FBT0D: Singular Fibres and Readout Transformations of the Relative–Phase Torus

FBT0D studies singular completions of the single active relative–phase bundle fixed by FBT0B v4.4, Krel ,→ X(6)Regπ −→ B(4)Reg, and keeps the strict principal branch distinct from affine torus families with nontrivial integral cycle monodromy. Generic codimension–two discriminant points are treated by an A1 Lefschetz slice. A special A3 collision requires instead a holomorphic transverse map IFBT = ατ ◦ πD + β whose complex Hessian has rank one and whose Jacobian algebra is C{ϵ}/(ϵ3). These intrinsic data give the normal form IFBT ∼ x4/4 + w2/2 + constant. After capping the genus–one, two–boundary–component Milnor fibre, a distinguished A3 chain descends to (a, b,±a). Its oriented boundary monodromy is ΦA3 = TaTbTa =0 1−1 0, Φ2 A3 = −I, Φ4A3 = I. Hence one isolated capped A3 block is incompatible with the complete puncture loop of the canonical product/principal branch, whose FBT0B v4.4 audit gives ρcan = I. It belongs instead to the affine mapping–torus branch with linear part Φ±1 A3 . The main new analytic result is a closed collar certificate. The Wang sequence gives H1(EΦ;R) = R[dθ], H2(EΦ;R) = R[Ωk], H2(EΦ;Z) = Z ⊕ Z/2Z. Equal fibre area removes the real cohomology obstruction; matched real or torus–valued affine flux removes the fibre-period component. After absorbing the remaining base primitive, the difference of the marked collar forms is db Λ, with zero marked fibre periods. The cutoff deformation uses the full identity d(χbΛ) = χ dbΛ + dχ ∧b Λ. A large exact base-area term is localised on the cutoff transition region, where it supplies the uniform Thurston nondegeneracy estimate; it vanishes on the Milnor core and on the final common overlap. On that overlap the removal path is explicitly split, Ωk +􀀀κ + (1 − s)Kdr ∧ dθ, 0 ≤ s ≤ 1, and is symplectic because κ > 0. Thus G1–G3—marked monodromy, area and real affine flux—derive the relative–Moser collar certificate; no separate G4 hypothesis remains for the local filling. The integral Z/2 torsion is invisible to this de Rham argument and is retained in the optional prequantum line–with–connection condition. DCQ12 supplies a separate upstream selection interface. On a selected DCQ8 label sheet with protected nonzero winding, its classifying chain Qrel4χ±i −−→ Z/4 m7→ΦmA3 −−−−−→ SL(2,Z) and the DCQ11 minimum |w| = 1 select ρDCQ(∂D) = Φ±1A3 . FBT0D then turns that boundary class into the unique marked affine-bundle class and its symplectic A3 filling. If the exterior target remains ρcan = I, cocycle closure forces the inverse compensator, whose global realisations belong to FBT07D. Because DCQ12 also proves that no nonzero primitive character is fully S3-invariant, this result does not claim an unpolarised canonical product branch spontaneously chooses one sign or changes its bundle type.