Local Strong Hyperbolicity in a Patched Single-Medium Effective Theory for Emergent Gravity

We isolate and sharpen the local partial-differential-equation content of a patched single-medium effective theory for emergent gravity with fields $(g_{\mu\nu},u^\mu,n^A,\chi)$. The global superfluid response law $\Fpat(y)$, with rigid-branch asymptotics $\Fpat(y)\sim y$ and deep-branch asymptotics $\Fpat(y)\sim \tfrac23 y^{3/2}$, is kept conceptually distinct from the deep-branch regularization $\Feps(y)=\tfrac23[(y+\epsF^2)^{3/2}-\epsF^3]$ used only inside the Cauchy analysis. After generalized harmonic gauge fixing of the metric sector, freezing $a_{0,\mathrm{eff}}$ inside the PDE core, adding an inertial completion $\tfrac{\tauN}{2}(u\!\cdot\!\nabla n)^2$ for the nodal sector, and requiring $\Jeps=\sqrt{y+\epsF^2}>0$, we obtain a first-order reduced system with block-triangular principal symbol. Under explicit patched-admissibility conditions, the symbol admits real characteristics and a uniformly complete eigenbasis, so the reduced system is strongly hyperbolic in the standard local sense. For compatible data in $H^s$, $s\ge 5$, we obtain local existence, uniqueness, continuous dependence on the data, and propagation of the gauge and internal constraints. The result is deliberately local and patched: it does not address global existence, boundary value problems, symmetric hyperbolicity in full generality, or non-perturbative ghost questions. The point of the paper is narrower and, in our view, stronger: the patched branch defines a mathematically controlled local Cauchy problem.