Paper 11: The Necessity of Dimension — Both-Endpoint Theorem, Survival Interval [0,4], Mod 8 Selection Theorem, Developmental Selection, and Four-Legs Theorem

In the preceding papers of the reciprocal dual model, the spatial dimension d = 4 (the lattice Z4) was an input. This paper organizes, as several mutually independent selection mechanisms, the extent to which d = 4 is necessary under the three assumptions (νλ = 1, the zero point 1/2, and the asymmetric one bit). First, the both-endpoint theorem: the condition for the half-wavelength censorship that protects composites (Paper 9) is, via the closed form of the maximal anharmonic shift of the container spectrum, Δν1(d) = (√d − 1)2/2, equivalent to Δν1(d) ≤ 1/2 ⟺ d ∈ [0, 4], and saturation (equality) holds exactly at the two endpoints d ∈ {0, 4} only. For d ≥ 5 the shift of the lowest mode exceeds half a quantum, a continuous decay channel opens, and no stable composite can exist. The interior (0 < d < 4) is stable with genuine margin, and this is the existence condition for a developmental path that "grows from zero with 4 as the ceiling." Second, the mod 8 selection theorem: since the dressed label of a d-axis lattice satisfies 4s = Σ(2|ki|+1)2 ≡ d (mod 8), the label is an integer only for d ≡ 0 (mod 4), and it is an odd integer (the foundation of the classification theorem for single states, Paper 5) only for d = 4 (d = 8 yields even labels and single states disappear). Sectors that can carry records (an integer ledger) are restricted to 4 axes — a second selection mechanism independent of the censorship ceiling. Third, fixing the normalization: hierarchical relativity forces the censorship criterion to be level-local (the ladder unit of each composite itself), and the comparison ladder ℓ + (d−1)/2 is the unique canonical equally spaced ladder by the exact identity of the spherical spectrum ωℓ2 = L2 − ((d−1)/2)2. The remaining normalization freedom reduces to the single condition of the curvature coupling ξ of the wave operator, and we take minimal coupling ξ = 0 as the default (we also make explicit that with conformal coupling the dimension selection disappears). Fourth, the four-legs theorem (conditional necessity): if a stable composite exists, then its state space must be discrete (the discrete spectrum of a compact container), bounded without boundary, of positive curvature, and with d ≤ 4. Flat infinite and negatively curved spaces have continuous spectra, hence no discrete ladder for the censorship to protect, and cannot support stable composites. Fifth, developmental selection and the selector from below: since the capacity at fixed budget S grows rapidly with dimension as N(d)(S) ≈ Vd Sd/2, dimensional growth is driven by the same counting principle as splitting. On the other hand, the actualization of an axis carries a zero-point cost, giving the budget bound d ≤ 4S, and the only dimension in which the minimal cell (pure zero point) coincides with the dual fixed point s = 1 is d = 4 (the self-dual seed). The ceiling from above (censorship) and the selector from below (self-duality) coincide at d = 4. These constitute a convergence of several independent selection mechanisms; this paper makes explicit the preconditions of each mechanism and identifies the remaining tasks for removing dimension from the inventory of axioms. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus two figures (PNG).