A power law for the free energy in two dimensional percolation

Consider bond percolation on the square lattice and site percolation on the triangular lattice. Let κ(p) be the free energy at the zero field. If we assume the existence of the critical exponents for the three arm and four arm paths and these critical exponents are −2/3 and −5/4, respectively, then we can show the following power law for the free energy function κ(p): κ(p) = +(1/2− p) for p < 1/2 κ(p) = −(1/2− p) for p > 1/2, where δ(x) goes to zero as x → 0. Note that the critical exponents for four arm and three arm paths indeed are proven to exist and equal −5/4 and −2/3 on the triangular lattice and the above power law for κ(p) therefore holds for the triangular lattice. Note that the above power law for κ(p) implies κ(p) is not third differentiable at the critical point of the triangular lattice. This answers a long time conjecture that κ(p) has a singularity at 1/2 since 1964 affirmatively.