From Primordial Turbulence to the Structured Vacuum

This work resolves the tension between continuous field theories and discrete vacuum hypotheses by demonstrating that the structured, discrete vacuum is the asymptotic, radiation-exhausted ground-state attractor of non-linear primordial turbulence rather than an axiomatic lattice imposed by hand. Continuous classical Navier–Stokes and Yang–Mills fields undergo a deterministic, four-stage evolutionary cascade that dynamically generates a topologically protected discrete scaffold. In the first stage, the forward enstrophy cascade compresses continuous circulation into thin shear layers that undergo Kelvin–Helmholtz roll-up into tubular filaments. Mutual Crow instabilities between counter-rotating filament pairs then drive periodic collisions and localized pinch-off at the viscous scale, fragmenting the continuous vorticity field into a Lagrangian gas of divergence-free solenoidal vorton dipoles. In the second stage, close-range collisions between vortons trigger Aksman–Novikov reconnection events that alter mutual linking numbers, causing violent impulsive rebounds that radiate excess kinetic and chromomagnetic energy into the far field as acoustic and non-Abelian bremsstrahlung. In the third stage, topologically trivial, unlinked modes cross-diffuse and annihilate completely, dissipating their entire energy into radiation. Under Taylor's principle of selective decay, enstrophy dissipates orders of magnitude faster than conserved topological helicity, culling zero-helicity structures and relaxing the surviving helical configurations into force-free Beltrami states where non-linear vortex stretching extinguishes identically. In the fourth stage, amorphous vorton foams are destabilized by continuous zero-energy deformation modes, requiring the network to eliminate all continuous moduli to achieve permanent stability. By the Baker–Heegner–Stark theorem, eliminating continuous deformation moduli uniquely anchors the vacuum to the terminal class-number-one imaginary quadratic field with discriminant minus one hundred sixty-three. Governed by Euler's lucky prime forty-one, the frozen vortons assemble into the forty-first triangular shell of eight hundred sixty-one nodes, forming the distance-transitive Johnson graph J(42,2) with a uniform valency of eighty and an invariant discrete spectral gap of forty-two. This frozen ground state provides a physical cutoff that guarantees global enstrophy boundedness in Navier–Stokes fluid dynamics and establishes a strictly positive, topologically protected mass gap in Yang–Mills gauge theory.