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An explicit rational certificate shows the two-point linear programming bound for sphere packing in dimension 36 exceeds the best known packing density by a factor of at least 32.91.

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2026-08-02 06:57 UTC pith:JR32MUAV

load-bearing objection A solid, carefully verified dual LP bound in dimension 36; worth refereeing seriously.

arxiv 2607.11319 v2 pith:JR32MUAV submitted 2026-07-13 math.MG math.NT

A dual linear programming bound for sphere packing in dimension 36

Rifat Jumagulov This is my paper
classification math.MG math.NT MSC 52C1711F1111F3090C05
keywords sphere packinglinear programming bounddual boundmodular formseta-quotientseventual positivityexact arithmeticdimension 36
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes a certified lower bound on the two-point linear programming bound for sphere packing in dimension 36. The bound exceeds the density of the best packing currently known in that dimension — center density 2^18/3^10 — by a factor of at least 32.91, i.e. the LP bound is at least 146.1036. The witness is an explicit pair of weight-18 modular forms for the level-24 group, one the transform of the other, satisfying coefficient nonnegativity conditions that, through a known reduction, yield the LP bound. This is the first such dual bound in any dimension above 32. The certificate is exact and machine-checkable: all coefficients are verified nonnegative up to n=800 by exact rational arithmetic, and for larger n via an explicit tail estimate whose constants are certified with interval arithmetic.

Core claim

The paper's central claim (Theorem 1.1) is a lower bound B for the two-point linear programming optimum in dimension 36: B = 32.91044... × 2^18/3^10 > 146.1036, with explicit rational value and certified decimal enclosures. The witness is a modular form g of weight 18 for Γ0(24) whose Fourier coefficients satisfy a0=1, a1=...=a9=0, an≥0 for all n≥10, and the coefficients of its transform ̃g satisfy bn≥0 for all n. Via a previously established reduction, such a pair yields a dual feasible point for the LP program with value B. The form is found inside an explicit 29-dimensional subspace by exact rational linear programming with constraint generation, and its full coefficient nonnegativity is

What carries the argument

The load-bearing object is a pair (g, ̃g) of weight-18 modular forms for Γ0(24) satisfying coefficient conditions: unit constant term, nine vanishing initial coefficients, and all subsequent coefficients nonnegative on both forms. A known reduction, taken as a black box, converts such a pair into a tempered distribution dual to the LP program, giving the value b0 (2/√24)^18 (√10/2)^36; b0 is an explicit rational number. Two technical tools carry the proof: an exact rational linear program solved by constraint generation, which produces a rational vertex whose expansions stay nonnegative far beyond the enforced range; and a lift-aware weighted coefficient bound (Lemma 4.1) that bounds cuspida

Load-bearing premise

The paper relies on a previously established reduction, invoked without proof, that turns a modular form with the required coefficient nonnegativity into a dual-feasible distribution for the LP program; if that reduction is incorrect, the constructed forms would not imply the stated bound.

What would settle it

Recompute the Fourier coefficients a_n and b_n of the two certified forms for any n much larger than 800 (for instance n=10^6) using independent exact arithmetic; any negative coefficient among them would refute Theorem 1.1. Alternatively, verify the asserted interval enclosure B ∈ [146.1036734821, 146.1036734822] by evaluating the published rational numbers.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • No two-point auxiliary function of the standard LP type can certify the currently best known packing in dimension 36 as optimal; a proof of optimality would need a different method or a stronger bound.
  • The two-point LP optimum in dimension 36 is now known to be at least 146.1036734821 (center density), with the first certified dual bound above dimension 32.
  • If an independent upper bound on the true maximum packing density in dimension 36 were ever shown to fall below 146.1036, the LP bound would be strictly not sharp there.
  • The certificate is fully machine-checkable from the published exact data, so the theorem can be verified without relying on the construction's heuristic steps.
  • The constraint-generation formulation and lift-aware coefficient bound are transferable tools for eventual-positivity verifications in future dimensions.

Where Pith is reading between the lines

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

  • The large ratio (32.9) between the LP bound and the record density suggests the record packing is far from optimal; the paper does not itself construct a denser packing, but the gap leaves room for one.
  • The failure of the naive coefficient bound here — its crossover falls at about n≈1372, beyond the verified window — hints that other eventual-positivity proofs in higher dimensions may need the same weighted bookkeeping; the paper's Lemma 4.1 is a ready template.
  • A natural next step is dimension 40, where the same pipeline is expected to work but requires a larger basis; the paper does not complete that certificate.
  • If the invoked black-box reduction from modular forms to dual distributions is ever called into question, the present certificate would need to be re-derived from first principles; until then, the bound stands on that reduction.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper constructs an explicit dual-feasible point for the Cohn–Elkies linear program in dimension 36: a pair (g, \tilde g) of weight-18 modular forms for Γ0(24). It proves the coefficient conditions a0=1, a_n=0 for n<T, a_n≥0 and b_n≥0 by exact rational scans to n=800 and a tail argument combining per-residue-class Eisenstein lower bounds with a lift-aware Deligne-type bound on the cuspidal part. Applying the Cohn–Triantafillou reduction as a published black box, it concludes δ_LP(36) ≥ B = 146.1036734821..., which exceeds the center density 2^18/3^10 of the Kschischang–Pasupathy packing by a factor 32.91. Thus no Cohn–Elkies auxiliary function can certify that packing as optimal. The certificate is exact rational, with certified interval enclosures for the tail constants and extensive independent cross-checks.

Significance. If correct, this is a substantial advance: it is the first certified dual LP bound in dimension greater than 32 and gives a strong quantitative statement about the Cohn–Elkies program in dimension 36. The paper is unusually careful on the rigor side: an exact rational LP vertex, exact coefficient scans, guarded Eisenstein floor constants, certified interval enclosures for C_w, an independent Arb re-verification of the critical interval solve, and a calibration of the modular-form interface against the published d=12 values. The constraint-generation formulation and the lift-aware Deligne constant are of independent methodological interest. The only external input is the Cohn–Triantafillou reduction, which the authors explicitly identify and calibrate; I found no circularity and no fitting disguised as prediction. The stress-test concern about hidden circularity does not, on reading, land.

minor comments (4)
  1. [§4.2, Lemma 4.1] The ramified-prime case p^2|M is asserted without a reference: the statement that for trivial character a_f(p)=0 and hence a_f(p^r)=0 is standard in the local newform theory, but it is not immediate. Please add a citation (e.g., Atkin–Lehner or Li) or a short justification. This is a clarity issue rather than a mathematical error; I did not find a problem with the resulting divisor bound.
  2. [§4.3] Typo: 'for non any residue class' should read 'for n in any residue class'.
  3. [Table 2] The gate row 'ranks 29,10' is ambiguous. It would be clearer to list separately the exact rank-9 check for (a1,...,a9) and the exact rank-10 check for (a0,...,a9).
  4. [Abstract and §1] The phrase 'best packing currently known' is a time-sensitive external claim. The theorem itself only requires the Kschischang–Pasupathy center density, so please attribute the 'best known' statement to [8] and to the Cohn sphere-packing table, or soften it, to avoid conflating the theorem with a mutable factual claim.

Circularity Check

0 steps flagged

No significant circularity: the certificate is self-contained, and the one external dependency is independent published work.

full rationale

The paper's central claim, Theorem 1.1, is established by exhibiting an explicit dual-feasible pair of modular forms and verifying the required coefficient inequalities (1) exactly: nonnegativity is checked by exact arithmetic up to n=800 and proved for all larger n via Lemma 4.1 with certified interval constants. The value B in (4) is the objective value of the solved exact rational LP, not a fitted or pre-chosen target; the constants c_E and C_w are derived from the constructed vertex and rigorously bounded, and no quantity called a prediction is used as evidence. The only load-bearing external ingredient is the Cohn–Triantafillou reduction [4, Prop. 2.1 and §§2–3], quoted explicitly as invoked as a black box; that is a published result by different authors, does not depend on the present paper, and is independently calibrated by reproducing the d=12 values. No self-citation chain, uniqueness claim, or ansatz-by-citation is used to force the conclusion. The certificate is machine-checkable with exact-data gates and independent re-verifications, so the derivation is self-contained apart from the acknowledged external theorem, which is legitimate independent support rather than circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claim rests on the Cohn–Triantafillou reduction as a published black-box theorem, on standard modular-form bounds (Deligne, local ramified bounds), and on elementary inequalities. The only hand-chosen parameter is T=10; the search subspace V is a harmless restriction since only feasibility is needed. There are no invented entities and no circular fitting of constants.

free parameters (1)
  • T (cutoff index) = 10
    The Cohn–Triantafillou construction requires the first T-1 coefficients of g to vanish. T=10 is chosen by hand; the theorem depends on this choice, but feasibility of the certificate is verified unconditionally for this T.
axioms (5)
  • domain assumption Cohn–Triantafillou Prop. 2.1: coefficient conditions (1) imply existence of a dual-feasible distribution with value B(g) as in (2).
    Invoked in §2 as a black box; the paper does not reprove it, only reproduces the d=12 values as calibration.
  • standard math Deligne's bound: |a_f(p)| ≤ 2 p^{(k-1)/2} for normalized newforms at unramified primes, and the divisor bound on Hecke eigenvalues used in Lemma 4.1.
    Deep theorem used to bound cuspidal coefficients in §4.2, Lemma 4.1.
  • standard math Local bound at ramified primes: for p || M and trivial character, |a_f(p)| ≤ p^{(k-2)/2} and a_f(p^r)=a_f(p)^r; for p^2 | M, a_f(p)=0.
    Stated in Lemma 4.1 with reference to [14, §5.8]; not proved in the paper but standard.
  • standard math σ0(n) ≤ 2√n and ζ(17) < 1 + 2^-17 + 2^-16/16.
    Elementary bounds used in §4.1 and §4.3.
  • standard math The dimension formula and Sturm bound for M_18(Γ0(24)).
    Used in §3.1 to certify the span of candidates and in §4 for validation.

pith-pipeline@v1.3.0-alltime-deepseek · 11901 in / 20267 out tokens · 167534 ms · 2026-08-02T06:57:02.089110+00:00 · methodology

0 comments
read the original abstract

We construct an explicit dual-feasible point for the Cohn--Elkies linear program in dimension $36$, built from the space of weight-$18$ modular forms for $\Gamma_0(24)$ following the method of Cohn and Triantafillou. The certificate shows that the two-point linear programming bound on the sphere packing density in dimension $36$ exceeds the density of the best packing currently known -- the Kschischang--Pasupathy packing, of center density $2^{18}/3^{10}$ -- by a factor of at least $32.91$. In particular, no Cohn--Elkies auxiliary function can certify the best known packing in dimension $36$ as optimal. To our knowledge this is the first such dual bound in any dimension above $32$, extending the table of Cohn--Triantafillou ($d=12,16,20,28,32$), Li ($3\le d\le 13$), and de~Courcy-Ireland--Dostert--Viazovska ($d=6$). The certificate is rigorous and machine-checkable, with exact rational data and certified interval bounds: the dual point is a rational vector, coefficient nonnegativity is verified by exact arithmetic up to $n=800$, and eventual positivity of the two relevant $q$-expansions is proved via an explicit Deligne-type tail bound whose constant is certified with outward-rounded interval arithmetic. Two methodological points may be of independent interest: a constraint-generation (cutting-plane) formulation of the exact rational LP, which is what makes an exact vertex whose Eisenstein data supports the tail argument reachable; and a sharpened, lift-aware form of the Deligne bookkeeping constant that discounts deep oldform lifts, without which the finite verification in dimension $36$ fails (the crossover moves past the verified window).

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Reference graph

Works this paper leans on

15 extracted references · 8 linked inside Pith

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