PAPER-DCQ6: From (CP1)3 to a Canonical 4+2 Geometric Decomposition: Symplectic Reduction and Relative-Phase Structure toward the FBT Arena

In earlier papers of the Discrete–Continuous–Quantum (DCQ) programme, the phaseencoded embedding of the six-bit configuration space into the Grassmannian Gr(3, 6) led to a compact six-dimensional symplectic manifold (N, ω) ≃󰀃 CP1󰀄3 equipped with a Hamiltonian T3 action. In the Fracture–Berry–Tension (FBT) framework, however, the basic geometric arena is usually formulated as a six-dimensional symplectic structure with a distinguished 4+2 organisation: a four-dimensional external sector together with a two-dimensional dual-phase sector. The aim of the present paper is to provide a mathematically controlled bridge between these two descriptions. We show that the product symplectic manifold 󰀃CP1󰀄3 naturally carries a diagonal phase redundancy. Reducing by the corresponding diagonal U(1) through the Marsden–Weinstein procedure produces a four-dimensional symplectic reduced space M4(d) = μ−1 diag(d)/U(1)diag, where μdiag is the diagonal moment map. The natural reduced coordinates are two relative action variables and two relative angle variables, which endow M4(d) with a canonical symplectic form. The original six-dimensional manifold N can then be reorganised, at least locally and on suitable regular strata, into a 4+2 geometric decomposition consisting of this fourdimensional reduced sector together with a two-dimensional relative-phase sector. In this sense, the present paper does not create a six-dimensional space by reduction; rather, it identifies within the DCQ symplectic manifold the precise geometric ingredients from which the FBT 4+2 arena arises. We further clarify the status of the relative-phase torus, distinguish it from the fully curvature-coupled dual-phase sector used in later FBT papers, and explain in what sense the present construction supplies the precursor geometry of that sector. The result is a rigorous route from the DCQ six-dimensional symplectic core to the structural decomposition employed throughout the FBT framework.