On the Support Geometry and Complexity of LABS

LABS are a fundamental combinatorial optimization problem arising in sequence design and statistical physics, yet the formal complexity of the standard fixed-geometry problem remains unresolved. We study how zero-energy complexity changes when support geometry is included as part of the input and then progressively restricted. For unary standard LABS, we establish membership in $\text{NP}$ and a sparsity/Mahaney consequence: Karp NP-hardness would imply $\text{Ptime}=\text{NP}$. For Pair-Set-LABS and Masked-LABS, zero-energy feasibility is polynomial-time decidable when every pair group or masked row has at most three terms and is NP-complete when four terms are allowed. NP-completeness persists when active supports are restricted to contiguous intervals.