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

This paper establishes the minimal geometric backbone of the Fracture–Berry–Tension (FBT) framework. The construction begins from two structural postulates. FP1 introduces a minimal noncommutative triadic tension core whose cyclic commutation relations have the local algebraic form of an su(2)-type structure. FP2 states that physical observability is realised through an admissible finite-dimensional coherent-state compression rather than through the full tension algebra itself. The semiclassical readout manifold is defined constructively as the coherent-state orbit associated with three independent phase-response channels. The canonical minimal compact realisation 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 curvature channels satisfy ω(1)FS ∧ ω(2)FS ∧ ω(3)FS ̸= 0, which forces the minimal real readout dimension to be six. On the regular toric locus, the geometry carries a Hamiltonian T3-phase frame. The three local phases are not individually intrinsic: a simultaneous diagonal shift changes the common phase reference while preserving all phase contrasts. The intrinsically readable phase content is therefore the abstract contrast quotient Qrel = U(1)3/ΔU(1) ∼=T2, which is identified as the minimal dual-phase readout sector. The present version deliberately does not claim that FBT0A derives a physical 4+2 decomposition or a four-dimensional spacetime. It is restricted to the first-principle origin of the six-dimensional coherent carrier, the abstract relative-phase quotient, and the covariance and topological constraints of local phase readout. The abstract quotient Qrel does not automatically act on the original carrier. Whenever an actual relative subtorus Krel ⊂ T3 is selected and acts freely and properly on a regular locus, its quotient defines a four-dimensional orbit space Brel =Mfree6 /Krel. This orbit space is used only as the base of a conditional principal Krel-bundle. It is not identified in this paper with physical spacetime. Connection, curvature, holonomy, and Chern classes then encode the globally consistent content of locally represented phase data. Chern integrality constrains admissible phase sectors, while discrete phase locking requires an additional admissibility rule selecting a discrete holonomy subgroup. The resulting foundational chain is FP1 + coherent FP2 =⇒ (CP1)3 =⇒ dimRM6 = 6=⇒ T3 phase frame =⇒ Qrel=⇒ readout covariance and Chern–holonomy constraints. FBT0B develops the actual relative-torus action, four-dimensional orbit base, connection, and global torus geometry; FBT0D studies conditional singular completions and discriminant loci; and FBT0E proves the typed DCQ1 phase-orbit and block–Grassmannian compatibility. FBT0A supplies the first-principle coherent-readout origin on which those companion layers are built.