Understanding Taproot Formulas

Can someone help me get a grip on taproot formulas? So, if we have a funding transaction that sends to a taproot address like [OP_1 push_bytes_32 KEY], and then a spending transaction that includes the witness part with [param] [param] ... [SCRIPT] [CBLOCK] Is there an easy way to connect SCRIPT, CBLOCK, and KEY? Like, would it be something along the lines of SHA256(SCRIPT || CBLOCK) == KEY?

The Control Block specifies a key and merkle branch hashes. You hash the script to get a leaf hash which is then combined with the merkle branch hashes until you get the root hash, which is applied as a tweak to the internal key in the control block. There's no simple formula. It's all specified in BIP 341 in the Script Validation Rules section.

OK, let us take the simplest P2TR transaction ever encountered in main/testnet/etc chain 0) Take transaction txid which spends output xxxx:0 1) Take the taproot script from witness-part on input, hash it sha256. We got the a1=yyyy 2) ? 3) ?

1. Check the control block is 33 + 32m bytes long, in this instance, it should be exactly 33 bytes. 2. Set leaf_version to be the first byte of the control block bitwise and with 0xfe 3. Extract p as the 2nd to 33rd bytes of the control block. 4. Set P by interpreting p as the x coordinate for a point on secp256k1 where the y coordinate is even. 4. Compute the hash kof the script as follows: sha256(sha256("TapLeaf") || sha256("TapLeaf") || leaf_version || compact_size(script) || script), where || is concatenation 5. Compute t as sha256(sha256("TapTweak") || sha256("TapTweak") || p || k) and interpret it as an integer. 6. Let Q = P + t * G where G is the generator for secp256k1 7. Check the x coordinate of Q matches the key in the output script 8. Check that the first bit of the first byte of the control block matches the parity of Q's y coordinate: control_block[0] & 1 == y(Q) mod 2 When the script tree has multiple possible scripts, there's more in the control block and hashing steps involving combining a merkle branch to calculate a merkle root.