This paper extracts and reformulates the geometric core underlying the early papers of the Discrete–Continuous–Quantum (DCQ) correspondence. Its main claim is that the natural continuous completion of the phase-encoded binary construction is not merely a three-torus of phase angles, but a compact real six-dimensional symplectic manifold N ≃ (CP1)3. Each factor represents the projective completion of the two-dimensional complex carrier associated with one binary pair, and the total product carries the canonical product K¨ahler form. Within this six-dimensional symplectic manifold, the previously studied pure-phase family appears as a distinguished Lagrangian torus T3phase ⊂ (CP1)3, while the 64 binary configurations form a finite distinguished subset obtained by selecting four special phase points on each factor. Thus the correct geometric chain is H6 ∼= Q ⊂ T3phase ⊂ (CP1)3 ⊂ Gr(3, 6). We show that the symplectic form is integral, compute the total symplectic volume (2π)3, describe the natural Hamiltonian T3-action and its momentum polytope, and explain how diagonal phase reduction produces a four-dimensional Marsden–Weinstein reduced space. We also distinguish this four-dimensional reduced sector from the fixed-action relative-phase torus: the latter is a two-dimensional phase-readout skeleton, not an additional independent factor of the reduced four-dimensional quotient. Finally, we clarify the compatibility of this six-dimensional completion with the Grassmannian embedding of DCQ1. The prequantum line bundle on (CP1)3 is the pullback of the determinant-line geometry underlying the Berry–Chern construction on Gr(3, 6). The paper does not claim that naive geometric quantization of (CP1)3 alone reproduces all later representation-theoretic or readout structures. Rather, it establishes the six-dimensional symplectic core as the common geometric stage on which the discrete, continuous, reduced, and prequantum layers of the DCQ correspondence are organized.
ZHAI Xingyun (Tue,) studied this question.