PAPER-FBT0B: Moment–Map Reduction and Global Torus Structure of the Dual–Phase Sector

FBT0A establishes the first-principle and coherent-state origin of the minimal six-dimensional readout geometry of the Fracture–Berry–Tension framework. In its canonical compact realization, the readout isMcoh6 = (CP1)3, equipped with a product Fubini–Study symplectic form and a Hamiltonian T3-phase frame. The present paper is a technical companion to FBT0A. It does not repeat the firstprinciple derivation of the six-dimensional readout. Instead, it studies the moment-map reduction, relative-phase quotient, Liouville torus structure, and global torus fibration associated with the regular dual-phase sector. For brevity, we denote the coherent-state readout (CP1)3 established in FBT0A by B6 (where B stands for “base readout”). This notationavoids confusion with other six-dimensional manifolds that may appear in later papers. A central clarification of this paper is the distinction between three related but mathematically different torus structures: T3 as the full Liouville phase fibre on the six-dimensional readout, T2rel = T3/ΔU(1) as the relative dual-phase torus, and T2 as the Liouville fibre of a rank-two integrable system on a four-dimensional reduced carrier. This distinction removes a common dimension-counting ambiguity: a rank-two moment map on a six-dimensional manifold has four-dimensional regular level sets, not two-dimensional tori. A two-dimensional Liouville torus arises only after diagonal Marsden–Weinstein reduction to a four-dimensional effective carrier, or equivalently as the quotient of a full T3 phase fibre by the diagonal common phase. The main chain established in this paper isB6 = (CP1)3 =⇒ T3 Hamiltonian phase frame =⇒ T2rel = T3/ΔU(1) =⇒ μ−1Δ (c)/ΔU(1) =⇒ rank-two Liouville T2 We then formulate the global principal T2-bundle structure of the relative-phase sector and describe the possible monodromy representation ρ : π1(Breg) −→ SL(2, Z), which records the twisting of the integral homology lattice of the torus fibres. Thus FBT0B provides the symplectic-reduction and global-torus counterpart to FBT0A: FBT0A identifies the dual-phase torus as the minimal relative phase readout, while FBT0B explains its Hamiltonian, reduced, and globally twisted geometry.