Reversible Digit-Shell Grammars for Products of Exponential Numeration Families

This paper establishes an exact scale self-similarity in the positional expansions of products formed from exponential numeration families. Increasing both factor indices by a multiplicative-order period transforms every sufficiently large canonical product word by adjoining fixed low- and high-order digit shells and applying a fixed finite-state subtraction transducer to the interior. This rewrite is reversible and yields a product-only parser that recovers the latent scale indices within the declared family. For repeated-digit factors, the interior transformation reduces to the deletion of a single digit; for repunits, both shells have explicit formulas in every base \(b≥ 3\). These results connect exponential identities, carry propagation, canonical digit representations, and reversible language structure.