Quantization and Minimal Matter Sector in the Harmonic Universe Model: A Modal Tower of Resonances

We develop a minimal quantum-effective formulation of the Harmonic Universe Model by quantizing a five-dimensional field theory defined on a product space M 4 × K, where the internal dimension K is taken either as a finite interval X ξ = [−`, `] (with explicit boundary conditions) or as a compact circle S 1 (with periodicity). We show that both choices constitute consistent realizations of the same modal framework, differing only in the spectral quantization conditions and degeneracy structure. Linearizing around a stable vacuum and performing a mode decomposition in ξ, we obtain an effective four-dimensional descriptionin which the eigenvalue problem on K generates a discrete tower of assive excitations (a resonance tower) with model-dependent selection rules determined by overlap integrals along the internal coordinate. We then introduce a minimal fermionic sector in five dimensions and derive its four-dimensional reduction, emphasizing the conditions under which chiral, localized low-energy modes arise in the interval (or orbifold-like) realization. A minimal vector/gauge sector is discussed at the effective level, sufficient to define consistent couplings between fermionic modes and the resonance tower within a controlled cutoff Λ. The resultingconstruction yields a compact set of falsifiable predictions: (i) the existence and spacing of resonance states controlled by ` −1 or R −1 , (ii) characteristic coupling hierarchies induced by internal-wavefunction overlaps, and (iii) benchmark signatures for laboratory and high-energy probes. This paper provides the minimal quantum-and-matter completion of the classical modal backbone and establishes the experimental handles needed for subsequent cosmological and celestial-mechanics extensions.