A general theory of what quadratically sharded validation *is*

Thinking of compatibility for extending to exponential/super-quadratic sharding, may you please talk to that? So you have an SMC in each shard, and a threshold committee(s) e.g. like the linked-committees in signature-linked binary trees above, where the threshold committees are in each shard except for the bottom one (to aggregate the votes from the committee in the shards one level below it).