Large finite irreducible matrix semigroups with two generators

For every integer m ≥ 1, we construct two zero-one matrices of order 20m whose semigroup under ordinary multiplication is finite, consists of zero-one matrices, is irreducible over the rationals, and has at least binom(2m,m)^m elements. One generator is idempotent and the other has nilpotency index five. Consequently, if M₂(n) denotes the largest cardinality of a finite irreducible semigroup of n × n rational matrices generated by at most two matrices, then limsup log M₂(n)/n² ≥ log(2)/200 as n tends to infinity. The construction combines independently writable balanced subsets, a matrix-unit irreducibility argument, and a five-layer encoding. The results are presented as candidate mathematics. The originality assessment is provisional, and the full 1978 Kim–Roush article has not been obtained or read.