An addition to this awesome idea:
Fragments can be used to open a d1sk from ANY season (excluding partnerships, event d1sks, and current season d1sks), this gives utility to assets collecting dust and would encourage people to connect with Illuvium Beyond.
Combine 'X' Illuvitars or 'X' accessories to be broken down into a fragment (this self balances the value of the fragments and d1sks being given away), and gives extra utility to an overabundance of non desirable illuvitars (similar to Aarons idea of burning illuvials in the future to create dust).
To keep development for this idea cheap and fast to integrate, the fragments should be given at the end of each run as a non tradeable currency that would appear in your inventory (not taking up any inventory slots, just show the #), like @twin locust said, 1 d1sk fragment for s1, 2 for s2, and 3 for s3.
Finally, d1sks should be redeemable from Sanctum Mesa, where the user can setup/change their PFP without leaving the game (In-game Illuvidex at Sanctum Mesa would go hard af), and redemption shouldn't count for any type of points, milestones, or airdrops.
This agile, efficient idea absolutely rocks, we just have to find the numbers that make sense for all the mentioned value exchanges. Shoutout @grave bone , I hope you start putting your ideas out there more. Very well spoken and easy to understand.