The Magnus Matrix Tree IV: Basis-Compressed Kinetic Operators, Projected Rank-Two Transfers, and Online Selector Admission

We develop a basis-compressed operator layer for exact kinetic distance aggregation. Let x1, ..., xn be labelled coordinates in strict current order, let Cij = |xi - xj|, and fix two selector bases P in R^(n×r) and Q in R^(n×s). Rather than maintaining individual contractions, the method maintains the complete projected distance operator G = P^T C Q. Every online selector pair u = Pα and v = Qβ is then answered by α^T G β, so the kinetic state depends on the selector spans rather than on the number of admitted selectors. Inside one order cell, each label has an exact matrix-valued coordinate gradient. When two adjacent labels exchange order, only the two corresponding gradients change, by opposite copies of a rank-at-most-two transfer. Coupled with the blocked implicit crossing scheduler, a range translation producing k crossings is processed in O((sqrt(n) + k + 1)(rs + log n)) time after O(nrs) compilation. The manuscript also gives online selector admission and deletion, batched coefficient contraction, exact one-column basis expansion, a symmetric quadratic specialization, and identifiability bounds showing that the projected operator has rs observable algebraic degrees of freedom in the general bilinear case. The construction upgrades selector epochs from a list of maintained observations to a dynamically maintained subspace operator.