Transaction Cut Through

Transaction Cut-Through, or Merging Transactions Across Blocks, is one of the excellent features of Mimblewimble protocol, it contributes to a compressed blockchain history. To quote the original here:

Now, we have used Dr. Maxwell's Confidential Transactions to create a noninteractive version of Dr. Maxwell's CoinJoin, but we have not seen the last of marvelous Dr. Maxwell! We need another idea, transaction cut-through, he described in [8]. Again, we create a noninteractive version of this, and to show how it is used with several blocks.

We can imagine now each block as one large transaction. To validate it, we add all the output commitments together, then subtracts all input commitments, k*G values, and all explicit input amounts times H. We find that we could combine transactions from two blocks, as we combined transactions to form a single block, and the result is again a valid transaction. Except now, some output commitments have an input commitment exactly equal to it, where the first block's output was spent in the second block. We could remove both commitments and still have a valid transaction. In fact, there is not even need to check the rangeproof of the deleted output.

The extension of this idea all the way from the genesis block to the latest block, we see that EVERY nonexplicit input is deleted along with its referenced output. What remains are only the unspent outputs, explicit input amounts and every k*G value. And this whole mess can be validated as if it were one transaction: add all unspent commitments output, subtract the values k*G, validate explicit input amounts (if there is anything to validate) then subtract them times H. If the sum is 0, the entire chain is good.