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A fun math aside, on the idea of splitting a large zk proving workload between multiple provers.

Suppose you have N provers, and you have a proving workload that you split into N parts (so, one part per prover). You require provers to pre-register, but registration is open-access.

Suppose you have a constant fault rate (eg. 1/5 of registered provers fail). Provers expect to complete in one round (eg. 3s). If one prover fails, other provers have to come in and re-prove that load. How many rounds does it take for the entire workload to get proven?

Answer: log*(N)

(yes, that's the iterated-log function)

Why:

In the first round, you go from N unproven workloads to N/5 unproven workloads

In the second round, each remaining workload gets assigned 5 provers, so per-workload failure rate becomes 1 in 5^5. So you go to N / 5 / 5^5 unproven workloads

In the third round, each remaining workload gets assigned ~5^5 provers, so failure rate is 1 in 5^(5^5). So you go to N / 5 / 5^5 / 5^(5^5) unproven workloads

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