Digit-Reversal Differences in Arbitrary Bases: Spectra, Multiplicities, Renormalization, and Limit Laws

We study the difference between a fixed-width base-\(b\) digit string and its digit reversal. A signed mirror-difference representation yields an injective description of the directed difference spectrum and leads to exact formulas for spectra, inverse reconstruction, fiber multiplicities, generating functions, and finite probability distributions. For genuine \(k\)-digit integers, the leading-digit restriction produces a single boundary modification of the unrestricted model. The normalized difference admits an exact renormalization recursion whose stationary operator is a \(1/b\)-contraction in Wasserstein distance. Its unique fixed point is the difference of two independent uniform random variables on \([0,1]\), hence has the universal triangular density \((1-|x|)_+\), independently of the base. The same fiber weights \(b-|x|\) that govern finite multiplicities form the refinement mask of this limiting density, giving a direct connection between the finite combinatorial theory and the asymptotic probability law. A Fourier-product representation telescopes to the corresponding sinc-squared characteristic function. For genuine \(k\)-digit integers the leading-digit constraint acts as a single boundary operator and produces an explicit asymmetric trapezoidal limit law. Quantitative coupling estimates give exponential convergence in the number of digit pairs.