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

We construct a unified scalar-tensor cosmology within pure induced gravity, in which a single real scalar field Φ with a Z2-symmetric potential VJ (Φ) =λ0/4 (Φ²−Φ0²) ²+Vc and non-minimal coupling ξΦ²R replaces the Einstein–Hilbert term, dynamically generating the Planck mass via ⟨Φ⟩=MP/√ξ. Conformal transformation to the Einstein frame yields a Starobinsky-type slow-roll plateau, giving the robust functional predictions nₛ≃1−2/N and r≃8/ (β²N²) with β=2/√ (6+1/ξ), where the e-fold number N is confined to the physical window 48, 55 by independent BBN and Planck nₛ constraints. A Dirac fermion ψ with a Yukawa coupling −gΦψ̄ψ serves as cold dark matter: its mass m_ψ=gΦ0 originates from the same VEV, it is gravitationally produced at the end of inflation, and it is absolutely stabilized by a discrete Z2^ψ gauge symmetry together with a conformally vanishing χψψ vertex—a structural consequence of the induced-gravity coupling F=ξΦ². The potential minimum furnishes a frozen cosmological constant with w=−1 (up to residual-condensate corrections ≲10⁻¹⁰), calibrated to Ω_Λ=0. 683. Fifth-force bounds are satisfied: the conformal identity Ω=Φ/Φ0 decouples the scalar from ψ at tree level, while the scalar's large mass m_χ∼10¹³ GeV suppresses couplings to Standard Model matter. The χ→hh and χ→ψψ̄ tree-level reheating channels are structurally closed; reheating proceeds via gravitational and conformal-anomaly production, self-consistently selecting the fiducial N≃50 (Tᵣeh∼10⁹ GeV). The Standard Model embedding is RG-stable (Δξ≲10⁻⁶). The key falsification window is a LiteBIRD-class measurement r>0. 01 at 95% CL: throughout N∈48, 55, r≲0. 0053 at ξ=11. 1 (up to O (1) systematic uncertainty from the transient post-inflationary phase), so such a detection would exclude the entire physically allowed model class. The w0=−1 prediction is independently falsifiable by DESI/Euclid at >3σ.