Results on the energy of planar configurations, on cardinal interpolation with Gaussian damping, and the universal optimality of the triangular lattice, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

The energy of a lattice — Lemma 1.1. A lattice of covolume one in the plane has density one, and its energy for any nonnegative interaction is the sum of the interaction over the nonzero lattice vectors, finite or not.
The linear programming bound for configurations of density one — Theorem 3.1. If a continuous even function with suitable decay has a nonnegative Fourier transform, then for every locally finite planar configuration of centred density one the lower limit of the averaged sums of the function over pairs is at least the transform at zero minus the function at zero.
The bound and lattices — Proposition 4.1. The linear programming bound is at most the energy of every lattice of covolume one, with equality exactly when the function touches the interaction at the nonzero lattice points and its transform vanishes at the nonzero dual lattice points.
Gaussian optimality from a sharp pair — Theorem 4.2. If a sharp pair exists for a Gaussian interaction, every configuration of density one has at least the energy of the triangular lattice for it.
The moment problem on the unit interval — Theorem 2.3. A sequence all of whose iterated differences alternate in sign is the sequence of moments of exactly one finite positive Borel measure on the unit interval.
Completely monotone functions as mixtures — Theorem 3.1. A completely monotone function on the closed half-line is the integral of the functions v^t against a finite positive measure on the unit interval, of total mass the value at zero and without mass at zero.
From Gaussians to completely monotone interactions — Theorem 4.1. A planar configuration of density one that has at least the energy of the triangular lattice for every Gaussian interaction has at least its energy for every completely monotone interaction of the squared distance, in the extended nonnegative reals.
Universal optimality from sharp Gaussian pairs — Theorem 5.1. If a sharp pair exists for every Gaussian, the triangular lattice has the least energy among planar configurations of density one for every completely monotone interaction of the squared distance.
Residues of the triangular form — Lemma 1.1. Every integer of the form j^2 + j l + l^2 is congruent to 0, 1, 3, 4, 7 or 9 modulo 12.
The dual of the triangular lattice — Lemma 1.2. The dual lattice of the triangular lattice of covolume one is the lattice turned by a right angle.
The Fourier transform of a planar Gaussian — Lemma 3.1. The Fourier transform of exp(-pi beta |x|^2) on the plane is exp(-pi |xi|^2 / beta) divided by beta.
Gaussian duality — Theorem 4.1. If the normalized statement holds for every parameter k at least 2.36, there is a radial sharp pair for every Gaussian.
The Fourier coefficients of the sine product — Proposition 1.1. The product of the squares of 2 sin(pi (s - a)/12) over a = 0, 1, 3, 4, 7, 9 is a trigonometric polynomial with frequencies l/6, l from -6 to 6, and coefficients 5, -1 + i sqrt 3, 1 + i sqrt 3, -2, (-1 + i sqrt 3)/2, -1 - i sqrt 3, 1 for l = 0 to 6 and their conjugates for negative l.
The sine product at its zeros — Proposition 1.2. At each of its real zeros the sine product has a double zero with leading coefficient above 7/4 and relative next coefficient below 53/25 in absolute value; both are given in closed form for the six residue classes.
Cardinal functions — Theorem 2.2. For a summable coefficient list the cardinal function is entire, real on the real axis, has value and derivative at each node determined by the two coefficients of that node, and is the sum of a multiple of the sine product and the integral of exponentials with frequencies between -1 and 1 against a density of integral at most 36 times the sum of the absolute values of the coefficients.
Uniform bounds for cardinal functions — Corollary 2.3. The j-th derivative of a cardinal function is bounded on the real axis by pi^j (25 |C| + 36 W), with W the sum of the absolute values of the pole coefficients, wherever the nonzero coefficients sit.
The Fourier transform of a planar Gaussian with a complex parameter — Proposition 1.2. For a complex number gamma with positive real part, the Fourier transform of exp(-pi gamma |x|^2) on the plane is exp(-pi |xi|^2 / gamma) divided by gamma.
Fourier pairs from cardinal functions — Theorem 3.2. For a cardinal function p there is an entire function K, real on the real axis and decaying exponentially on the half-line with all derivatives, such that exp(-pi h s) p(s) and exp(-pi h s) K(s), with s = (sqrt 3 / 2)|x|^2 and h = 2/5, are Fourier transforms of each other on the plane.
The coupled pair — Corollary 3.3. For two coefficient lists the functions H_1 = p_1 + K_{p_2} and H_2 = p_2 + K_{p_1} give an admissible radial function exp(-pi h s) H_1(s) with Fourier transform exp(-pi h s) H_2(s); if they satisfy the contact and sign conditions, the normalized statement holds.
The norm of the jet operator — Proposition 4.1. The operator S that sends a coefficient list to minus the jet list of its transformed function maps into summable lists, with the sum of absolute values at most 4000 times that of the input list.
The transformed kernel on two frequency ranges — Lemma 1.1. For frequencies of absolute value at most 5/6 the kernel decays at the rate 0.70; between 5/6 and 1 it is smaller in size and decays at the rate 0.187 + 2.48 (1 - |t|).
Refined bounds for the transformed function — Theorem 3.1. The transformed function of a cardinal function and its derivatives are bounded on the positive half-line by a term with the rate 0.70 plus a term with the rate 0.187 whose part coming from the pole coefficients carries a factor 1/s.
The tail bound for the jet operator — Theorem 3.2. The sum of the absolute values of the output of the jet operator at the nodes n >= r is at most U_r(w, c) = 800 (w + c) e^(-0.70 r)/(1 - e^(-0.70)) + 5.3 (2.54 w/r + 2c) e^(-0.187 r)/(1 - e^(-0.187)), where w bounds the pole coefficients and c the constant entry of the list.
The first rule of Fejér — Theorem 1.2. For even N the rule with the points cos(pi (v + 1/2)/N) and the weights o_v gives the average over [-1, 1] of every polynomial of degree at most N - 1; the weights are at least 1/(N(N - 1)) and add up to 1.
The error of the rule for holomorphic functions — Proposition 2.1. For a function holomorphic and bounded by A on the closed disc of twice the radius of the interval, the average over the interval and the rule differ by at most 4A 2^(-N).
The quadrature error for the spectral integrals — Theorem 3.3. With 384 points on each of the six frequency intervals, the values and scaled derivatives y^a p^(a)(s)/a! and y^a K^(a)(s)/a! for a list with one unit entry at a node up to 168, for 0 <= s <= 168 and |y| <= 3/2, differ from their quadrature forms by less than 10^-31.
The jet matrix and its quadrature form — Lemma 1.1. On lists that vanish beyond the node 168, every entry of the matrix of the jet operator at the nodes up to 168 differs from its quadrature form by less than 2 x 10^-31.
The residual of the reference columns — Lemma 3.1. The reference columns satisfy X_1 - D X_2 = g_1 - D^512 g_1 and X_2 - D X_1 = g_2 - D^512 g_2 on the nodes up to 168.
Norm bounds for the finite matrices (computed) — Proposition 4.1. The matrix D^64 has column-sum norm below 0.004, the two finite geometric sums have norms adding up to less than 36, and the columns of the two reference matrices have total l^1 norm below 2.3 (columns 1 to 5) and 5.6 (columns 6 to 10). Established by a computation in ball arithmetic; the program is printed in the lesson.
The size of the reference lists — Corollary 4.2. For every parameter k at least 2.36 the two reference lists have total l^1 norm below 4.8 and constant entries of total absolute value below 0.02, and D^512 has norm below 0.004^8.
The inverse of the finite block — Lemma 2.2. The block of the interpolation operator on the nodes up to 168 is invertible, with inverse of norm less than 40.
The interpolation operator is invertible — Theorem 2.4. The operator (a_1, a_2) -> (a_1 - S a_2, a_2 - S a_1) is a bijection of the space of pairs of summable lists on the nodes.
Exact interpolation — Theorem 3.1. For every k at least 2.36 the interpolation system has exactly one solution with the chosen extra entries; it differs from the reference lists by less than 2 x 10^-8 in total l^1 norm, and the two lists have total norm less than 5.
The functions with contacts at all nodes — Corollary 3.2. For every k at least 2.36 there are entire functions H_1, H_2, real on the real axis, with H_1 = T_k, H_1' = T_k', H_2 = 0, H_2' = 0 at every node, such that exp(-pi h s) H_1(s) is an admissible radial function with Fourier transform exp(-pi h s) H_2(s); the normalized statement follows if H_1 <= T_k and H_2 >= 0 on the half-line.
The quotient for the exact and the reference lists — Proposition 2.1. On every half-gap the quotients of the functions of the exact lists and of the reference lists, after the value and the slope at the centre are removed, differ by less than 0.00011.
The polynomial of degree 40 — Proposition 2.2. On every half-gap the quotient of the reference function differs by less than 10^-6 from the polynomial of degree 40 whose coefficients are the quadrature forms of its scaled derivatives at the centre.
Lower supports of the target — Proposition 2.3. For k between two consecutive endpoints K < J of the list 2.36, 2.65, 3.2, 4, 6, infinity, the quotient of T_k on a half-gap is at least an explicit polynomial depending only on K, J and the half-gap, minus 10^-6.
Reduction of the sign conditions on the bounded range — Theorem 3.1. If the polynomials Phi_i(x; k) are at least 0.001 on [0, 1] for all 91 half-gaps and both indices, then T_k - H_1 and H_2 are nonnegative on [0, 89.5].
Containing parameter boxes — Lemma 1.1. For k between two consecutive endpoints K < J, a linear expression in the ten numbers q(k) is at least the bracket [l]_{K,J}, the smaller of two values of a concave lower envelope formed from the corners of a box that contains the curve q(k).
Bernstein coefficients — Lemma 2.1. A polynomial of degree at most 40 is on [0, 1] at least the smallest of its Bernstein coefficients of degree 40, which are the sums of binomial(d, r)/binomial(40, r) times its coefficients.
The inequalities on the half-gaps (computed) — Proposition 3.1. For the 91 half-gaps, both indices, the five parameter intervals and the 41 Bernstein indices, the lower bounds for the Bernstein coefficients of the polynomials Phi_i are larger than 0.001. Established by a computation in ball arithmetic; the program is printed in the lesson.
The sign conditions on the bounded range — Theorem 3.2. For every k at least 2.36 the functions of the exact interpolation satisfy H_1 <= T_k and H_2 >= 0 on [0, 89.5].
A quadratic lower bound for the sine product — Lemma 1.2. For every real s and a zero m of the sine product nearest to s, P(s) is at least 0.68 (s - m)^2.
The sign conditions beyond 89.5 — Theorem 3.4. For every k at least 2.36 the functions of the exact interpolation satisfy H_1 <= T_k and H_2 >= 0 on the half-line from 89.5.
The normalized statement — Theorem 4.1. For every real k at least 2.36 there are entire functions H_1, H_2, real on the real axis, such that exp(-pi h s) H_1(s) is an admissible radial function with Fourier transform exp(-pi h s) H_2(s), with H_1 <= T_k and H_2 >= 0 on the half-line and with equal values and first derivatives of H_1 and T_k, and of H_2 and 0, at every node.
Universal optimality of the triangular lattice — Theorem 4.2. For every completely monotone interaction of the squared distance, every configuration of density one in the plane has at least the energy of the triangular lattice of covolume one. Three steps of the proof are finite computations with rigorous error bounds, with the programs printed in the lessons.
Gaussian energies of periodic configurations — Lemma 1.2. For a periodic configuration C of density one, H_C(t) = (1/N) sum of P_t(a_i - a_j) - Psi_t(0) equals the energy of C for the Gaussian exp(-u/(4t)) divided by 4 pi t; it depends only on the set C, and H_C(t) >= H_A(t) for the triangular lattice A and every t > 0.
Fourier coefficients of the periodic kernel — Proposition 3.2. The periodic kernel K_{s,L}, defined as a heat integral with sigma = 1 - s/2, is integrable over a cell and has the Fourier coefficients (4 pi^2 |xi|^2)^(-sigma) at the nonzero points of the dual lattice and 0 at 0.
The singular part of the periodic kernel — Proposition 3.3. K_{s,L}(x) - g_s(x)/q_s has a limit at 0, where g_s = |x|^-s (0 < s < 2) or -log|x| (s = 0) and q_s = 2^(2-s) pi Gamma(1-s/2)/Gamma(s/2) (q_0 = 2 pi); the limit is given by a heat integral over the nonzero lattice points.
The periodic energy as an integral of Gaussian energies — Theorem 4.1. For 0 < s < 2 the periodic energy of a periodic configuration C of density one is q_s/Gamma(1-s/2) times the integral over t of (H_C(t) - 1) t^(-s/2); for s = 0 the same with a correction 1/(4 pi t) for t > 1 and a constant. It depends only on the set C.
Comparison across period lattices — Corollary 4.2. For periodic configurations C, D of density one with any period lattices and 0 <= s < 2: V_s(C) - V_s(D) = q_s/Gamma(1-s/2) times the absolutely convergent integral of (H_C(t) - H_D(t)) t^(-s/2).
The triangular lattice minimizes the periodic Riesz and logarithmic energies — Theorem 5.1. For 0 <= s < 2, every periodic configuration of density one in the plane has at least the periodic energy of order s of the triangular lattice of covolume one. The proof uses the theorem of the course for all Gaussians.

Reading order

Thirteen lessons with complete proofs. The first three prove the theorem from one named statement, the normalized statement of the third lesson. The lessons from the fourth to the ninth construct the functions of that statement, and the last three prove its two sign conditions. Three statements, in the eighth, eleventh and twelfth lesson, are finite sets of inequalities between explicitly defined numbers and are established by computations in ball arithmetic; the lessons print the programs, about 240 lines in all, which run for about two minutes. The thirteenth lesson derives from the theorem that the triangular lattice has the least periodic energy for the interactions |x|^-s with 0 < s < 2 and -log|x| among all periodic configurations of density one. Not in the course: the renormalized energy of configurations that are not periodic, which the last section of OpenAI's text defines through fields, and the jellium problem; that section derives them with further theorems of the literature.