An Unconditional Proof of the Unboundedness Conjecture in OEIS A358200

This paper establishes an unconditional proof of the unboundedness conjecture recorded in OEIS A358200, concerning the frequency rankings of ratios of consecutive prime gaps. Let \(p_n\) denote the \(n\)-th prime, let \(d_n=p_{n+1}-p_n\), and define \(r(n)=d_{n-1}/d_{n-2}\) for \(n\ge3\). Using Pintz's theorem that consecutive prime-gap ratios satisfy \(\limsup_{n\to\infty}r(n)=\infty\) and \(\liminf_{n\to\infty}r(n)=0\), we prove that infinitely many distinct ratios occur. Consequently, the OEIS sequence \(a(n)\) has infinitely many first-occurrence terms satisfying \(a(n)=n\), establishing \(\limsup_{n\to\infty}a(n)=\infty\). Furthermore, infinitely many new record-high and record-low ratios occur, strengthening the original conjecture. The proof is unconditional and requires neither the Riemann Hypothesis nor any other unproved conjecture. The result is a deduction from established prime-gap theorems rather than a claim of a new prime-gap estimate.