Induced-Gravity Cosmology: Inflation, Dark Energy, and Dark Matter from a Common Scalar Origin

We construct a scalar-tensor cosmology in pure induced gravity: a real scalar Φ with Z₂ potential V_J(Φ) = (λ₀/4)(Φ² − Φ₀²)² + V_c and coupling ξΦ²R replaces the Einstein–Hilbert term, generating M_Pl via ⟨Φ⟩ = M_Pl/√ξ. Conformal transformation yields a Starobinsky-type slow-roll plateau. With locked N ≡ ln(a_end/a_*) and exact potential slow-roll, ξ = 11.1 gives N = 50 ⇒ λ₀ ≈ 6.70 × 10⁻⁸, r ≈ 0.00425, n_s ≈ 0.962; anomaly T_reh ≈ 10⁹ GeV maps to N ≈ 50.7. Planck 2σ selects N ≈ 45–56 (r ≤ 0.0052; capped at N_max ≈ 55.6 by instantaneous reheating); LiteBIRD r > 0.01 would exclude the model class. A Dirac fermion ψ with −gΦψ̄ψ is a cold-dark-matter candidate: m_ψ = gΦ₀ shares the VEV; stability relies on an assumed anomaly-free discrete Z₂^ψ plus the conformally vanishing Yukawa vertex. Matching Ω_DM with the exact mode-equation spectrum on the exact homogeneous background (transition and oscillating-condensate era) selects g ≈ 1.5 × 10⁻⁷ (m_ψ ≈ 1.1 × 10¹¹ GeV ≈ 6.6 × 10⁻³ H_inf), cold kinematics (λ_fs ∼ 10⁻¹⁹ Mpc); absolute normalization is closed at the homogeneous level (convergence/unitarity/conformal battery), with transition-shape systematic (×1.6) and a 3D preheating cross-check remaining. The potential minimum gives a frozen cosmological constant w = −1 (residuals ≤ 10⁻¹⁰); dark energy is calibrated, not predicted (V_c fixed by Ω_Λ = 0.683, 10¹¹⁰ hierarchy unexplained); confirmed DESI DR2 preference for w₀ ≠ −1 would exclude it and ΛCDM alike. Tree-level χ → h h̄ and χ → ψψ̄ close; reheating proceeds via conformal anomaly. SM embedding is RG-stable (Δξ ≤ 10⁻⁶); w₀ = −1 is shared with ΛCDM; the sharp falsification handles remain inflationary.