PAPER-FBT0D: Singular Fibres and Readout Transformations of the Dual–Phase Torus

FBT0B studies the regular moment-map reduction and global torus-fibration structure of the dual-phase sector of the Fracture–Berry–Tension framework. Its main object is the regular relative torus fibration T2 ↪→ XRegπ −→ BReg, together with connection, Chern, holonomy, and monodromy data. The present paper is a technical bridge that asks what happens when this regular fibration is regarded as the regular part of a larger singular completion π : X → B. The new object is the discriminant locus Δsing = B \ BReg, over which the relative T2-fibre may degenerate, one or more cycles may collapse, the action–angle chart may fail, or the integral cycle basis may undergo nontrivial monodromy. The paper formulates the singular-fibre readout transformation principle: regular connectiontype data over BReg may fail to extend smoothly across Δsing, and the failure is read downstream as discrete transition data, phase-locking data, monodromy residue, or thimble/Floer wall-crossing data. In schematic form, regular smooth torus readout =⇒ singular-fibre transformation =⇒ discrete/topological/locked readout. The paper deliberately separates three levels. First, the canonical (CP1)3 product model of FBT0A/FBT0B naturally has toric boundary degenerations, where circle orbits collapse at the poles of the CP1 factors. Second, more general singular completions or perturbed reduced integrable systems may contain focus–focus, saddle–saddle, or other non-toric singular fibres. Third, physical interpretations such as effective mass channels, confinement-like behaviour, particle excitations, or force unification are downstream readouts, not theorems of the present paper. Thus FBT0D does not replace FBT0B. It completes its regular torus picture by introducing the optional singular-completion layer:FBT0B: regular T2 fibration =⇒ FBT0D: singular completion and readout transformation. Its role is to provide a controlled geometric interface for later thimble, Floer, phase-locking, mass-channel, and monodromy-based readout papers. A complementary ambient interpretation is supplied by FBT0E. While FBT0D studies singular completions of the regular dual-phase torus fibration, FBT0E embeds the same coherent carrier (CP1)3 into the Grassmannian Gr(3, 6) as a block–Segre locus. This Grassmannian picture is not required for the definition of singular completion, but it provides an ambient Pl¨ucker–Schubert–cluster language in which selected discriminant interfaces, including A3-type local models, may later be compared.