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, extracted first from Li-type settings, 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. 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 (Z), \ rather than only for sequences \ (\Yₙ\\). This permits a uniform treatment of different kinds of RH-equivalent data. As fully worked examples, we treat the Li coefficients both in their direct zero-sum form and in their \ (\) -binomial representation. We also discuss the Bombieri--Weil positivity framework in a scalar-valued reformulation parallel to the Li setting, by treating the admissible test function as an index and absorbing the universal quantifier into the criterion-bearing predicate. In this form, the unresolved tail again appears on the zero side, while the global positivity requirement is carried by the full family indexed by admissible test functions. Finally, we indicate Tur\'an-type criteria as a plausible further testing ground for the same structural mechanism. 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.