{-40}\lfloor10^{40}w\rfloor\] for real \(w\), and apply it componentwise to complex numbers. Define an approximation \(e_*(u)\) to \(e^{i\pi u}\) by the following finite recursion: \[ v_0=\left[\sum_{q=0}^{32}\frac{(ip_*u/1024)^q}{q!}\right]_{40}, \qquad v_{a+1}=[v_a^2]_{40}\ (0\le a<10),\qquad e_*(u)=v_{10}. \tag{84}\] Real and imaginary parts of \(e_*(u)\) give the cosine and sine replacements when \(u\) is real.

A fixed rational evaluation

We now apply the fixed primitives to the full \(N=256\) rule, using rational coefficients and rational \(C,m,y\) in the finite domain of Lemma 16. After defining the rounded sums, we prove their uniform approximation error. No matrix, residual, or Bernstein values are asserted by that error theorem.

In (71), replace each cosine \(\cos(\pi u)\) by \(\Re e_*(u)\), with \(u=\alpha_v\) or \(u=2a\alpha_v\) as appropriate, and then apply \([\ ]_{40}\) to the final weight \(o_v\) only. Denote the results by \(o_v^{\#},\tau_{jv}^{\#}\). In the displayed coefficient list for \(P_l\), replace \(b\) by \(b_*\); call the resulting numbers \(P_l^{\#}\). Evaluate (72) by replacing \(\pi\) by \(p_*\), \(P_l\) by \(P_l^{\#}\), and every exponential by \(e_*\), at the approximate point \(t=\tau_{jv}^{\#}\). Call the resulting density \(f_j^{\#}(t)\). For either these points or the exact atom points \(t=t_j\), use \[ \lambda_*=[i/(b_*(t+ih))]_{40},\qquad z_*=[-B/(t+ih)-ih]_{40}, \tag{86}\] and set \[ E_l(t)=e_*(tm)\frac{(ip_*ty)^l}{l!},\qquad G_l(t)=\lambda_*e_*(z_*m)\frac{(ip_*z_*y)^l}{l!}. \tag{87}\] The rational moment sums are therefore \[ \begin{aligned} U_{p,l}^{\#} &=\Re\left\{\frac16\sum_{j=1}^6\sum_{v=0}^{255} o_v^{\#}f_j^{\#}(\tau_{jv}^{\#})E_l(\tau_{jv}^{\#}) +C\sum_{j=0}^6e_jP_j^{\#}E_l(t_j)\right\},\\ X_{p,l}^{\#} &=\Re\left\{\frac16\sum_{j=1}^6\sum_{v=0}^{255} o_v^{\#}f_j^{\#}(\tau_{jv}^{\#})G_l(\tau_{jv}^{\#}) +C\sum_{j=0}^6e_jP_j^{\#}G_l(t_j)\right\}. \end{aligned} \tag{88}\] No additional floors are used. For \(Q_n^{\#},D_n^{\#}\) use the product and cotangent formulas defining \(Q_n,D_n\), with the same sine and cosine replacements and residue \(a=n\bmod12\in A\); cotangent means the ratio of the replaced cosine to the replaced sine.

For the finite pseudocode in Section 6.3, the rational prescription uses the same measure order: the \(6\cdot256\) quadrature pairs followed by the seven atom pairs. Put \(\#\) on the prescribed quantities and make exactly the replacements above; no other rounding is performed.

Propagation to finite certificates

We record stability estimates before applying any numerical margin. The first controls the two operator coordinates under errors in the moments and node constants. The second controls Bernstein sums from errors in the signed-\(y\) moments.

To propagate a moment radius, abbreviate \(X_\ell=X_{p,\ell}(m,1)\) and write \(\mathcal S_{m,c},\mathcal S_{m,d}\) for the two components of (76), with hats for their rational approximations. Suppose \(|X_0|,|X_1|\le H\), \(|D_m|\le d\), and rational approximations have errors at most \(\epsilon,\epsilon,\eta_Q,\eta_D\) in \(X_0,X_1,Q_m,D_m\), respectively. If both exact and approximate \(Q_m\) exceed \(q>0\), expanding the numerator and then the reciprocal in (76) gives \[ \max\{|\widehat{\mathcal S}_{m,c}-\mathcal S_{m,c}|, |\widehat{\mathcal S}_{m,d}-\mathcal S_{m,d}|\} \le \frac{(d+1+\eta_D)\epsilon+H\eta_D}{q} +\frac{(d+1)H\eta_Q}{q^2}. \tag{89}\] For example, the perturbed second numerator differs by at most \((d+\eta_D)\epsilon+H\eta_D+\epsilon\), and the exact numerator is bounded by \((d+1)H\); this proves the inequality, which also dominates the first entry’s error. The Taylor budgets and the separated and common-kernel first-block estimates below use \(q=7/4\) and \(d=2.12\). The separate-rounding comparison uses \(q=7/4\) and \(d=4.3\). The rational lower bound \(Q_m>197192/112500\) proved after (10) exceeds \(7/4\) by \(1268/450000\), so \(\eta_Q\le10^{-25}\) also ensures that the approximate \(Q_m\) exceeds \(q\).

An independent rational check of the first block

The optional program checks the \(6\times6\) block on \(J=\{1,3,4\}\) by exact rational arithmetic. It uses the cosine rule (71) with \(N=64\) and the sum over \(1\le a\le31\), and otherwise the rounded exponential, kernels and basis densities specified above. There are no atom terms for these basis inputs. The following error bound makes this smaller check independent of the full Arb calculation.

If every entry of this \(6\times6\) matrix has error below \(\varepsilon=10^{-12}\), its induced column-norm error is below \(6\varepsilon\). The program accordingly enlarges the computed defect norm by \(\|W_\eta\|_1\,6\varepsilon\), then verifies that the result is below \(.003\) and that the resulting inverse bound is below \(32\). This is the same perturbation-of-identity argument as in Section 6.4. It checks both small inverse certificates without asserting that the full \(N=256\) rational prescription has been executed.

Comparison of first-block majorants

The preceding \(N=64\) proof separates the value and derivative integrands. Two coarser arguments instead use one majorant for both. They concern the same basis inputs \(n\in J\), row nodes \(m\in J\), \(y=1\), and orders zero and one, with no atoms. The table compares their disk and node bounds with the resulting entry errors: \[\begin{array}{c|c|c|c} \text{route}&\text{disk majorant}&|D_m|&\text{entry error}\\ \hline \text{separated}&M_0<24028,\ M_1<468015&<2.12&<7\cdot10^{-14}\\ \text{common kernel}&M=10^6&<2.12&<4.1\cdot10^{-13}\\ \text{separate rounding}&M=750000&<4.3&<5.23\cdot10^{-13} \end{array}\] The separated route uses the explicit density bound \(D_0\) from Section 7.4. The two common routes use density and exponential bounds \((1800,2.8)\) and \((2500,3)\), respectively.

Rational-center Taylor integration

For a finite input in the real moment domain of Lemma 16, let \(F^U_{j,\ell}\) and \(F^X_{j,\ell}\) be the density (72) multiplied by the respective factors in (73). The input is supported on nodes at most \(100\), has coefficient-list norm at most \(3\) and \(|C|\le1\), and the parameters satisfy \(0\le m\le100\), \(|y|\le3/2\), and \(\ell\in\{0,\ldots,30\}\). The coefficients, \(C,m,y\) are rational when a rational enclosure is requested. The six centers and the two radii are \[ c_j=\frac{2j-1}{12},\qquad R=\frac16,\qquad a=\frac1{12} \quad(1\le j\le6). \tag{98}\] Each \(F^U_{j,\ell}\) and \(F^X_{j,\ell}\) is analytic on a neighborhood of \(|t-c_j|\le R\): the input list is finite and the only rational-kernel pole, \(-ih\), is outside that disk. This analytic assertion is made for finite inputs. An arbitrary \(\ell^1\) list need not give a complex-frequency density analytic on these disks.

Here are explicit rational operations for obtaining the coefficient enclosures in this lemma. Write \(\mathcal B(a,r)=\{w\in\mathbb C:|w-a|\le r\}\), where \(a\in\mathbb Q(i)\) and \(r\in\mathbb Q_{\ge0}\), and put \(\langle a\rangle_1=|\Re a|+|\Im a|\). The following enclosure operations use rational centers and radii: \[\begin{align*} \mathcal B(a,r)+\mathcal B(b,s)&\subseteq\mathcal B(a+b,r+s),\\ \mathcal B(a,r)\mathcal B(b,s)&\subseteq \mathcal B\bigl(ab,\langle a\rangle_1s+\langle b\rangle_1r+rs\bigr),\\ \mathcal B(a,r)^{-1}&\subseteq\mathcal B(a^{-1},r/\ell_0^2) \quad\text{if }0<\ell_0\le |a|-r. \tag{101}\end{align*}\] In the last line \(a^{-1}\in\mathbb Q(i)\), and the condition on a rational \(\ell_0\) can be checked by squaring rational quantities. The bound follows from \(|w^{-1}-a^{-1}|=|w-a|/(|w||a|)\). These operations allow repeated use of the same enclosure; possible dependence between its occurrences does not invalidate inclusion.

Constant exponentials also admit arbitrarily small rational radii. Use the rational bounds \(\mathcal T_n,\mathcal E_n\) from (85). If \(w\in\mathcal B(w_0,r)\), choose rational \(q\ge|w_0|\), for example \(q=\langle w_0\rangle_1\), and nonnegative integers \(n,s\) with \(q<n+2\) and \(q+r<s+2\). Then \[ e^w\in\mathcal B\left(\sum_{k=0}^n\frac{w_0^k}{k!}, \mathcal T_n(q)+r\mathcal E_s(q+r)\right). \tag{102}\] Indeed, the first radius bounds the complex Taylor tail at \(w_0\). The integral first-difference formula for \(e^w\) bounds the change from \(w_0\) to \(w\) by \(r e^{q+r}\), giving the second radius. Both preconditions in (85) are explicit; no estimate of a positive exponential by a truncated sum alone is being used. For bounded \(q\), \(\mathcal T_n(q)\to0\) as \(n\to\infty\).

For clarity, the coefficient recurrences needed for the present integrands are as follows. If \(a(x)=\sum a_kx^k\) and \(b(x)=\sum b_kx^k\), multiplication is convolution, \[ (ab)_k=\sum_{v=0}^k a_v b_{k-v}. \tag{103}\] If \(a_v\in\mathcal B(\widehat a_v,e_v)\) and \(b_v\in\mathcal B(\widehat b_v,f_v)\), a rational radius for the center \(\sum_{v=0}^k\widehat a_v\widehat b_{k-v}\) is explicitly \[\sum_{v=0}^k\left(\langle\widehat a_v\rangle_1f_{k-v} +\langle\widehat b_{k-v}\rangle_1e_v+e_vf_{k-v}\right).\] For a reciprocal series \(q(x)=a(x)^{-1}\) with \(a_0\ne0\), and for an exponential series \(E(x)=e^{g(x)}\), the recurrences are \[\begin{align*} q_0&=a_0^{-1},& q_k&=-q_0\sum_{v=1}^k a_vq_{k-v}\quad(k\ge1),\\ E_0&=e^{g_0},& E_k&=\frac1k\sum_{v=1}^k v g_v E_{k-v}\quad(k\ge1). \tag{104}\end{align*}\] The second line follows by comparing coefficients in \(E'=g'E\). Use (102) for \(E_0\) and [ind:ball-operations] in these recurrences. Along with the displayed convolution radius, this specifies a rational radius for every coefficient, not merely a formal power series.

More particularly, \(d_j=c_j+ih\in\mathbb Q(i)\) and \[ r_0=d_j^{-1},\quad r_{k+1}=-r_k/d_j,\qquad \lambda_k=ir_k/b,\quad z_0=-Br_0-ih,\quad z_k=-Br_k\ (k\ge1) \tag{105}\] are the coefficients of \((t+ih)^{-1}\), \(\lambda(t)\), and \(z(t)\) at \(t=c_j+x\). The \(r_k,z_k\) are rational complex numbers; an enclosure for \(b^{-1}\) gives the radii for \(\lambda_k\). On piece \(j\), put \[L_{n,j}=\sum_{v=j}^6P_v e^{i\pi t_vn},\qquad H_{n,j}=\sum_{v=j}^6t_vP_v e^{i\pi t_vn}.\] For \(\Phi_n(x)=e^{-i\pi n(c_j+x)}\), use \(\phi_{n,0}=e^{-i\pi nc_j}\) and \(\phi_{n,k+1}=(-i\pi n)\phi_{n,k}/(k+1)\). The density is then the finite series expression \[ f_j(c_j+x)=2\pi\sum_n\Phi_n(x) \left\{-\pi c_n\bigl(H_{n,j}-(c_j+x)L_{n,j}\bigr)+id_nL_{n,j}\right\}. \tag{106}\] The constant exponentials in \(L_{n,j},H_{n,j},\phi_{n,0}\) are enclosed by (102). For a series \(v(x)\) equal to \(c_j+x\) or \(z(c_j+x)\), the divided powers obey \[ V_0(x)=1,\qquad V_{\ell+1}(x)= \frac{V_\ell(x)\,i\pi y v(x)}{\ell+1}. \tag{107}\] Finally apply the exponential recurrence to \(g=i\pi m(c_j+x)\) or \(g=i\pi m z(c_j+x)\), and multiply the resulting series by the density, the divided power, and, in the transformed case, \(\lambda\). Truncating these recurrences after degree \(D\) gives every coefficient through \(D\) exactly at the level of formal series; omitted higher coefficients cannot enter a lower convolution coefficient.

The seven possible atoms are handled separately. At \(t_j\), enclose their masses \(e_jCP_j\) and the appropriate factor in (73) by the same rational operations and constant exponential bound. If their respective centers and radii are \((a_j,r_j)\) and \((g_j,s_j)\), their product has radius \[ \eta_j=\langle a_j\rangle_1s_j+\langle g_j\rangle_1r_j+r_js_j. \tag{108}\] The sum \(\eta=\sum_{j=0}^6\eta_j\) is added to the six-piece error; there is no Taylor integration tail for an atom. When \(C=0\), all these terms vanish.

All of these radii can be refined effectively. Alternating rational arctangent sums in Machin’s identity enclose \(\pi\) to arbitrarily small width, and rational bisection of the positive square root of \(3/4\) does the same for \(b\). The kernel denominators have \(|c_j+ih|\ge h=2/5\), and \(b>4/5\). Taking real and imaginary parts of the constant exponential enclosures also encloses cosine and sine with the same radius. The nonzero sine denominators in (9) have modulus greater than \(1/4\). Consequently sufficiently refined enclosures certify fixed positive lower bounds for the moduli of all denominators used above or in computing \(Q_m,D_m\), and separately a positive lower bound for \(Q_m\) at pair division. For each fixed \(D\) there are finitely many operations in (103)–(107). Their radius formulas tend to zero when their input radii tend to zero, while (102) has radius tending to zero by refining its input and increasing its two truncation orders. Thus one may refine the rational inputs and exponential orders, recompute the displayed rational radii, and stop when any prescribed finite collection of positive coefficient and atom budgets holds. The convergence just proved ensures that this rational stopping test terminates. This is a computability statement for each specified finite set of inputs, not a claim that any of its moment sums have been evaluated.

We next make sufficient budgets explicit. Put \(h_D=\lfloor D/2\rfloor+1\). For rational \(\delta>0\), the conditions \[ e_{j,2r}\le \frac{\delta(2r+1)12^{2r}}{h_D} \quad(0\le2r\le D) \tag{109}\] on each piece and moment imply that the weighted coefficient-radius sum in (99) is at most \(\delta\).

We now combine these integration budgets with the pair estimate (89) and the Bernstein estimate from Section 7.3.1.

For a matrix with \(r\) rows and entry error at most \(\varepsilon\), the induced column-norm error is at most \(r\varepsilon\), and multiplication by \(W_\eta\) adds the factor \(\|W_\eta\|_1\). In particular \(r=6\) for the \(J\) row block and \(r=12\) for the rows on \(E\cap[7,16]\). In a tabulated residual, the other summands in (80) are exact rationals, so its error is just the corresponding pair error. Finally, for the signed half-gaps of Lemma 10, the two moment errors below \(10^{-14}\) give a power-coefficient error below \(8\cdot10^{-14}\) and a Bernstein-coefficient error below \(2.32\cdot10^{-12}\), as proved in Section 7.3.1. These bounds specify the margins required in a Taylor-based comparison; they do not themselves assert the matrix, residual, or Bernstein values printed in Appendix 6.

Direct node-Taylor representation

There is also a local Taylor representation for the direct cardinal function that does not integrate its spectral density. It applies to the full \(\ell^1\) class of (12). Write \(\mathcal N_0=\{0\}\cup\mathcal N\) and, for \(m\in\mathcal N_0\), put \[ \rho_m=\min_{\substack{n\in\mathcal N_0\\n\ne m}}|m-n|. \tag{110}\] The consecutive gaps in one period of \(\mathcal N_0\) are \(1,2,1,3,2,3\). Thus the minimum exists and is positive: \(\rho_m=1\) for \(m=0\) and for residues \(0,1,3,4\), while \(\rho_m=2\) for residues \(7,9\). This is a guaranteed radius to the nearest other node; it may be smaller than the analytic radius for a particular list whose corresponding coefficients vanish.

For finite rational input, the constants in this direct formula can be enclosed by the rational operations above. For an arbitrary \(\ell^1\) input the formula is an exact convergence statement; a computable error for truncating its node sum additionally requires a bound for that list’s tail. It represents the direct function \(p\) only. The transformed function \(K_p\) still has the kernel representation (20), and this local identity neither places \(K_p\) in the cardinal class nor replaces the exact-derivative certificates for the tabulated functions.