How much data does it take to prove a given transaction exists on the blockchain? As I understand the SVP protocol, a client can be reasonably certain that a transaction has been accepted on the blockchain if they know it is a member of a certain block, via the Merkle path, and a reasonable number of blocks have been mined on top of that block.

How much total data would that be, roughly?  I guess if you were beginning with 0 knowledge, you would need all of the block headers, beginning from the genesis block?  But once you are convinced you are looking at the head of the right chain, you could discard all of the ancient block headers, and just save a recent one as a trusted starting point to evaluate subsequent transactions?



I guess the premise of my question is faulty, as a fake transaction could always be embedded in a fake chain, if enough computing resources were available.  So, there is no amount of data that insures a transaction is valid, without access to other information, e.g., what is the longest chain currently in existence.

A better question for my purposes would be how much would the electricity cost to create one fake block header at today's difficulty level?  Figuring that blockchain currently pays around $60K to mine one block, I would guess it must be in that order of magnitude, though smaller since we are removing the race condition.
In theory, you would only need the data from the block of the transaction, and the ability to prove it the transaction still exists after at least 6 confirmations.

The more confirmations (blocks mined beyond a transaction) the lower probability it becomes possible to reverse. Generally speaking, 6 confirmations is enough to identify a transaction is irreversible.

Here's a quote from Mastering Bitcoin by Andreas Antonopoulos:


  By convention, any block with more than six confirmations is considered irrevocable, because it would require an immense amount of computation to invalidate and recalculate six blocks. 


Block vary in size, but in SegWit transactions, the entire block can be a maximum  ~2 MB.

These days there are many online tools that can verify this without needing the entire blockchain.
The bitcoind JSON RPC API also works fairly well in providing transaction data. (Cannot be manipulated like with an online API)