Paper 4: Splitting of Reciprocal Dual Cells and Hierarchical State Structure — A Minimal Observational Model of Internal State Capacity, Volume Gap, and Duality Breaking from νλ=1

Starting from the minimal reciprocal dual condition λnνn = 1 placed between wavelength components and frequency components, this paper reconstructs, as a single minimal observational model, the wavelength space, the frequency space, the 4-dimensional lattice cell counting, and the closed 4-dimensional structure treated in the three preceding observational papers. When the possible existence of a minimal conjugate width δmin is added to the reciprocal dual condition, a state is treated not as a mathematical point but as a conjugate cell with a minimal width. The composite frequency radius R² = Σ νn² can then be read as an energy-like scale determined from the internal frequency components: R = 1 is observed as the minimal conjugate cell, R = 2 as eight first-neighbor quasi-ground states, and R = 3 as a large internal state shell containing 128 additional states (N₀(1) = 1, N₀(2) = 9, N₀(3) = 137). Comparing the 4-dimensional hyperball volume V₄(R) = (π²/2)R⁴ with the fully-inscribed cell volume N₀(R), the filling fraction shows an asymptotically increasing tendency with local fluctuations, while the absolute volume gap ΔV(R) = V₄(R) − N₀(R) grows at order R³ as a boundary-layer quantity. From this observation, a toy model is introduced that reads a single high-R state as a decomposition into many finite-R' states. Finally, an observation of duality breaking is described: in the initial state the roles of ν and λ are undifferentiated, and through the splitting into finite-R' cells and the minimal conjugate width, the frequency side differentiates into dense internal states while the wavelength side differentiates into a sparse extension. This paper does not replace or modify standard physical theory; it does not derive spacetime, mass, energy, interactions, the Standard Model, quantum gravity, or a grand unified theory, and it does not identify 137 with the fine-structure constant. The λn and νn treated here are model variables for observing the dual geometric structure of wavelength space and frequency space, and the conclusion is limited to the observation that the minimal conjugate cell, internal state capacity, volume gap, finite-R' decomposition, duality breaking, and curved self-similar hierarchical structure can be read in a chained manner within the same observational model. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (SVG).