DCQ6: From the DCQ Symplectic Core to FBT Relative-Torus Geometry

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, the basic geometric arena is usually organised locally in a 4+2 form: a four-dimensional effective sector together with a two-dimensional dual-phase or phase-readout 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 reduced symplectic space M4(d) = μ−1 diag(d)/U(1)diag, where μdiag is the diagonal moment map. A central clarification of this revised version is that the relative phase coordinates appearing in the reduced space must not be counted twice. The reduced four-dimensional sector is locally described by two relative action variables and two relative angle variables: (R1,R2, φ1, φ2). The two relative angles (φ1, φ2) are therefore part of M4(d). Separately, when one fixes the three action variables and looks only at the angular readout torus, quotienting the original T3 phase torus by the diagonal phase gives a two-dimensional relative-phase torus T3/U(1)diag ∼= T2. This T2 is a phase-readout skeleton and a precursor of the FBT dual-phase sector; it is not an additional independent complement to M4(d). Thus the present paper does not claim that a new six-dimensional manifold is created by reduction. Rather, it identifies two related structures already latent in the DCQ symplectic core: a four-dimensional Marsden–Weinstein reduced sector and a two-dimensional fixedaction relative-phase readout torus. Together they explain the structural origin of the 4+2 language used later in the FBT framework, while avoiding a literal double-counting of the relative phase variables.