The Asymptotic Resolution Dichotomy in Li's criterion

The Bombieri--Lagarias decomposition writes the Li coefficients in the form \[ \lambda_n = \bar\lambda_n + \tilde\lambda_n, \qquad \tilde\lambda_n = -\sum_{j=1}^n \binom{n}{j} \eta_{j-1}, \] where the trend $\bar\lambda_n$ is explicit, while the oscillatory part is a binomial transform of the local coefficients $\eta_k$ of $-\zeta'/\zeta$ at $s = 1$. Motivated by work of Ma\'slanka and Coffey, many attempts to approach the Riemann Hypothesis through Li's criterion seek to bound $\tilde\lambda_n$ by estimating the auxiliary sequence $(\eta_k)$. We show that this class of strategies is subject to a structural barrier at three levels. First, a \emph{closure principle}: the exact evaluation of the binomial sum for $\tilde\lambda_n$, using the power-sum representation of the $\eta_k$, reproduces precisely the Bombieri--Lagarias formula---exact computation goes in circles and produces no new information. This result is the theoretical complement to the obstruction theorems that follow: while the closure principle concerns exact evaluation, the remaining results concern approximate evaluation. Second, a \emph{finite-enrichment barrier}: if RH is false, then a finite stage either already contains an off-critical zero, or some deeper finite enrichment does. Once such a zero is resolved, the corresponding finite Li-mode sum contains exponentially growing modes along an infinite subsequence. Thus non-detection at any fixed finite stage cannot certify global Li positivity. Third, the \emph{Asymptotic Resolution Dichotomy} (ARD): the explicit finite enrichments converge uniformly to the complete coefficient sequence $(\eta_k)_{k\geq1}$, yet this uniform coefficient accuracy is not preserved under the family of Li binomial transforms. A high unresolved zero may be uniformly tiny in every $\eta_k$ while becoming exponentially large at sufficiently high Li index. The missing condition is therefore not merely greater coefficient accuracy but global control of the unresolved spectral tail. In the natural modal formulation, exclusion of all expanding unresolved modes is exactly RH on that unresolved spectrum. Sections~7--8 illustrate this distinction using estimates of $\eta_k$ and Stieltjes constants.