On the Geometric and Spectral Inevitability of Ultrahyperbolic (3 + 3) Spacetime: Mathematical Foundations, Temporal Eigenvalue Spectra, and Unified Gauge Phenomenology

The geometric formulation of fundamental physical interactions has traditionally presumed a pseudo-Riemannian manifold of Lorentzian signature (1, 3). However, deep-seated math- ematical obstructions—such as non-renormalizable ultraviolet divergences in perturbative quantum gravity, the arbitrary multiplicity of the three Standard Model fermion gener- ations, and the ad-hoc introduction of chiral parity violation—strongly suggest that the temporal sector possesses non-trivial topology and dimensionality. In this work, we analyze the mathematical consistency and phenomenological potency of an ultrahyperbolic manifold M3,3 ∼= R3 T × R3 X equipped with the canonical pseudo-Euclidean metric ds2 = 3 X a=1 dt2 a − 3 X i=1 dx2 i = dτ 2 −dσ2. Tracing the historical trajectory of multi-temporal geometries—from early higher-dimensional Kaluza–Klein formulations and Dirac-Bars two-time gauge physics to recent spectral developments— we show that three temporal dimensions (DT = 3) form a critical algebraic threshold. Draw- ing upon and expanding the framework established by Kletetschka (2025), we demonstrate that the temporal Laplace–Beltrami operator ∆T = P3 a=1 ∂2/∂t2 a under the temporal sym- metry group SO(3, T) yields discrete mass eigenvalue spectra ∆T ψn = m2 nψn of the form mn = m0 exp(αn/β), generating exact generational mass ratios (1 : 4.5 : 21.0) across leptons and quarks without requiring arbitrary Yukawa hierarchies. Furthermore, we explore the curious mathematical properties of the (3+3) signature: (i) the exterior Clifford algebra Cℓ3,3(R) ∼= Mat(8, R) provides a purely geometric origin for the chiral current Jµ = ¯ψγµ(1 −γ5)ψ via the volume form γ5 = it1t2t3γx, naturally encoding maximal parity violation and electroweak mixing angles (sin2 θ12 = 0.307, sin2 θ23 = 0.546, sin2 θ13 = 0.0220); (ii) the extended multi-frequency Green’s function G(k, ω) = [k2 + ω2

@article{deceuster2026_73-on-the-geometric-,
  title = {On the Geometric and Spectral Inevitability of Ultrahyperbolic (3 + 3) Spacetime: Mathematical Foundations, Temporal Eigenvalue Spectra, and Unified Gauge Phenomenology},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.21939787},
  url = {https://peterdeceuster.uk/papers/73-on-the-geometric-and-spectral-inevitability-of-ultrahyperbolic},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/exp1.pdf}
}