Paper 2026/2846

The isodual constant of the LLL potential: exact values and structure

Abstract

The LLL potential $\Psi^2(B)=\prod_k\det\mathrm{Gram}(b_1,\dots,b_k)$, minimized over the bases of a lattice $L$ and normalized by its determinant, defines an isodual lattice invariant $\eta(L)$. Its supremum $\eta_n$ over $n$-dimensional lattices, introduced in ePrint 2007/105, is the staircase case of the generalized Hermite constants of Watanabe and Meyer, for which explicit values were known only for $n\le4$. We prove that $\eta_5=6$, attained only by $D_5$ and its dual: the flag analogue of Korkine and Zolotarev's value of $\gamma_5$. The natural reduction to four-dimensional lattices, through a function $\Phi_5$, does not suffice: the global maximum of $\Phi_5$ is $2^{16}/(3^4\cdot5^3)\approx6.4727$, attained not at $D_4$ but at an even $135$-modular lattice $M_*$, which we have not found in the literature. A second stage, over the lifts of the remaining region, completes the proof; both stages are computer-assisted, with about $9\times10^9$ leaves, each re-proved in interval arithmetic by a separately written checker; code and data are public. We also prove $\eta_8=2304$, attained by $E_8$, and show that $E_6$, $E_7$, $E_8$ and their duals are strict local maxima of $\eta$ (for $n=6,7$ these are the conjectured extremal lattices). Asymptotically, $\log\eta_n=(\frac14+o(1))\,n^2\log n$. For lattice reduction we introduce the $\Psi$-Hermite constant $\mu_m$, the largest first Gram–Schmidt norm in a potential-minimal basis of an $m$-dimensional lattice, and bound it through $\eta_m$. This gives $\mu_m=\gamma_m$, Hermite's constant, for $m\le5$, and $\mu_m\le\exp(O(\log^2 m))$ in general.

Note: Follow-up to ePrint 2007/105. Code: https://github.com/nickhg/isodual_constant (Zenodo DOI 10.5281/zenodo.23743844). Certificate data: https://huggingface.co/datasets/nickhg1971/isodual-constant-certificates (DOI 10.57967/hf/11260).