The 196560 auxiliary-function conjecture for the Leech lattice

Yaoran Yang; Yutong Zhang

The 196560 auxiliary-function conjecture for the Leech lattice

T0 review · 0 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves the 2009 auxiliary-function conjecture for the Leech lattice by constructing an explicit one-parameter family of radial Schwartz functions, the admissible members of which form exactly a closed ray.

desk verdict Proves the Cohn–Kumar 196560 conjecture via a clean quotient argument; the main open risk is the explicit but unverified coefficient extraction in Tables 1–2. read the letter →

arxiv 2608.12094 v1 pith:Z36R3S63 submitted 2026-08-12 math.NT math.CAmath.MG

Yutong Zhang , Yaoran Yang This is my paper · ORCID

classification math.NTmath.CAmath.MG MSC 42A3811F2711H3152C17

keywords FourierinterpolationLeechlatticeradialSchwartzfunctionauxiliaryPoissonsummationspherepackingmodularformsthetaseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a 2009 conjecture about the Leech lattice: there exists a radial Schwartz function g on $R^{24}$ whose direct side is nonpositive outside radius $\sqrt$(6), whose Fourier transform is nonnegative everywhere, which is positive at radius 2, and whose Poisson-summing ratio (hat g(0)-g(0))/g(2) equals 196560, the number of shortest Leech vectors. The proof constructs g as g_C = a_2 - C b_2, a one-parameter family built from the first two radial Fourier interpolation basis functions, and shows the admissible parameters are exactly a closed ray [C*, infinity). This matters because such an auxiliary function is the missing input that makes the linear-programming optimality argument for the Leech lattice fully explicit, with no numerical search. The same basis also recovers every nontrivial Leech-shell coefficient.

What carries the argument

The central object is the radial Fourier interpolation basis in dimension 24: Schwartz functions a_n, b_n, tilde a_n, tilde b_n indexed by n >= 2 that interpolate the value and first radial derivative of a function and of its Fourier transform at the radii sqrt(2n). Only the first value-derivative pair a_2, b_2 is used, with b_2 identified as a fixed multiple of the dimension-24 sphere-packing magic function, which supplies the strict signs b_2(r)>0 for r>2 away from nodes and hat b_2(r)<0 for r>=0 away from nodes. The argument is carried by the removable quotients R(r)=a_2/b_2 and hat R(r)=hat a_2/hat b_2: their boundedness converts the sign conditions into C >= sup, and their asymptotic limits are computed by exact coefficient extraction from the interpolation kernel, an S-cusp expansion at z=it, and a Laplace-Bessel transfer to large radii.

What would settle it

Evaluate the paper's own explicit Laplace-integral formulas for b_2 and hat b_2 at a single non-node point, say r=3 for b_2 and r=1/2 for hat b_2; if b_2(3) were not positive, or hat b_2(1/2) not negative, the sign reduction would fail and the ray [C*, infinity) would not be admissible.

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Extended reading notes

Core claim

The paper's central claim is that the 2009 auxiliary-function conjecture for the Leech lattice is true, with a complete description of the admissible functions in its natural affine family. For every real C at least C*, where C* is the maximum of the suprema of the smooth quotient functions R(r)=a_2(r)/b_2(r) on r >= $\sqrt$(6) and hat R(r)=hat a_2(r)/hat b_2(r) on r >= 0, the function g_C = a_2 - C b_2 is a radial Schwartz function on $R^{24}$ satisfying g_C(r) <= 0 for r >= $\sqrt$(6), hat g_C(r) >= 0 for r >= 0, g_C(2)=1, and (hat g_C(0)-g_C(0))/g_C(2)=196560. Poisson summation over the self-dual Leech lattice gives exactly this identity, because the interpolation node data force g_C to vanish on every nonzero Leech shell except the shortest one and force hat g_C to vanish on every nonzero Leech shell. The paper also shows the admissible set in the affine line is exactly the closed ray [C*, infinity), with C* > rho = (43+240 log 2)/15.

Load-bearing premise

Everything rests on the sign anchor—b_2 > 0 on r > 2 except at nodes and hat b_2 < 0 on r >= 0 except at nodes—which is inherited from the previously constructed sphere-packing magic function and not reproved here.

Editorial extensions

If this is right

The complete admissible set in the affine family is [C*, infinity), so validity is not a single lucky function but a whole ray of valid auxiliary functions.
Every member g_C with C >= C* proves the 196560 identity through Poisson summation, with no hidden numerical verification.
The explicit asymptotics R(r), hat R(r) -> rho with opposite exponential corrections imply C* > rho and pin down the direction from which the quotients approach their limit.
The same interpolation basis yields exact Leech theta coefficients N_m = (65520/691)(sigma_11(m) - tau(m)).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because C* is a maximum of two explicit smooth quotients with known limits and corrections, one could compute rigorous two-sided bounds on C* by evaluating those quotients on a finite grid plus a tail estimate; the paper does not attempt this.
The quotient strategy should transfer to other dimensions with an interpolation basis and a signed anchor function, such as the analogous setting in dimension 8, yielding an exact ray of auxiliary functions for the corresponding ratio conjecture; the paper only treats dimension 24.
The exact coefficient cancellations at the cusp suggest a general modular identity governing the n=2 coefficients; testing whether similar finite identities hold for n>2 could make shell-coefficient recovery uniform.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: it's the real thing—it proves the 196560 auxiliary-function conjecture—but the load-bearing finite computation is exactly where I'd want a referee to spend time.

What's new: the affine ansatz g_C = a_2 - C b_2, the reduction of the conjecture to boundedness of the two quotients, and the exact S-cusp expansion showing both quotients tend to ρ = (43 + 240 log 2)/15 with opposite-side exponential corrections. That last bit is nice: it gives C* > ρ explicitly. The corollary recovering every Leech-shell coefficient from interpolation is a clean bonus.

What's good: the sign reduction is clean and uses only the sign anchor from [4] plus removability of quotient singularities. The cusp-to-Laplace transfer in Lemma 6.1 is a solid piece of analysis: the Bessel function ratio handles the exponential decay properly and the Gaussian tail bounds are correct. The modular-form machinery is used with the right amount of care, and the authors are explicit about which parts come from [3] and [4]. There's no fitting: the ratio 196560 comes out of Poisson summation, not from tuning coefficients.

The soft spot is the one the stress-test flags: Sections 4–5 rest on a sign-sensitive coefficient extraction. Tables 1–2 and identities (4.38)–(4.41) are finite but long, with branch-dependent logs and a parity substitution. A single scalar error would change the leading constants and break the boundedness proof. The authors disclose that part of the extraction was developed with an AI language model and that no machine check was run. That is not a sin, but it makes independent verification important. The good news: everything needed is in the paper, so a referee (or the authors) can check the tables with a computer algebra system in an afternoon. The valuation argument at the end of Section 4 correctly explains why the truncation is exact, so this is a human-verifiable proof, not an appeal to a black box.

I don't see a circularity problem: the external inputs are independent published theorems, and the new contribution is the quotient analysis. The theorem itself is a real open-problem resolution, significant for the Fourier-interpolation and sphere-packing communities. The paper deserves a serious referee, and the referee should be asked to verify Tables 1–2 and the signs in (5.7)–(5.8). I'd accept subject to that verification.

Referee Report

0 major / 4 minor

Summary. The paper constructs radial Schwartz functions on R^24 that prove the 2009 Cohn-Kumar auxiliary-function conjecture for the Leech lattice. Working with the radial Fourier interpolation basis of [3], the authors form the one-parameter family g_C = a_2 - C b_2. They identify b_2 with -16380 times the sphere-packing magic function of [4], which supplies strict signs for b_2 and its Fourier transform. The main technical work is an exact analysis of the quotients a_2/b_2 and hat-a_2/hat-b_2 on their respective half-lines: after proving that all apparent singularities are removable, the paper computes the common limit rho=(43+240 log 2)/15 through an S-cusp expansion of the n=2 interpolation coefficients and obtains an explicit first exponential correction. It follows that the admissible parameters form a nonempty closed ray [C*, infinity), and every member satisfies the required sign conditions and the ratio identity (hat-g(0)-g(0))/g(2)=196560 by Poisson summation over the Leech lattice. A corollary recovers every nontrivial Leech-shell coefficient from the interpolation basis.

Significance. This is a substantial result: it settles a conjecture that has been open since 2009 and gives a clean, parameter-free construction from Fourier interpolation rather than a numerical search. The proof has several strong features: the target value 196560 enters only through Poisson summation and is not an input; the sign anchor is the independent sphere-packing function from [4]; the asymptotic analysis is carried out by explicit finite identities; and the paper honestly discloses which inputs come from [3] and [4]. I spot-checked representative entries in the coefficient tables and found them consistent with the displayed formulas. The absence of a machine-checked certificate is not, in my view, a defect, because the coefficient extraction is presented as finite identities in formal power series rather than as an opaque computer calculation.

minor comments (4)

[Tables 1-2] The blank entries in Tables 1 and 2 are easy to misinterpret in a text rendering; please ensure that the final typeset version has explicit column alignment (for example, by repeating the power of x in each header or by using a vertical layout) so that each coefficient can be checked unambiguously.
[Section 4.3, Eqs. (4.29)-(4.30)] The branch choices for L and L_S are described as those used in [3], but since the S-cusp calculation is sign-sensitive, it would help to state the branch convention in a self-contained sentence before these expansions are used.
[Section 6, Eq. (6.7)] The statement that the leading constant in the Bessel asymptotics 'simplifies exactly to 1' would benefit from a one-line derivation, since the powers of a and the r^{-23/2} factor are not immediately obvious from the preceding display.
[Section 8] Given the disclosure that portions of the symbolic extraction were developed with an OpenAI language model, I suggest adding a sentence clarifying whether the displayed identities (4.38)-(4.41) and Tables 1-2 were subsequently re-derived and checked independently by the authors; this would directly address reproducibility concerns.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 196560 identity is derived via Poisson summation, signs come from independent published external work, and no fitted parameter is renamed as a prediction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The target ratio 196560 is not an input: after the sign conditions are established, Poisson summation over the self-dual Leech lattice reduces to equation (7.2) because the interpolation data force g_C to vanish on all nonzero shells except the first, where g_C(2)=1; hence the ratio is forced as a theorem, not assumed. The sign anchor in Proposition 2.2 is imported from Cohn-Kumar-Miller-Radchenko-Viazovska's dimension-24 sphere packing paper [4], which is independent external work not authored by Zhang/Yang, and it is not fitted to the Cohn-Kumar conjecture. The interpolation basis from [3] is likewise external. The boundedness of the quotients is proved by exact coefficient extraction and Laplace-Bessel transfer; no parameter is fitted to the target ratio and no 'prediction' is a renamed fit. The only citation-dependent steps that could raise circularity concerns are citations to independent published theorems with stated assumptions, so they constitute real evidence under the review rules. The long coefficient extraction is an internal correctness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on two deep external theorems (interpolation [3] and the magic function [4]) plus standard mathematical facts. There are no fitted parameters and no newly postulated entities; the affine parameter C is a free variable, not a fitted value.

assumptions (6)

domain assumption Fourier interpolation theorem for radial Schwartz functions on R^24 from Cohn-Kumar-Miller-Radchenko-Viazovska (references [3], equations (2.1)-(2.3) and Proposition 5.4).
Used to define basis functions a_2 and b_2 and the integral representations (6.1)-(6.4); assumed without proof.
domain assumption Sphere-packing magic function f_sp in dimension 24 exists with the sign and zero properties used in Proposition 2.2, including the positivity of f_sp-hat on [0, sqrt(2)] via the continuation identity (2.12) from [4].
Provides the sign anchor b_2 = -16380 f_sp and the strict sign properties in Proposition 2.2.
domain assumption Large-t coefficient bounds from [3, Eq. (5.22)]: alpha_2(it) = t e^{4 pi t} + O((1+t)^2 e^{2 pi t}), beta_2(it) = 1/(4 pi) e^{4 pi t} + ..., and similar tilded bounds.
Needed in Proposition 6.2 to apply the Laplace-Bessel transfer and control integrals as t approaches infinity.
standard math Standard Poisson summation for Schwartz functions on the self-dual Leech lattice Lambda_24.
Used in the proof of Theorem 1.1 to derive the 196560 identity.
standard math Theta series of the Leech lattice belongs to M_12(SL_2(Z)) and equals E_12 - (65520/691) Delta.
Used in Corollary 1.2 to derive explicit shell coefficients.
standard math Hadamard's lemma for smooth functions with double zeros and standard Laplace-Bessel integral representation of modified Bessel functions.
Used in Lemmas 3.1 and Section 6 for quotient extensions and asymptotics.

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Pith. "Pith review of The 196560 auxiliary-function conjecture for the Leech lattice." pith.science (2026). https://pith.science/paper/Z36R3S63

@misc{pith2026260812094,
  author       = {Pith},
  title        = {Pith review of: The 196560 auxiliary-function conjecture for the Leech lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z36R3S63}},
  note         = {Machine review of arXiv:2608.12094}
}

abstract

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g\colon\R^{24}\to\R$ satisfying $g(r)\leq0$ for $r\geq\sqrt6$, $\widehat g(r)\geq0$ for $r\geq0$, $g(2)>0$, and $(\widehat g(0)-g(0))/g(2)=196560$. We construct such functions from the radial Fourier interpolation basis in dimension $24$. If $a_2,b_2$ denote the basis functions dual to value and radial-derivative interpolation at radius $2$, then $g_C=a_2-Cb_2$ has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies $b_2$ and supplies its strict signs. We prove that the removable quotients $a_2/b_2$ and $\widehat a_2/\widehat b_2$ are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an $S$-cusp expansion. Both quotients tend to $(43+240\log2)/15$; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [4]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska,The sphere packing problem in dimension24, Ann. of Math. (2)185(2017), no. 3, 1017–1033

work page 2017

[2] [3]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska,Universal optimality of theE8 and Leech lattices and interpolation formulas, Ann. of Math. (2)196(2022), no. 3, 983–1082

work page 2022

[3] [1]

Cohn and A

H. Cohn and A. Kumar,Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2)170(2009), no. 3, 1003–1050

work page 2009

[4] [2]

Cohn and N

H. Cohn and N. Elkies,New upper bounds on sphere packings. I, Ann. of Math. (2)157(2003), no. 2, 689–714

work page 2003

[5] [5]

M. S. Viazovska,The sphere packing problem in dimension8, Ann. of Math. (2)185(2017), no. 3, 991–1015

work page 2017

[6] [6]

Cohn and S

H. Cohn and S. D. Miller,Some properties of optimal functions for sphere packing in dimensions8 and24, preprint, arXiv:1603.04759 (2016)

arXiv 2016

[7] [7]

J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices and Groups, third ed., Grundlehren der mathematischen Wissenschaften, vol. 290, Springer-Verlag, New York, 1999

work page 1999

[8] [8]

Radchenko and M

D. Radchenko and M. Viazovska,Fourier interpolation on the real line, Publ. Math. Inst. Hautes Études Sci.129(2019), 51–81

work page 2019