Which sig threshold is reasonable?

Occasionally, sigs or sig limits are shifted by $\log\frac{K}{2}$, $K$ denoting the number of time series data points. Which sig threshold is the true one?

In fact both versions are correct, but they apply to different questions. The version without $\log\frac{K}{2}$ refers to the probability that an amplitude level (a peak) at a given frequency and phase occurs by chance. The version including $\log\frac{K}{2}$ corresponds to the probability that the highest out of $\frac{K}{2}$ independent peaks occurs by chance. According to the sampling theorem, the DFT of $K$ data points (a system with $K$ degrees of freedom) produces $\approx\frac{K}{2}$ independent frequencies in Fourier space, if the sampling is equidistant. Although there is no explicit prescription where to find a set of independent frequencies for non-equidistant sampling, the system will still have $K$ degrees of freedom, and the statistical considerations will be essentially the same.

A simple experiment makes the situation clearer: we roll a dice and obtain the result ``4''. The probability that that such an experiment returns at least ``4'' is, of course, $50$%. This refers to the examination of an individual peak without respect to all the others in the spectrum. If we roll $10$ dices, the probability for at least one showing ``4'' or more is dramatically higher, namely $>99.9$%. This refers to examining the highest out of $10$ peaks. The increasing probability of obtaining such a result by chance corresponds to a decreasing significance of the result.