Compare your observed correlation against the domain's background dependence, not against zero
Both tests use the Fisher transformation \(z = \operatorname{atanh}(r)\), which has sampling variance \(\approx 1/(n-3)\).
Classical test asks: is \(r\) different from zero?
$$z_{\text{classical}} = \frac{|\!\operatorname{atanh}(r)|}{\sqrt{1/(n-3)}}$$Crud-aware test asks: is \(r\) unusual relative to the domain's background?
$$z_{\text{crud}} = \frac{|\!\operatorname{atanh}(r)|}{\sqrt{\sigma_{\text{crud}}^2 + 1/(n-3)}}$$The denominator of the crud-aware test includes both sampling noise and background variance. A correlation must exceed the crud scale to be declared significant, not just exceed zero.
The crossover sample size where sampling variance equals crud variance is:
$$n^* = \frac{1}{\sigma_{\text{crud}}^2} + 3$$Below \(n^*\), both tests behave similarly. Above \(n^*\), the classical test starts flagging crud as "significant."