Reading Action and Path Integrals from Local Evolution

We present a stepwise reconstruction of actions from local evolution equations. After time discretization, terms in the approximate equations prescribe nodal derivatives of an unknown function. Integrating these conditions, summing the interval contributions, and taking the continuum limit shows how the dynamics determines the terms and coefficients of the action. Detailed Newtonian, relativistic, and Schrödinger examples establish the connection between vanishing nodal derivatives and vanishing first variation in every admissible direction. Maxwell theory, the Standard Model with right-handed neutrinos, and general relativity extend the construction to spatial derivatives, coupled fields, and compatibility multipliers. Fourier transformation, split short-time evolution, and Gaussian integration give the Feynman path integral by repeated composition; its phase contains the same h(T − V) as the Newtonian construction. Our contribution is a common sequence of explicit calculations for these established actions and propagation formulas, making the passage from local dynamics to global stationarity and path phases directly verifiable. Accompanying Lean proofs check selected key calculations. AI assistance and verification scope ChatGPT and Codex were used to assist with English translation, checking Lean code, and selected derivations in the Standard Model section. The author developed the research approach and supplied the original Newtonian manuscript, and personally checked the literature and mathematical derivations. The author did not independently inspect the Lean proof code; the reported verification status is based on the execution results obtained during the AI-assisted workflow. Formal verification is limited to the selected statements described in the manuscript. The author takes responsibility for the content of this preprint.