Hidden coordinates and selection rules in the Fourier entropy-influence conjecture

For a Boolean function $f$ and a coordinate $i$, the erasure entropy $E_i(f)$ is the conditional entropy of the event $i\in U$ given $U\setminus\{i\}$, where $U$ is drawn from the spectral distribution $\hat f^2$. The Fourier entropy-influence conjecture is equivalent to every non-constant $f$ having a coordinate with $E_i(f)\le c\,\mathrm{Inf}_i(f)$ for an absolute $c$. We study a family of monotone functions with a hidden coordinate: a variable that enters only through one point below a majority threshold. Its ratio $E_i/\mathrm{Inf}_i$ tends to infinity, yet the sum of the erasure entropies over all coordinates is at most twice the total influence, and at most five times the total influence after every other variable is replaced by a self-dual monotone function on fresh variables. These functions show that no rule that picks the coordinate of smallest influence, of largest influence, of largest or of smallest normalised level-two Fourier weight, or the variable lying in the most shortest minterms, can serve in the equivalence, even for monotone functions. We also show that the two singleton identities of monotone functions do not by themselves bound the erasure entropy, and we give the exact maxima of $\min_iE_i/\mathrm{Inf}_i$, $H/I$ and $\sum_iE_i/I$ over all monotone functions of at most seven variables.