The Purpose of the Object
A formula was proven before anyone knew what it counted — and the question, once found, had to be asked in order before this paper could answer it.
Notes from the frontier.
Cut to the Same Measure
Corrected, the higher Dyson systems meet an object from Jacobi's theory at exactly the same elliptic law — and the correction is finite enough to write down in full.
The Weight Comes Last
The leading terms cancel modulo q, and the degree left behind is settled by whether an addition in base q carries — the reduction Evan Chen and Ken Ono use to prove Thakur's three hypotheses, open since 2009.
The Address Before the Room
One family is a third the size of the other — but that third is a doorway, not the matching that has to be built behind it.
The Same Ground
In the right coordinate, Kaprekar's process is not a different puzzle in every base — it is one operation: doubling.
The Figure and the Remainder
A multiplicity that reads as the blur of cancellation comes into focus, in natural families, as an exact count of bounded Dyck-path structures.
The Reveal
In the hardest window for lattice triangles, a density-1 result rules out almost every candidate.
Infinitely Rare
When can Ramanujan's tau function take a prime value? Almost never.
Axplorer
Democratizing the search for mathematical constructions.
Releasing AXLE
Our Infrastructure for Mathematical Reasoning at Scale.
Proof of Concept
An autonomous, verified proof on an open conjecture linking syzygies to Ramanujan’s identities
Primes: Almost surely, partially, regular
A 350-year-old question, a new theorem, and a proof written by a machine
