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 (gμν,u^μ,nA,χ). The global superfluid response law (y), with rigid-branch asymptotics (y)~ y and deep-branch asymptotics (y)~ 23 y3/2, is kept conceptually distinct from the deep-branch regularization (y)=23[(y+²)3/2-³] used only inside the Cauchy analysis. After generalized harmonic gauge fixing of the metric sector, freezing a0,eff inside the PDE core, adding an inertial completion /2(u\!·\!∇ n)² for the nodal sector, and requiring =√y+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 Hˢ, s≥ 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.