Odd perfect numbers: the largest prime factor exceeds 10^9 and the second largest exceeds 10^6

We prove that every odd perfect number is divisible by a prime exceeding 10^9, improving the bound 10^8 of Goto and Ohno. The proof follows the method of Hagis and Cohen, Jenkins, and Goto and Ohno. Its computational part is organised so that every intermediate claim is either an exact inequality checked in integer arithmetic or a finite list that can be regenerated independently; the implementation reproduces all previously published bounds before being applied at 10^9. We also prove that the second largest prime factor of an odd perfect number exceeds 10^6 (the previous bound, 10^4, is due to Iannucci, 1999), by a finite-component method: exponent bounds for every prime below the target, obtained from tables of cyclotomic values and a lattice lemma, leave finitely many possible divisor-sum components, and a case analysis on the special prime refutes them all. Every computational step of the second proof is re-derived by independently written programs. This manuscript combines and supersedes the records 10.5281/zenodo.23284648 and 10.5281/zenodo.23288365.