FGG conjecture for QAOA on the ring of disagrees

The FGG conjecture asks whether, for even cycle size $N$ with $2p+2\leq N$, depth-$p$ QAOA achieves exactly the ratio $\frac{2p+1}{2p+2}$. Farhi, Goldstone, and Gutmann predicted the value numerically and proved the case $p=1$.

Known results

• FGG proved $p=1$.
• Wang et al. proved $p=2$ using an exact fermionic analysis.
• Mbeng, Fazio, and Santoro derived the general momentum-mode decomposition and numerical saturation through $p=128$, but did not prove attainability when $2p+2\leq N$.

Claimed machine-verified proof

The manuscript claims a universal lower bound and an explicit polynomial-interpolation construction attaining residual energy $\frac{1}{2p+2}$, thereby proving the conjecture. It says Claude Fable 5 generated the decisive argument; independent mathematical confirmation is not yet reported.