PAPER-FBT0E: The Continuous Block–Grassmannian Extension of the DCQ1 Carrier

Abstract

DCQ1 constructs a phase-encoded embedding of the six-bit polarity space H6 = {±1}6 into the Grassmannian Gr(3, 6). The present paper does not introduce a second six-dimensional carrier independent of DCQ1. Rather, it extracts the continuous block-geometric completion implicit in the DCQ1 construction. The finite DCQ1 chain H6 −→ μ34⊂ U(1)3 is extended to the compact coherent carrier (CP1)3, and this carrier is then embedded into Gr(3, 6) by the block direct-sum map. Thus FBT0E should be read as a continuous Grassmannian ambient reinterpretation of the DCQ1 carrier, not as a second independent construction of the same six-dimensional geometry. The guiding chain is H6 −→ μ34⊂ U(1)3 ⊂ (CP1)3 ↪→ Gr(3, 6). The construction starts from the DCQ1 block decomposition C6 = V1 ⊕ V2 ⊕ V3, Vi ≃ C2. Then (CP1)3 = P(V1) × P(V2) × P(V3) embeds into Gr(3, 6) by (ℓ1, ℓ2, ℓ3) ↦−→ ℓ1 ⊕ ℓ2 ⊕ ℓ3. Under the Pl¨ucker embedding Gr(3, 6) ↪→ P(∧3C6), this point is represented by [v1 ∧ v2 ∧ v3], 0 ̸= vi ∈ ℓi. Equivalently, the embedding factors as the Segre embedding (CP1)3 ↪→ P(V1 ⊗ V2 ⊗ V3) ⊂ P(∧3C6). The ambient Pl¨ucker carrier has dimension dim ∧3C6 = 20. Relative to the DCQ1 block decomposition, it splits as ∧3C6 = (V1 ⊗ V2 ⊗ V3) ⊕⨁︂ i̸=j(∧2Vi ⊗ Vj), with dimension split 20 = 8 + 12. The 8-dimensional block sector contains the pure coherent DCQ carrier, while the 12-dimensional mixed sector supplies ambient Pl¨ucker directions for exchange, deformation, and boundary interfaces. The same 20-dimensional Pl¨ucker carrier also admits a complementary representationtheoretic decomposition 20 = 10 + 10 after choosing the spinorial structure Spin(6) ≃ SU(4), C6 ≃ ∧2C4. This 10+10 split is not a refinement of the 8+12 block split. The two decompositions arise from different auxiliary structures and serve different downstream readout purposes. The central conclusion is deliberately limited: (CP1)3 ↪→ Gr(3, 6) is the continuous block–Grassmannian extension of the DCQ1 phase-encoded carrier. The paper does not identify the whole Grassmannian with the FBT carrier, does not derive the Standard Model, does not compute scattering amplitudes, and does not prove CKM/PMNS data.