Formulation and Scale Verification of "Effective Output" Based on Information Thermodynamics — Mathematical Anatomy of Organizational Cognitive Limits, the 5 Phases of Complexity, and Phase Transitions —

[English Abstract] This study elucidates the "productivity paradox"—where despite the elimination of information generation costs due to advancements in AI technology, maintenance and coordination costs in modern organizations increase non-linearly, leading to stagnant productivity—from the perspectives of information thermodynamics and linear algebra. We redefine labor and decision-making as energy conversion processes and formulate a "complexity penalty" in dimensions of watts (W) based on Landauer's principle and the human biological metabolic limit (20W). In response to the explosive increase in input information volume (N), we derive five thermodynamic phase transition phases through which a system progresses from optimal equilibrium, through a "parasitic dissipative structure," to functional cessation. We mathematically prove that the macroscopic optimal organizational scale follows P_opt ∝ N^(2/3). Furthermore, in the micro-geometric section, individual skills are redefined as projection operators from high-dimensional tensors to low-dimensional manifolds. We formulate the explosive increase in dissipation heat caused by the "phase difference (orthogonality)" between the unconstrained output of AI and the highly localized cognitive manifolds of experts, proving the paradox that more highly skilled experts exceed their metabolic limits earlier, triggering a cascade of organizational collapse. This paper presents the geometric requirements of orthogonalization of information structures and prior dimensionality reduction as the sole conditions for saving the system from thermodynamic collapse (vacuum decay). [日本語要旨] AI技術の進展により情報の生成コストが消失したにもかかわらず、現代の組織において維持・調整コストが非線形に増大し生産性が停滞する「生産性パラドックス」を、情報熱力学および線形代数学の観点から解明する。本研究は、労働や意思決定をエネルギー変換プロセスとして再定義し、Landauerの原理と人間の生体代謝上限(20W)に基づく「複雑性ペナルティ」をワット(W)の次元で定式化する。入力情報量(N)の爆発的増大に対し、系が最適均衡から「寄生的な散逸構造(回避経済相)」を経て機能停止に至る5つの熱力学的相転移フェーズを導出し、マクロな最適組織規模が P_opt ∝ N^(2/3) に従うことを数理的に証明する。さらにミクロ幾何学編では、個人のスキルを高次元テンソルから低次元多様体への射影演算子として再定義する。AIの無制約な出力と専門家の極在化した認知多様体間の「位相差(直交性)」が引き起こす散逸熱の爆発的増大を定式化し、高度な専門家ほど早期に代謝限界を超過して組織崩壊のカスケードを引き起こす逆説を証明する。本論は、系を熱力学的崩壊(真空崩壊)から救うための唯一の条件として、情報構造の直交化と事前次元削減の幾何学的要件を提示する。