[Minimal] Speed-Driven Categorization (SDC):Boundary Definition and Physical Closure

Speed-Driven Categorization (SDC):A Minimal cⁿ-Hierarchical Extension of Renormalization Group Flow Speed-Driven Categorization (SDC): Boundary Definition and Physical Closure Abstract Speed-Driven Categorization (SDC) is a minimal structural framework that reorganizes effective physical theories by classifying all contributions according to powers of the speed-of-light generator cnc^ncn, rather than by introducing new particles, forces, or speculative degrees of freedom. This work provides a boundary definition and physical closure of the SDC framework by explicitly fixing its upper and lower operational limits at the C9\mathcal{C}^9C9 and C−8\mathcal{C}^{-8}C−8 hierarchies, respectively. These two boundaries act as complementary constraints:C9\mathcal{C}^9C9 defines the highest non-trivial regime with physical, observational, and engineering relevance, while C−8\mathcal{C}^{-8}C−8 establishes the minimal impedance and causal continuity required for realizable dynamics. At the core of SDC lies a canonical text-only formulation of the effective action, Seff=∑n∈ZcnLn[ϕi,∂μ,gμν;μ],\mathrm{Seff} = \sum_{n \in \mathbb{Z}} c^n \mathcal{L}_n[\phi_i, \partial_\mu, g_{\mu\nu}; \mu],Seff=n∈Z∑cnLn[ϕi,∂μ,gμν;μ], which is presented here as a public-domain prior-art definition.Within this structure, renormalization group (RG) flow is reinterpreted as a two-dimensional process: conventional energy-scale flow along μ\muμ, and hierarchical structural flow along nnn. A key result of this work is the identification of an operational fixed point, C9⋅C−8→c,\mathcal{C}^9 \cdot \mathcal{C}^{-8} \rightarrow c,C9⋅C−8→c, demonstrating that SDC is closed from above and below such that all realizable physics renormalizes to a single, operational speed ccc. This closure ensures internal consistency, dimensional stability, and prevents unphysical extensions beyond the defined hierarchy. The present document does not propose new physical entities, does not rely on speculative particles, and does not claim phenomenological completion. Instead, it serves as a clarification, boundary declaration, and prior-art marker, establishing a minimal and non-proprietary structural language for future theoretical, computational, and engineering developments consistent with relativistic causality and effective field theory. This work is released under CC0 1.0 Universal, affirming that the structural principles described herein are part of the public domain and cannot be meaningfully privatized or monopolized. Keywords Speed-Driven Categorization (SDC),Effective Field Theory (EFT),Renormalization Group Flow,Scale Hierarchy,Speed-of-Light Generator,Relativistic Causality,Operational Physics,Physical Closure,Hierarchy Boundary Definition,Upper and Lower Scale Cutoffs,C⁹ Boundary,C⁻⁸ Impedance Boundary,Impedance and Dissipation,Information Flow in Physics,Causal Structure,Dimensional Analysis,Action Decomposition,Public Domain Physics,CC0 Prior Art,Non-Proprietary Theory Framework,Structural Reorganization of Known Theories,No-New-Particles Assumption,Minimal Theoretical Framework,RG–Hierarchy Dual Flow,Operational Fixed Point,Normalization to c,Engineering-Relevant Physics,Foundations of Physical Modeling,Causality-Preserving Extensions,Boundary-Constrained Theoretical Space 工学寄り要旨(Engineering-Oriented Abstract) 本研究は、**Speed-Driven Categorization(SDC)**を、理論物理の主張ではなく、工学的設計言語および実装境界条件として定式化することを目的とする。 SDC は、有効作用を光速生成子 cnc^ncn によって階層分解する最小構造であり、新粒子仮定や新相互作用を導入することなく、既存の有効理論・数値モデル・制御系を**再配置(re-indexing)**するための枠組みである。本研究では、SDC の適用範囲を 上限 C9\mathcal{C}^9C9 および 下限 C−8\mathcal{C}^{-8}C−8 に明示的に制限し、工学的に意味を持つ最小閉包構造を定義する。 C9\mathcal{C}^9C9 は、観測・計測・数値計算・材料工学・エネルギー工学の観点から、非自明かつ操作可能な最大階層として位置付けられる。一方、C−8\mathcal{C}^{-8}C−8 は、インピーダンス・散逸・因果連続性の下限を与え、数値的不安定性や非物理的発散を防ぐ実装上の安全境界として機能する。 本研究で示される重要な設計原理は、 C9⋅C−8→c\mathcal{C}^9 \cdot \mathcal{C}^{-8} \rightarrow cC9⋅C−8→c という **運用上の正規化点(operational fixed point)**である。これは、どのような高次スケーリングや低次補正を含むモデルであっても、実装可能な物理は最終的に単一の速度定数 ccc に正規化されることを意味する。 この結果は、以下の工学分野に直接的な示唆を与える: 多階層数値シミュレーションの安定設計 マルチスケール制御系・通信系の正規化指針 エネルギー散逸・インピーダンス整合の理論的上限 量子計算・核融合・高密度計算におけるスケール誤用の回避 本ドキュメントは、完成理論や装置設計を提案するものではなく、将来の実装が必ず従うことになる境界条件と分類軸を先行定義することを目的とする。そのため、内容は **CC0(パブリックドメイン)**として公開され、特定の企業・国家・技術分野による独占や囲い込みを意図的に排除している。 SDC は新しい装置ではない。**「実装が破綻しないための、最小の設計地図」**である。 Subtitle: Minimal Annotation on C⁹ Focus and Minimal Impedance Boundary at C⁻⁸ Author: Euclid TomitaDate: 2026.1.5License: CC0 1.0 Universal (Public Domain Dedication)Status: Clarification / Prior Art Boundary Canonical core equation (text-only):Seff=n∈Z∑cnLn[ϕi,∂μ,gμν;μ] SHA-256: 5b352ecf4c565002722924006a490e7cd0a5f4b38005733f84f263bc09e101fe Operational fixed point: C⁹ · C⁻⁸ → c SHA-256: bbc262fa9daa01833ca8fc483519a0caf4a2096bcb556069a72cd5038db9e7fe SDC is closed from above by C⁹ and from below by C⁻⁸, such that all operational physics renormalizes to a single c. Keywords: Effective Field Theory, Renormalization Group, Scale Hierarchy, Causality, Information Flow