Is it true that for every $k$ there exists $n$ such that$$\prod _{0\le i\le k}(n-i)\mid \left(\genfrac{}{}{0ex}{}{2n}{n}\right)?$$

Erdos #396: verified REDUCTION. For every k, the conjecture (exists n with prod_{0<=i<=k}(n-i) | C(2n,n)) follows from long consecutive runs in the governor set G={m : m | C(2m,m)}. Chain (GPT-Pro, Opus-verified): Kummer per-prime form prod|C(2n,n) iff for all p sum_i v_p(n-i)<=C_p(n); carry-invariance lemma (q odd prime, q|m, 0<=j<=(q-1)/2 => C_q(m+j)=C_q(m)) [verified 0/3000 violations]; small-prime barrier (in a k-witness every block term passes the governor test at primes q>=2k+1); reduction proposition (n..n-k in G + finite small-prime check => n in E_k); the small-prime set S_k has density 1 (Pomerance Lemma 2), so small primes are NOT the obstruction. Density: E_k subset {n: P+(n),P+(n-1)<=sqrt(2n)} giving upper-log-density <= (1-log2)^2 ~ 0.094 (Teravainen, arXiv 1710.01195) < governor density 0.11425 (Ford-Konyagin, arXiv 1909.03903) -- so #396 is a genuinely correlated-pattern problem, not the k=0 governor problem. Data corrected to OEIS A375077 (witnesses k<=14; k=5,6,7 = 3648841,7979090,101130029 independently re-verified). The carry-invariance lemma + barrier are independently formalized in the Dehorty Lean project (github jdehorty/erdos396). OPEN CORE: prove G contains runs of length k+1 for every k (Ford-Konyagin give positive density but not runs).

{"reduces_to":"consecutive runs in G={m | C(2m,m)}","verifier":"scripts/verify_396_reduction.py","carry_invariance":"verified 0/3000","witnesses":"A375077 k<=14; k=5,6,7 re-verified","governor_runs...

depends on (the wall): scripts/verify_396_reduction.py

scripts/verify_396_reduction.py + citation checks (OEIS/arXiv/GitHub)