Partition interactions and sharp binary rigidity

A partition interaction is a signed measure, and the choice of that measure matters before any norm or Fourier observation is applied. We first bound the difference between normalized partition interactions by defects of proper marginals. This identifies their common pure stratum while distinguishing coincidence of all operators from coincidence of one particular pair. At order four, the difference between the Lancaster and Streitberg measures is a sum of three products of pairwise defects. For four balanced signs the Fourier norm of the Streitberg interaction equals the magnitude of the joint cumulant. A short second-moment argument gives its sharp bound two and the eight common-sign equality laws. The relative-sign distribution gives exactly the total variation distance to this entire equality family. An exact local expansion then proves that the optimal leading stability coefficient is one quarter. An explicit family shows why this leading coefficient is not a global linear bound. The classical partition identities provide the framework; the quantitative conclusions concern the extremal geometry of balanced four-sign laws.