Shear Representations, Socle Recovery, and a Local Symplectic Torsion Module

We study representations of a rank-two elementary abelian group equipped with a compatible Levi action. In a specified range of degrees, symmetric powers with a shear action have an explicit socle filtration and a cyclic presentation. A recovery theorem shows that a finite representation with prescribed fixed vectors and first multiplication rank is one of these truncated shear modules. A cohomological criterion reduces the verification of the first multiplication map to excluding one constituent. For a specified local coefficient system in an unnormalized principal series of GSp4 (Q₈₃), we identify the actual integral source, face maps and target lattice with complete finite coordinates. Integral cohomology and exact target certificates establish the fixed-space and character conditions of the recovery criterion. The resulting residual kernel has dimension 820, socle layers 40, 39, . . . , 1, and an explicit cyclic group-algebra presentation. This is a local result for the specified coefficient system; it makes no assertion about a global connecting-map image or other primes.