PAPER-DCQ2: Emergent Dimensionality, Phase–Orbit Geometry, and Morse–Thimble Structure

Paper DCQ1 constructed a phase-encoded embedding of the six-bit configuration space

H6 = {±1}6

into the complex Grassmannian Gr(3, 6), together with metric compatibility, a finite phasesector embedding

H6 −→ μ34 ⊂ U(1)3,

and two distinct carrier layers: the 20-dimensional Pluecker/Fock carrier Λ3(C6), and the separate 24-dimensional pure Bose–Fermi readout carrier

RBF = Sym3(C4) ⊕ Λ3(C4).

The present paper develops the next geometric layer of the DCQ programme. First, the continuous completion of the three bit-pair phase blocks gives a six-dimensional phase-orbit submanifold

N ≃ (CP1)3 ⊂ Gr(3, 6),

containing the 64 embedded discrete code states as a finite μ34-labelled subset. Second, the determinant Berry line bundle restricts to N with curvature class

󰀗Ω2π󰀘N = (1, 1, 1) ∈ H2((CP1)3; Z),

so each CP1 factor carries one unit of Berry–Chern flux. Third, a diagonal U(1) Marsden–Weinstein reduction of N gives an effective four-dimensional symplectic quotient
Cδ = μ−1   diag(c)/U(1)diag, dimR Cδ = 4.

Finally, the paper formulates an adapted Morse-theoretic structure in which the 64 discrete states are treated as preferred minima of a smooth potential on N. The associated Picard–Lefschetz discussion is presented as a formal complexified thimble ansatz, not as a complete analytic construction of complex integration cycles. The result is a pre-dynamical geometric framework linking finite phase data, Berry–Chern topology, symplectic reduction, and semiclassical expansion.