FBT0E: The Block-Grassmannian Ambient Geometry of the DCQ Carrier

DCQ1 assigns the six-bit polarity space H6 = {±1}6 to three fourth-root phase labels and then to block-decomposable three-planes in C6. FBT0A independently selects the coherent six-real-dimensional carrier Ncoh = (CP1)3. The purpose of FBT0E is to prove that these two constructions are compatible after the required phase-orbit map and block convention are stated explicitly. It does not infer the continuous carrier from the finite set by set-theoretic inclusion. Fix the DCQ1 block decomposition C6 = V1 ⊕ V2 ⊕ V3, V1 = Span{e1, e4}, V2 = Span{e2, e5}, V3 = Span{e3, e6}. For each block choose the phase orbit si(eiθ) = Span(︁ei + eiθei+3)︁∈ P(Vi). The correctly typed compatibility chain is then H6g−−→ μ34↪→ T3 s −−→ (CP1)3 ιblk −−−→ Gr(3, 6). The finite DCQ1 map equals the restriction of this composite to μ34, up to the fixed basis convention. The block map ιblk(ℓ1, ℓ2, ℓ3) = ℓ1 ⊕ ℓ2 ⊕ ℓ3 is a closed algebraic embedding. Under the Plücker embedding it is exactly the Segre embedding into P(V1 ⊗ V2 ⊗ V3). The ambient Plücker vector space decomposes as ∧3C6 = (V1 ⊗ V2 ⊗ V3) ⊕⨁︂i̸=j(∧2Vi ⊗ Vj), with dimension split 20 = 8 + 12. The number 12 counts a complementary linear coordinate sector in the ambient Plücker representation. It is not the dimension of the normal bundle and is not identified with a gauge Lie algebra. The intrinsic tangent audit instead gives dimC T Gr(3, 6) = 9, dimC T(CP1)3 = 3, rankC N = 6. With standard equal normalisation, the Grassmannian Kähler form pulls back to the sum of the three Fubini–Study forms. General independently weighted forms Σ︁i λiω(i)FS require a separately chosen product polarisation and are not all the pullback of one fixed canonical Grassmannian form. The embedding is equivariant for the actual acting relative subtorus Krel ⊂ T3, including the symmetric choice KA2 = ker(z1z2z3). It therefore transports the FBT0B regular action into the Grassmannian ambient space, but it does not create the four-dimensional orbit base or a second reduced carrier. Finally, Plücker, Schubert, and cluster structures are retained only as ambient comparison languages. A vanishing Plücker coordinate is not by itself a singular fibre. An interface with FBT0D requires an extension of the FBT fibration, a matching of the relevant incidence stratum with the nonregular locus, and a transverse Lefschetz-type local model. The paper therefore separates the proved block geometry from conditional singular, cluster, and representation-theoretic interfaces.