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This paper develops the first quantitative treatment of structure growth within the gravitating vacuum framework, in which quantum vacuum energy ρ_vac and the cosmological constant Λ are treated as physically distinct quantities. The cosmological constant drives homogeneous expansion as geometry; vacuum energy gravitates locally according to E = mc² and general relativity, with its density depending on the local matter density through the ansatz ρ_vac(ρ_m) = Λ₀ − α ρ_m. We construct the minimal self-consistent unit of this model — the void–cluster cell — consisting of a spherical void with unsuppressed vacuum energy bounded by a matter shell hosting a gravitationally bound cluster. Two vacuum phases are identified: in the expanding void interior (w = −1), vacuum energy drives accelerated expansion and repels matter; in bound structures, the vacuum's dynamical contribution is geometrically suppressed by a factor (r/r_vac)³, derived from the de Sitter–Schwarzschild metric. The background Friedmann equation is identical to ΛCDM (vacuum is sequestered from cosmological dynamics), ensuring automatic compatibility with CMB, BAO, and the expansion history. Only the Poisson equation for perturbations is modified, with effective gravitational coupling G_eff = G(1 + 2α). The central finding is a duality between two observational tensions: for α > 0 (vacuum suppressed by matter), growth is enhanced at all epochs, naturally explaining the massive high-redshift galaxies observed by JWST but worsening the S₈ tension; for α < 0 (the sign favored by running vacuum model fits), growth is suppressed, alleviating S₈ but making the JWST puzzle harder. A chi-squared fit to seven fσ₈(z) measurements identifies α = −0.003 as the best-fit value (χ² = 4.49 vs ΛCDM χ² = 6.08, Δχ² = −1.6), yielding S₈ = 0.806 — a 3% reduction from the Planck-predicted 0.831 in the direction of weak-lensing observations (KiDS: 0.759, DES: 0.776). This value corresponds exactly to the published running vacuum model parameter ν ≈ 10⁻³. A vacuum phase transition at z ≈ 0.7 is proposed as a speculative mechanism to resolve both tensions simultaneously, with the cosmic web forming as gravitationally bound structure during the pre-transition high-energy vacuum epoch and surviving as a fossil structure in the accelerating universe. Void galaxies are interpreted as debris stripped from filaments by post-transition tidal forces. The scenario requires N-body simulations for quantitative verification and is presented as a direction for future work, not as an established result. The paper includes a complete peer review with six major issues addressed: the (1+z)³ vacuum scaling is reframed as a motivated assumption with a mode-counting physical argument; the bound-phase equation of state is derived from de Sitter–Schwarzschild geometry rather than heuristic analogy; the growth equation is presented as an effective model with explicit assumptions and error estimates; a chi-squared statistical comparison with observational data is provided; the phase transition scenario is clearly separated as speculative; and CMB compatibility is demonstrated quantitatively. This is Paper #8 in the research program "What If the Vacuum Gravitates Locally? Separating Cosmic Expansion from Quantum Vacuum Energy." Keywords: vacuum energy, structure growth, S₈ tension, cosmic voids, perturbation theory, dark energy, dark matter, growth factor, integrated Sachs–Wolfe effect, void–cluster model, cosmological constant problem, running vacuum model, JWST early galaxies, cosmic web, phase transition License: Creative Commons Attribution 4.0 International (CC BY 4.0) Resource type: Preprint Related identifiers: IsPartOf: Research Program "What If the Vacuum Gravitates Locally?" References: doi:10.5281/zenodo.18896536 (Paper #4: Matter-Dependent Vacuum Energy Density) Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO) General Relativity and Quantum Cosmology (gr-qc) Authors: Boris Kriger (ORCID: 0009-0001-0034-2903) Affiliations: ¹ Information Physics Institute, Department of Theoretical Astrophysics and Cosmology ² Institute of Integrative and Interdisciplinary Research, Toronto, Canada