プレプリント / バージョン7

TAD理論とその応用

履歴を可視化する数理フレームワーク

##article.authors##

DOI:

https://doi.org/10.51094/jxiv.1593

キーワード:

Time Allocation Dynamics(TAD)、 二時間構造モデル、 履歴依存ダイナミクス、 保存則・エントロピー発展、 変分逆問題・逆設計、 Kovacs効果、 自然言語モデル(Transformer)、 ロジスティック成長、 待ち行列理論、 生物進化論

抄録

本論文は、入力時刻と観測時刻の二時間構造 (t,T) を明示的に導入した履歴分布 g(t,T) を状態変数とする Time Allocation Dynamics(TAD)理論を体系化する。履歴依存型の入出力過程を、単一の保存構造の下で統一的に記述することを目的とし、在庫鮮度モデルを出発点として離散系から連続系へ拡張し、割当・減衰・スケール変換の三機構を含む一般的発展方程式を導出する。さらに、この二時間構造に固有の保存則(会計恒等式)とエントロピー発展式を与えることで、履歴効果を力学的かつ情報論的に同時に扱う枠組みを確立する。加えて、変分法による逆設計問題の定式化およびラプラス解析に基づく再現・予測手法を整備し、TAD の数理的骨格を明確に示す。
本理論の射程を検証するため、物理学、人口動態、教育・記憶、マーケティング、経済理論、自然言語モデル(Transformer)、待ち行列理論、生物進化論に適用し、既存理論の再導出と、履歴変数を中心に据えることで得られる新たな解釈および検証可能な帰結を提示する。とりわけ、ロジスティック成長モデルの作用原理解釈、物理学におけるKovacs 効果の他分野への展開、履歴データベース(TAD-DB)の最小構造と運用プロトコル、ならびに履歴分布の可視化枠組み(TAD 地形学)を提案する。
以上により、TAD は従来個別理論として扱われてきた履歴依存現象を、共通の保存則と発展方程式の下に再編成し、解析・設計・予測を統一的に扱うための基盤理論を提供することを示す。

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本研究は筆者個人の業務外活動として独立に実施したものであり、勤務先を含むいかなる組織からも資金・データ・設備等の提供を受けていません。本稿は一般理論の提示を目的とし、特定企業の業務・製品・見解を代表するものではありません。利益相反はありません。

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公開済


投稿日時: 2025-10-04 12:15:52 UTC

公開日時: 2025-10-28 01:25:43 UTC — 2026-04-02 11:47:05 UTCに更新

バージョン

改版理由

各章の数理的完成度の向上、待ち行列理論及び生物進化論への応用を新たに追加した改訂版である。変更点は以下の通りである。 第1章「在庫鮮度モデルの実務への応用」 ⇒「研究背景と現場課題」「問題設定と在庫配分にもとづく記述枠組み」の小節を追加 第3章「在庫鮮度モデルの実務への応用」 ⇒全体的に再構成 第8章「物理学の既存理論への接続」 ⇒「線形応答理論・生存時間解析・TADの構造対応」の小節を追加 ⇒「境界幅極限と即時割当の描像」の小節を追加 第9章「人口動態モデルへの応用」 ⇒「SIRモデルの一般状態空間(グラフ)上の TAD 結合による再構成」の小節を再構成 第14章「待ち行列理論への応用」 ⇒新章として新規追加 第15章「生物進化論への応用」 ⇒新章として新規追加 第17章「Kovacs効果のTADによる統一的理解:物理から他分野系へ」として再構成 第18章「TAD-DB(履歴データベース)の導入とその応用」 ⇒「TAD-DBの一般化と履歴再構成」の小節を追加
研究分野
学際科学