PAPER-FBT0D: Berry–Chern Quantization on the Relative Dual–Phase Torus

This paper clarifies the Berry–Chern quantization mechanism associated with the relative dual-phase torus in the Fracture–Berry–Tension framework. The upstream geometric input comes from FBT0A and FBT0B. FBT0A constructs the canonical coherent-state readout B6 = (CP1)3 with its Hamiltonian T3 phase frame. FBT0B then distinguishes the full phase torus T3, the relative dual-phase quotient T2rel = T3/ΔU(1), and the reduced T2 Liouville fibres appearing after diagonal Marsden–Weinstein reduction. The present paper works with the relative dual-phase torus Σ2 := T2rel = T3/ΔU(1) ∼= T2. The main claim is deliberately modest and precise. If the relative dual-phase torus Σ2 carries a genuine Hermitian Berry line bundleL → Σ2with unitary connection and real Berry curvature Ω, then standard Chern–Weil theory gives 1/2π󰁝Σ2Ω = 〈c1(L), Σ2〉∈ Z. Thus the discreteness of the integrated Berry curvature is not introduced by an operator postulate; it is the first Chern number of a line bundle over the relative dual-phase torus. A central clarification is that this result does not follow from compactness alone. Compactness and orientation provide the stage for global flux integration, but integrality requires the existence of a Berry line bundle whose curvature represents an integral cohomology class. A second clarification is that this Berry-line-bundle Chern number over Σ2 is distinct from the torus-bundle Chern classes studied in FBT0B. The two structures are compatible, but they live at different geometric levels. Finally, the paper explains that the Berry curvature used in the FBT framework may be viewed as the antisymmetric component of a broader quantum-geometric readout tensor. Its real part defines a local distinguishability metric on readout states, while its imaginary part gives the Berry curvature whose integral produces the Chern–holonomy constraint. The paper also explains how fibrewise Berry quantization may be compatible with a global prequantum line bundle on the six-dimensional readout when the total symplectic class is integral. This global extension is treated as an additional compatibility structure, not as a replacement for the fibrewise Berry–Chern theorem.