Zeros of Optimal Functions in the Cohn-Elkies Linear Program

Nina Zubrilina

arxiv: 1906.11112 · v1 · pith:RPWE3UUSnew · submitted 2019-06-26 · 🧮 math.NT · math.MG

Zeros of Optimal Functions in the Cohn-Elkies Linear Program

Nina Zubrilina This is my paper

Pith reviewed 2026-05-25 15:19 UTC · model grok-4.3

classification 🧮 math.NT math.MG

keywords sphere packingCohn-Elkies linear programzeros of functionsroot lengthsoptimal functionslinear programming bounds

0 comments

The pith

Optimal functions in the Cohn-Elkies program have root lengths whose separations are bounded above in every dimension yet must cluster arbitrarily closely over long intervals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves theorems describing the possible locations of zeros for any optimal function in the Cohn-Elkies linear program in arbitrary dimensions. It shows that distances between these root lengths are bounded from above for all n at least 1, while for n at least 2 the distances are not bounded from below. It further shows that for any fixed length C and any small epsilon the roots must include an interval of length C containing points no farther than epsilon apart. These location constraints hold independently of any explicit construction of the functions. The results also include a method for improving certain non-optimal functions.

Core claim

We prove that distances between root lengths are bounded from above for n ≥ 1 and not bounded from below for n ≥ 2, and that the root lengths have to be arbitrarily close for arbitrarily long, that is, for any C, ε > 0, there is an interval of length C in which the root lengths are at most ε apart.

What carries the argument

The positivity and Fourier-transform sign conditions that any optimal function in the Cohn-Elkies linear program must satisfy, which constrain the possible locations of its zeros.

If this is right

Any optimal function's zeros are forced to lie within bounded gaps in every dimension.
In dimensions two and higher the zeros can accumulate in tight clusters separated by larger gaps.
No optimal function can have all its roots spaced uniformly or with a positive minimum separation in higher dimensions.
The clustering property holds regardless of whether the function comes from modular forms or another construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The clustering forces any attempt to list all possible zero sets to account for dense subsets inside long intervals.
The improvement technique for non-optimal functions may be applicable whenever a candidate function violates the upper bound on root separations.
The results suggest that explicit constructions in new dimensions will need to produce zeros that become dense in this controlled way.

Load-bearing premise

Optimal functions exist in every dimension and the locations of their zeros are controlled solely by the sign conditions of the linear program.

What would settle it

An optimal function in some dimension whose root lengths have separations that are unbounded above or that fail to enter arbitrarily close clusters inside intervals of arbitrary length.

read the original abstract

In a recent breakthrough, Viazovska and Cohn, Kumar, Miller, Radchenko, Viazovska solved the sphere packing problem in $\mathbb{R}^8$ and $\mathbb{R}^{24}$, respectively, by exhibiting explicit optimal functions, arising from the theory of weakly modular forms, for the Cohn-Elkies linear program in those dimensions. These functions have roots exactly at the lengths of points of the corresponding optimal lattices: $\{\sqrt{2n}\}_{n\geq 1}$ for the $E_8$ lattice, and $\{\sqrt{2n}\}_{n\geq 2}$, for the Leech lattice. The constructions of these optimal functions are in part motivated by the locations of the zeros. But what are the roots of optimal functions in other dimensions? We prove a number of theorems about the location of the zeros of optimal functions in arbitrary dimensions. In particular, we prove that distances between root lengths are bounded from above for $n \geq 1$ and not bounded from below for $n \geq 2$, and that the root lengths have to be arbitrarily close for arbitrarily long, that is, for any $C, \varepsilon > 0$, there is an interval of length $C$ in which the root lengths are at most $\varepsilon$ apart. We also establish a technique that allows one to improve a non-optimal function in some cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives conditional constraints on zero locations that any optimal Cohn-Elkies function must obey in general dimensions.

read the letter

The paper establishes theorems on the locations of zeros for optimal functions in the Cohn-Elkies linear program across arbitrary dimensions. Specifically, it shows that distances between root lengths are bounded from above for n greater than or equal to 1, but not bounded from below for n greater than or equal to 2, and that the roots must cluster arbitrarily closely within arbitrarily long intervals. This is the new part. Prior work focused on constructing optimal functions in dimensions 8 and 24 using modular forms, with zeros at the lattice point lengths. Here the author derives these separation and clustering properties directly from the defining conditions of the linear program: the positivity of the function and its Fourier transform. The paper does well in isolating these structural constraints without needing an explicit function. It also sketches a method to improve non-optimal functions in some cases, which might have some utility. The main limitation is that these results are conditional on the existence of an optimal function. Explicit optimal functions are known only in 8 and 24 dimensions, so for other n the statements describe what would be required if one existed. The stress-test note correctly identifies this: the derivations treat the LP sign conditions as sufficient to control zero locations, independent of further structure. That holds up on the abstract, and the paper does not claim to resolve existence. The arbitrary clustering property in particular seems to rely on the conditions allowing accumulation points or dense sets in some way. Without the full proofs it is hard to assess how tight the arguments are, but the claims appear logically derived from the program definition. This work is for specialists in the sphere packing linear programming method. Someone already thinking about the Cohn-Elkies program will find the zero constraints useful as additional properties any candidate must satisfy. It deserves a serious referee because the results are specific mathematical statements that can be verified or refuted based on the proofs. I recommend sending this to peer review rather than desk rejecting it.

Referee Report

1 major / 1 minor

Summary. The manuscript proves theorems on the locations of zeros of optimal functions for the Cohn-Elkies linear program in arbitrary dimensions n. It establishes that distances between root lengths are bounded above for n ≥ 1 and unbounded below for n ≥ 2, that roots must cluster arbitrarily closely within intervals of arbitrary length, and presents a technique for improving certain non-optimal functions.

Significance. If the results hold, they supply general structural constraints on zero locations derived solely from the positivity and Fourier-transform sign conditions of the linear program, without relying on explicit constructions (known only for n=8,24). This could inform searches for optimal functions in other dimensions.

major comments (1)

[Abstract and Introduction] Abstract and statements of the main theorems: the claims about root-length gaps and clustering are stated unconditionally for arbitrary n, yet they presuppose the existence of optimal functions (whose zeros are then controlled by the LP conditions). Existence is established only for n=8 and n=24; the theorems therefore require an explicit conditional formulation (e.g., “if an optimal function exists, then …”) to be load-bearing for the central claims.

minor comments (1)

[Section 1] Notation for the Cohn-Elkies functional and the sequence of root lengths r_k could be introduced with a dedicated preliminary subsection to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We agree that the statements of the main results should be formulated explicitly as conditional on the existence of an optimal function, and we will revise the abstract and introduction to make this clear.

read point-by-point responses

Referee: [Abstract and Introduction] Abstract and statements of the main theorems: the claims about root-length gaps and clustering are stated unconditionally for arbitrary n, yet they presuppose the existence of optimal functions (whose zeros are then controlled by the LP conditions). Existence is established only for n=8 and n=24; the theorems therefore require an explicit conditional formulation (e.g., “if an optimal function exists, then …”) to be load-bearing for the central claims.

Authors: We agree with the referee that the theorems are conditional on the existence of an optimal Cohn-Elkies function in dimension n. The results derive structural properties of the zeros solely from the positivity and Fourier-transform sign conditions that any optimal function must satisfy. While existence is currently known only for n=8 and n=24, the statements are intended to apply in any dimension where an optimal function exists. We will revise the abstract and the statements of Theorems 1.1–1.3 (and the surrounding discussion in the introduction) to include the explicit qualifier “assuming an optimal function exists, then …” so that the conditional nature is unambiguous. This is a straightforward clarification that does not alter the proofs or the scope of the results. revision: yes

Circularity Check

0 steps flagged

No circularity: theorems derive consequences from LP optimality conditions

full rationale

The paper derives upper bounds on root-length gaps, lower unboundedness, and clustering properties directly from the positivity and Fourier-transform sign conditions that define optimality in the Cohn-Elkies linear program. These are stated as theorems conditional on the existence of optimal functions in arbitrary dimensions, with no reduction of any claimed prediction or result to a fitted parameter, self-citation chain, or definitional equivalence. The abstract and described claims treat the LP constraints as the starting point for independent analytic consequences about zero locations, without invoking prior author work as a uniqueness theorem or smuggling an ansatz. This is the standard non-circular pattern of extracting properties from a set of inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a pure proof paper operating inside the established Cohn-Elkies framework; no numerical fitting, new physical entities, or ad-hoc constants are introduced.

axioms (1)

domain assumption Standard analytic properties of the Cohn-Elkies linear program, including positivity of the test function and sign conditions on its Fourier transform.
All stated theorems are derived from this pre-existing framework.

pith-pipeline@v0.9.0 · 5776 in / 1303 out tokens · 31559 ms · 2026-05-25T15:19:45.100846+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that distances between root lengths are bounded from above for n ≥ 1 and not bounded from below for n ≥ 2, and that the root lengths have to be arbitrarily close for arbitrarily long
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2 … Let S ⊆ Rn be a compact set of measure μ(S) > 1. Then there exist z ∈ (S−S) ∩ {x : ∥x∥ ≥ r} such that f(z)=0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

New upper bounds on sphere packing s

Henry Cohn and Noam Elkies. New upper bounds on sphere packing s. I. Ann. of Math. (2), 157(2):689–714, 2003

work page 2003
[2]

Universally optimal distribution of points on spheres

Henry Cohn and Abhinav Kumar. Universally optimal distribution of points on spheres. J. Amer. Math. Soc. , 20(1):99–148, 2007

work page 2007
[3]

Miller, Danylo Radchenko , and Maryna Via- zovska

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko , and Maryna Via- zovska. The sphere packing problem in dimension 24. Ann. of Math. (2) , 185(3):1017–1033, 2017

work page 2017
[4]

Sphere packing bounds via spherica l codes

Henry Cohn and Yufei Zhao. Sphere packing bounds via spherica l codes. Duke Math. J. , 163(10):1965–2002, 2014

work page 1965
[5]

A formal proof of the Kepler conjecture

Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat Dang, John Harrison, Le Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat Than g Nguyen, Quang Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai An Ta, Nam Trung Tran, Thi Diep Trieu, Josef Ur ban, Ky Vu, and Roland Zumkeller. A formal proof of t...

work page 2017
[6]

The sphere packing problem in dimension 8

Maryna Viazovska. The sphere packing problem in dimension 8. Ann. of Math. (2) , 185(3):991–1015, 2017. 9

work page 2017

[1] [1]

New upper bounds on sphere packing s

Henry Cohn and Noam Elkies. New upper bounds on sphere packing s. I. Ann. of Math. (2), 157(2):689–714, 2003

work page 2003

[2] [2]

Universally optimal distribution of points on spheres

Henry Cohn and Abhinav Kumar. Universally optimal distribution of points on spheres. J. Amer. Math. Soc. , 20(1):99–148, 2007

work page 2007

[3] [3]

Miller, Danylo Radchenko , and Maryna Via- zovska

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko , and Maryna Via- zovska. The sphere packing problem in dimension 24. Ann. of Math. (2) , 185(3):1017–1033, 2017

work page 2017

[4] [4]

Sphere packing bounds via spherica l codes

Henry Cohn and Yufei Zhao. Sphere packing bounds via spherica l codes. Duke Math. J. , 163(10):1965–2002, 2014

work page 1965

[5] [5]

A formal proof of the Kepler conjecture

Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat Dang, John Harrison, Le Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat Than g Nguyen, Quang Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai An Ta, Nam Trung Tran, Thi Diep Trieu, Josef Ur ban, Ky Vu, and Roland Zumkeller. A formal proof of t...

work page 2017

[6] [6]

The sphere packing problem in dimension 8

Maryna Viazovska. The sphere packing problem in dimension 8. Ann. of Math. (2) , 185(3):991–1015, 2017. 9

work page 2017