An amplitude-independent ratio between the lensing and clustering modifications in quasi-static Horndeski gravity

In the quasi-static limit of Horndeski gravity with unit tensor propagation speed, the Planck-mass-normalised deviations of the lensing and clustering functions obey ΔΣ̂ / Δμ̂ = (α_B + α_M) / (α_B + 2α_M), identically in the amplitude of the α functions: the background combination β₁ cancels before any expansion, and neither the kineticity nor the sound speed enters. We derive the relation in two independent ways, from the β-basis coefficients and from the small-scale limits of Pogosian and Silvestri. The ratio depends on r ≡ α_M/α_B alone and inverts on each branch, so a measured value selects a unique r; in particular it equals 2/3 if and only if α_M = α_B ≠ 0, which on the subclass G₅ = 0, G₄ = G₄(φ) is the Lagrangian condition X G₃ₓ = 2 G₄,φ. Read in reverse, and given r, the relation returns the running Planck mass from the unnormalised functions, M*²/M_Pl² = (1 − R)/(Σ − Rμ). We show that the value 1/3 quoted for the potential-response ratio is the same statement in another convention, while the unnormalised ratio carries no fixed value. The relation is verified four ways: against a Boltzmann extraction with the α functions set directly, at four values of r, to better than 0.1% for k ≳ 0.05 Mpc⁻¹; by measuring the departure from the quasi-static limit, which scales as (aH/k)^2.02; on a covariant branch where α_M = α_B is an output of the Lagrangian, where the full linear equations give ΔΣ̂/Δμ̂ = 0.66657 ± 0.00008 and the quasi-static coefficients 9 and 6 to 0.1%, with the ratio unchanged to 0.1% across six potential slopes and both signs of the coupling; on an independently implemented enhanced-early-gravity model over the 95% region of its Planck+BAO constraints, where −1.45 ≲ r ≲ −1.15, the general inversion recovers the Planck mass to 0.03% and the 2/3 form misreads it by 0.1–4.4%. The coefficient of the (aH/k)² correction vanishes exactly at r = 0 and factorises as r/(1+2r) times the difference of the fractional corrections to clustering and slip, which we verify at ten values of r. We also record an extraction pitfall that produces a spurious common offset of the size of the signal. Code and data: doi:10.5281/zenodo.22897960 Keywords Horndeski gravity modified gravity quasi-static approximation effective field theory of dark energy weak lensing redshift-space distortions effective Planck mass gravitational slip scalar-tensor theories hi_class