Quebbemann 64维格中向量代表的隐藏帕斯卡对称性与矩约束
On the Hidden Pascal Symmetry and Moment Constraints of Vector Representatives in Quebbemann's 64-Dimensional Lattice
AI总结：
该研究针对Quebbemann 64维格的移位向量，揭示其隐藏帕斯卡对称性，通过放宽边界得到闭式核心，生成范数平方为20的最优移位向量，并关联Craig格的重复差分范式。
AI中文摘要：
本文研究格的A类构造的陪集代表元（移位向量）所遵循的底层代数与组合结构，重点关注J.H. Conway和N.J.A. Sloane的经典著作《球填充、格与群》第8章第3段给出的方程（14）和（15），这组对偶方程定义了解析生成64维Quebbemann格（Q64）的边界条件。我们证明，在算术或几何级数约束下求非零解时，由于π的超越性，会出现结构坍缩为平凡零向量的情况；相反，通过将这些边界放宽至唯一坐标配置，我们揭示了由帕斯卡三角交替系数支配的精确闭式代数核心。此外，我们通过连续到离散投影实现能量最小化模型，得到了欧几里得范数严格为整数的最优移位向量，其范数平方||z||²=20.000000。最后，我们将该公式与Craig格Aₙₘ的重复差分范式建立关联，展示了Θ级数的谱分量如何被这些二项式结构自然过滤。
英文摘要：
In this paper, we investigate the underlying algebraic and combinatorial structures governing the coset representatives (shift vectors) for Construction A of lattices, with a particular focus on equations (14) and (15) presented in Paragraph 3, Chapter 8 of the seminal work by J.H. Conway and N.J.A. Sloane, "Sphere Packings, Lattices and Groups". These dual equations define the boundary conditions for the analytical generation of the 64-dimensional Quebbemann lattice (Q64). We prove that seeking non-zero solutions constrained by arithmetic or geometric progressions yields a structural collapse to the trivial zero vector due to the transcendental nature of π. Conversely, by relaxing these bounds to unique coordinate configurations, we uncover an exact, closed-form algebraic core governed by the alternating coefficients of the Pascal triangle. Furthermore, we implement an energy-minimization model via continuous-to-discrete projection that yields an optimal shift vector with a strictly integer Euclidean norm ||z||^2 = 20.000000. Finally, we bridge this formulation to the Repeated Differences paradigm of Craig's lattices Anm, showing how the spectral components of the Θ-series are naturally filtered by these binomial structures.
