On the Hidden Pascal Symmetry and Moment Constraints of Vector Representatives in Quebbemann's 64-Dimensional Lattice

Abstract

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 {\pi}. 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 {\Theta}-series are naturally filtered by these binomial structures.