Generalized DNS: Heterogeneous Discrete Structural Configurations

This paper introduces Generalized DNS as a structural sub-program of the SHA-DNS framework, generalizing DNS from a numeral system to a class of heterogeneous, history-dependent discrete structural configurations. The framework is organized around four structural axes: Heterogeneity + Occupancy + Cost + History Core constructions: 1. A configuration space D_GDNS consisting of candidate positions, occupancy indicators, local structural parameters, and structural sites. 2. A history-enriched state space X = H × D_GDNS, with formation trace, compressed history, and hardened structure distinguished. 3. The threefold distinction between vacancy (∅), occupied zero (0_T), and occupied value (x), replacing the classical implicit identification of vacancy and occupied zero. 4. A cost functional C = K(H), with cost contributions as derived quantities and the separation between hardened structure and cost. 5. A generalized carry operator C_{i→j}: D_GDNS ⇀ D_GDNS allowing non-adjacent layer transitions, with an induced structural activation rule and an activation cost ΔC_act ≥ 0. 6. The soft-state / structural-history non-equivalence principle: identical current total soft content does not imply identical hardened structure, together with an S-H coherence constraint. 7. An intended specialization structure in which classical numeral systems and DNS appear as special cases of Generalized DNS. Continuous representations (Fourier, Taylor, splines) are positioned as observation interfaces, not as the ontology of the framework. The paper includes a foundational formalization agenda, a set of open problems, and a set of future extensions organized as the progressive relaxation of current constraints: - Dynamic connectivity (P8) - Bidirectional structural evolution (P9) - Probabilistic and quantum extensions of occupancy (P10) - Generative structural relaxation (P11) This paper is programmatic. It defines the core structure and lists open problems. It is a sub-program of the SHA-DNS framework, and is intended to serve as a higher-level structural framework for Directed Layered Bases. Version: v2.4 (September 2026) Series: Generalized DNS / SHA-DNS Sub-Program