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

Let \(p>3\), \(k=\overline{\mathbb{F}}_p\), and \(K=\mathbf{Q}_p\). For the reduced symplectic Emerton--Gee component indexed by the regular non-Steinberg Serre weight \(\sigma_{p,d,c}=F(p-2+d,p-2;c)\), where \(1\le d\le p-2\) and \(c\) is an arbitrary central-weight representative, we establish a geometric restriction injection \[ \Gamma(\mathcal{C}_{\sigma_{p,d,c}},\mathcal{O})\hookrightarrow k[D^{\pm1},q^{\pm1}]. \] The corresponding spherical mod-\(p\) Hecke algebra is \(k[U,V,Z^{\pm1}]\). 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 \(\mathrm{GL}\) 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 \(\mathrm{GL}\) 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.