{"schema":"vela.problem-packet.v0.1","problem":1094,"statement":"For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.","status":"open","seam":"sealed","closureRoutes":[{"type":"witness","verifierKind":"binom_exception_enum","note":"extended exception enumeration, frozen-verified"},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_2d4edfce58578092","banked":"the 14 ELS exceptions enumerated provably-complete to k<=40 (binom_exception_enum)","open":"extend the complete enumeration / prove finiteness of the exception set for all k.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_6b897cfb0c6ce7b8","kind":"reduction","claim":"Erdős #1094 (LPF C(N,k) <= max(N/k,k) for N>=2k): UNCONDITIONAL explicit cutoff + finite-certificate decidability, Opus-verified. Slot machinery: g_{k,r}=gcd(lcm(1..k), r*C(k,r)), G_k=max_r g_{k,r}. Lemma 4 (verified on all 14 ELS exceptions + 400 random Kummer-defined cases, 0 violations): if C(N,k) is Kummer-defined then the k-smooth part a_r of every top-block term x_r=N-k+r divides g_{k,r}. Lemma 3: every counterexample is Kummer-defined and its UNIQUE multiple-of-k slot x_{r0} is k-smooth, so x_{r0}|g_{k,r0} and N=x_{r0}+k-r0 <= G_k+k-1. THEOREM: N > G_k+k-1 => LPF(C(N,k)) <= max(N/k,k) (no counterexample), with G_k <= k*C(k-1,floor((k-1)/2)) ~ 2^{k-1}sqrt(2k/pi) — an explicit, prime-distribution-free certificate recovering the ELS93 2^k sqrt(k) phenomenon. For each fixed k, #1094 is decidable by a finite divisor search (choose r0, x|g_{k,r0} with k|x, set N=x+k-r0, test the top-block certificate). The genuine residual is the Kummer-defined positive-deficiency region below G_k where every nonsmooth cofactor b_r is a prime > max(N/k,k) — a prime-cofactor lemma, NOT a #1093 deficiency lemma. Supersedes att_bbe2fa7f9f61c08b. Credits ELS93/ELS88.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_ac7890f6b3bf663a","kind":"verified_witness","claim":"Erdős #1094: PROVABLY-COMPLETE finite verification — no exception beyond the 14 ELS for k<=60, Opus-confirmed. Overnight Codex enumerated ALL candidates via the divisor certificate (x|g_{k,r}=gcd(lcm(1..k),r*C(k,r)), k|x, N=x+k-r, no N-cap) for k<=60: 2,216,413 candidates, 89 Kummer-defined, 0 extras beyond the 14 ELS exceptions. Opus independently reproduced k<=40 with a from-scratch enumeration: exactly the 14 ELS in range, 0 extras, 0 missing. STRONGER than a range-capped search (provably complete, not N<=bound), but it EXTENDS the verified range only; it does NOT close #1094 (the full theorem still needs the prime-cofactor lemma for k>k0).","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_2f6ea04fc508e9a5","kind":"reduction","claim":"Erdős #1094: the prime-cofactor lemma reduces to a named ANCHORED semismooth-count bound, Opus-verified. Unconditional inputs: Kummer-defined => prod a_i=k!, each a_i|lcm(1..k); Granville-Ramare => N>exp(c(log^3 k/loglog k)^{1/2})>k^2 (large k) so y=N/k; a composite nonsmooth cofactor would have a prime factor <=N/k (else b_i>(N/k)^2>N), so the lemma reduces to ruling out every nonsmooth b_i being 1 or a single prime >N/k; deficiency d<=log(k!)/log(N-k+1)=(1+o(1))k log k/log N (verified (N-k+1)^d<=k! on all 14 ELS). The lemma FOLLOWS from S(N,k)=#{i: x_i=a*q, a<k, q>N/k prime} << k log k/log N (anchored to Kummer-defined positive-deficiency blocks): then k<=S+d<=(C+1+o(1))k log k/log N, contradicting Granville-Ramare since k log k/log N <= (k/c)(loglog k/log k)^{1/2} -> o(k). The OBSTRUCTION: standard Sylvester-Schur/Selmer/Brun-Titchmarsh do NOT prove this; the UNANCHORED version is FALSE (N=L_k t+k gives x_i=i*(L_k/i t+1) with cofactor>N/k for all i<k, so k-1 slots can be a*q form — verified). The proof must use the divisor anchor (some x_{i0}|L_k or |G_k). Sharpened obstruction map; conditional cutoff k0 derivable if the anchored bound holds.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_d12f184a5c5389d6","kind":"reduction","claim":"Erdős #1094 Range B (2k<=N<k^2) is UNCONDITIONALLY CLOSED for all sufficiently large k, Opus-verified. In Range B max(N/k,k)=k, so #1094 fails iff C(N,k) is good (Kummer-defined). By Konyagin's #1095 bound g(k) >> exp(c(log k)^2) > k^2 eventually, there is a finite k0 with no good C(N,k) for N<k^2, k>k0 — no abc needed. Exact Kummer local density Delta_k <= exp(-(log2/2+o(1))k/log k) (primes k/2<p<=k, delta_p=(1-1/p)(2-k/p)) verified numerically, so k^2 Delta_k -> 0. Residual: the short-interval CRT discrepancy in [2k,k^2) (modulus exp((2+o(1))k) >> k^2, the level barrier) is not supplied by standard sieve. Corrects the digit-domination detail (ell_p=1+floor(log_p k); p>sqrt k needs N mod p>=b AND floor(N/p) mod p>=a). (62,6) is outside Range B; Range B holds 13 of the 14 ELS exceptions. Partial theorem.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_bbe2fa7f9f61c08b","kind":"verified_witness","claim":"Erdős #1094 (least prime factor of C(n,k), n>=2k): the Kummer cheap verifier lpf(C(n,k)) = smallest prime p with k NOT digit-dominated by n base p is verified (0 mismatches vs brute force, n<150). Exhaustive search (k<=200, n in [2k,20000], ~3.9M binomials) finds EXACTLY the 14 ELS88 exceptions and no others; a second independent method (direct factorization) reproduces exactly those 14 over the box k<=30,n<=320. Largest exceptional n=284 (k=28); no exception for any k>=29 (to k=60). Obstruction map: an exception forces R1 (small primes p<=k: k digit-dominated by n in every base) AND R2 (middle primes k<p<=n/k: n mod p >= k); the binding constraint at large n is R1 (the small-prime digit-domination side, tying #1094 to #1095), since every large-n R2-survivor is killed by a prime p<=k. Banked obstruction_map, ELS88/ELS93 prior art.","grade":"obstruction_map","gateStatus":"verified","superseded":true}],"velaLean":[{"file":"lean/Vela/Erdos1094Finiteness.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos1094Finiteness.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}