Triptyque Conceptuel: The Unifying Equation as a Theorem: Proof from the Kinetic Operator and CMB Consequences

Description Version 18 of the Triptyque Conceptuel establishes Theorem T: in Phase I (σ ≫ σ_min, UV regime), the equation of motion of the complex scalar field ψ = √ρ · eiS on curved spacetime is (□_g + R/6)ψ + ∂V/∂ψ* = 0, where □_g + R/6 is the Yamabe operator, the unique conformally covariant second-order differential operator in four dimensions. This was Conjecture U in V17; Version 18 proves it. The proof uses the Seeley–DeWitt heat kernel expansion. The fluctuation operator K = −Ω²(σ)□_g + M²_eff is brought to canonical form by the conformal transformation g̃_μν = Ω² g_μν and the field redefinition χ = Ω δψ, after which the Parker–Toms theorem establishes ξ = 1/6 as the unique UV fixed point, independently of Ω². Theorem T constrains the form of the equation for any V(|ψ|). The parameter m enters through the specific Triptyque potential V(σ, m) and the self-consistency constraint κ = 3m², which is equivalent to the universal invariant η_V · K = 8/3 (Theorem K, V17), thereby singling out d = 3 spatial dimensions through the self-consistency condition. In the standard case m = 1, all exact invariants belong to ℤ[√2]. Under Theorem T, the effective potential in the Einstein frame develops a Starobinsky plateau, yielding n_s = 1 − 2/N* ≈ 0.965 and r = 12/N*² ≈ 0.004, compatible with Planck 2018 at 0.3σ and testable by LiteBIRD. The background phase velocity Ṡ₀ = L/(Ω²σ₀²a³) → 0 in Phase I as a³ grows, decoupling the amplitude and phase modes at leading order. Under Theorem T, a unified description of inflation, dark energy (w = −1, V_min = √2 − 1), and dark matter (⟨w⟩ = 0, stable via U(1) symmetry) is obtained from a single field and a single parameter. Numerical checks are provided by Script 59.