The Universal Decimal Scaling Property: Fibonacci, Mersennes, and Arbitrary Bases

We report a computational discovery concerning integer representations as sums from fixed generating sets: certain additive bases exhibit decimal scaling stability, meaning that representations of an integer (N) can be systematically transformed into representations of for all . Through extensive computational experiments, we document this phenomenon for Fibonacci numbers, Mersenne exponents, and a variety of heterogeneous dense sets. We identify persistent structural features of scaling chains, including term persistence, predictable migration of dominant summands, and sublinear growth in representation length. We prove a rigorous scaling theorem for Fibonacci numbers, formulate a general logarithmic coverage heuristic explaining the observed behavior, and state conjectures characterizing broad classes of scaling-stable additive bases. These results suggest a new dynamical perspective on additive representations and open several avenues for further theoretical and computational investigation.