The goal of the design is to give the network a means of automatically gauging how decentralized the full node distribution is. It's essentially a 'proof of UTXO_set storage'. Some parts of it got inspiration from gmaxwell's 'proof of storage' concept, found here:

https://bitcointalk.org/index.php?topic=310323.0

The idea:

Incentivize users to attach a 'proof of UTXO_set storage' with their txs, by charging users that don't a higher tx fee.

The method by which a user creates this proof is to first create a unique 'signed UTXO_set' for every single privkey with which they want to sign Bitcoin txs. This would require the user to create and store multiple versions of the UTXO_set - as many as the number of privkeys they want to create bitcoin txs with, and this is one of the reasons this proposal is impractical.

A signed UTXO_set is a hash tree where every leaf is signed by the same privkey.

To provide a proof of UTXO_set storage, the user attaches to their Bitcoin transaction a truncated merkle root of the signed UTXO_set generated with the same privkey used to sign the Bitcoin transaction. Every single signature in the tx requires a corresponding truncated merkle root for the transaction to escape the higher tx fee.

The hash of the block in which the tx is first confirmed, and the tx hash, are used as inputs to a RNG function that generates an R that is used to select one of the leaves in the UTXO_set. In the following block, the user must present a merkle branch linking the signed UTXO to the previously presented merkle root, to prove they had signed the UTXO with the same private key used to sign the tx at the time that the tx was confirmed. If they do not provide a valid merkle branch, they forfeit the UTXO_set_proof bitcoin deposit they attached with their tx.

To ensure users are keeping their UTXO_set up to date, at every block N at which a difficulty adjustment occurs, the valid UTXO_set becomes N-1440. This means users have ~2 weeks to generate new 'signed UTXO_sets' at every difficulty adjustment period, so that they are ready use when the next diff adjustment happens.