The Einstein Window in a Patched Single-Medium Effective Theory for Emergent Gravity: Reduction Theorem, Gravitational-Wave Corollary, and Confirmatory Protocol

We formulate the Einstein window of a patched single-medium effective theory for emergent gravity in a way that is operational both mathematically and observationally. The window is defined by five explicit smallness parameters measuring habitat dressing, superfluid suppression, infrared dissipation, coarse-graining error in the nodal sector, and non-geodesicity of the medium frame. In that window the patched field equations reduce to the Einstein equations with cosmological constant and matter coarse-grained from the nodal sector, with a controlled remainder $\|\Delta_{\mu\nu}\|\le C\epsilon_{\mathrm{red}}$. A central point of the construction is the corrected treatment of the superfluid small parameter: in the Newtonian branch it is $|\mu-1|$, equivalently $\aeff^{\,2}/|a|^2$, not $|a|^2/\aeff^{\,2}$. We derive the local weak-field, FRW, and gravitational-wave corollaries and then isolate the gravitational-wave channel as the first confirmatory-ready benchmark. Because the public data and analysis stack for the benchmark are frozen but not yet executed in this manuscript, the observational part is presented as a blind confirmatory protocol rather than as a post-fit numerical claim. The paper is therefore theorem-and-protocol by design: it closes the reduction statement and the admissible test grammar without pretending that the confirmatory run has already been performed.