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

This paper establishes the minimal geometric backbone of the Fracture–Berry–Tension (FBT) framework. Starting from two structural postulates—an underlying noncommutative tension algebra (FP1) and a minimal observability principle (FP2)—we derive the smallest admissible geometric readout manifold as a six-dimensional symplectic manifold. The paper has four precise aims. First, we show that the minimal geometric readout compatible with three independent phase-response channels is necessarily six-dimensional. Second, we show that on a regular open locus this geometry admits a canonical local 4+2 decomposition into a four-dimensional effective sector and a two-dimensional phase sector. Third, we identify the phase sector as the relative-phase quotient U(1)3/ΔU(1) ∼= T2, thus explaining the emergence of the dual-phase torus. Fourth, we show that the global nontriviality of this torus sector is encoded by Chern classes and holonomy, yielding a topological locking principle for admissible physical phases. At the same time, the paper clarifies the meaning of the three words in the name Fracture–Berry–Tension. Tension refers to the minimal noncommutative triadic core of the theory; Berry refers to the geometric phase data carried by the dual-phase sector; and Fracture refers to the fact that observability is intrinsically local, so that the full geometry becomes accessible only through local 4+2 readout charts rather than through a single global decomposition. The purpose of the present paper is therefore not to derive the full physical content of the FBT programme, but to establish its minimal geometric architecture: FP1 + FP2 =⇒ (M6, ω) =⇒ local 4+2 =⇒ Σ2 ≃ T2 =⇒ Chern–holonomy locking.