This paper systematically deduces the core mathematical structures of stochastic analysis starting from two fundamental principles: the Information Conservationaxiom (A1) and the Finite-Step Computability axiom (A2). Research shows thatcontinuous-time stochastic processes exemplified by Brownian motion, Itô stochastic integrals, stochastic differential equations, and path integrals (Feynman integrals) are not ad hoc mathematical tools for describing random phenomena in thephysical world, but rather inevitable mathematical emergences of underlying discrete, probabilistic information processes under macroscopic continuous limits. Weprove that the intrinsically generated symmetric Bernoulli branching process withinthe A1/A2 framework, via scaling limits (Donsker’s invariance principle), uniquelyconverges to standard Brownian motion; in order to define dynamics on this process, the construction of the Itô integral becomes a necessary choice under the dualconstraints of causality and the continuity of the martingale structure; while theoperation of summing over all possible historical paths according to their information weights naturally leads to the path integral formulation in the continuouslimit. This paper further proposes quantitative experimental test schemes basedon the ”volatility smile” in financial options markets and the ”sign problem” inquantum Monte Carlo simulations, thereby establishing a falsifiable bridge betweenstochastic mathematical structures and the physical world.