The Natural Constant e: A Complete Derivation of the Topological Origin of Spontaneous Evolution in Polar Gradient Fields

Project: W≡0 Global Topological Unified Field Theory Construction ProjectStage: Phase 4 · Mathematical Preliminaries Layer Abstract In classical mathematics, the definitions of the natural constant via limits and exponential self-consistency are formal inductive statements lacking support from first principles at the topological ontological level, and cannot explain its innate necessity as a global continuous evolution scale. Based on the W≡0 global conservation axiom and the polar gradient flow evolution equation locked in Phase 3 and restated in the Phase 4.01 toolkit, this paper completes a fully closed-loop derivation of the topological origin of within the pure mathematical framework of Phase 4. This paper supplements the topological symmetry reduction from the global gradient flow equation to the first-order linear evolution equation, and completes compliant simplification based on the linear response setting of the locked system. It adopts a primitive construction path of "defining logarithms by definite integrals and exponential functions by inverse functions" to completely eliminate the risk of circular reasoning. Through Euler discretization, a rigorous correspondence between continuous differential systems and discrete recursive sequences is established, and a complete proof of the monotonic bounded convergence of the sequence is supplemented. The strong causal proofs of number-theoretic properties are revised to interpretations of topological ontological consistency, strictly observing the boundary specifications of Phase 4. DOI errors in references are corrected, and terminology and formatting are unified throughout the text. The results show that is the unique intrinsic base for the continuous evolution of polar gradient fields, and its existence, uniqueness, and limiting value are uniquely determined by global conservation constraints and topological evolution rules. This paper finally achieves equivalence and self-consistency between the topological origin definition and all classical mathematical definitions, perfecting the trinity mathematical system of innate global constants 0, , and , fully meeting the permanent locking standard for the mathematical preliminaries layer of Phase 4. Keywords: natural constant ; polar gradient flow; topological evolution; differential equation; primitive construction of exponential function; limit convergence; topological ontological interpretation