Repeated-Inertia Obstructions to Spectral Mod-p Satake for GSp4

Let \(p>3\), \(k=F̄_p\), and \(K=Q_p\). For the reduced symplectic Emerton--Gee component indexed by the regular non-Steinberg Serre weight \(σp,d,c=F(p-2+d,p-2;c)\), where \(1≤ d≤ p-2\) and \(c\) is an arbitrary central-weight representative, we establish a geometric restriction injection \[ Γ(C_{σp,d,c},O) k[D±1,q±1]. \] The corresponding spherical mod-\(p\) Hecke algebra is \(k[U,V,Z±1]\). Consequently the global-function ring has transcendence degree at most two, whereas the Hecke algebra has transcendence degree three; the two are not isomorphic even as abstract \(k\)-algebras. The proof uses the full, non-split \(GL\) source in Lee's construction of symplectic components, invariance of unframed Siegel extensions under scalar contraction, extension of the split Levi map from a schematically dense open, and the repeated-inertia symmetry combined with Lee's non-Steinberg \(GL\) Satake injection. We make explicit why the endpoint \(d=p-2\) is permitted. No exact presentation of the symplectic global-function ring, and no categorical Hecke action, is asserted.