The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra?

Principal manuscript, Theorem 1.1: (i) a finite lattice is representable iff it admits a finite colored-graph witness; (ii) no Turing machine decides representability from the order table, so not every finite lattice is representable; (iii) there is a nonrepresentable lattice of minimum size, larger than 7, specified by a finite description involving a Busy Beaver constant (its size is not computed); (iv) recognizing full subgroup intervals of finite groups is undecidable. The companion constructs a finite lattice that is no interval $[D,G]$ in a finite group, hence (Palfy-Pudlak) a nonrepresentable one. No explicit small counterexample is exhibited.

Principal manuscript, Theorem 1.1: (i) a finite lattice is representable iff it admits a finite colored-graph witness; (ii) no Turing machine decides representability from the order table, so not every finite lattice is representable; (iii) there is a nonrepresentable lattice of minimum size, larger than 7, specified by a finite description involving a Busy Beaver constant (its size is not computed); (iv) recognizing full subgroup intervals of finite groups is undecidable. The companion constructs a finite lattice that is no interval $[D,G]$ in a finite group, hence (Palfy-Pudlak) a nonrepresentable one. No explicit small counterexample is exhibited.