Why Geometric Closures Cannot Certify the Collatz Conjecture A Hidden-State Obstruction via Factor Maps

This record provides a theoretical analysis identifying a structural obstruction common to geometric and topological approaches to the Collatz problem. We show that the accelerated Collatz map is inherently non-invertible, and that any model defined solely on the visible integer coordinate necessarily constitutes a non-injective factor of the full dynamical system. As a consequence, geometric closures, invariant curves, or apparent periodic structures observed in such factor systems cannot logically certify global convergence or the absence of nontrivial cycles in the original dynamics. The obstruction is formalized using the language of factor maps and extensions, with explicit identification of the missing hidden state (symbolic itinerary / inverse-limit information). This work does not propose a proof of the Collatz conjecture. Instead, it delineates a sharp boundary between illustrative geometric intuition and proof-grade reasoning, clarifying what any future successful approach must explicitly address.