Reframing P Vs. NP via Structural Descent: SBI, Alankar Chains and the Emergence of Temporal Complexity Theory

This paper introduces Temporal Complexity Theory (TCT), a generalization of classical computational complexity in which problem difficulty is modeled as a function of evolving knowledge states rather than fixed representations. Building on the Structural Descent Framework (SDF), we formalize problem solving as a trajectory through a dynamically evolving theory-space, where admissible representations, invariants, and transformations are updated via Structural Bayesian Inference (SBI) and operationalized through Alankar chains. Within this framework, we define three complementary notions of complexity: absolute complexity, corresponding to classical worst-case cost; knowledge-conditioned complexity, which depends on the current knowledge state; and retrospective complexity, capturing the minimal cost of solution from a structurally optimal representation. This decomposition explains the pervasive empirical phenomenon of hindsight simplicity, wherein historically difficult problems become trivial once appropriate structural representations are discovered. A central contribution is the introduction of the coordination invariant, a representation-dependent measure of irreducible coupling among problem constraints, formalized via graph-theoretic, communication, and circuit complexity perspectives. We show that this invariant lower-bounds computational complexity and governs the emergence of epistemic plateaus (resolvable through structural updates) and ontic plateaus (potentially irreducible). From this perspective, the P versus NP problem is reframed as a question of global coordination compressibility: whether all instances admit representations in which constraint interactions can be reduced to polynomial structure. While this reformulation does not resolve the problem, it isolates the structural hypothesis underlying computational hardness and provides a unified account linking complexity theory, learning dynamics, and scientific discovery. More broadly, TCT establishes a principled framework in which computation, representation, and knowledge evolution are treated as a single coupled process, shifting the study of complexity from static analysis to dynamic structural accessibility.本文介绍了时序复杂度理论（Temporal Complexity Theory，TCT），该理论是经典计算复杂度的推广，其将问题难度建模为演化知识状态的函数，而非固定表征的函数。本文基于结构下降框架（Structural Descent Framework，SDF），将问题求解形式化为一条穿过动态演化理论空间的轨迹，其中可容许表征、不变量与变换通过结构贝叶斯推理（Structural Bayesian Inference，SBI）进行更新，并通过Alankar链付诸实践。在此框架下，我们定义了三种互补的复杂度概念：绝对复杂度，对应经典的最坏情况开销；知识条件复杂度，取决于当前知识状态；以及回溯复杂度，刻画了从结构最优表征出发求解问题的最小开销。这一分解解释了普遍存在的事后简化经验现象：即历史上棘手的问题，一旦发现合适的结构表征，便会变得极易解决。本文的核心贡献之一是引入了协调不变量（coordination invariant），这是一种与表征相关的问题约束间不可约耦合程度的度量，通过图论、通信复杂度与电路复杂度的视角进行形式化定义。我们证明，该不变量为计算复杂度提供了下界，并支配了认知平台期（epistemic plateaus，可通过结构更新解决）与本体平台期（ontic plateaus，可能不可约）的出现。从这一视角出发，P与NP问题（P versus NP problem）被重新表述为全局协调压缩性的问题：即所有实例是否都存在可将约束交互简化为多项式结构的表征。尽管这一重构并未解决该问题，但它分离出了计算难度背后的结构假说，并为关联复杂度理论、学习动力学与科学发现提供了统一的阐释框架。更广泛地说，时序复杂度理论构建了一个原则性框架，将计算、表征与知识演化视为单一的耦合过程，将复杂度研究从静态分析转向动态结构可及性研究。