Half of M31's dwarfs lie in a plane with 1:10 aspect ratio, most sharing one rotation sense, and the plane is lopsided toward the Milky Way (Ibata 2013; Conn 2013). One such structure is a rare outlier in simulations; sharing a Local Group with the Milky Way's VPOS multiplies the coincidence beyond credibility (Pawlowski 2018).
Stochastic accretion onto a triaxial halo cannot draw a thin co-rotating disk of satellites, cannot make it lopsided toward a neighbor, and cannot do it twice in one group. Each feature must be a separate accident, and the model's defense, rare alignments plus selection effects, thins with every confirmed velocity.
SCT reads the M31 plane as the second imprint of the geometry that made the first. M31 and its dwarfs condensed from one debris field carrying one deposited J = μ(b × v_rel) (P31), so its satellites co-rotate in a thin plane for the same reason the Milky Way's do (recid 130): co-planarity and co-rotation are one conserved initial condition, not an assembled coincidence (P32). Two planes in one group stop being a multiplied improbability because the two hosts are not independent draws: they are sibling structures of the same Local Group deposition, their satellite systems inheriting related J components from the shared parent geometry (P58, P59).
The lopsidedness is the shared context showing through: a debris field organized by one parent event distributes its fragments with the asymmetry of that event's geometry, so plane members concentrating toward the sibling host is the kind of correlated structure stochastic infall forbids and common descent expects. The registered program-level prediction stands behind both planes: co-rotating satellite structures around essentially every adequately sampled host, with plane normals of neighboring hosts correlated, exactly what the MW-M31 pair exhibits. The debris-field mechanics are in Paper 6, From Chaos to Cosmic Collisions, with the hierarchy-wide inheritance in Paper 5, From Chaos To Corotating Hierarchies.
Keystone economy: P31 and P32 carry both planes; P58 and P59 explain why they share a group and lean toward each other. Nothing is paid twice.
Full proper motions of the M31 dwarfs carry the kill: if the complete dynamical census dissolves the co-rotation into a projection artifact, with orbital poles scattered as stochastic accretion predicts, the shared-stamp reading fails for the strongest claimed case. The pair-level test is registered: SCT requires the MW and M31 plane normals to be correlated through the shared Local Group geometry; demonstrated statistical independence of the two planes at high significance would break the sibling-deposition account even with both planes real.