A Kernel-Checked Exclusion Certificate for Erd\H{o}s Problem 647

Abstract

Erd\H{o}s problem 647 asks whether any $n > 24$ satisfies $\max_{m<n}(m + \tau(m)) \le n + 2$, where $\tau$ is the divisor-count function. Computational searches have excluded solutions up to $10^{12}$ by direct sieve and up to roughly $9.17 \times 10^{18}$ within a modular reduction whose Lean component relies on native_decide; those computations sit outside any proof kernel. We give the first exclusion checked end to end by one: no solution exists with $24 < n \le 10^9$, proved in Lean 4 with axiom closure exactly {propext, Classical.choice, Quot.sound} -- no sorry, no native_decide, no problem-specific axiom. The proof replays a chain of 6,685,922 factorization witnesses whose excluded intervals concatenate across $(24, 10^9]$; it needs no primality facts beyond primes below 1024, and it is the finite, fully proved form of a domination-interval argument whose asymptotic step was the identified gap in a withdrawn January 2026 claim on this problem. The generation pipeline is cross-checked by two further independent implementations, the compiled development replays through the standalone lean4checker, and two from-source verification legs -- Lean toolchains compiled from source by gcc and by clang, mathlib rebuilt with no cache -- reproduce the committed certificates byte for byte, with olean digests identical across three builds on two architectures. Our range is three to ten orders of magnitude below the computational frontiers we cite; the contribution is the trust base, not the range.