A No-Go Result for Pure Conformal Coupling in Single-Field Solutions to the H0 and S8 Tensions

We examine whether a single scalar field with a pure Jordan-frame conformal coupling Ω(ϕ) = 1+ξϕ2/M 2 Pl can simultaneously generate an Early Dark Energy phase that raises the CMB-inferred Hubble constant and a late-time modification of gravity that lowers S8. Using the standard quasi-static effective Newton constants for this coupling, we show that in both the massless and the constant-mass limits any value of the dimensionless field amplitude x = ξϕ2/M 2 Pl large enough to produce an appreciable reduction of σ8 forces a corresponding reduction of the CMB lensing amplitude. Under the simple scaling Cℓ ϕϕ ∝ [G (2) eff ]2Pm, even a modest x ≃ 0.05 already yields a multi-sigma effective-amplitude discrepancy with the Planck measurement Alens = 1.008 ± 0.016. A pure conformal coupling with constant (or vanishing) effective mass is therefore strongly disfavoured as a simultaneous solution of the two tensions. A strongly scale-dependent mass or a more general scalar-tensor structure remains the only logical possibility within this framework; neither has been constructed explicitly. The present note is a phenomenological argument under stated approximations, not a full Boltzmann exclusion. PACS: 98.80.-k, 04.50.Kd, 95.36.+x, 98.80.Es