DNS Nested Springs: Generation and Evolution, Paper 1 (Generation)

This paper is the generation paper of the Dissipative Numeral System (DNS) series. It constructs a finite, deterministic, traceable nested-spring prototype in which DNS ledgers are projected into a geometric point sequence through a fixed example mapping, and the resulting sequence is translated into classical recursive-spring parameters. What the paper establishes. A level-wise classification of DNS cost into (advance cost, curvature cost)_n at each level n, where the advance cost plays the role of the helix lead and the curvature cost governs the turn. A structured zero ⊚ with non-cancellable direction markers and chirality: opposite direction markers do not cancel, and mirroring yields a distinct object with a new ID and a preserved source ID. An encapsulation operator E and an envelope operator W with explicit cylinder and planar branches, together with an operational closure test separated ontologically from geometric closure. A discrete first–last closure condition for the equal-step equal-turn case, with explicit degenerate and non-coprime examples. A two-tier translation interface with classical recursive springs: Tier 1 maps ledgers to geometric parameters by an example mapping; Tier 2 uses the classical spring parametrization. The translation is exact as an example correspondence, but is not an equivalence of theories. What the paper does not establish. It does not claim that classical spring geometry necessarily follows from DNS rules; it does not claim that the dual-axis cost reduces to the scalar cost of DNS Paper 0; it does not claim that ⊚^(n) exhausts the structured-zero class 0_(n) of Paper 0. The cluster curvature-cost component κ_α is an explicit caller input; a generative mechanism for it (curvature sedimentation or otherwise) is a candidate direction only and is left to future work. Verification model. The eleven assertion rows of the worked example are verified by hand; the program-side forms are specifications for a future discrete implementation (E.2b). A continuous-limit visualization script and its run output are included. The freeze checklist of Appendix D.9 and the run output of Appendix E.3 are both complete in this version. Position in the DNS series. This paper consumes the structured-zero and generation principles of DNS Paper 0 (Axiomatic Foundations) and Paper 0.5 (Structural Zeros: From Label to Generative Skeleton), and precedes Paper II (The Decomposition Operator). It does not consume the decomposition, geometry, dynamics, multi-dimensional, differential, or probabilistic content of Paper II onward. Boundaries with each later paper are stated in Appendix C.4. Status. Version 1.0. The theoretical layer, the hand calculation of Appendix D, the two-tier translation of Appendix F, and the continuous-limit visualization of Appendix E.2a are complete. The discrete generation implementation and program-side verification are left to future work (E.2b).