Dissipative Numeral System, Paper II The Decomposition Operator and the Algebra of Structural Memory

Building upon the axiomatic foundation established in DNS Paper 0 (DOI: 10.5281/zenodo.21310817), this paper introduces the structural decomposition operator as the necessary backward operation corresponding to costed formation. The operator is defined on the free R-module of structured zeros. To provide a rigorous foundation for non-invertibility, we formalize the formation map as a transformation between resource and structure state spaces, and introduce two axioms: Threshold Forgetfulness (formation does not preserve precise resource values) and Residual Non-Recoverability (decomposition cannot recover consumed thresholds). We prove a non-invertibility theorem showing that the structural projection of formation is not injective, decomposition is not a left inverse of formation, and the composition is independent of input resource values. The classical limit C_n = 0 is shown to recover ordinary positional notation. This work provides a formal algebra of structural memory with a rigorously established irreversibility, completing the foundational operator calculus of DNS.