This paper develops the first intrinsic curvature analysis of the Fracture–Berry–Tension (FBT) carrier geometry established in the foundational papers FBT0A and FBT0B 1, 2. Starting from the six-dimensional symplectic manifold (B6,Ω) endowed with a principal T2-bundle structure over a four-dimensional base M4, we study the decomposition of the total observable curvature relative to the connection-induced splitting TB6 = H ⊕ V. The first main result is that the observable curvature admits a natural three-channel decomposition Ω = Ωspace + Ωcoupl + Ωphase, corresponding respectively to the horizontal, mixed, and vertical sectors of the 4+2 geometry. The second main result is that this decomposition induces a natural three-component scalar readout: the total Liouville volume κ0, the reduced dual-phase area κ1, and the local mixed-channel density κ2. Here κ0 and κ1 are global symplectic invariants, whereas κ2 is a local scalar density measuring the relative strength of horizontal–vertical coupling. The third main result is that κ2 is the unique local scalar quantity canonically extracted from the mixed curvature channel relative to the Liouville form. This makes it the natural geometric seed of scale-dependent effective readout. Finally, we show that Liouville flow provides the natural symplectic notion of scale variation for the mixed channel. In this sense, scale dependence in the present framework is read not from metric conformal rescaling but from the behaviour of the mixed-channel density relative to Liouville dilation. The paper is geometric in scope. It does not assume a specific mass-generation model, nor does it rely on the algebraic gate interpretation developed separately in FBT01B4. Its purpose is to establish the curvature-channel decomposition and the three-κ system as the basic scalar symplectic readout of the minimal FBT geometry.
ZHAI Xingyun (Sun,) studied this question.