PAPER-FBT0B: Global Relative–Phase Torus Geometry of the Coherent Carrier

FBT0A establishes the coherent six-dimensional carrier Bcoh6 = (CP1)3 and its regular Hamiltonian three-phase frame. FBT0B v4.4 develops the global relative-phase geometry needed by the active four-dimensional FBT architecture. The revision removes the diagonal Marsden–Weinstein branch from the active framework. The FBT four-dimensional carrier is now unique: it is the orbit base of an actual relative two-torus action, Krel ,→ X(6)Regπ −→ B(4)Reg. The abstract contrast torus and the acting torus are distinguished. The intrinsic phasecontrast group is Qrel := T3ph/ΔU(1), but this quotient group does not automatically act on the original six-dimensional carrier because the diagonal circle is not the kernel of the Hamiltonian T3-action. An action requires either a marked integral splitting or an explicitly chosen relative subtorus. On the permutation symmetric branch the natural choice is KA2 = {(z1, z2, z3) ∈ U(1)3 : z1z2z3 = 1}. It is an actual A2-lattice two-torus, but the comparison map KA2 → Qrel has kernel KA2 ∩ ΔU(1) ∼= Z/3. Thus KA2/μ3 ≃ Qrel; the familiar relative-phase quotient and the permutation-symmetric acting torus are isogenous rather than canonically identical. A theorem shows that no S3-equivariant integral direct splitting of the phase lattice exists. On every regular branch on which the selected Krel-action is free and proper, the orbit space B(4)Reg is a smooth four-manifold and the projection is a principal two-torus bundle. A connection gives the covariant horizontal–vertical splitting required downstream by FBT01A, FBT27A and FBT33A. This 4 + 2 organisation is bundle-theoretic: it is not produced by a bare coordinate regrouping, is not generally a global product, and does not by itself identify the base with physical Lorentzian spacetime. The symplectic audit is explicit. Relative torus orbits are isotropic for the parent Fubini–Study form, while contraction of that form with a relative fundamental vector field is generically nonzero. Consequently the parent symplectic form is not basic and does not descend to a nondegenerate four-dimensional symplectic form on B(4)Reg. On the standard toric branch the quotient instead carries a natural rank-two Poisson structure, with two relative action variables as Casimirs. The downstream Lorentzian structure is therefore a separate construction of FBT08A, not the continuation of an unused symplectic reduction. Finally, principal and affine torus geometries are separated. Translation- valued transition functions define a principal Krel-bundle and ordinary Chern data. Nontrivial cycle-frame monodromy requires the enlarged structure group Krel ⋊ SL(2,Z), an affine torus bundle, a cycle local system and twisted Chern class. A principal reduction exists precisely when the linear monodromy is trivial. This separation supplies the type-correct global input for the local-system formulation used in FBT33A. The revision also states the geometric reason that the affine branch can survive the reduction to four dimensions. The absence of an absolute common phase produces the contrast quotient, whereas the absence of an absolute integral basis for the two relative cycles produces the possible SL(2,Z) frame cocycle. Quotienting the fibre removes its pointwise phase coordinates but does not remove the cocycle by which local cycle frames are compared. The latter descends to the four-dimensional base as the integral cycle local system Lcyc :=􀀀R1(ϖρ)∗Z∨≃ {H1(T2x ;Z)}x∈B(4)Reg, ϖρ : Tρ → B(4)Reg. Accordingly, the principal branch is the globally cycle-framed sector ρ = I, while the affine branch is an optional topologically nontrivial sector with ρ ̸= I. Nontrivial affine monodromy is not forced by the product carrier or by common-phase unobservability alone; it requires nontrivial global gluing, a non-simply-connected regular base, or an optional singular completion of the kind studied in FBT0D. The active orbit projection that defines B(4)Reg remains the principal Krel-bundle above. A nontrivial affine object over the same base must be typed separately as an optional torus-family or cycle-readout layer Tρ → B(4)Reg; it is not a second construction of the fourdimensional base and cannot be identified with the principal orbit bundle unless its linear monodromy is trivial. Version v4.4 completes the canonical-branch monodromy audit. If a closed loop crosses affine transition maps y 7→ Miy + ti in order, its linear monodromy is the ordered product Mγ = MN · · ·M1. On the canonical product branch every Mi = I, because the global A2 cycle frame is fixed. Therefore ρcan(γ) = I for every closed loop γ. This is compatible with nontrivial torus-valued connection holonomy and nonzero principal Chern data; neither produces linear cycle monodromy. In particular, the canonical product branch cannot itself be the closed-loop boundary of one isolated capped A3 block, whose FBT0D boundary monodromy is conjugate to Φ±1A3 with ΦA3 =􀀀 0 1−1 0̸= I. The corresponding nontrivial affine branch is boundary-selected additional gluing data, not hidden monodromy of the canonical product carrier. The active emergence chain is therefore X(6)Reg/KrelFBT08A−−−−−−−→ (B(4)Reg, [g]Lor) FBT15A←−−−−−−−loc.Ws FBT33A←−−−−−−− S(6) 7→ S(4). FBT0B supplies the single global orbit-base provenance; it does not supply a second fourdimensional carrier.