State the claim. The kernel decides whether it is proved.
A panel of models attacks your problem with the literature, SageMath, PARI/GP and an SMT solver, then formalises the result in Lean 4 against Mathlib. Lean's kernel either accepts the proof or it does not, and no amount of confident prose changes that. When it fails you get the exact goal that remains, which is usually where the informal argument was hand-waving.
A checker that cannot be persuaded
Lean 4 with Mathlib type-checks the final statement, and a proof that leans on sorry, native_decide or a fresh axiom is rejected rather than counted. Alongside it: SageMath, PARI/GP, Z3, CVC5 and the OEIS, so a construction can be computed and identified before anyone tries to prove anything about it.
A failed proof is a finding
When formalisation fails, the goal Lean could not close is handed to whichever seat is best placed to attack it: just that goal, not the whole history. A rejected proof names the gap precisely, which is more than most informal arguments ever do.
Nothing is proved twice
Every lemma the panel establishes goes in a shared ledger with its proof, so it is never re-derived and dead ends are never retried. Long problems pause and resume without losing work: close the tab and come back tomorrow.