A live geometry of numbers laboratory

The Ellipsoid That Kept Touching

Build an exact lattice packing from its Gram matrix in every dimension from 1 through 24, then enter a constrained ellipsoid chamber inspired by Boaz Klartag's peer-reviewed 2026 proof. The browser derives shortest vectors, covolume, ball volume, packing density, contact equations, and the finite lattice search cutoff live, while the unknown universal constant stays visibly unknown.

published result Announced on arXiv in 2025, accepted and published in Inventiones Mathematicae in 2026. DOI 10.1007/s00222-026-01412-w

Layer one · the packing benchGive a lattice room to breathe

A Gram matrix tells the browser every squared distance inside a lattice. Move the dimension control. The page proves a shortest vector by complete Cholesky branch-and-bound, halves its length for the sphere radius, derives the fundamental-cell volume from the determinant, and divides one ball by one cell.

Compare as

Computing the selected Gram matrix.

The live text description of the selected lattice will appear here.

Construction

computing

exact Gram data

Shortest squared norm

computing

Sphere radius

computing

half the centre distance

Gram determinant

computing

covolume is its square root

Ball volume

computing

derived from Γ

Covered fraction

computing

construction lower bound

Computing Vn(r) / √det(G).
Computing a rigorous comparison.
Open the selected Gram matrix and the exhaustive search certificate

Construction data will appear here.

These are concrete constructions, not values of Klartag's theorem. The theorem's constant is not numerically evaluated in the paper, so its bound belongs on the page as algebra until a reader explicitly chooses a hypothetical value.

The polynomial factors, with the hidden constants exposed

Every row below has the common exponential factor 2-n removed. Ratios are between the displayed leading expressions only. They are not finite-dimension density estimates.

Computing symbolic ratios.

Layer two · the contact chamberKeep every point you touch

A line labelled n² does not explain a packing. Here is the local mechanism. In two dimensions the symmetric matrix A = [[a,b],[b,d]] has three degrees of freedom. Its ellipsoid is xᵀAx < 1. Any nonzero integer point inside is forbidden. A point on the boundary becomes a linear constraint, and future seeded steps are projected to preserve it.

A certified ℤ²-free ellipsoid

Checking the initial ellipsoid.

The live text description of the ellipsoid and its lattice contacts will appear here.

det(A)

computing

positive means positive-definite only with positive trace

Ellipsoid area

computing

π / √det(A)

Certified cutoff

computing

from the minimum eigenvalue

Contact points

computing

opposite points counted separately

Independent constraints

computing

one per opposite pair at most

Motion left

computing

three symmetric-matrix parameters minus rank

Computing the determinant-volume identity.

This chamber verifies its own path and invariants. It is not a replay of the high-dimensional stochastic proof. Klartag's process uses Brownian motion in the full symmetric-matrix space and probabilistic estimates over random lattices. The chamber only makes the determinant, exclusion, contact, and degrees-of-freedom mechanism operable.

The check

Two independent live certificates meet here. The packing bench recomputes a Gram determinant, a globally shortest lattice vector, a sphere radius, a ball volume, and a density. The chamber derives a finite lattice cutoff from the minimum eigenvalue, tests every integer point inside it, and recomputes every contact equation.

Selected lattice

computing

Ball formula

computing

Klartag theorem

computing

Rogers reference constant

computing

Symmetric-matrix count

computing

Current ellipsoid

computing

Choices and approximations

  • The Klartag constant cK is unknown. Its slider is a reader-chosen hypothetical value, never a published estimate.
  • The Rogers slider is also a comparison choice. The separate check prints the published proof value 2/e live.
  • Terms written o(1) are asymptotic statements, not finite-dimension error bars.
  • The construction catalogue is frozen to the Nebe-Sloane records retrieved on 29 July 2026. The page uses named exact constructions, not a claim to list the best construction in every dimension.
  • Floating arithmetic is used after exact integer Gram data are loaded. The offline verifier recomputes determinants with exact integer arithmetic.
  • The two-dimensional seeded walk is a free pedagogical choice. It preserves its own contacts and exclusion invariant but does not simulate the theorem's probability law.

Still open

  • The exact unrestricted packing optimum is known only in dimensions 1, 2, 3, 8, and 24. It remains open in all other dimensions.
  • The exact lattice optimum is also unknown in general.
  • The lower scale c n² 2-n remains exponentially separated from the asymptotic upper scale 2-(α + o(1))n, where α is about 0.5990.
  • Klartag does not numerically evaluate the universal constant produced by the proof.
  • The paper conjectures an analogous lattice-free conclusion for every origin-symmetric convex body of the stated volume scale.
  • The paper does not supply an explicit efficiently computable family attaining the theorem's scale.

The offline check lifts the shipped browser functions from this file, sweeps them against separately written reference implementations, verifies the published construction invariants, and confirms that a deliberately mutated interior-point inequality is caught. Run node research/the-ellipsoid-that-kept-touching/verify-the-ellipsoid-that-kept-touching.mjs.

Primary recordSources and status

Boaz Klartag. “Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid.” Inventiones Mathematicae 244, 1251-1279. Published 4 March 2026. Theorem 1.1 supplies the all-dimensions lattice bound; equations 5, 6, 9, and 10 supply the contact and ellipsoid apparatus. DOI 10.1007/s00222-026-01412-w; arXiv:2504.05042.

peer-reviewed

C. A. Rogers. “Existence theorems in the geometry of numbers.” Annals of Mathematics 48, 994-1002, October 1947. Historical source for the linear polynomial scale; Klartag records that this proof yields c ≥ 2/e. DOI 10.2307/1969390.

peer-reviewed

Akshay Venkatesh. “A Note on Sphere Packings in High Dimension.” International Mathematics Research Notices 2013(7), 1628-1642. Published online 7 March 2012. Source for the sparse-sequence lattice limsup bound. DOI 10.1093/imrn/rns096.

peer-reviewed

Marcelo Campos, Matthew Jenssen, Marcus Michelen, and Julian Sahasrabudhe. “A new lower bound for sphere packing.” Submitted 15 December 2023. Source for the unrestricted asymptotic (1/2 - o(1)) n log n 2-n scale. The current arXiv record lists no journal reference. arXiv:2312.10026.

preprint

Nihar Gargava and Maryna Viazovska. “Mean Value for Random Ideal Lattices.” Submitted 22 November 2024, revised 10 December 2025. Source for the factor-two improvement to Venkatesh's sparse ideal-lattice construction. arXiv:2411.14973.

preprint

Henry Cohn and Yufei Zhao. “Sphere packing bounds via spherical codes.” Duke Mathematical Journal 163, 1965-2002, 2014. Source for the Kabatiansky-Levenshtein exponential upper scale and its constant-factor improvement. arXiv:1212.5966; DOI 10.1215/00127094-2738857.

peer-reviewed

Maryna Viazovska. “The sphere packing problem in dimension 8.” Annals of Mathematics 185, 991-1015. Published online 12 April 2017. Proves optimality of the E8 packing. DOI 10.4007/annals.2017.185.3.7.

peer-reviewed

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska. “The sphere packing problem in dimension 24.” Annals of Mathematics 185, 1017-1033. Published online 12 April 2017. Proves optimality of the Leech lattice packing. DOI 10.4007/annals.2017.185.3.8; arXiv:1603.06518.

peer-reviewed

Gabriele Nebe and Neil J. A. Sloane. Catalogue of Lattices, RWTH Aachen. The exact A2, E8, and Leech Gram records used as versioned construction data report determinants and minimal norms that this page recomputes rather than trusts. Retrieved 29 July 2026.

data catalogue