Paper 5: The Cell–Standing-Wave Dictionary and the Quantization of the Radius — The Wave Identity of 4-Dimensional Lattice Counting, Structural Identification of the Zero Point, the Coherence Condition, and a Classification Theorem for R²

For the fully-inscribed cell count N₀(R) on the 4-dimensional lattice treated in the preceding Papers 1–4 (N₀(1) = 1, N₀(2) = 9, N₀(3) = 137), this paper establishes three rigorous results. First, the dictionary theorem: the composite-frequency cutoff obtained by adding a per-axis zero-point shift +1/2 to the real scalar standing-wave basis on the unit 4-torus T⁴ coincides exactly with N₀(R). That is, lattice cell counting and standing-wave mode counting are not two separate explanatory mechanisms but dual representations of one and the same structure. Through this identification, the minimal conjugate width δmin² = 1/2, treated as an illustrative value in Paper 4, is structurally identified as the zero-point quantum. Furthermore, the leading coefficient 16π/3 of the volume gap closes quantitatively as the Weyl deficit of the effective boundary generated by the zero-point shift (exactly twice the Dirichlet boundary of the unit box). Second, the coherence condition: the representative value rmax² of a cell takes only odd integer values (by mod 8 arithmetic and Jacobi's four-square theorem), and the condition for a splitting to participate in the counting reduces to all parts being odd. Under this condition, the advantage of splitting with respect to the volume-gap indicator eliminates all four exceptions that existed for unrestricted partitions, and becomes an exception-free theorem (exhaustive check of the 890 odd partitions). A Z₂ selection rule follows as well, by which the parity of the number of fragments coincides with the parity of R². Third, the classification theorem for R²: the realizable R² are only the odd integers for single states and only the positive integers for composite states (subject to the Z₂ selection rule); non-integers are not realizable. Once this quantization is accepted, the closed inequality (full inclusion of the boundary shell) on which N₀(3) = 137 depends is forced intrinsically rather than being a choice of normalization, and the value 105 appearing in the symmetric smoothing limit is definitively positioned as a "diagnostic value of the continuum embedding." In addition, we show that the non-overlap of cells is the same fact as L² orthogonality, and that the shell sequence has a closed form connected to Jacobi's divisor-sum function. All results of this paper are accompanied by exhaustive computer verification or machine-precision numerical verification, and reproduction scripts are provided in a public repository. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (PNG).