Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry

Abstract

We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.