How to factor 2048 bit RSA integers with less than a million noisy qubits

The 2025 revision cited on Shor's algorithm, beside the 2021 Gidney and Ekerå estimate. The abstract says a 2048-bit RSA integer could be factored in less than a week by a quantum computer with less than a million noisy qubits. It states the same physical assumptions as the 2019 arXiv version of the earlier estimate: a square grid of qubits with nearest-neighbor connections, a uniform gate error rate of 0.1 percent, a surface-code cycle of 1 microsecond, and a control-system reaction time of 10 microseconds. This is a cost-model preprint, not an experiment that has factored a 2048-bit integer.