All-level NANUQ circularity and exact displayed-split support

Original NANUQ is circular decomposable at every finite level and with any number of blobs in the binary semi-directed LSA, galled, outer-labeled planar class (at least four taxa). Its positive split support equals the union of displayed-tree splits. This answers the source's level-two multi-blob and level-three circularity questions. A six-label reduction, exact finite certificate and composition identity remove the level bound. The local four-score family has exact universal anchor-positivity domain $s=o=1$, $1/2<=a<=1$, $0<=c<=a$; five labels are the sharp testing cutoff. Contiguous multicopy trees admit an extreme-copy reduction. Raw-matrix error $<1/2$ is optimally correctable by rounding; exact support uses a supplied correct order. Planarity cannot be dropped for exact support. Full all-level Lean verification, network uniqueness, statistical inference and unrestricted ARGs are not claimed.

Original NANUQ is circular decomposable at every finite level and with any number of blobs in the binary semi-directed LSA, galled, outer-labeled planar class (at least four taxa). Its positive split support equals the union of displayed-tree splits. This answers the source's level-two multi-blob and level-three circularity questions. A six-label reduction, exact finite certificate and composition identity remove the level bound. The local four-score family has exact universal anchor-positivity domain $s=o=1$, $1/2<=a<=1$, $0<=c<=a$; five labels are the sharp testing cutoff. Contiguous multicopy trees admit an extreme-copy reduction. Raw-matrix error $<1/2$ is optimally correctable by rounding; exact support uses a supplied correct order. Planarity cannot be dropped for exact support. Full all-level Lean verification, network uniqueness, statistical inference and unrestricted ARGs are not claimed.