Finite-Resolution Contraction, Quantization Operators, and Persistent Remainders

We analyze a class of real-valued contraction-quantization operators defined by the composition of linear contraction and finite-resolution truncation. For a fixed resolution depth n and contraction factor λ ∈ (0,1), we define the operator Ωₙ and prove an exact decomposition of the contraction map into a quantized component and a bounded remainder. This decomposition, termed the Omega Identity, is structural and holds identically rather than asymptotically. We establish uniform bounds on the remainder, characterize the induced cell structure on ℝ, and analyze the recursive dynamics of residuals under iteration, showing geometric decay once the residual falls below the quantization threshold. Stability conditions are derived for finite-precision arithmetic, including resolution limits compatible with IEEE-754 double precision and bounds on coherent iteration depth. All results are formulated strictly within real analysis, operator theory, and numerical stability, without invoking physical interpretation.