General Relativity as the Infrared EFT of a Proto-Field:\\ Induced Gravity, PN Equivalence, and Compact-Object Phenomenology

I formulate a concrete “GR as induced IR–EFT’’ construction in which the effective geometry is a rank–one disformal update of Minkowski,$g^{\rm eff}{\mu\nu}=\eta{\mu\nu}+\alpha,\partial_\mu\Phi,\partial_\nu\Phi$,and the Einstein–Hilbert term arises at one loop on $(\mathcal M,g^{\rm eff})$. The induced coefficient is$\tfrac{1}{16\pi G_{\rm ind}}=\tfrac{N_{\rm sp}\Lambda^2}{32\pi^2}\big(\xi_{\rm eff}+\tfrac16\big)$(minimal spectators: $\xi_{\rm eff}=0\Rightarrow\tfrac{1}{16\pi G_{\rm ind}}=\tfrac{N_{\rm sp}\Lambda^2}{192\pi^2}$); I adopt the minimal case unless stated otherwise. The $\Lambda^2$ term is scheme–dependent up to local counterterms, whereas the curvature–squared logarithms (and their nonlocal kernels) are universal. These logs fix finite range scales $(\ell_2,\ell_0)$ that enter only through $(1+\ell^2\Box)$ factors in the linearised operator. Under five explicit assumptions—(A1) no extra massless field, (A2) finite ranges $\ell_{2,0}$, (A3) asymptotic flatness with no incoming massive modes, (A4) slow–variation for the heat–kernel expansion, (A5) universal coupling to $g^{\rm eff}{\mu\nu}$—the operator factorises into GR times $(1+\ell^2\Box)$, and in the infrared window $k\ell{2,0}\ll1$ the theory is \emph{measurement–equivalent} to GR: the conservative PN sector matches through 2PN, the dissipative sector through 2.5PN (quadrupole flux), and there is no dipole channel.<br>For static, spherically symmetric C$^\prime$ backgrounds I solve $\nabla^2\Phi=4\pi,\zeta,\rho$ with $\Phi(\infty)=0$ to obtain a monotone lapse $N=e^{2\Phi}$ and use $A(r)=1+\alpha,\phi'(r)^{2}$ as a controlled interior weight. The photon–sphere condition is lapse–only ($rN'=2N$), yielding a phase boundary $\zeta_{\rm crit}=[\max_r M(r)/r]^{-1}$. As a result, the shadow diameter $\theta_{\rm sh}\simeq 2b_{\rm crit}/D_L$ depends only on $N$, while strong–lensing time delays and echo spacings depend on $\sqrt{A/N}$ and thus probe $\alpha$. In the eikonal ringdown, $\Omega_{\rm ph}=\sqrt{N}/r_{\rm ph}$ and $\Lambda_{\rm ph}=\sqrt{\big(2N-r^{2}N''\big)/(2A,r^{2})}$ (evaluated at $r_{\rm ph}$) separate lapse versus interior sensitivities.<br>For slow rotation I implement the Hartle–Thorne scheme with $j=(NA)^{-1/2}$ at $O(\Omega)$ and a stabilised interior/exterior match at $O(\Omega^2)$ to extract $(I,Q)$ and the invariants $(\bar I,\bar Q)$; in the minimal C$^\prime$ family the fitted quadrupole is prolate across the scanned compactness. The observational programme is therefore a joint–constraint triangle: \emph{shadow} fixes $\zeta$ (lapse), \emph{strong–lensing delay} fixes $\alpha$ (interior weight) at fixed $\zeta$, and \emph{inspiral phasing} bounds the mass–scaled $\alpha$ via the 2.5PN log. Within $r\gg\ell_{2,0}$ these tests over–constrain the model while preserving PN/PPN equivalence with GR.