In this paper, we show that if we have a sequence of Hadamard triples {(Nn,Bn,Ln)} with Bn⊂{0,1,..,Nn−1} for n=1,2,..., except an extreme case, then the associated Cantor-Moran measureμ=μ(Nn,Bn)=δ1N1B1⁎δ1N1N2B2⁎δ1N1N2N3B3⁎...=μn⁎μ>n with support inside [0,1] always admits an exponential orthonormal basis E(Λ)={e2πiλx:λ∈Λ} for L2(μ), where Λ is obtained from suitably modifying Ln. Here, μn is the convolution of the first n Dirac measures and μ>n denotes the tail-term.We show that the completeness of E(Λ) in general depends on the “equi-positivity” of the sequence of the pull-backed tail of the Cantor-Moran measure ν>n(⋅)=μ>n((N1...Nn)−1(⋅)). Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of {ν>n}. This result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research, whose focus has been specifically on #Bn≤4.Using the Bourgain's example that a sum of sine can be asymptotically small, we show that, in the extreme case, there exists some Cantor-Moran measure such that the equi-positive condition fails and the Fourier transform of the associated ν>n uniformly converges on some unbounded set.