Complex Hopf Fibrations as Prequantization Geometries: Equivariant Spectra, Higher-Form Curl, and Controlled Kaluza–Klein Reduction

This paper develops the complex Hopf fibrations S1→S2n+1→CPnS¹ S^2n+1 CPⁿS1→S2n+1→CPn as rigorous prequantization geometries at the intersection of bundle topology, contact and Kähler geometry, representation theory, spectral analysis, geometric quantization, and controlled Kaluza–Klein reduction. With explicit normalizations, the paper derives the Chern–Weil and Boothby–Wang structures of the Hopf bundle; identifies circle-weight sectors with sections of the line bundles O (k) O (k) O (k) ; computes their complete scalar spectra and multiplicities, including canonical Berger deformations; and relates the lowest spectral band to Borel–Weil theory and the Spincᶜc index. It also distinguishes finite-dimensional bundle automorphisms from local gauge symmetries and derives the representation congruences associated with the central quotient U (N) ≅ (SU (N) ×U (1) ) /μNU (N) (SU (N) U (1) ) /NU (N) ≅ (SU (N) ×U (1) ) /μN. Beginning with the Hopf Beltrami field on S3S³S3, the paper constructs a parity-sensitive theory of higher-form curl on odd-dimensional manifolds and exhibits canonical middle-degree eigenforms on spheres of dimension 4r+34r+34r+3. For product compactifications, it derives exact scalar and differential-form Kaluza–Klein towers and a heat-kernel factorization supporting mathematically well-defined one-loop zeta determinants for positive Euclidean Hessians. It further formulates consistency conditions for chiral extensions, including anomaly tests, and discusses KAM-stable invariant tori, Bott-integrable Reeb flows, and hyperbolic sets as controlled dynamical observables. A central methodological feature is the explicit separation of intrinsic geometric theorems from model-dependent field-theoretic consequences. The results establish a constructive foundation for using Hopf geometry in mathematical physics without attributing unsupported particle-physics predictions to topology alone.