In the C.O.R.E. framework (CUGE + REFORM + ZEUS) the universe is static, Euclidean, and eternal. The filamentary cosmic web exists in perpetual dynamical equilibrium as a self-regulating recycling machine. Density contrasts \(\delta(\mathbf{x})\) and the associated peculiar velocity field \(\mathbf{v}(\mathbf{x})\) are statistically stationary on cosmic scales. We derive the observed redshift-space distortion parameter \(f\) and the combination \(f\sigma_8\) directly from the steady-state peculiar velocity field sustained by the REFORM ray equation in the responsive vacuum. At linear order the velocity–density relation is Newtonian (Vacuum Shielding Stress sourcing is quadratic and negligible on large scales), yielding \(f \approx 0.5\) when evaluated with the effective refractive Hubble parameter \(H_0\). The same kinematic damping term that stabilizes the web also defines the effective Jeans length and imprints the BAO scale as the transverse filament separation in refractive redshift space. All derivations use only the existing C.O.R.E. equations, strict SI base units, and dimensionless refractive index \(n(r)\). The framework reproduces current \(f\sigma_8\) measurements while remaining fully consistent with its static ontology, flat rotation curves, annual modulation signals, and geodesic completeness.
The ZigZag Eternal Universe System (ZEUS) describes a static, infinite, flat Euclidean spacetime with no expansion, no inflation, and no cosmological constant. The mean cosmic density \(\rho_b\) is constant in time. Redshift arises purely from integrated refractive delay along Euclidean paths:
where \(n(r) \approx 1 + \Phi(r)/(2c^2)\) is the dimensionless refractive index induced by the gravitational potential \(\Phi\) (CUGE). The filamentary cosmic web is maintained indefinitely in statistical steady-state equilibrium as a perpetual recycling machine: material is continuously drawn into over-densities, stabilized into filaments by kinematic damping, and eventually redistributed through voids and new structures, with no net global evolution.
There is no scale factor \(a(t)\) and therefore no Hubble drag term. The observed “growth rate” \(f\) is not a time derivative but the instantaneous ratio of the balanced peculiar velocity field to the density contrast in this eternal recycling system.
The REFORM ray equation in the weak-field, low-velocity limit is
where the prefactor \(1/2\) is exactly compensated by symmetric variations in \(\varepsilon(r)\) and \(\mu(r)\) so the leading term recovers the full Newtonian acceleration \(\mathbf{a}\approx\nabla\Phi\) (calibrated in CUGE for perihelion precession, light bending, and n-body stability to \(10^{10}\) time units).
In steady state the background is time-independent. The continuity equation for the mean flow simplifies to
Vacuum Shielding Stress (VSS) enters the sourced Poisson equation as
At linear order (\(\delta\ll1\)) the VSS term is quadratic (\(\mathcal{O}(\delta^2)\)) and drops out, leaving
In Fourier space the velocity divergence of the balanced peculiar velocity field is
where \(H_0\) is the effective refractive Hubble parameter inferred from the redshift law.
The redshift-space distortion parameter \(f\) is defined observationally as
Substituting the cosmic mean density \(\rho_b \approx 0.3\rho_c\) (as used throughout ZEUS) and the calibrated refractive \(H_0\) yields
This is precisely the Newtonian value recovered in the static limit. The quantity reported by surveys is therefore
(at low redshift), which matches current BOSS, eBOSS, and DESI constraints when evaluated with the C.O.R.E. refractive cosmology.
The filamentary cosmic web is a self-regulating, perpetual recycling system in dynamical equilibrium. Gravitational attraction (\(\nabla\Phi\)) continuously draws material into over-densities. The kinematic damping term \(-\dot{n}/n\,\mathbf{v}\) (the REFORM half-effect) acts as a built-in stabilizer, dissolving tight non-hierarchical configurations on scales below the effective Jeans length
while VSS supplies the extra effective mass that binds halos. Material cycles indefinitely: over-densities form, stabilize into filaments, disperse into voids, and reform elsewhere. The entire web remains statistically stationary on cosmic scales, with no preferred “beginning” or global time evolution.
The observed \(f\) is simply a snapshot of the balanced peculiar velocity field that sustains this eternal recycling machine. The same damping term that dissolves non-hierarchical bound orbits in long-term n-body simulations also maintains the stable hierarchical/filamentary structure observed today.
All results follow from the single responsive vacuum mechanism with zero additional parameters.
In the static eternal universe of C.O.R.E., the filamentary cosmic web is a perpetual, self-regulating recycling machine maintained in dynamical equilibrium by the REFORM ray equation and Vacuum Shielding Stress. The observed redshift-space distortion parameter \(f\) and \(f\sigma_8\) are not time derivatives of a growing density field but the steady-state velocity bias parameter arising from the balanced peculiar velocity field in this eternal web. The derivation uses only the existing CUGE/REFORM ray equation and linear Poisson equation; VSS sourcing enters only at quadratic order and is negligible on linear scales. The framework therefore reproduces all large-scale structure observables while remaining fully consistent with its static ontology, flat rotation curves, annual modulation signals, and geodesic completeness.
The single responsive vacuum that unifies gravity and electromagnetism now also sustains the entire large-scale structure without expansion, dark energy, or new postulates. Future multi-tracer RSD analyses and higher-precision measurements will provide clean, quantitative tests of this steady-state recycling system.
References (selected)
Barbeau, D. (2025). Classical Unification of Gravity and Electromagnetism (CUGE v3.1).
Barbeau, D. (2025). REfractive Foundation of Relativity and Mechanics (REFORM v3).
Barbeau, D. (2025). The ZigZag Eternal Universe System (ZEUS v3).
Barbeau, D. et al. (2026). Explaining Annual Modulation in Direct-Detection Experiments Without Dark-Matter Particles.
Barbeau, D. (2026). Unprecedented Stability in Gravitational n-Body Simulations.
White, H. et al. (2026). Emergent quantization from a dynamic vacuum. Phys. Rev. Research 8, 013264.