One Vessel, Three States — Conditions for the Selection of Three: Layered Three Selected by Vessel Geometry and Clustered Three Selected by Planar Contact(一器三態論――三が選ばれる条件:器の形が選ぶ層の三と、平面の接触が選ぶ束の三)

This preprint investigates conditional mechanisms selecting three in two distinct settings: internal layers in a localized structure and finite clusters on a plane. For the layered case, a three-component field projected onto three p-type vessel modes links vessel geometry to cubic anisotropy. The tested vacuum paths support N_layer = L − 1; a cubic vessel gives A/B = 9/4 < 3 and selects six axial vacua associated with three layers. In a spherical, SO(3)-invariant vessel, explicit fixed-axis cubic anisotropy is forbidden, while a spontaneous l = 4 route and possible additional constrained local minima remain unresolved. For planar clustering, SALR (short-range attraction and long-range repulsion) models favor a broad three-cluster window among the configurations examined; measured non-additive terms in a two-component type-1.5 Ginzburg–Landau model penalize quartets more strongly than trios. Connections to Lotum Field Theory (LFT) are discussed without treating its Jeans-type branch as an established SALR mechanism. These are model-dependent results, not a universal derivation of the number three. Analytical results, numerical support, candidate formulas and untested routes are distinguished. Project-internal AI-assisted reviews do not constitute external journal peer review. 本プレプリントは「三」が選ばれる条件を、局在構造の内部に重なる「層の三」と、平面上に集まる「束の三」に分けて検討します。三つのp型モードへの射影を前提に、器の形と立方異方性を結び付け、検証した真空経路では N_layer = L − 1 を支持します。立方体の器では A/B = 9/4 < 3 が六軸の真空を選び、三層につながります。一方、球形かつSO(3)対称の器では固定軸の明示的な立方異方性は生じませんが、自発的なl = 4秩序や別の制約付き局所極小の可能性は未解決です。平面上のSALR型相互作用では、調べた配置の範囲で三個のクラスターの成立窓が広く、二成分type-1.5 Ginzburg–Landau場で測定した非加算項は三個より四個を強く罰します。LFTとの接続も、未導出の部分を区別して議論します。自然界全体に通用する「三」の証明ではなく、前提を明示した条件付きの解析・数値研究です。 Files: English and Japanese PDFs and Markdown sources, version v0.5 (manuscript dated 22 September 2026). The Japanese manuscript is authoritative where the two language versions differ./日英PDFとMarkdown原稿を収録。言語間の差異がある場合は日本語版を正本とします。