Environmental Transition Cosmology (ETC): Mathematical Foundations of Linear Perturbations, Joint Cosmological Fit, and Structure Growth Dynamics (Paper 1)

Environmental Transition Cosmology (ETC): Mathematical Foundations of Linear Perturbations and Structure Growth (Paper 1) This paper establishes the mathematical foundations of linear matter perturbations in ETC. By replacing the cosmological constant (ΛΛΛ) with an environmental energy exchange rate Γ(z)=γ0H(z)Γ(z) = γ_0 H(z)Γ(z)=γ0H(z), we derive the modified perturbation equations. A joint fit against 31 Cosmic Chronometers and 18 fσ8fσ_8fσ8 benchmarks demonstrates that ETC resolves the S8S_8S8 tension (σ8,0≈0.779σ8,0 ≈ 0.779σ8,0≈0.779) and accommodates higher local expansion rates (H0≈71.4H_0 ≈ 71.4H0≈71.4) without dark energy. This serves as Paper 1 of the foundational ETC series.