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REVIEW 2 major objections 4 minor 12 references

On two counterexamples in the geometry of numbers

T0 review · 2 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Two product and height optimization principles for lattices fail in dimensions 8 and 9.

desk verdict Exact rotated-E8 counterexample kills the Cassels–Zong product formulas; the height claim on Λ9 is only floating-point but cleanly presented. read the letter →

arxiv 2607.11695 v1 pith:EFTU4TIQ submitted 2026-07-13 math.MG math.SP

Nihar Gargava This is my paper · ORCID
classification math.MGmath.SP MSC 11H0652C1711M41
keywords geometryofnumberscriticaldeterminantlatticepackingCartesianproductflattorispectralheightE8Korkine-Zolotareff
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

The paper shows that two long-standing optimization principles for lattices do not hold in every dimension. First, the densest packing of a product of two convex bodies need not equal the product of their densest packings: a carefully rotated copy of the E8 lattice packs two four-dimensional balls more densely than the product of the best four-dimensional lattice packings, and the same construction also beats the product of the best unrestricted four-dimensional sphere-packing densities. Second, among nine-dimensional flat tori of unit volume, the lattice that maximises the shortest nonzero vector is not even a local minimiser of the spectral height; an explicit one-parameter deformation lowers the height while preserving volume. The first counterexample is exact; the second is numerical but reproducible to ordinary floating-point precision. Together they demonstrate that product formulae and packing-height coincidences that hold in low dimensions can fail once the dimension reaches eight or nine.

What carries the argument

A concrete orthogonal projection of E8 onto a pair of complementary four-planes defined by a symmetric integer matrix H with H^{2}=5I; after rotation the projected roots realise a block-norm minimum larger than that of D4⊕D4, producing the improved product packing density.

What would settle it

Recompute the spectral height of the given one-parameter family of nine-dimensional Gram matrices with rigorous interval arithmetic or arbitrary-precision cut-offs; if the derivative at zero is non-negative or the height at ε=10^{-3} is not smaller, the second counterexample collapses.

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Extended reading notes

Core claim

For the Euclidean unit ball B in four dimensions the lattice packing density of the product body B imes B is at least (π^{4}/1024)(1+1/√5)^{4}, which is strictly larger than the product of the individual lattice packing densities π^{4}/256. The same lower bound, combined with a known rigorous upper bound on the unrestricted packing density of B, also yields a strict inequality for unrestricted congruent packings. Independently, an explicit determinant-preserving path through Gram matrices of the densest nine-dimensional lattice produces a negative derivative of height at the origin and a strictly smaller height value at a nearby parameter, showing that this densest lattice is not a local hei

Load-bearing premise

The height calculation relies on ordinary floating-point summation of the spectral zeta derivative over finitely many lattice and dual vectors, without interval-arithmetic certification of the sign.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper gives two counterexamples in the geometry of numbers. For the product problem of Cassels (critical determinants) and Zong (lattice/translative packing densities), it exhibits a rotated copy of E8 that packs B_2^4 imes B_2^4 more densely than the product of the D4 packing densities, yielding the strict inequality δ_L(B imes B) ≥ (π^4/1024)(1+1/√5)^4 > π^4/256 = δ_L(B)^2 (and the analogous critical-determinant form); the same lower bound plus the Cohn–de Laat–Salmon upper bound on δ(B_2^4) also falsifies the unrestricted and translative product formulae. Independently, for the Sarnak/Chiu height-minimization conjecture on unit-volume flat tori, it constructs an explicit determinant-1 path G_ε through Gram matrices of the Korkine–Zolotarev lattice Λ_9 (the unique densest lattice in dimension 9) along which numerical evaluation of the spectral zeta derivative shows a negative derivative at ε=0 and a strict drop in height at ε=10^{-3}.

Significance. The results settle long-standing questions of Cassels and Zong in the negative already for four-dimensional balls, and supply the first evidence that densest lattices need not minimize height (already in dimension 9). The lattice-product counterexample is fully rigorous and elementary once the algebraic identities H^{2}=5I and rᵀHr=±2 on E8 roots are verified; the paper supplies machine-checkable ancillary scripts for the latter and an explicit rotation. The height calculation is reproducible via the same ancillaries and an explicit one-parameter family. These strengths (exact identities, open verification code, clean separation of the two independent constructions) make the work a solid contribution to packing and spectral geometry of lattices, even though the second claim remains numerical.

major comments (2)
  1. [§3.3–3.4, Remark 2] Sections 3.3–3.4 and Remark 2: the second central claim (that Λ_9 is not a local height minimizer, so the Sarnak/Chiu conjecture fails) rests entirely on ordinary double-precision summation of the spectral-zeta derivative over lattice and dual vectors with finite cut-offs. The reported values dh/dε|ε=0 ≈ -0.2035 and h(G_{10^{-3}})-h(G_0)≈-2.03 imes10^{-4} are large enough that the sign is almost certainly correct, yet the paper explicitly declines interval-arithmetic certification. Because this is load-bearing for one of the two advertised counterexamples, either supply rigorous error bounds (interval arithmetic, explicit tail estimates, or multiprecision with a priori remainder control) or rephrase the result strictly as strong numerical evidence rather than a definitive counterexample.
  2. [§3.2, Theorem 15] Theorem 15 and the paragraph preceding (11): uniqueness (up to isometry) of the densest unit-covolume lattice in dimension 9 is taken from the preprint [10]. The height calculation itself does not need uniqueness, but the logical step “the unique densest lattice is not a local height minimizer ⇒ the conjecture fails” does. If [10] remains unpublished, the claim should be weakened to “this particular densest lattice is not a local height minimizer” (still sufficient to cast serious doubt on the conjecture, but not a complete disproof).
minor comments (4)
  1. [§3.3] The numerical height evaluation (cut-offs, number of vectors retained, convergence checks) is only sketched; a short paragraph or table in the ancillary documentation listing the precise truncation radii used for the reported digits would improve reproducibility.
  2. [§2.5] In the unrestricted comparison of §2.5 the elementary bounds π>3.14 and √5<25/11 already suffice; the more precise Cohn table value can be relegated to a remark so that the argument is self-contained without external floating-point data.
  3. [Acknowledgements, title page] Typographical: “dimen-sions” (Acknowledgements), and the future date “July 13, 2026” on the title page should be checked against the arXiv stamp.
  4. [§3.3] The matrix M defining the perturbation A is presented without motivation; a one-sentence remark that it was chosen so that the first-order change in λ_1 of the dual is positive while that of the primal is negative would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: both counterexamples are independent constructions from known lattices, not tautological reductions of their inputs.

full rationale

The lattice-product counterexample constructs an explicit rotation of E8 via a concrete integer matrix H with H^{2}=5I, verifies the root-projection identity rᵀHr=±2 by finite enumeration, and obtains a density strictly larger than the product of the known D4 densities; the comparison is elementary arithmetic and does not encode the target inequality by definition. The unrestricted version merely multiplies a published external upper bound (Cohn–de Laat–Salmon) by itself and compares it with the same explicit lower bound. The height counterexample starts from the independently established densest lattice Λ9 (Dutour Sikirić–van Woerden), builds an explicit trace-zero path Gε that preserves det=1, and evaluates the standard spectral-zeta derivative numerically; the sign of the derivative is an output of that summation, not an input fitted to force a decrease. No parameter is fitted to height data, no uniqueness theorem of the present author is invoked, and no self-citation carries the load of either claim. The only soft spot is ordinary floating-point accuracy (already flagged by the authors), which is a rigor gap, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The product counterexample rests only on classical facts about E8 and D4 plus one external upper bound; the height counterexample rests on the recent classification of the densest 9-dimensional lattice and on floating-point evaluation of a standard spectral series. No free parameters are fitted to data; the single ad-hoc matrix A is an explicit, hand-chosen direction of descent.

free parameters (1)
  • perturbation matrix A (scaled M) = explicit 9×9 integer matrix given in §3.3
    The symmetric trace-zero matrix M (and its normalization A) is chosen by hand so that the directional derivative of height is negative; it is not derived from a uniqueness or optimality principle.
assumptions (5)
  • standard math D4 realizes the maximal lattice packing density of the 4-ball (Korkine–Zolotareff 1872)
    Used to identify the product value π^4/256 as the lattice product density (eq. (5)).
  • standard math E8 is even unimodular of minimal norm 2 with 240 roots of the stated form
    Standard realization (Conway–Sloane) used for the rotated packing and for the exact root-projection lemma.
  • domain assumption Λ9 is the unique densest unit-covolume lattice in dimension 9 (Dutour Sikirić–van Woerden arXiv:2508.20719)
    Invoked to conclude that a local height decrease for Λ9 falsifies the Sarnak–Chiu conjecture (Theorem 15).
  • domain assumption Unrestricted 4-dimensional sphere-packing density ≤0.6361073321551329 (Cohn–de Laat–Salmon)
    Used only for the unrestricted product counterexample (eq. (10)); the lattice counterexample does not need it.
  • ad hoc to paper Floating-point evaluation of the spectral zeta derivative with finite cut-offs correctly determines the sign of dh/dε and the inequality h(G_{10^{-3}})<h(G_0)
    Explicitly unproved; authors decline interval arithmetic (Remark 2).

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Pith. "Pith review of On two counterexamples in the geometry of numbers." pith.science (2026). https://pith.science/paper/EFTU4TIQ

@misc{pith2026260711695,
  author       = {Pith},
  title        = {Pith review of: On two counterexamples in the geometry of numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFTU4TIQ}},
  note         = {Machine review of arXiv:2607.11695}
}
read the original abstract

We give counterexamples to two optimization problems in dimensions eight and nine. 1. The Cartesian-product problem posed by Cassels for critical determinants and later formulated by Zong for lattice packings and for packings allowing translations but not rotations: whether the corresponding product inequalities are always equalities. 2. A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector. The first counterexample is exact and also disproves the natural product formula for unrestricted congruent packings. The second is numerical but within reasonable floating-point accuracy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 1 linked inside Pith

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