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Autor Principal:	Li, Y.Y.N.
Formato:	Recurso digital
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Publicado:	Zenodo 2025
Acceso en liña:	https://doi.org/10.5281/zenodo.17658705
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We propose a quantitative law of intelligent computation, the \emph{Principle of Least Structural Action} (PLSA). Given a reasoning trajectory $\psi(t)$ produced by a solver, algorithm, or large language model (LLM), we define a structural Lagrangian \[ L(\psi,\dot\psi) = \alpha K(\psi) + \beta S(\psi) + \gamma C(\psi), \] where $K$ is a compression-based incompressibility index, $S$ is structural curvature along the trajectory, and $C$ is a correctness potential that penalizes constraint violations. PLSA asserts that physically realized intelligent systems approximate trajectories that minimize the structural action \[ \mathcal A[\psi] = \int_0^T L(\psi,\dot\psi)\,dt. \] On top of this microscopic principle we identify a computable, Kolmogorov-style \emph{order parameter} for reasoning, \[ \phi(x) := \log\bigl(1+\lambda_K(x)\bigr), \qquad \lambda_K(x) =\frac{K_{\mathrm{solution}}(x)-K_{\mathrm{problem}}(x)}        {K_{\mathrm{problem}}(x)}, \] where $K_{\mathrm{problem}}$ and $K_{\mathrm{solution}}$ are gzip-compressed lengths of the problem encoding and the joint problem--trace encoding, respectively. Here $\log$ denotes the natural logarithm. The quantity $\lambda_K(x)$ measures normalized structural overhead of the reasoning trace relative to the bare problem, and $\phi(x)$ is a smoothed, monotone transform that behaves well across many orders of magnitude. We instantiate PLSA empirically on random 3-SAT using a CDCL solver and LLM chain-of-thought (CoT). On a mixed dataset of $60$ CDCL traces (for $n\in\{20,50,100\}$ and $\alpha\in\{3.5,4.0,4.26,4.5\}$) and $8$ CoT traces (for $n=10$), a simple global Structure--Time Law \[ \log_2 T(x)\approx \alpha_T\,\phi(x) + \gamma_T \] explains a large fraction of the variance in wall-clock runtime. A least-squares fit yields \[ \log_2 T(x)\approx 9.741\,\phi(x) - 12.196, \qquad R^2\approx 0.785. \] Thus a \emph{single} Kolmogorov order parameter $\phi(x)$ accounts for approximately $79\%$ of the log-runtime variance across reasoning systems ranging from an efficient CDCL solver to a token-level LLM CoT engine. We further show that classical 3-SAT hardness phenomena---the phase transition near clause--variable ratio $\alpha\approx 4.26$---admit a clean structural interpretation: the CDCL order parameter $\phi(x)$ exhibits a systematic shift with $\alpha$, decreasing from about $0.16$ at $\alpha=3.5$ to about $0.09$ at $\alpha=4.5$, while the runtime $\log_2T$ increases (becomes less negative), consistent with harder formulas inducing denser, more information-efficient traces. Finally, in a toy bounded-curvature model we prove an explicit exponential lower bound \[ T \;\ge\; 2\sqrt{D}\,e^{K^\*/2}, \] where $K^\*$ is an effective structural hardness and $D$ is a structural distance to the solution manifold. This provides a rigorous example of how structural incompressibility forces exponential time scaling. We conjecture that PLSA and the Kolmogorov order parameter $\phi(x)$ constitute a universal framework for describing reasoning cost, phase transitions, and the emergence of general intelligence across algorithms, LLMs, and biological systems.

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