  Weak Arthur packets have long been instrumental in the study of the unitary
dual and automorphic spectrum of reductive Lie groups, and were recently
introduced in the p-adic setting by Ciubotaru - Mason-Brown - Okada.
  For split odd orthogonal and symplectic p-adic groups, we explicitly
determine the decomposition of weak Arthur packets into Arthur packets that
arise from endoscopic transfer. We establish a characterization of the Arthur
packets that partake in such decompositions by means of ramification properties
of their constituents.
  A notion of weak sphericity for an irreducible representation is introduced:
The property of containing fixed vectors with respect to a (not necessarily
hyperspecial) maximal compact subgroup. We show that this property determines
the weak Arthur packets in a precise sense.
  As steps towards this description, we explore alignments between
Langlands-type reciprocities for finite and p-adic groups, and their dependence
on the geometry of the unipotent locus of the dual Langlands group.
  Weak sphericity is shown to match with Lusztig's canonical quotient spaces
that feature in the geometric theory for Weyl group representations, while the
fine composition of weak Arthur packets is found to be governed by the
partition of the unipotent locus into special pieces.
