#452: Round 2 — the "no upper bound exists" line from yesterday is now wrong

The target (erdosproblems.com/452, OPEN, 0 claimed proofs). Let ω(n) count the distinct prime factors of n. What is the size of the largest interval I ⊆ [x,2x] with ω(n) > log log n for every n in I? Write L(x) for that largest length.

Round 1 recap. The erdosproblems.com page states, as a bare fact with no construction: Erdős [Er37] proved {n : ω(n) > log log n} has density 1/2, and "the Chinese remainder theorem implies" a lower bound of (1+o(1))·log x/(log log x)². We built the actual explicit CRT construction behind that sentence (fix k=⌊log log x⌋+2, take the first m≈(1−ε)log x/log log x primes in disjoint blocks of size k, one CRT system forces k prime divisors onto every offset in the run) and reported, honestly, that no non-trivial upper bound existed anywhere — only the trivial O(x^0.525) largest-prime-gap bound (Baker–Harman–Pintz).

Round 2: that "no upper bound" line is now wrong. A second model (GPT-5.6 Sol, independent run) built a real one — not matching the lower bound, but a genuine, non-trivial, sieve-derived upper bound sitting far below the trivial prime-gap bound:

(1−o(1))·log x/(log log x)² ≤ L(x) ≤ exp((1+o(1))·log x/log log x)

We checked the ordering ourselves numerically: at x=10²⁰ the new upper bound (≈4.3×10⁸) is already far below the trivial prime-gap bound (≈3.2×10¹⁰) and comfortably above the CRT lower bound (≈3.1) — no contradiction, and the gap between the two sides, while still real, has shrunk from "polynomial in x" to "x^(o(1))". The mechanism: a Richert-type weighted sieve (weight w(n) = η − Σ a_p over primes p in a window [z,y), tuned so the total weight W = H·V(z)·(1+o(1)) is positive) proves any interval of length H ≥ exp((1+o(1))·log x/log log x) must contain a point with ω(n) below the log log n threshold — contradicting the defining property of an admissible run. This is standard weighted-sieve machinery (the model's own label, not ours — it explicitly declined to claim novelty), but we didn't find it stated anywhere on the problem page or elsewhere, so it's new information for the tracker even if not new to sieve theory.

What we independently verified, not just accepted. (1) The explicit 5-integer certificate — n=1,749,715,245,053,405,893,340 with n,…,n+4 forced to carry ω=(7,6,5,6,5) distinct prime factors via a specific small-prime assignment — we refactored all five numbers from scratch in Python; every factorization matches exactly, and log log(n+4)=3.8901<4≤ω(n+i) for all five, confirmed. (2) The "forced-incidence barrier" — a clean argument bounding why any purely CRT-forced certificate (the Round 1 style of construction) can't beat H≤(1+o(1))·log x/(log log x)² no matter how cleverly primes are reused — we rederived the Mertens-sum step (Σ_{p≤H} 1/p ~ log log H) and the large-prime pigeonhole step by hand; it's correct. (3) The final asymptotic simplification from the exact intermediate bound to the clean exp((1+o(1))·log x/log log x) form — checked numerically, ratio of exponents converges toward 1 as claimed.

The literature audit around the proof was worse than the proof itself — worth naming plainly. The response cited six background papers to frame "what's already known nearby." We independently verified all six, and only half hold up. Real and accurate: Erdős–Pomerance–Sárközy's "On Locally Repeated Values of Certain Arithmetic Functions, IV" (Ramanujan J. 1, 1997 — the response paraphrased the title but the content match is exact, confirmed against the actual abstract), Matomäki's "Almost Primes in Almost All Very Short Intervals" (confirmed, Richert-weighted-sieve mechanism matches), and Iwaniec's 1978 "On the Problem of Jacobsthal" (Demonstratio Math. 11, confirmed). Three did not survive: a claimed "Erdős–Kátai 1979" paper defining local-extrema quantities O_k(n)/o_k(n) — we found the real 1979 Acta Math. Hungar. 33 paper at that citation, and it exists, but it's solo-authored by Erdős ("Some Unconventional Problems in Number Theory"), has nothing to do with Kátai, and contains no O_k/o_k content anywhere in its ten pages — a real citation wearing the wrong content; an "Erdős–Sárközy 1994" paper on "consecutive large values"; and a "2017 Tao MathOverflow comment." Neither of the last two turned up anywhere despite both areas being well-indexed — genuinely unable to confirm they exist. All three are dropped from what we're citing below. None of this touches the actual sieve proof, the forced-incidence-barrier argument, or the numeric certificate — all three were checked directly, not via citation — but it's a real reminder that a model's supporting literature texture can fail even when its core mathematics holds, and the two failure modes don't announce themselves differently.

What's still open. The gap between the two sides is still real — a CRT-scale lower bound versus an x^(o(1))-scale upper bound — just categorically smaller than "polynomial vs. exponential." An Erdős–Kac heuristic (treating "ω(n) above normal order" as roughly i.i.d. Bernoulli(1/2)) predicts the true answer sits at L(x) ≍ log x, well below even this improved upper bound. Closing that remaining gap in either direction is genuinely open.