The erasure entropy of a forced coordinate under every bias

A monotone Boolean function Fk,t, built from a majority function on k inputs with one hidden coordinate x by substituting threshold functions of $2t$ fresh variables, has x as the only variable lying both in a shortest minterm and in a shortest maxterm. The ratio Eₓ/Infₓ of the erasure entropy to the influence at x was known to be unbounded when k and t grow together, and to grow linearly in k for $t=1$. We show that for every fixed t≥1 it equals (1+o(1))log₂(1/Infₓ)=(1+o(1))(k-1)log₂(1/Pₜ), where Pₜ is the probability that a binomial variable with parameters $2t$ and 12 is at least t. More generally the same holds for the underlying majority function under every rational product bias P∈(12,1), the phase of the relevant coefficients advancing by 2P-1/2P, an irrational multiple of π. At P=12 that angle is π2, and the behaviour changes: there the coefficients are values of Krawtchouk polynomials at the middle of their range, half of them vanish, and the ratio is 12log₂(1/Infₓ)-12log₂log₂(1/Infₓ)+O(1), which we derive from an exact single-sum formula.