Odd perfect numbers have at least two prime factors exceeding 10^6: statement and verification record

Statement of a computational theorem with its verification record. Theorem: every odd perfect number has at least two distinct prime factors exceeding 10^6; in particular its second largest prime factor exceeds 10^6. The previous bound on the second largest prime factor, 10^4, is due to Iannucci (Mathematics of Computation 68, 1999). The result uses the published bound 10^8 on the largest prime factor (Goto and Ohno, Mathematics of Computation 77, 2008) and does not depend on the author's separate 10^9 result. Method: assuming exactly one prime factor at or above 10^6, explicit bounds on the exponents of all primes below 10^6 leave finitely many candidate components sigma(q^(n-1)); a counting argument over the large prime factors of these components closes every case (case A by a supply-and-demand count over the special prime, case B by a capacity count over the roots of the components). The computation at 10^6 ended with the verdict that no odd perfect number has exactly one prime factor at or above 10^6; the same method reproduces the bounds 10^4 and 10^5 before being applied at 10^6. Verification: the case analysis was re-verified by four programs written separately from the search, sharing no code with it (case A; completeness of the component lists; the product components; case B), each confirming with zero problems on 8 and 9 October 2026. The deposited sheet records the SHA-256 fingerprints of the component file and the verification logs. The manuscript with the full argument is in preparation; this deposit establishes the date of the result.