Openai 2026 universal optimality triangular lattice

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corollary_8_1: The claimed triangular minimum of the renormalized field energy with unit background for the logarithm and the Riesz kernels $|x|^{-s}$, $0<s<2$, on compatible tori and among density-one configurations; unverified here.

theorem_1_1: The claimed universal energy minimality of the unit-covolume triangular lattice among planar configurations of centered-disk density one, for every smooth completely monotone function of squared distance; unverified here.

OpenAI, Universal optimality of the triangular lattice, OpenAI Math Release preprint, September 23, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_universal_optimality_triangular_lattice.pdf, and the release's TeX bundle sits in the same folder.

@misc{OAI:Universal-optimality-of-the-triangular-lattice-September-23-2026,
  author = {{OpenAI}},
  title = {{Universal optimality of the triangular lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf}{OAI:Universal-optimality-of-the-triangular-lattice-September-23-2026}},
  year = {2026}
}

The release's root README states that its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all of them have Lean formalizations, and that "Some of the unformalized results could have issues." The manuscript's own README adds no sentence about authorship or human assistance; its Verification section describes two finite checkers held in the release's verification/ folder beside the paper. The manuscript names no author beyond "OpenAI" and carries the date September 23, 2026. These are the source's own statements, recorded here as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalogue (lean/formalization.yaml) does not name this manuscript. The release's page for this family of manuscripts (lean/docs/090.md) lists a formalization of the same energy conclusion, the density-one triangular lattice minimizing lower energy per particle for every nonnegative completely monotone function of squared distance with infinite energies allowed, together with sharp Gaussian minorants and a planar packing certificate, and attributes it to the atomic-certificate and circle-packing companions named below, rather than to this manuscript or to the Coulomb companion; the comparator statement files it names are TriangularEnergy.lean, TriangularGaussian.lean, AtomicGaussian.lean and PlanarPacking.lean, read statically from the release's catalogue; not built, replayed or audited for fidelity in this repository. Whether those statements match Theorem 1.1 as this manuscript states it was not compared here. The renormalized and jellium results of Section 8 have no listed formalization, and no Lean file in the release is a proof of any Erdős problem.

The library holds cards for the three companions of the same family. An atomic certificate for triangular-lattice universal optimality is described by the manuscript as an independent proof of the same energy conclusion by a different auxiliary construction (finite atomic columns, a node set omitting 24, damping $h=17/50$ against $2/5$ here); the present proof does not depend on it. A sharp Fourier certificate for planar circle packing supplies the framework of cardinal formulas, Gaussian transforms and finite sign certificates (its Sections 2--4) that the manuscript says it adapts; the needed formulas are re-proved here and no packing bound is imported. Triangular minimality for planar Coulomb renormalized energy is a direct logarithmic result that the manuscript says overlaps its Corollary 8.1 at $s=0$ under a different field normalization.

Read status: claims checked for Theorem 1.1, Theorem 2.1, Proposition 6.1, Lemma 6.2, Proposition 7.2 and Corollary 8.1, read clause by clause in the TeX source (sections/01-uniform-gaussian-theorem.tex, sections/02-cardinal-fourier.tex, sections/06-gaussian-energy-transfer.tex, sections/07-shifted-mixtures.tex and sections/08-renormalized-jellium.tex of the release's TeX bundle; PDF pp. 2, 6, 33--34, 37 and 41) on 2026-10-07; the proofs, Sections 2--5, Appendix B and the two arithmetic appendices A and C, were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Section 1, Introduction (pp. 1--5). Defines the unit-covolume triangular lattice $A$, centered disk density one ($N_R/(\pi R^2)\to1$ for the point count $N_R$ in the closed disk $B_R$) and the lower energy $E_g(\mathcal C)=\liminf_R N_R^{-1}\sum_{x\ne y\in\mathcal C_R}g(|x-y|^2)$ over ordered pairs, and states Theorem 1.1: for smooth completely monotone $g\ge0$ and every such configuration, $E_g(\mathcal C)\ge\sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A)$ in $[0,\infty]$. Section 1.1 reviews the lattice-only results (Rankin, Cassels with his corrigendum, Ennola, Diananda; Montgomery's theta theorem), the Cohn--Kumar conjecture, the dimension 8 and 24 theorem of Cohn, Kumar, Miller, Radchenko and Viazovska, and the restricted planar results of Faulhuber, Shafkulovska and Zlotnikov, Hardin and Tenpas, and Leblé; it states that the theorem identifies the minimum value and not all minimizers, and that the per-potential sharp auxiliary function the Cohn--Kumar conjecture asks for is not claimed. Section 1.2 outlines the Gaussian route: a radial Schwartz minorant $f_\alpha\le G_\alpha$ with $\widehat f_\alpha\ge0$, equality at nonzero lattice points and vanishing transform at nonzero dual points, for each $G_\alpha=e^{-\pi\alpha|x|^2}$. Section 1.3 names the antecedents (Beurling--Selberg constructions, Radchenko--Viazovska interpolation, Talebizadeh Sardari's planar non-uniqueness, Cohn--Elkies, the companion manuscripts). Section 1.4 previews the renormalized and jellium consequences.
• Section 2, Cardinal interpolation and Fourier transformation (pp. 5--12). Fixes $b=\sqrt3/2$, $h=2/5$, $B=4/3$, the radial coordinate $s=b|x|^2$, and the node set $\mathcal N=(12\mathbb Z+\{0,1,3,4,7,9\})\cap(0,\infty)$, a strict periodic superset of the shell values $j^2+j\ell+\ell^2$ (24 is a node and not a shell); notes $A^*$ is a rotation of $A$. States Theorem 2.1 (Normalized Gaussian Fourier pair): for every real $k\ge2.36$ there are entire $H_1,H_2$ and a real radial Schwartz $f$ with $f(x)=e^{-\pi hb|x|^2}H_1(b|x|^2)$, $\widehat f(\xi)=e^{-\pi hb|\xi|^2}H_2(b|\xi|^2)$, $H_1\le T_k$ and $H_2\ge0$ on $[0,\infty)$ for $T_k(s)=k^{-1}e^{-k(s-1)}$, with value and first derivative contacts $H_1=T_k$, $H_1'=T_k'$, $H_2=H_2'=0$ at every node. Builds the sine product $P$ with double zeros at the nodes, cardinal functions with double-pole and simple-pole terms and their compactly supported spectral measures (Lemma 2.2), Gaussian damping and the transformed kernel $K$ (Lemma 2.4, Fourier pairs), and the jet operator $S$ with its tail bound $U_r(w,c)$ (Lemma 2.5).
• Section 3, Positive quadrature and finite data (pp. 13--22). Replaces the six density integrals by a positive Fejér rule with $N=384$ points on each density interval, for the 84 nodes up to $M=168$ (Lemma 3.1, quadrature error below $10^{-31}$), fixes the three extra coordinates, builds ten reference columns by finite geometric sums of the quadrature block $D$, the containing parameter boxes with endpoints $2.36,2.65,3.2,4,6,\infty$ (Lemma 3.2), Bernstein rows on half-gaps up to $s_0=89.5$, and states Proposition 3.3 (Finite certificate): norm bounds on $D^{64}$, $V_\pm$ and the columns, and strict lower bounds for 37,310 Bernstein comparisons and ten tail comparisons. Its proof is a computer check: the release's verification/numeric_balls.py encloses the exact arrays in Arb ball arithmetic at 256 bits and tests the stronger cutoffs of Tables 1--3; the manuscript flags this as the computer-assisted step.
• Section 4, Exact interpolation (pp. 22--26). Proposition 4.1: for every $k\ge2.36$ a unique pair of summable positive-node lists with the fixed extras gives the exact jets of Theorem 2.1, with combined norm below 5 and correction below $2\cdot10^{-8}$ from the reference lists; proved by inverting the finite block (norm below 40) and a Schur-complement Neumann series on the tail. Notes the contrast with Talebizadeh Sardari's non-uniqueness on the actual shells.
• Section 5, Signs (pp. 26--32). Lemma 5.1 (exponential supports for the target), Lemma 5.2 (signs on $[0,s_0]$ through deleted quotients, Bernstein enclosure and parameter boxes with margin $.000888$), Lemma 5.3 ($P(s)\ge.68(s-m)^2$ at a nearest zero), Lemma 5.4 (signs for $s\ge s_0$: low-node rational part above $.002$, remainder with double zeros and second derivative below $.0002$); proof of Theorem 2.1 assembled on p. 32.
• Section 6, From Gaussian pairs to energy comparisons (pp. 33--35). Proposition 6.1 (Linear programming with centered disk density): for real even Schwartz $f$ with $\widehat f\ge0$ and every locally finite $\mathcal C$ of centered disk density one, $\liminf_R N_R^{-1}\sum_{x\ne y\in\mathcal C_R}f(x-y)\ge\widehat f(0)-f(0)$, also for any nonnegative $\Phi\ge f$; proof by Cauchy--Schwarz against area measure on $B_{(1+\varepsilon)R}$, after Cohn and de Courcy-Ireland and Cohn and Zhao. Lemma 6.2 (Gaussian duality) converts Theorem 2.1 into a sharp pair for every $\alpha>0$: $\alpha\ge1$ by $k=\pi(\alpha/b-h)\ge2.36$ and scaling, $0<\alpha<1$ by a Fourier complement after Cohn and Miller. Poisson summation over $A$ gives display (6.6), the Gaussian energy inequality for every $\alpha>0$.
• Section 7, Shifted moments and positive mixtures (pp. 35--39). Lemma 7.1: a smooth $F\ge0$ on $[0,\infty)$ with $(-1)^jF^{(j)}\ge0$ is $\int_{[0,1]}v^t\,d\rho(v)$ for a positive measure of mass $F(0)$ with no atom at 0 (a finite-difference proof of the Hausdorff--Bernstein--Widder representation). Proposition 7.2: the Gaussian inequality for every $\alpha>0$ implies Theorem 1.1's inequality, by applying the lemma to the shift $g(\varepsilon+t)$, Fatou and Tonelli, and removing the shift in the lattice sum alone. The completion (pp. 38--39) checks that $A$ has density one and attains the bound, including when the lattice sum is infinite.
• Section 8, Renormalized and jellium energies (pp. 39--43). Fixes the field-first conventions of Petrache and Serfaty with $\kappa_0=2\pi$, $\kappa_s=4\pi$, the extension weight $|y|^{s-1}$ for $0<s<2$, the truncated field energy $\mathcal W_s$ with centered-square upper limit then cutoff removal, the configuration infimum $W_s$, the canonical periodic energy $W_{s,L}$ and the ordered-pair finite jellium energy with its thermodynamic limit. States Corollary 8.1: for $0\le s<2$ the classes of $A/(nA)$ minimize $W_{s,nA}$ among $n^2$ distinct points of $\mathbb R^2/(nA)$, $A$ minimizes $W_s$ among density-one configurations, and the jellium minimum is $W_s(A)/\kappa_s$. Lemma 8.2 (Heat comparison across period lattices) and the proof rest on the Gaussian inequality (6.6), Proposition B.1, and the scalar thermodynamic identities of Lewin, Lieb and Seiringer and of Lauritsen.
• Appendix A, Scalar bounds (pp. 44--48). Lemma A.1 encloses $\pi$ and $\sqrt3/2$ by 80-digit rationals (Machin's formula); rational proofs of the threshold $\pi(2/\sqrt3-2/5)>2.36$ and of the scalar error budgets in the table (A.4), the target-tail bounds and the half-gap remainders; describes the release's verification/arithmetic_bounds.py (512-bit Arb balls and exact rationals), which checks these budgets and the roundoff majorants of Appendix C, and a degree-2000 rational route the programs do not implement.
• Appendix B, Periodic approximation for the field energy (pp. 48--61). Proposition B.1 (Normalized cubic approximation): the truncation limit of the field energy exists in $\mathbb R\cup\{+\infty\}$, its infimum $m$ over compatible fields is finite, and square-periodic simple configurations (period $2R_j\mathbb Z^2$, $4R_j^2$ points per cell) have canonical values tending to $m$. Lemmas B.2 (cutoff control), B.3 (a stationary minimizing law with vanishing close-pair defect, using a pointwise ergodic theorem of Lindenstrauss), B.4 (screening without new close pairs, after Petrache and Serfaty's Section 6), then reflection, projection and removal of the cutoff.
• Appendix C, An alternative rational verification procedure (pp. 61--68). A floor-rounded rational procedure with roundoff bounds (Lemma C.1) whose conclusion holds only if its tests pass; the manuscript states it is not executed by the supplied programs.
• References (pp. 68--70): 33 entries, including the three companion manuscripts of the release.

The release's verification/ folder for this manuscript holds, by its README, the two finite checkers numeric_balls.py (37,310 Bernstein and ten tail comparisons, plus the norm tables) and arithmetic_bounds.py (the scalar budgets), an input-binding manifest and a pinned dependency; the README says they check finite arithmetic and not the analytic reductions, and do not run the Appendix C construction. The folder is not copied here and nothing in it was run here.

Bears on

• Problem 991: does not apply. The problem asks whether the $n$-point maximizers of the product of pairwise distances on $S^2$, the minimizers of logarithmic energy, have cap discrepancy $o(n)$. The manuscript says nothing about the sphere, about discrepancy, or about the asymptotics of spherical logarithmic energy; Corollary 8.1 at $s=0$ is a planar energy statement, that the triangular lattice minimizes a renormalized logarithmic field energy among density-one configurations, with no spherical or discrepancy content. The claim is unverified here and leaves the page's status, which rests on its own acceptance evidence, untouched.
• Problem 662: does not apply. The problem compares counts of pairs at distance at most $t$ in a finite set with minimum separation one against the triangular lattice. Theorem 1.1 covers smooth completely monotone functions of squared distance among configurations of centered-disk density one; a threshold indicator is not completely monotone, the hypothesis is a density and not a separation, and the manuscript makes no claim about distance counts. Nothing here supports or contradicts either variant the page records; the claim is unverified here and the page's status rests on its own evidence.