Hierarchically Nested Dimensions Theory (HNDT) (Version 1.2)

Hierarchically Nested Dimensions Theory (HNDT) (Version 1.2) Version: 1.2Publication date: June 2026Authors: Yingkui Lu (corresponding), Yifei LuDOI: Zenodo 20587416 Abstract summary This paper presents Version 1.2 of the Hierarchically Nested Dimensions Theory (HNDT), building upon the axiomatic core of V1.1. New contributions include: (i) a Hongdao‑field isomorphism that defines the manifestation degree λλ via normalized electromagnetic field strength, providing a field‑theoretic complement to the existing phase‑space definition; (ii) empirical, data‑driven formulas for computing λλ for material systems (mass ratio to the observable universe) and for civilizations/organisms (power ratio relative to a Dyson sphere), including detailed taxonomic tables; (iii) an extensive mathematical formalization supplement (categorical formulation, variational derivation of λλ dynamics, coupling λλ to Einstein equations, mapping of θθ modes to gauge groups, first‑principles numerical calculation of λλ, and reformulation of the global closure condition as a conservation law). All axioms from V1.1 remain unchanged. What’s new in version 1.2 (compared to V1.1) Hongdao‑field isomorphism (Section 2.3.2) λman=B/Bmax⁡λman=B/Bmax, λint=H/Hmax⁡λint=H/Hmax, λman=μλλintλman=μλλint Provides a direct link between λλ and measurable electromagnetic quantities, strictly enforcing λ∈(0,1)λ∈(0,1). Empirical computation of λλ (Section 2.3.1 & Appendix 11.8) Material Hongdao: λm=Msystem/MUλm=Msystem/MU (mass ratio) Observational Hongdao: λobs=Pciv/P⊙ηmax⁡⋅λ⊙λobs=ηmaxPciv/P⊙⋅λ⊙ (power ratio) Includes taxonomic tables for elementary particles, stars, galaxies, and civilizations. Mathematical formalization supplement (Section 11) Categorical formulation of the dimensional hierarchy with projection functors. Variational derivation of λλ dynamics from an Onsager‑Machlup action. Coupling λλ to Einstein equations as a dynamical scalar field (dark energy interpretation). Speculative mapping of θθ modes to gauge groups (e.g., Z23↪E8Z23↪E8). First‑principles numerical calculation of λλ for hydrogen using phase‑space volume. Reformulation of ∑λint=∑λman∑λint=∑λman as a conservation law (Hongdao current). Minor improvements Clarified the distinction between material and observational Hongdao. Strengthened the Boltzmann‑brain argument with explicit exponential suppression P∝e−CλP∝e−Cλ. Added “plain words” summaries after technical subsections for accessibility. All V1.1 axioms, definitions, and scaling laws remain unchanged; the update is fully backward‑compatible. Licence Creative Commons Attribution 4.0 International (CC BY 4.0) How to cite Yingkui Lu, & Yifei Lu. (2026). Hierarchically Nested Dimensions Theory (HNDT): A Foundational Axiomatic Framework for the Structure of Existence (Version 1.2). Zenodo. [DOI will be inserted after publication] Contact luyingkui2026@outlook.com Additional notes This is a theoretical/foundational study; no empirical datasets are included. The authors welcome collaboration on further formalization, empirical testing, and development of the framework. Preprint deposited on Zenodo for rapid dissemination and community feedback. Future journal submission (e.g., International Journal of Theoretical Physics or similar) is intended.