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
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)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、教育版 Kovacs 効果に関する予測、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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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 — 2026-02-19 04:21:34 UTCに更新
バージョン
改版理由
数理的完成度の向上、物理分野での新たなテーマの追加、新しい概念としてのTAD地形学の提示を行った改訂版である。変更点は以下の通りである。 第6章「TADの構造拡張と普遍的性質」 ⇒TADの基本設定に内在する暗黙の前提(公理的整理)を追加 第7章「TADによる事象の再現(予測)・逆設計への応用」 ⇒「逆設計における目的関数最適化(変分法)」の節に最小相対エントロピー正則化(測度KL)を追加し、全体的に再構成 第8章「物理学の既存理論への接続」 ⇒「物理学におけるTADの第一原理と第二原理」節を追加し、これまでの知見を再構成 ⇒「生物物理学への応用:タンパク質ターンオーバー」節に「生成・分解結合による自己維持型ターンオーバー」小節を新たに追加 ⇒「酵素拡散増大(Enzyme Enhanced Diffusion:EED)のTADによる再現と解釈」節を新たに追加 ⇒「時間発展する揺らぎ関係と定常極限」節を新たに追加(Jarzynski恒等式、第二法則型不等式、揺動散逸関係等) 第13章「自然言語モデル(LLM:Transformer)への応用」 ⇒「検証プロトコルと実装上の位置づけ(試行可能性の整理)」の小節を新たに追加 第17章「TAD地形学」 ⇒新章として新規追加 第18章「全体総括と将来展望」 ⇒「将来展望:理論的深化と学際的展開」節を新しい知見とともに再構成 ⇒「TAD理解のための直感的メタファー」節を新規追加ライセンス
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