T+\varepsilon$ the $2-$core has density $\ge 1+ \varepsilon.$ $T$ is determined as the solution of an analytical equation.

The expected number of solutions of the $2-$core is $d^{n-m} ,n$ the number of variables, $m$ the number of equations. When the $2-$core has density $\ge 1+ \varepsilon$ whp. no solution exists. This holds in particular when the density of $F(n, p)$ itself is $\ge 1+\varepsilon.$ The formulas considered here always have density $<1.$ In seminal work Dubois and Mandler [@DUMA] consider equations $\mod 2:$ $x_1 \, +\, \dots \, + \,x_k=a,$ $0 \le  a\le 1, k=3.$ They show satisfiability whp. when the $2$-core has density $\le 1-\varepsilon.$ For larger $k\ge 15$ a full proof for this result is given in [@MITZ], Appendix C . Thus $T/n^{k-1}$ is the threshold for unsatisfiability in this case.

It is a natural conjecture that the same threshold applies to equations as discussed initially (and to some other types.) However, it seems difficult to prove the conjecture in some generality. One of the difficulties seems to be that we have $2$ parameters $k$ and $d.$ We make some progress towards this conjecture. We show it for equations $\mod 3. $ (The result is for $k>15,$ but 