PAPER-FBT0A: First Principles of the Fracture–Berry–Tension Framework

This paper establishes the minimal geometric backbone of the Fracture–Berry–Tension (FBT) framework. The starting point consists of two structural postulates. The first postulate, FP1, introduces a minimal noncommutative triadic tension core whose cyclic commutation relations have the local algebraic form of an su(2)-type structure. The second postulate, FP2, is an observability principle: a physical readout is not the full tension algebra itself, but a finite-dimensional admissible compression whose semiclassical geometry is obtained through a coherent-state readout. The main improvement of the present version is that the semiclassical readout manifold is no longer left as a formal “semiclassical spectrum.” Instead, it is defined constructively by the coherent-state orbit associated with the phase-response channels selected by the compression. The canonical minimal compact model is Mcoh6 = CP11 × CP12 × CP13 , equipped with the product Fubini–Study symplectic form ω = λ1ω(1)FS + λ2ω(2)FS + λ3ω(3)FS , λk > 0. The three factors supply three independent Berry phase-response channels, and their curvature forms satisfy ω(1)FS ∧ ω(2)FS ∧ ω(3)FS ∕= 0. This gives a constructive realization of the minimal six-dimensional symplectic readout. The paper then shows that on a regular open locus the geometry carries a Hamiltonian T3-phase frame. Removing the unobservable common phase leaves the relative-phase torus U(1)3/ΔU(1) ∼= T2, which is identified as the dual-phase sector of the FBT framework. The local 4+2 readout is therefore understood as a relative T2-phase fibration over a four-dimensional effective sector. A related Marsden–Weinstein reduction at fixed diagonal moment level μ−1Δ (c)/ΔU(1) provides the corresponding four-dimensional reduced effective carrier.Finally, the paper explains how a globally nontrivial dual-phase sector is constrained by connection, curvature, holonomy, and Chern classes. Chern integrality constrains admissible phase sectors, while a genuinely discrete Chern-locking rule requires an additional admissibility condition selecting a discrete holonomy subgroup. The resulting foundational chain is thereforeFP1 + coherent FP2 =⇒ (CP1)3 =⇒ dimRM6 = 6 =⇒ T3 phase frame =⇒ T2 relative phase =⇒ Chern–holonomy