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Combining with Parts I-III, we provide an algorithm and verify the resulting approximation factor satisfies $\\gamma \\le 21 < q/2=1665$ for ML-KEM-1024, with a success probability $\\ge 0.99$. We apply a tower decomposition of the Principal Ideal Problem (PIP) through the chain $\\Q \\subset \\Q(\\zeta_8) \\subset \\cdots \\subset \\Q(\\zeta_{2^k})$ which yields a polynomial-time quantum algorithm costing $O(n^3 \\log^2 n)$ gates, $O(n^2 \\log n)$ qubits, and $\\mathrm{poly}(n)$ classical bit operations. 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