An Archimedean Diagonal Non–Coverage Principle for Anchored Indexed Systems

We study canonically anchored congruence systems on the index axis, in which residue classes are uniquely determined by arithmetic divisibility and admit no freedom of choice. While such systems are known to be non–covering on the infinite index axis, many arithmetic problems involve finite intervals together with involutive symmetries that pair indices diagonally. We introduce an archimedean refinement of the anchored framework showing that finite truncation and inversion do not restore coverage. Using a square–root threshold linearized in index space, we prove that only finitely many congruence moduli can obstruct a given finite interval. We then show that if a finite family of canonically anchored congruence classes were to cover such an interval, periodic extension would force coverage of the entire index axis, contradicting global non–coverage. As a consequence, we obtain a diagonal non–coverage theorem: for every finite interval and every involutive pairing of indices, there exists at least one diagonal pair that avoids all canonically anchored obstructions below the square–root threshold. The result is purely structural, relies only on anchoring, exact CRT structure, and archimedean finiteness, and is independent of any particular arithmetic application.