PAPER-FBT01B: Curvature Readout and Horizontal–Vertical Coupling

This paper revises the curvature analysis of the Fracture–Berry–Tension (FBT) carrier geometry in light of the updated foundational structure established in FBT0A and FBT0B. In the current formulation, the minimal six-dimensional carrier is the coherent-state symplectic readout B6 ≃ (CP1)3, with product Fubini–Study symplectic form ΩB = λ1Ω(1)FS + λ2Ω(2)FS + λ3Ω(3)FS . FBT0B further distinguishes the full Hamiltonian T3-phase frame, the relative dual-phase torus T2rel = T3/ΔU(1), and the four-dimensional diagonal Marsden–Weinstein reduced carrier Mred 4,c = μ−1Δ (c)/ΔU(1). The purpose of the present paper is to clarify the curvature readouts that arise on top of this geometry. A central point is that the word “curvature” refers to several related but distinct objects: the parent coherent-state symplectic form ΩB, the curvature FA of a relative T2-bundle connection, and the effective dual-phase Berry or prequantum curvature Ωeff_phase on the reduced phase torus. These objects should not be identified without specifying the geometric level on which they live. Relative to a chosen local 4+2 readout splitting TB6 = H ⊕ V, the parent symplectic form admits a horizontal–vertical decomposition ΩB = ωH + η + ωV , where η ∈ H∗ ∧ V ∗ is the mixed component. In the undeformed product-factor splitting of (CP1)3, this mixed component may vanish. In the relative-phase readout splitting associated with FBT0B, however, mixed action–phase terms generally appear. Thus the mixed channel is not a universal topological invariant; it is a readout-splitting dependent measure of horizontal–vertical coupling. The paper therefore replaces the earlier “three-κ system” by a more cautious scalar readout convention: the global Liouville capacity Kvol =󰁝B613!Ω∧3B , the reduced phase flux Kphase =1/2π󰁝Σ2Ωeffphase, and the local mixed density 1/2η ∧ η ∧ ωH = ρmix13!Ω∧3B . The first two are global scalar readouts, while ρmix is a local, connection-dependent density. This distinction provides a more stable geometric foundation for later scale-dependent or effective-field interpretations.