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 underlying Benford probabilities, and these widths are interpreted as multiplicative traversal times. The paper shows that Benford behavior degenerates continuously as the base parameter b→1, where positional notation collapses and leading digits cease to exist. Simulations confirm the constructive and geometric mechanisms. This work is the first installment in a series developing a structural and geometric framework for multiplicative behavior. Paper 1 introduces the labeling geometry underlying Benford’s Law; subsequent papers develop the master operator, empirical geometry, transverse invariants, symmetry interactions, and broader structural consequences.