As far as I understand this "protocol", you have to register an element, i.e. a pattern (e.g. a sequence of numbers in the blockchain) in an Ordinals inscription and then you will be able to build a token upon it. Again afaik, the supply which can be minted depends on the occurrance of the pattern registered as an "element".

I may be wrong but it seems that if you discover that in the current block the pattern associated to the "element" is present, you can mint a certain number of the token. This limits the supply somewhat but everybody can still mint the tokens.

Reading this $NAT minting guide and this post on X it seems indeed that the token minting process isn't even related to mining; instead everybody can try to "mint a block" with an Ordinals inscription referencing the $NAT pattern. Thus it seems the natgmi site is misleading, the tokens seem not to be directly credited to miners but instead some tokens somehow are "redirected" from minters to miners (they write literally: "directing newly generated $DMT-NAT tokens to miners who win blocks" on natgmi.com). In no place of the website this "redirecting" process is explained in detail.

In any case, the "Digital Matter Theory" thingy seems to be just another technobabble "project" to try to pull out some more value off the BRC-20 protocol and Ordinals. It is not necessary in any way to credit tokens to miners, this can be achieved completely without Ordinals, simply creating a "minting" process you can only accomplish if you have mined a recent block and sign a message with the same private key than the address you mined to (which is easy, but may require extensions of current token protocols).