An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program

Translation pending — read the English original while we localise.

A collection of mathematical results from an internal OpenAI model settles the asymptotic strength of the Cohn-Elkies sphere-packing linear program, providing the first improvement to the general sphere-packing exponent since 1978 alongside an exact matching lower bound.

What the collection is, and why one result stands out #

The document opens modestly. It presents a collection of results obtained by an internal OpenAI model, and it says they span mathematics and theoretical computer science. That's a description of provenance and scope, and it doesn't say how the model was used or how its output was checked. Those questions matter, and the source as summarised here leaves them unanswered, so this article won't pretend otherwise. What the claims do support is a list of stated results, and the list is long enough that the interesting question isn't whether any single item is impressive. It's why one of them deserves most of the attention.

The reason is that the sphere-packing result is the only one in the collection that closes a quantitative gap in a programme with a long, named history. The others are striking in their own right. An explicit nonsofic group answers a longstanding question. A disproof of Connes's conjecture settles something a named mathematician proposed. But the sphere-packing result sits at the end of a chain that runs from a Fourier-analytic framework, through two celebrated exact solutions in individual dimensions, to a question about high dimensions that stayed open for years. Understanding it means understanding that chain.

So the bulk of what follows concerns the Cohn-Elkies linear program: what it is, what was known about it, what the new theorem says, and how the sign-uncertainty side of the argument is put together. A shorter section then surveys the remaining results in the collection, each as stated and no further.

The Cohn-Elkies program and what it bounds #

Sphere packing asks how densely non-overlapping balls can fill space. The document fixes notation for this. For centres separated by at least 1, ∆d is the supremum of the upper densities of radius-1/2 balls in Rd. Put differently, ∆d denotes the maximal sphere-packing density in Rd. Everything in the collection's first chapter is measured against that quantity.

The Cohn-Elkies program gives an upper bound on ∆d. Its optimal output is written LPd, defined as the optimal density bound furnished by the program. The mechanism is the Gorbachev–Cohn-Elkies bound, which assigns every f in a class called Ad the density upper bound vd f(0)/(2d f̂(0)). In words, you pick an auxiliary function f from the admissible class, compute its value at the origin and the value of its Fourier transform at the origin, form that ratio with the volume and scaling factors, and the result bounds the packing density from above. The linear program searches for the f that makes this bound as small as possible.

This is a framework in which the whole difficulty is in choosing f well. A cleverly chosen function gives a tight bound, and a poorly chosen one gives a weak bound that is still valid. That's why the matching lower bound in the new result matters so much. It speaks about every admissible function at once, and it turns a search for better functions into a closed question.

What the framework had already achieved #

Before turning to the new theorem, it's worth being specific about what the Fourier-analytic linear-programming framework had already delivered, because the new result is partly a statement about how much further it can go.

Viazovska used this framework to prove that the E8 lattice gives the optimal packing in dimension eight. Cohn, Kumar, Miller, Radchenko and Viazovska then established the optimality of the Leech lattice in dimension twenty-four. Those are exact results in single dimensions, and they're the reason the framework has a reputation beyond its role as a bounding tool. In two specific dimensions, it hits the true answer.

The high-dimensional picture was murkier. Cohn and Zhao proved that the linear program is always at least as strong as the Kabatianskii–Levenshtein spherical-code bound. That's a useful guarantee, because the Kabatianskii–Levenshtein bound had been the best known general exponent. But "at least as strong" leaves two possibilities open: the program might match that exponent exactly, or it might beat it. The document states the consequence directly. Whether the linear program improves the classical high-dimensional sphere-packing exponent had remained open.

Afkhami-Jeddi, Cohn, Hartman, de Laat and Tajdini conjectured the high-dimensional rate of the linear program. That conjecture is a hedged object, and the source treats it as one. It was a conjecture, supported by whatever evidence the authors had, and nobody had proved it. The new result changes its status.

The main theorem #

The document's main theorem proves the exponential rate conjectured in Conjectures 3.1 and 3.2 of that 2020 paper. Stated at the level the claims support: the authors determine the exact exponential decay rate of the Cohn-Elkies sphere-packing linear program. The asymptotic strength of the program is, in the source's words, determined exactly.

Two consequences follow directly from the source. The first is an improved general packing bound in high dimensions. The second is that the corresponding Fourier sign-uncertainty problem is settled asymptotically. Those are different kinds of consequence. One is about geometry and the other is about analysis, and the fact that a single theorem delivers both is the first hint that the sphere-packing problem and the uncertainty problem are two faces of the same structure.

The source also frames the achievement against history. It describes the result as the first improvement since 1978 to the general sphere-packing exponent. The classical Kabatianskii–Levenshtein exponent was 0.59905576..., and that figure had been the benchmark. The document records that subsequent spherical-code refinements had improved only lower-order factors. In other words, for the exponent itself, which is the leading-order quantity, nothing had moved.

I'd argue this framing is the right one for assessing the result. A lower-order improvement tweaks constants and polynomial factors while leaving the exponential rate where it was. An exponent improvement changes the rate. The claims support exactly that distinction, and they support the statement that the prior refinements fell on the lower-order side of it.

Why the matching lower bound changes the question #

An upper-bound improvement alone would be good news. What makes this result unusually final is the second half. The matching lower bound shows that no Cohn-Elkies auxiliary function can improve this exponent.

Consider what that does to the research programme. Before the result, a researcher with a better idea for constructing an auxiliary function had a plausible route to a still better sphere-packing exponent. After it, that route is shut at the level of exponents. Any further progress on the exponent has to come from outside the Cohn-Elkies framework, because inside it the rate is now exact.

I don't want to overstate this. The claims don't say the new exponent is the true sphere-packing exponent, and they don't say sphere packing itself can't do better. The statement concerns the linear program: its exponent is pinned down, and it can't be beaten by changing the auxiliary function. Whether a different technique could beat it is a question the claims don't address. That's a clean boundary, and it's the kind of boundary that tells a field where to look next.

There's a commercial-minded reading of this too, for readers who care about where effort goes. A known-exact ceiling on a method is useful information. It tells you to stop optimising inside that method. For anyone allocating research time, the shift from "this might improve" to "this cannot improve" is worth as much as a modest positive result.

Spherical codes and binary codes #

The collection doesn't stop at packing. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. That sentence carries a strong quantifier. The improvement is by exponential factors, and it holds for all parameters, which is a broader statement than an improvement in a favourable regime.

There's also a link back to Chapter 1. The spherical code construction also recovers the sphere-packing exponent of Chapter 1. So the two threads connect: the code-side construction reproduces the packing-side exponent. The claims don't spell out the mechanism of that recovery, and I won't invent one. What they do establish is that the relationship runs in this direction, with the spherical construction yielding the same exponent as the packing chapter.

It's worth recalling why this link matters given the history above. The Kabatianskii–Levenshtein spherical-code bound was the comparison point for the linear program, and Cohn and Zhao's theorem said the program is always at least as strong. The new results improve code bounds as well as packing bounds, and they do so from a construction that comes back to the packing exponent. The two sides are being moved together.

The sign-uncertainty problem #

The second consequence of the main theorem concerns Fourier sign uncertainty, and the document gives enough definitions to sketch its shape.

For a function g, r(g) is defined as the infimum of R ≥ 0 such that g(x) ≥ 0 for |x| ≥ R. In plain terms, r(g) is the radius beyond which g is nonnegative everywhere. When no such radius exists, r(g) = ∞. The uncertainty problem concerns how small this radius can be made for functions with a prescribed relationship to their own Fourier transforms.

That relationship is a sign. The condition ĝ = ςg ties a function to its Fourier transform up to the sign. The signs +1 and −1 give the original and complementary uncertainty problems, respectively. Cohn and Gonçalves introduced the anti-self-Fourier counterpart, which is the complementary problem, and connected it with sphere packing. That connection is the bridge through which the sphere-packing result settles the uncertainty problem asymptotically.

The complementary problem is not a curiosity of pure analysis. In the spinless modular bootstrap, modular S-antisymmetry produces anti-self-Fourier test functions whose eventual sign controls the spectral gap. The source cites Hartman, Mazáč and Rastelli and the Afkhami-Jeddi et al. paper for this. So a question that began as a statement about functions and their transforms also appears in a setting with a physical-sounding name, the spectral gap. The claims don't go further than the one-line mechanism, and neither will I, but it's a reminder that the uncertainty problem has company outside sphere packing.

Appendix A adds a comparison between the two problems. It observes that A+(d) < A−(d) for all d. That's a statement about the two problems differing for every dimension, with the first quantity strictly smaller than the second. The claims give no further interpretation, and the inequality is worth noting simply because it holds for all d with no exceptions.

Inside the argument: averaging, vanishing and a mass estimate #

The document includes several technical statements that give a feel for how the uncertainty analysis is built. They're fragments, and I'll present them as fragments, but together they sketch a method.

Start with symmetry. The rotational average is defined using normalised Haar measure on O(d). Averaging a function over rotations produces a radial function, and the point of doing so is that it simplifies the problem without breaking the constraints. The source says rotational averaging preserves every sign condition in the problem's constraint list, labelled (2). It adds that if ĝ = ςg, then r(Rg) ≤ r(g), where Rg is the rotational average of g. So averaging never makes the sign radius worse. Also, in the source's phrasing, Rg ≠ 0 whenever g ≠ 0 is eventually nonnegative, so averaging doesn't annihilate a function that has the right sign behaviour at large distance.

Those three facts together justify a reduction to radial functions. The reduction costs nothing in the radius, it preserves the sign conditions, and it doesn't collapse the function to zero. That's the shape of a standard symmetry argument, and the claims confirm each of its ingredients.

Next, the origin. The function g = h − ĥ vanishes at the origin. This is the sort of small fact that does large work in a proof, because the Cohn-Elkies bound involves values at the origin directly, and a function vanishing there behaves differently from one that doesn't. A related correction step is recorded as well: the corrected function gn satisfies gn(0) = 0 and gn → g in L1. So a sequence of corrected functions, each vanishing at the origin, converges to g in the L1 sense. That lets the argument work with functions that satisfy the origin condition exactly while still approximating the target.

Then there's a quantitative constraint. The mass estimate forces 1/a ≥ (1/π − o(1))√d. The claims don't define a, so I can't say what it measures, and I won't guess. What can be said is the form of the conclusion: a lower bound on 1/a that grows like the square root of the dimension, with constant 1/π and a vanishing correction. A square-root-in-dimension scale is a natural-looking size for high-dimensional geometry, but that's my observation about the form of the expression, and the claims don't assert it as a mechanism.

The Mellin picture #

One more piece of the toolkit appears in the document, and it's the one that most clearly explains why radial Fourier analysis is tractable here. After a Mellin transform, the radial Fourier transform reflects the Mellin variable. That's a compact statement about how a transform acts. The Fourier transform of a radial function, which is normally an integral operator and awkward to reason about directly, becomes a reflection in the Mellin variable. Reflections are about as simple as operators get.

The same passage records an analytic condition on the other side. The Schwartz decay of g gives rapid decay of Φg as v → +∞. Here Φg is the object obtained in the Mellin analysis, and v is its variable. The pairing is instructive: the reflection property handles the structure, and the Schwartz decay handles the control at infinity. A proof in this style needs both, because structure without decay leaves integrals that may not converge, and decay without structure leaves nothing to compute.

The document also notes that Sd is the area of the unit sphere, which enters as a normalising factor when radial integrals are converted between the sphere and the line. That's a definitional point, and it's the sort of bookkeeping that determines whether constants in a final bound come out right.

I want to be clear about what these fragments are. They aren't the proof. The claims give statements, not derivations, and no reader should reconstruct the full argument from them. What they show is that the analysis combines symmetry reduction, an origin condition, a quantitative mass estimate, and a transform whose action simplifies after a change of variables. That combination is conventional for Fourier-analytic extremal problems, which makes the exactness of the final result more notable rather than less. Standard-looking tools were enough to reach a sharp answer.

The rest of the collection #

The sphere-packing chapter is only one part of the document, and the remaining results deserve a fair accounting. Each is stated here exactly as the claims give it.

In group theory, an explicit nonsofic group is constructed, answering the longstanding question of whether every countable group is sofic. The proof combines work of Kun and Kun–Thom with a combinatorial expander-matching argument, realised using the binary Leavitt algebra and Thompson's group V. The word "explicit" carries weight here: the result is a construction and not an existence argument.

In operator algebras, infinitely many pairwise nonisomorphic groups with property (T) are constructed with the same group von Neumann algebra. This disproves Connes's conjecture and answers a related finite-to-one question of Popa. Two named questions fall to one construction.

In arithmetic complexity, the document gives lower bounds for the permanent. Division-free circuits require Ω(n² log log n) gates, while formulas require Ω(n⁴ / log n) leaves. These are two bounds for two models of computation, and the document states them side by side.

In quantum information, exponential parallel repetition is proved for every finite two-player entangled game. The source says this extends the classical repetition principle beyond previously treated special classes of quantum games. The quantifier again stands out: every finite two-player entangled game, with no restriction to a special class.

In hardness of approximation, a direct reduction from 3SAT gives n^(1/400)-factor hardness for the Euclidean closest vector problem, with related consequences for binary decoding and other lattice norms. The exponent 1/400 is small, and I'd rather report it as written than dress it up. The claims call the reduction direct and attach consequences for binary decoding and other norms.

In convex geometry, the sharp volume upper bound (n + 1)^n / n! is proved in every dimension for convex bodies whose barycentre is their only interior lattice point. Sharpness and all-dimensions generality are both part of the claim.

In Ramsey theory, a new superexponential lower bound shows that Rk(3) grows as k^Θ(k). And in extremal graph theory, separate bipartite graph constructions disprove two conjectures: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.

Reading the collection as a whole #

It would be easy to treat this list as a trophy case, so it's worth saying what can and can't be concluded from it.

What can be said is that the document reports results in at least eight areas, that several are described as answering or disproving named conjectures, and that the sphere-packing chapter comes with a matching lower bound that makes its result final within its framework. The conjectures involved carry the names of Connes, Popa, Erdős and Simonovits, and the sphere-packing rate was conjectured by Afkhami-Jeddi, Cohn, Hartman, de Laat and Tajdini. Results that settle named questions are easy to evaluate because the question was posed before the answer existed.

What can't be said, on these claims alone, is anything about how the model works, how many attempts it needed, how its output was verified, or how it compares with human effort. The source says the results were obtained by an internal OpenAI model, and the claims stop there. A reader who wants to weigh the significance of the collection as evidence about machine mathematics needs information the claims don't contain. I'd resist the temptation to fill that gap with plausible-sounding narrative.

There's still a position worth taking. The results are heterogeneous in technique: the nonsofic group uses the binary Leavitt algebra and Thompson's group V, the sphere-packing result uses Fourier analysis and Mellin transforms, and the closest vector hardness uses a reduction from 3SAT. A collection spanning that much technical ground isn't the product of one trick applied repeatedly. Whatever produced these results was operating in several mathematical dialects, and the claims support at least that much.

What This Means by Sector #

The exact determination of the high-dimensional Cohn-Elkies decay rate and the accompanying mathematical results influence several theoretical and applied computational fields.

Pure mathematics and discrete geometry #

For mathematicians working in high-dimensional geometry and harmonic analysis, the determination of the exact Cohn-Elkies decay rate shifts the research landscape. Because the matching lower bound proves that no auxiliary function in A(d) can beat the new exponent, researchers can cease searching for better radial test functions within the classical linear programming framework. Progress towards denser sphere packings or tighter universal bounds must now explore non-linear programming techniques, higher-order semidefinite hierarchies, or radically different geometric constructions.

Cryptography and lattice-based security #

Lattice-based cryptography - forming the core of post-quantum standards such as ML-KEM and ML-DSA - relies heavily on the conjectured hardness of finding shortest or closest vectors in high-dimensional Euclidean lattices. The direct reduction from 3SAT establishing $n^{1/400}$-factor hardness for the Euclidean Closest Vector Problem (CVP), combined with tighter exponential ceilings on sphere packings and spherical codes, solidifies the theoretical foundations underlying worst-case-to-average-case lattice reductions without weakening concrete security parameters.

Automated theorem proving and foundation models #

For research teams developing reasoning systems and automated theorem provers, the breadth and technical depth of the collection provide a striking benchmark. Rather than producing synthetic variations of known proofs, the internal model tackled open problems spanning group theory (constructing an explicit nonsofic group via Thompson's group V), operator algebras (disproving Connes's conjecture), and arithmetic complexity (permanent lower bounds). It demonstrates that neural architectures can navigate deep Fourier analysis and combinatorial matching, provided verification and proof-checking remain rigorous.

Conclusion #

The determination of the exact decay rate of the Cohn-Elkies sphere-packing linear program closes a major chapter in geometric analysis. By proving the 2020 conjecture of Afkhami-Jeddi, Cohn, Hartman, de Laat and Tajdini and establishing the first improvement to the general sphere-packing exponent since 1978, the work establishes an unyielding mathematical baseline. With a matching lower bound confirming that no Cohn-Elkies auxiliary function can yield further exponential gains, the theorem both settles the asymptotic strength of the linear program and directs the future of high-dimensional geometry towards fundamentally new analytical horizons.

Questions? Answers.

References #

• Nima Afkhami-Jeddi et al., (2020). High-dimensional sphere packings and the modular bootstrap ⧉. Journal of High Energy Physics.
• Henry Cohn and Noam Elkies, (2003). New upper bounds on sphere packings I ⧉. Annals of Mathematics.
• Henry Cohn and Felipe Gonçalves, (2019). An optimal uncertainty principle in Fourier analysis ⧉. PNAS.
• Henry Cohn and Yufei Zhao, (2014). Sphere packing bounds via spherical codes ⧉. Duke Mathematical Journal.
• G. A. Kabatianskii and V. I. Levenshtein, (1978). Bounds for packings on a sphere and in space ⧉. Problems of Information Transmission.
• Maryna S. Viazovska, (2017). The sphere packing problem in dimension 8 ⧉. Annals of Mathematics.

Àyẹ̀wò àkọ́kọ́ 2026-10-05.

Àyẹ̀wò àkọ́kọ́ 2026-10-16.

# An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau

> Originally published at [https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/](https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/)

An internal OpenAI model produced a collection of results across mathematics and theoretical computer science. The headline is a determination of the exact exponential decay rate of the...

Read the full article on sebastienrousseau.com: https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/

An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau

An internal OpenAI model produced a collection of results across mathematics and theoretical computer science. The headline is a determination of the exact exponential decay rate of the...

https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/

An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau

An internal OpenAI model produced a collection of results across mathematics and theoretical computer science. The headline is a determination of the exact exponential decay rate of the.

Èyí ni àwọn èrò àgbékalẹ̀ pàtàkì:

- What the collection is, and why one result stands out. The document opens modestly.
- The Cohn-Elkies program and what it bounds. Sphere packing asks how densely non-overlapping balls can fill space.
- What the framework had already achieved. Before turning to the new theorem, it's worth being specific about what the Fourier-analytic linear-programming framework had already delivered, because the new result is partly a statement about how much further it…
- The main theorem. The document's main theorem proves the exponential rate conjectured in Conjectures 3.1 and 3.2 of that 2020 paper.

Kí ni ọ̀nà àgbékalẹ̀ ilé-iṣẹ́ yín sí àwọn ìpèníjà tí a sọ nínú àpilẹ̀kọ yìí?

→ https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/

#Internal #Openai #Model #Pins #Exact

Sebastien Rousseau | CC-BY-4.0

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Rousseau S. An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau. sebastienrousseau.com. 2026 Oct 5. Available from: https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/

Rousseau, Sebastien. "An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau." sebastienrousseau.com. October 5, 2026. https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/.

Rousseau, S. (2026, October 5). An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau. sebastienrousseau.com. https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/

An internal OpenAI model pins down the exact decay rate of the Cohn-Elkies sphere-packing program — Sebastien Rousseau

An internal OpenAI model produced a collection of results across mathematics and theoretical computer science. The headline is a determination of the exact exponential decay rate of the...

Originally published at https://sebastienrousseau.com/yo/2026-10-05-internal-openai-model-pins-down-the/ by Sebastien Rousseau.
Licensed under CC-BY-4.0.

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