Every account credited alongside Ballard, sorted by one question: could it trade places with another account without anything looking different?
We took every Instagram post we've captured where Ballard shares the credit with other accounts: 492 posts, 172 accounts. The answer splits them cleanly. 73 are one of a kind, like the Sun and the Moon. 99 are interchangeable, like Starlink satellites in the same shell: swap any two and the constellation looks exactly the same. Ballard, his foundation and his own projects are the Sun and Moon. Most of the accounts credited beside him are satellites.
The math comes from a branch of algebra called group theory, and its key word happens to be orbit. The astronomy picture fits better than you'd expect.
Something no swap can move without being noticed. There's only one Sun; swap it with anything and you'd see at once.
A set of things that can be swapped among themselves with nothing changing. Thousands of identical satellites in one shell: swap any two and the network works the same.
The test is the name-tag test from our earlier page: if two accounts swapped name tags, would the record of who was credited with whom read any differently? If not, they're in the same orbit. SageMath, free software mathematicians use, checks every possible swap at once across a sign-in sheet of 172 accounts by 492 posts.
These accounts each have a pattern of credits nobody else shares. The structure depends on them.
Ballard is on every post because we chose his collaborations. What matters is who else is irreplaceable: his own foundation, his own projects, and a short list of regulars, Latter Day Chad among them.
Every large batch is made of accounts that appear exactly once, plugged into posts with the same shape. Any of them could have filled any of those seats.
On our earlier page we studied just seven group posts and found a trio of accounts, H, I and J, that could be swapped freely. This time Sage got all 492 posts, with no hint, and found the same three accounts forming their own orbit. A small input and a large one, the same answer.
Shuffling the satellites within each batch, in every combination, gives about 3.3 × 10³⁰ rearrangements that leave everything looking the same. That's far more than the number of stars in the observable universe. The batch of 14 alone can be ordered about 87 billion ways. In plain terms: across Ballard's collaborations, an enormous number of seats could have been filled by anyone in the same batch. Only 73 accounts carry real structure.
Online accounts usually get judged one at a time: does this one look real? This method looks at the shape of the whole network instead. It ignores what any account posts or says. It only asks who is credited with whom, then lets algebra sort the irreplaceable from the interchangeable. It joins three fields that rarely meet:
| It shows | It doesn't show |
|---|---|
| which accounts the network depends on (73) | who runs any account |
| which accounts are interchangeable seat-fillers (99, in 28 batches) | why anyone posts |
| that one-off guests arrive in batches with identical slots | that interchangeable means fake by itself |
| that a larger, independent run reproduces the H, I, J result | anything outside the posts we've captured |
Software agreeing is not the same as experts checking the reasoning. Corrections welcome.