PAPER-DCQ5: The Six-Dimensional Symplectic Core of the DCQ Correspondence

This paper extracts and reformulates the geometric core underlying the preceding DCQ trilogy. 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 a two-dimensional amplitude–phase sphere 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, while the 64 binary configurations form a finite distinguished subset obtained by selecting four special phase points on each factor. We show that the symplectic form is integral, compute the total symplectic volume (2π)3, describe the natural Hamiltonian torus action and its momentum polytope, and explain how diagonal phase reduction leads to an emergent four-dimensional reduced space. We also clarify the compatibility of this six-dimensional completion with the Grassmannian embedding of DCQ1 and show that its prequantum line bundle is precisely the pullback of the determinantline geometry underlying the Berry curvature construction. The paper does not claim that naive geometric quantization of (CP1)3 alone reproduces all later quantum representationtheoretic 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.