Prove the conjectural finite-rank formula for the opposite-fixed ratio at alpha equals 2

Prove the finite-rank identity $\mathcal R_{2,N}^{(\eta)}=((2N+1)(6N-7))/((4N-5)(4N-1))$ for the character-inserted pinned $C_N$ ensemble, and thereby establish its thermodynamic limit $\mathcal R_2^{(\eta)}=3/4$.

Background

The exact character-insertion representation reduces the opposite-fixed ratio to a discrete Selberg average, but the paper does not derive a closed all-rank evaluation of that inserted sum. Exact-arithmetic calculations verify the displayed rational expression through $N=15$.

The formula would imply the thermodynamic value $3/4$ and the corresponding defect contribution to the stabilizer Rényi entropy. Its status remains conjectural because verification at finite ranks has not been replaced by an analytic proof.