Paper 7: NTLA-O VI: Inverse Systems, Observer Tower, Inverse Limit, and Pro-Observer Identity

Inverse Systems, Observer Towers, Inverse Limits, and Pro-Observer Identity — Resolution Histories Beyond Limit Equivalence

NTLA-O series Paper 7. The preceding five papers establish four observational dimensions — Role, Locality, Resolution, Transport — where observation resolution is described by a family of progressively refined kernels. This paper formally establishes Observer Tower Theory: each layer's kernel produces a quotient, kernel inclusion gives a surjection, forming a standard inverse system. The first major result is a natural injection from the quotient of the kernels' intersection to the inverse limit. The title itself states the paper's core thesis: "limit identity does not imply tower identity" — reaching the same limiting kernel does not mean the entire resolution-refinement history (the pro-object) is the same object.