Universal Residual Seeds

Abstract

A residual representation is a genuine zeroth-order coordinate. All positive-depth perturbations above it may fill its formal deformation neighborhood, but they cannot move into a different residual class. This paper develops the corresponding universal object: a residual forest whose roots are residual representations, whose finite levels are lifts modulo powers of a uniformizer, and whose infinite branches are complete characteristic-zero objects. We prove the residual partition, no-crossing, inverse-limit reconstruction, finite-branch compactness, exact smooth and singular branch counts, packet decomposition of tangent and obstruction cohomology, a directed minimum-seed theorem, rational seed-zeta formulas, and finite companion-forest completion. The coefficient prime two is represented by a weighted forest in which characteristic-zero packets may merge modulo two and become visible only at later depth. We then define a prime-independent system seed through a common Frobenius spine, prove a compatible-carrier summand theorem under finite Frobenius separation, and reduce finite occupancy in a fixed semisimple carrier to the nonnegative integer trace of an exact joint projector. The paper does not prove that every admissible residual companion exists or that every finite Hecke packet is occupied. Those two existence statements remain the exact horizontal boundary beyond vertical deformation theory.