RLMT: The Conformally Coupled Layer-2 Scalar Is Exactly Dark Radiation: A No-Go Result for Early Modified Gravity from the Conformal Branch

Early modified gravity (EMG), a scalar field with non-minimal coupling F(σ)R/2, F = M_pl² + ξσ², and a quartic potential, is among the most successful proposals for raising the CMB-inferred Hubble constant. The layer-2 (L2) sector of R-layer Mode Theory (RLMT) contains a scalar of exactly this type, with the coupling fixed by conformal symmetry to ξ_R = 1/6 (ξ = −1/6 in the EMG convention). We show that in this case the background evolution is exactly that of dark radiation. For any quartic self-coupling λ ≥ 0 and any cosmological background, the Jordan-frame Friedmann equation reduces identically to 3M²H² = ρ_m + ρ_r + ρ_Λ + E/a⁴ with a conserved E. The apparent enhancement of the effective gravitational constant during radiation domination is this dark radiation, ΔN_eff ≃ 7.45 ε/(1 − ε) with ε = φ_i²/(6M²), and the quartic potential cannot produce an early-dark-energy-like injection. Big-bang nucleosynthesis then limits ε ≲ 0.05, i.e. an H0 gain of at most about 1.7%. Reproducing EMG requires a non-conformal coupling of the opposite sign, ξ_R ≈ −0.15. We examine four candidate mechanisms within RLMT (renormalization-group running, the induced-gravity branch of L2, frame choice, and a layer mixture) and find that none currently fixes this value. This is a negative result, recorded so that the EMG route is not re-opened on the basis of the conformal L2 coupling. The identity is verified symbolically (SymPy script included). A Japanese version of the note is included. This note follows "RLMT: RLMT and the Hubble Tension: Primordial and Late-Time Ω-Layer Tests against Planck, DESI DR2 and Pantheon+, with Methodological Corrections" (version 2).