The Asymptotic Resolution Dichotomy as a Structural Obstruction to RH Proofs

We formulate an abstract obstruction principle for proof strategies toward the Riemann hypothesis that proceed by finite resolution, truncation, enrichment, or surrogate reconstruction of a global criterion-bearing object. The guiding observation, introduced first in the Li/\(\eta\) setting in The Asymptotic Resolution Dichotomy for the Li criterion, is that the essential difficulty is not merely quantitative. Rather, unresolved RH-violating structure remains available in the tail unless it is excluded by additional input of RH strength. To capture this phenomenon, we introduce the *asymptotic resolution dichotomy* (ARD). In its weak form, it states that any finite explicit resolution faces a structural alternative: either RH-violating structure is already visible at the resolved level, or the unresolved remainder still leaves open the possibility of criterion-destroying configurations. In its strong form, it states that any surrogate which appears to succeed where finite explicit resolution cannot must derive its additional force from hidden tail control, rather than from finitely resolved data alone. We develop the framework at the level of general target objects $Y=V(\mathcal Z),$ rather than only for sequences \(\{Y_n\}\). This permits a uniform treatment of different kinds of RH-equivalent data. As fully worked examples, we treat four RH-equivalent settings: the Li coefficients in their direct zero-sum form and in their \(\eta\)-binomial representation; the Bombieri--Weil positivity framework in a scalar-valued reformulation parallel to the Li setting; the Nyman--Beurling--B\'aez-Duarte $L^2$-approximation criterion; and the Speiser criterion for the zero-free half-plane of $\zeta'$. All four settings instantiate the same structural obstruction in analytically distinct realizations. Finally, we indicate Tur\'an-type criteria as a further prospective testing ground. The purpose of the paper is not to refute individual estimates one by one, but to isolate a general obstruction pattern. A global RH-equivalent object cannot, in general, be certified by finite resolution or surrogate compression alone. Whenever such a passage appears to succeed, the missing force must come from additional control of the unresolved remainder.