Global Functions and Nonextendable Spectral Germs on Ordinary GSp4 Components

We compute the global functions on an explicit family of ordinary Serre-weight components of the symplectic Emerton--Gee stack over \(\mathbf{Q}_p\). For symmetric ordered inertial exponents with adjacent gaps \(r,s,r\), we assume \(r,s\ge7\) and \(2r+s\le p-4\) (in particular \(p\ge29\)). The global ring is a polynomial algebra in two variables and an invertible similitude coordinate, and its restriction to the split torus agrees with the characteristic-\(p\) Satake transform. The proof combines Siegel and Klingen component morphisms with global integral prefix functions pulled back from a matched \(\mathrm{GL}_4\) component. A separate comparison concerns the divided Frobenius coordinates on suitable patched ordinary deformation germs. Under the stronger bounds \(r,s\ge8\) and \(2r+s\le p-8\) (in particular \(p\ge37\)), at specified points with trivial unramified diagonal parameters the intrinsic completed-global restriction is split injective. Its two coordinates and the two divided Frobenius coordinates form part of a regular parameter system on the framed residual ring. The corresponding formal fraction fields are linearly disjoint; every nonconstant meromorphic function in the prescribed local spectral coordinates is transcendental over the completed-global field. Thus the intrinsic global comparison and the prescribed local spectral map have different formal images. This second comparison is conditional on a patched input, namely the existence of a suitable globalization of the seed carrying a congruent patched pair at every patching place; whether the seeds studied here admit such globalizations is not established in this paper, and §S7.1 records the precise status of this input, including a candidate route through the globalization construction of Enns-Lee.