Benford's Law as a Labeling Artifact: A Geometric, Structural, and Constructive Explanation

Benford’s Law is often presented as a universal statistical regularity. This paper develops a geometric and structural reinterpretation: first‑digit behavior is not an intrinsic property of data but a consequence of the labeling geometry imposed by positional digit systems. The paper proves a constructive theorem showing that any strictly positive first‑digit distribution can be induced by a strictly increasing relabeling of the number line, provided the source distribution is atomless and has full support. If some digits are assigned zero probability, such laws cannot be achieved by bijections but can be realized by weaker monotone measurable maps. Digit‑band widths are identified as the geometric mechanism…