Emergent Quantum Entanglement from Soliton Moduli-Space Topology: From the Hydrodynamic Spacetime Lagrangian to |S|=2√2

We show that the hydrodynamic spacetime theory---a deterministic classical field theory in which spacetime is modelled as a causal viscous medium---contains within it the full structure of quantum entanglement, without any quantum-mechanical postulate. The mechanism is as follows. The theory admits topological solitons as particle-like excitations. In $1\!+\!1$D, we compute the Manton metric on the moduli space of two dark solitons and show it is non-product (g₁₂~-1.3\,d/ξ≠ 0); the Gauss constraint adds a non-decaying long-range inter-region coupling. In $3\!+\!1$D, the solitons are Skyrmions with an $SO(3)$ orientational moduli space whose fundamental group is π₁=Z₂ for odd baryon number. The Finkelstein--Rubinstein theorem then forces half-integer spin upon quantisation, yielding spin-12 fermions. Two well-separated Skyrmions produce a Hilbert space H=C²² that supports the singlet state with correlation E(a,b)=-cos(a-b) and CHSH value |S|=2√2. We show that the four necessary auxiliary conditions---the integrality condition of geometric quantisation, the Born rule, corrections beyond the product ansatz, and the physical interpretation of measurement---are either topologically protected, derivable from geometric quantisation and Gleason's theorem, or reducible to the SU(2) structure of the soliton--detector interaction. No quantum axiom is introduced at any stage.