Weber's conjecture (1886) governs three aspects of lattice-based cryptography: the solvability of the Principal Ideal Problem, the freeness of modules over rings of integers, and the tightness of worst-case-to-average-case reductions in Ring-LWE (R-LWE) and Module-LWE (MLWE). Existing verifications for $k \ge 9$ rely on Generalized Riemann Hypothesis (GRH). In this paper, we present the first unconditional proof for $k \le 12$. Our method combines the Fukuda-Komatsu computational sieve, inductive structure of the cyclotomic $\mathbb{Z}_2$-tower, and Herbrand's theorem.

翻译：韦伯猜想（1886）支配着基于格的密码学的三个方面：主理想问题的可解性、整数环上模的自由性，以及环LWE（R-LWE）和模LWE（MLWE）中最坏情况到平均情况归约的紧致性。现有对$k \ge 9$的验证依赖于广义黎曼假设（GRH）。本文首次给出了$k \le 12$的无条件证明。我们的方法结合了Fukuda-Komatsu计算筛法、分圆$\mathbb{Z}_2$-塔的归纳结构以及Herbrand定理。