Dissipative Numeral System, Paper III: Cost-Induced Relational Geometry on Hierarchical Structures

The convergence of number and geometry: cost-induced spatial emergence. Traditionally, mathematics has treated geometry as a pre-existing, neutral container for number systems. The choice of positional base remains a notation, divorced from the geometric metric. This paper (Dissipative Numeral System, Paper III) reverses this classical assumption: geometry is not a container, but an emergent structure generated by the resource costs required to organize numbers hierarchically. By introducing cost-bearing positional systems, we connect foundational geometry and number theory through a shared origin in structural cost, leaving extensible interfaces for various geometric paradigms. Key Highlights of Paper III: - Axiomatic Restructuring: We elevate the DNS point to a robust four-tuple containing Cost, Level, History, and Occupancy (P = (C, L, H, O)), replacing the classical featureless "bare point". - Relation-First Geometry: We introduce the DNS Edge encoding both symmetric distance and antisymmetric direction, and mathematically prove the Cost Recoverability Theorem—absolute costs can be uniquely reconstructed from boundary relations. - A Taxonomy of Induced Geometries: We demonstrate how different cost invariants systematically correspond to diverse geometric categories, including Riemannian-like, Finsler-like, Tropical, Fractal, and Scheme-inspired stratified structures. - A Calculable Toy Model: A concrete, three-tier decadic model (C₀=1, C₁=2, C₂=5) is explicitly calculated to show how "carrying" (upward aggregation) and "rupturing" (downward collapse) lead to mathematical irreversibility and path dependence. This work represents the core foundation of the DNS series, transitioning the framework from conceptual foundations to calculable mathematical theory. It shifts the mathematical paradigm from "how numbers represent quantity" to "how structural costs generate mathematical space." Future directions of the framework include: Lifecycle dynamics: introducing temporal maintenance costs, where structures may self-decay or reset if not sustained.- Deconstruction penalties: modeling irreversible breakdown, depreciation, and liquidation through explicit decommissioning costs.- Heterogeneous radices: extending to level-dependent bases, mapping variable coefficients (e.g., Taylor expansions) to branching rates.- Cross-disciplinary interfaces: inviting exploration in non-von Neumann computing, industrial digital twins, and multi-scale stochastic modeling. This revised version (v2.3) refines the dynamic preview section for improved clarity and thematic alignment with the subsequent paper in the series.