A Rigorous Proof of a Ramanujan Machine Identity for $-\pi \mathrm{/}4$ via Exact Recurrence Solving

We prove a polynomial continued fraction identity for the constant $-\pi \mathrm{/}4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a closed-form expression for the denominator sequence, $q_n=(-1{)}^n(2n-3)!!(n^2+n-1)$, and establish absolute convergence via a Wronskian telescoping argument. The limiting value is reduced by Abel summation to a Beta-function integral, which is evaluated in closed form through an elementary substitution and a single integration by parts, yielding the exact value $-\pi \mathrm{/}4$.