Benford's Law as a Substrate-Interface Signature: A Structural Account of the Additive–Multiplicative Mismatch

Benford's Law is a labeling artifact (Moser 2026, "Benford's Law as a Labeling Artifact: A Geometric, Structural, and Constructive Explanation," 10.5281/zenodo.18765897): the statistical shadow cast by the mismatch between additive digit labels and multiplicative reality. That paper established the geometric mechanism (digit-band widths as multiplicative traversal times), proved a constructive theorem (any strictly positive digit law is inducible by appropriate relabeling), and stated the underlying principle directly. This companion paper provides a substrate-level structural account of the principle. Drawing on the foundational construction of the standard number systems (Moser 2026, "The Number Line as the Shadow of Continuation," 10.5281/zenodo.20074572), which distinguishes substrate-feature tracings (within-instance content of additive character, producing N, R, C) from shadow operations (between-instance content of multiplicative character, including Q via rate-comparison between distinct continuation threads), the paper identifies Benford's Law as the projection signature at the contact between two structurally distinct kinds of substrate engagement. Under this identification, four properties of Benford's Law follow as structural consequences: domain-independent robustness, scale invariance (predicted as the privileged symmetry before observation), the structural axis Hill's mixture theorem implicitly samples over (multiplicative rate-structures), and the constraint-case failure mode (external bounds force between-instance content into within-instance shape). The empirical content of the labeling-artifact paper is unchanged. What changes is the structural location: Benford's Law sits at the interface where two structurally distinct kinds of substrate engagement meet.