Hierarchically Nested Dimensions Theory (HNDT): The Triadic Emergence of Spacetime, Mass, and Energy (v4)

Notes: This is Version 4 (V4) of the Hierarchically Nested Dimensions Theory (HNDT), building upon the conceptual framework of V1, V2, and the mathematical formalization of V3. V4 adds four major extensions: (1) a first-principles derivation of hierarchical coupling coefficients from a generating functional equation; (2) a Langevin-type dynamical equation for the manifestation degree λ; (3) Bayesian likelihood analysis using Planck 2018 CMB data and EHT M87* shadow data; (4) an effective coupling to Quantum Field Theory. All core postulates and formulas from earlier versions remain unchanged. The core postulate is a generative hierarchy of dimensions: D₀ ≺ D₁ ≺ D₂ ≺ D₃ ≺ D₄ where ≺ denotes dimensional subordination — lower dimensions are more fundamental, contained within higher dimensions, capable of upward penetration/transmission, while higher dimensions can only project downward. A continuous parameter λ ∈ 0, 1 (the manifestation degree) governs how properties manifest in higher dimensions. From these postulates, the theory derives seven foundational quantum phenomena (superposition, wave-particle duality, Heisenberg uncertainty, entanglement, tunneling, measurement collapse, and spin-statistics), corrects the Schwarzschild gravitational potential with hierarchical terms γₖ = 2^-k (k-1) /2 derived from first principles, and predicts a characteristic tanh² suppression in the CMB power spectrum at multipoles ℓ < 30. Bayesian analysis of Planck 2018 data yields ℓcrit = 32. 4 ± 5. 7 with a Bayes factor of ~280 in favor of HNDT over ΛCDM (limited to low-ℓ range). The predicted black hole shadow correction for M87* is 22% larger than GR, consistent with current EHT bounds within 1. 5σ. Two falsifiable predictions are presented: a correction to quantum decoherence times τdec = ħ/ (κ λₑnv m d²) (testable with matter-wave interferometry) and the tanh² CMB suppression (testable by LiteBIRD and CMB-S4). HNDT offers a logically parsimonious, mathematically consistent, and empirically testable alternative to geometric interpretations of dimensions, providing a new pathway toward unifying quantum physics and gravity. This version (V4) transforms the theory from a purely formal framework into an empirically calibrated, dynamically complete theory. The earlier qualitative frameworks (V1 and V2) and the mathematical formalization (V3) are fully superseded by this complete formulation.