Mathematical Principles of Information Dynamics ——The Universe as a Natural Philosophy of Automatic Control

Abstract

Why are mathematical conjectures—the Riemann Hypothesis, the Kakeya Conjecture, P vs NP—so extraordinarily difficult to solve? For centuries, countless mathematicians have tried to dismantle them using “manual deduction”, only to hit a wall. The author argues that the root cause is: these conjectures are inherently not “manual” but “automatic”. Behind them lies the same dynamical structure—the self‑organising evolution of an information field. Traditional mathematical tools attempt to capture a dynamic, closed‑loop feedback process with static logical chains, much like trying to drive an automatic car with a manual gearbox. This paper proposes a new cross‑disciplinary framework: Information Dynamics. Its core is the generalised Ginzburg–Landau equation, whose four operations (diffusion, anti‑diffusion, nonlinear compression, logarithmic potential) form the atomic instruction set of universal self‑organisation. By faithfully embedding this equation into the category of nonlinear automatic control, we translate the three great conjectures into standard control‑theoretic properties: Riemann Hypothesis ⇔ passivity (positive realness) of a control system; Kakeya Conjecture ⇔ zero measure of the reachable set; P vs NP ⇔ polynomial stabilisability. Significance for Physical AI:This work not only provides a new language for mathematical conjectures, but also directly gives birth to a new paradigm: Physical AI. Traditional AI (including deep learning) requires massive labelled data and backpropagation—it is “manual driving”. Physical AI, in contrast, lets the information field evolve autonomously under the GL equation toward a target state, without any training—it is “autonomous driving”. Prototype experiments, such as the prime density generator, the five‑dimensional single‑point Kakeya set, and linear‑time DNA assembly, have already validated the feasibility of this paradigm. Physical AI promises to become a general problem solver, directly handling images, video, sequences, and beyond, initiating a revolution from “computation” to “generation”. Traditional algorithms adopt a search paradigm, often with exponential complexity. Physical AI provides a control paradigm: encode the problem’s state space as an initial distribution of the information field, then let the GL equation automatically evolve as a closed‑loop feedback system towards a steady state. Information Dynamics defines the physical dynamics of information — that is, how the information field itself, as a physical entity, driven by specific laws (the generalized Ginzburg–Landau equation), spontaneously evolves from disorder to order, generating complex patterns, structures, and knowledge. It answers the question: How can orderly structures and mathematical truths emerge from the quantum vacuum? This paper is not a final proof, but a research programme that can be made rigorous. All assumptions (Hilbert–Pólya conjecture, existence of a continuous limit, etc.) are explicitly stated. Code and experimental data:The numerical experiments (prime density generation, five‑dimensional Kakeya set, DNA assembly) are distributed across several GitHub repositories of the author: Riemann Hypothesis information‑dynamics proof: https://github.com/hkaiopen/Riemann-ID Kakeya set GL construction: https://github.com/hkaiopen/Kakeya-ID DNA assembly: https://github.com/hkaiopen/ComputationalBiology-ID Because the code is scattered across multiple actively developed sub‑projects, no single archive is provided on Zenodo. Please visit the links above for the latest versions.