Branch-Defect Principle for Principal-Logarithm Arithmetic: Data-Driven Evidence from NOAA NDBC Wind Vectors

The principal complex logarithm Log z = ln |z| + i Arg (z) (with Arg ∈ (−π, π]) is single-valued but fails to be additive because arguments are wrapped at a branch cut [ 1 , 3 , 2]. This manuscript proposes two data-driven, practically testable notions attributed to S M Nazmuz Sakib: (i) the Sakib Branch-Defect Parallelogram Theorem, which packages the principal-branch “overflow” into an explicit nonnegative defect term for the log-energy functional E(z) = | Log z|2; and (ii) the Sakib Index (SI), a normalized scalar in [0, 1] that quantifies branch-instability risk for multiplying/dividing complex- valued observations derived from real-world directional data. A further S M Nazmuz Sakib Branch-Center Principle is introduced: a one-parameter family of principal branches indexed by a cut-center μ is optimized directly on data by minimizing empirical SI, yielding a data-adaptive principal-branch choice. We demonstrate these constructs on the open NOAA National Data Buoy Center (NDBC) standard meteorological dataset (Station 42040, January 2025), using wind direction and speed to generate complex-valued wind vectors. On this dataset, optimizing the branch center reduces the observed overflow rate from 34.7% to 23.9% and reduces mean SI by approximately 56%, suggesting immediate relevance to phase-sensitive computations and phase-unwrapping pipelines.