A Local-Global Dichotomy in Li's Criterion for the Riemann Hypothesis

The Bombieri--Lagarias decomposition writes the Li coefficients in the form\ₙ=ₙ+ₙ, ₙ=-₉=₁ⁿ nj₉-₁, the trend ₙ is explicit, while the oscillatory part is a binomial transform of the local coefficients ₖ of -'/ at s=1. Motivated by work of Maslanka and Coffey, many attempts to approach the Riemann Hypothesis through Li's criterion seek to bound ₙ indirectly by estimating the auxiliary sequence (ₖ). We show that a broad class of such -based strategies is subject to a structural barrier. In the zeta case we use the decomposition ₖ=Tₖ+Zₖ, where Tₖ is the explicit odd-integer contribution and Zₖ is the regularized power sum over non-trivial zeros. Finite enrichment means that Tₖ is taken exactly and that one adjoins the contributions of an arbitrarily large, but finite, symmetry-closed set of zeros. Our first main result is a closure, or route-invariance, principle: independently of whether enrichment is carried out through the coefficients ₖ, through zero power sums, or through equivalent generating-function data, it projects exactly to a finite truncation of Li's zero-sum representation\ₙ=_ (1- (1-1) ⁿ), the remaining uncertainty is canonically the complementary Li tail. Thus finite enrichment never produces a new structural object; it only removes finitely many Li modes. Our second main result is a barrier theorem. If the Riemann Hypothesis fails, then for every finite enrichment the complementary tail contains an off-critical symmetry quartet whose contribution grows exponentially along an infinite subsequence. Consequently no finite enrichment---even one incorporating the full explicit odd-integer part and arbitrarily many zeros---can yield the uniform control of ₙ required by domination strategies of the form ₙ |ₙ|. We formulate this as an asymptotic resolution dichotomy: either enrichment explicitly reveals an RH-violating mode, or the unresolved tail remains logically decisive. In particular, any black-box approximation that systematically outperforms explicit enrichment must already encode hidden tail control, hence information of essentially RH strength.