U.S. Library and Information Science (LIS) schools are organized differently: some are independent, and others share a parent unit with computing, communication, education, or the arts and sciences. We ask whether their research differs accordingly, using 18,048 publications (2013--2026) by 1,602 faculty members at 44 ALA-accredited schools. We grouped the 15,934 publications with abstracts into 15 research themes with document embeddings and hierarchical clustering, and classified venues by field. Faculty members' research profiles differ by school type in themes and, more clearly, in venues, but type accounts for only 2.2% and 5.3% of the variation; most lies among faculty of the same school. The differences run from libraries and archives to computational research. Schools housed with computing spread over the most themes and publish most on security, networks, and urban computing, on clinical and biomedical informatics, and in computer science venues; schools housed with education spread over the fewest and concentrate most on libraries, archives, and computing education; independent schools stay closest to the average. Over the period, schools moved away from library research and LIS venues, but seldom toward the venues of their parent unit's discipline. Research appears to follow a school's faculty more than its parent unit.
For masked rank-one least squares with unrestricted signs, gradients reveal weighted margins and the loss reveals the target's squared norm. We prove that this one scalar resolves the gradient ambiguity on bipartite cycle and length-three parallel-path blocks: every target fixed before sampling and nonzero on cycle edges is identified almost surely by one legal positive query drawn from an absolutely continuous law. On six-cycle flowers, open target and query sets retain $3^r$ bounded gradient candidates. A common algebraic witness excludes cross-block energy cancellation. Cactus masks admit explicit exception certificates and target-uniform finite-grid bounds; theta blocks admit an existence guarantee. For nondegenerate cactus reductions, an exact decoder returns complete actual real lists, with bit cost polynomial in input length, the cycle-degree product $D$ and the optional budget parameter $2^κ$. Energy-coordinate propagation certifies singletons without a product join. The product $D$ can grow exponentially in cycle count. Exponential certificate-degree and output bounds, and weak NP-hardness at a fixed uniform query, characterize distinct costs of exact decoding.