How does these two correlates or talk? I know its api but thats not my question. Here is the scenario, I built my Umbrel node and then install public pool for solo mining. I point my miner and I can see that the pool hashrate is already in the EH or PH wherein I only have 1 miner. I can also see the difficulty. So, questions are , 1) Is the pool difficulty in my solo public-pool dictated by the main Public Pool? 2) Does my solo public pool submits the share to the main Public Pool? 3) Who dictates the range of nonce in my pool? I know my miner will have the entire range of nonce to use but how does it sync with the main Public Pool or the bitcoin network so that there is dup nonce range?
No, the Bitcoin Blockchain does that! It is the rate the blockchain is being mined by all Pools!
Mempool will show you a nice graph:
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So, the pool(what pool is that) total hasrate is 846.8 and the bitcoin network has set a difficulty of 141.6T, isnt the pool is greater than the difficulty and should be hitting blocks?
No, a New Block is found when you find a block with a greater difficulty than 141.6T. Hashrate is just the total rate of the miners that are mining in that Pool or in this case of my Pic above, the total Hashrate of the whole Blockchain.