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 ℤ. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group 𝒢 under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to (ℚ/ℤ) × (ℚ/ℤ), with canonical decomposition into Prüfer p-groups and maximal cyclotomic value field ℚab. Extending to coordinate powers yields the polynomial phase group 𝒢poly, classified by integer-valued polynomials Int(ℤ). Adjoining addition yields the group algebra ℂ[𝒢poly], 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 𝒱H, we prove a polar decomposition into scalar envelopes and SU(2) spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime s2 = c2 t2 − |x|2 and the Lorentz group SO+(1,3). We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.
Related
- Dynamical C*-algebras and coarse geometry
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BibTeX
@misc{smith2026wave,
title = {{Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry}},
author = {Terence R. Smith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23233},
eprint = {2609.23233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23233},
}
RIS
TY - UNPB
TI - Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry
AU - Terence R. Smith
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.23233
UR - https://arxiv.org/abs/2609.23233
AB - 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 ℤ. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group 𝒢 under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to (ℚ/ℤ) × (ℚ/ℤ), with canonical decomposition into Prüfer p-groups and maximal cyclotomic value field ℚ^ab. Extending to coordinate powers yields the polynomial phase group 𝒢_poly, classified by integer-valued polynomials Int(ℤ). Adjoining addition yields the group algebra ℂ[𝒢_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 𝒱_H, we prove a polar decomposition into scalar envelopes and SU(2) spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime s^2 = c^2 t^2 − |x|^2 and the Lorentz group SO^+(1,3). We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.
ER -