Dissipative Numeral System, Paper VIII Structural Contraction and Residual Reset

Dissipative Numeral Systems (DNS) constitute a special class of structured numeral systems within the Discrete Network Structures (DNetS) background framework of the Soft-Hard Algebra (SHA) ontology. Within the DNS series, Dynamic Numeral Systems (DynNumSys) denote the evolving, carry-based realization of the dissipative numeral-system architecture. This paper formalizes a core interface between structural hardening and residual reset. We study a model of a core hardening map as a potentially non-injective transition. We define Structural Degeneracy, D(H)=∣h−1(H)∣D(H)=∣h−1(H)∣, and worst-case Hardening Information Loss, Lwc(H)=log⁡2D(H)Lwc(H)=log2D(H). We prove two results within the stated axiomatic framework: (1) a hardened state with a degenerate fiber cannot possess a deterministic, single-valued global left inverse; and (2) under finite-fiber, zero-error, worst-case exact-recovery assumptions, an auxiliary residual encoding requires capacity at least ⌈log⁡2D(H)⌉⌈log2D(H)⌉ bits. We also introduce a refined resource ledger Γ=(Γavail,Γlocked,Γdissipated)Γ=(Γavail,Γlocked,Γdissipated) and distinguish primitive structural dynamics from resolution and representation operators. The paper provides a minimal mathematical interface between SHA, DNetS, DNS, and Reset Dynamics. It does not claim to constitute a complete physical theory of entropy, time, or thermodynamic irreversibility; all results are relative to the definitions and assumptions explicitly stated here.