The Closed Loop of Precision and Its Blind Spots: Geometric Measures and Topological Residues in Quantum Mechanics

Quantum mechanics has achieved a prediction precision of one part in a trillion in local experiments (e.g., the electron anomalous magnetic moment ), but this extremely brilliant precision does not point to ontological transparency; instead, it is accompanied by long-standing measurement difficulties and a deadlock at the interface with quantum gravity. This paper points out that the precision of quantum mechanics originates from the fact that the "observation instrument" and the "theoretical model" share the same geometric slice, thereby forming a geometric closed loop. Within the local validity domain inside this closed loop, the sheaf property ("sheafiness") holds to a high degree; and its difficulties correspond precisely to the necessary failure of the sheaf property at the boundary of the closed loop (). This paper argues that the measurement problem and the conflict with quantum gravity are not "the theory has not yet been calculated correctly," but topological manifestations of the inability of the geometric slice to be glued at the boundary. By introducing the perspectives of sheaf theory and homotopy theory, we characterize this failure as topological aliasing caused by projection, and point out that the erased high-dimensional information is locked in the projection as homotopy kernel () residues. In order to completely avoid the historical predicament of "hidden variables" in single-layer ontology, this paper proposes a two-layer ontology, reconstructing the high-dimensional information as "topological residue." The path to breaking this predicament is not to patch the local geometry, but to establish a topological diagnostic framework targeting the boundary blind spots—DHT (Defect–Homotopy–Topos).