Give their customers a free bitcoin account (web wallet) on the Walmart server, no double spending allowed, like a normal bank account with no automatic overdraft.
Walmart runs a miner node, but any spending from the wallet to Walmart are held in escrow, not cleared on the blockchain immediately. Any other spending is celeared in run time with the regular public block chain.
Whenever the bitcoin transactions to Walmart fill a complete block, it is optimally packed, and Walmart submits it to the public block chain. This is equivalent to a branded custodial bitcoin.
But,but...
That means Walmart has secure elements in its server. Bu Walmart sells hardware walleys, why not cutnoaste that code on the server into the hardware wallets, do it in cooperation with Trezor.
Then, anyone can be a custodial bitcoin holder, spend them all you want your cash card knows not to double soed and any holder of bitcoin transactions can submit them to the block chain at their leisure.
So, the customers spend merchants collect up to a week or month of transactions with deferred calls to the ledger. When they have a batch of transactions, they get a cut rate deal from the regular miners, they are already partially packed.
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