DCQ4: The Six-Dimensional Symplectic Core of the DCQ Correspondence

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.