Dynamical Emergence of Vacuum Structure: Phonon–Roton Spectral Organization and Topological Condensation in Primordial Turbulence

The paper derives the discrete architecture of the vacuum from a scale-free, compressible primordial quantum fluid, demonstrating that spatial order emerges dynamically through spectral softening and topological condensation: Phonon-Roton Spectral Softening: Long-range dipolar velocity fields generated by solenoidal vortex rings induce intermediate-scale softening in the collective excitation spectrum, producing a stable roton minimum that selects a preferred spatial scale while precluding tachyonic vacuum decay. Analytical Scale Hierarchy: The ratio between the emergent lattice spacing and the microscopic vortex core radius is fixed to the inverse fine-structure constant through the self-consistent equilibrium between hydrodynamic circulation tension and conserved axial chiral twist charge, eliminating empirical fine-tuning. Helicity Invariance and Condensation: Total kinetic helicity is proven to be an exact constant of motion under compressible flow with quantum gradient pressure; acoustic radiation emitted during vortex filament reconnections carries away excess energy, driving Taylor relaxation into a stable, force-free Beltrami lattice. Continuous Band Embedding: Embedding the internal 861-node Johnson graph into a three-dimensional Bravais lattice broadens its discrete spectral levels into continuous energy bands, simultaneously generating a gapless acoustic Goldstone branch that defines the speed of light and a protected roton gap that structurally stabilizes the emergent vacuum against collapse.