I recently noticed (while playing around) that the product of a Laver matrix with a Hadamard matrix gives a very sparse matrix. In particular, all but logarithmically few rows are all zero. The nonzero rows all seem to occur near powers of 2, or multiples of large powers of 2. For instance, multiplying the 256x256 Laver table by the 256x256 Hadamard matrix, the only nonzero rows are

```
{1, 2, 4, 8, 9, 16, 32, 33, 64, 65, 96, 97, 128, 129, 160, 161, 192, 193, 224, 225, 256}
```

Is this a known phenomenon? Is there an explanation for it, other than the high periodicity of the Laver table?

In particular since all current proofs of the unbounded growth of the period of the first row of a Laver table rely on strong cardinal axioms, I wonder if understanding this phenomenon could help us understand it.