The Divergence Between Mathematics and Physics — A Demonstration of the Inevitability of Flux Conservation Based on Time Field Theory

Mathematics can define metric benchmarks outside a system, while physics never can. This is the fundamental divergence between mathematics and physics, and the logical starting point of Time Field Theory (TFT). Unlike mathematics, which can simply define an external benchmark quantity and finish construction, any physical benchmark quantity is itself a physical process inside the system and governed by physical laws. Therefore, it must satisfy a requirement absent in pure mathematics: after being substituted into the same set of equations, it can solve for itself as well as other physical quantities, and all operation results must be self-consistent. Starting from this fundamental distinction, this paper conducts rigorous proof on two levels of necessity: mathematical necessity — flux conservation is the joint result of path independence and source-free vacuum conditions; physical necessity — any physical benchmark quantity must measure itself in a self-consistent way, and flux conservation is the unique solution of this self-consistency under continuous field conditions. Flux conservation is neither "assumed" nor "discovered", but "deduced". It is the fundamental mark that distinguishes physically observable universes measurable by internal observers from purely mathematical possible worlds.