{"schema":"vela.problem-packet.v0.1","problem":1093,"statement":"For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i<k$ such that $n-i$ is $k$-smooth, that is, divisible only by primes $\\leq k$. Are there infinitely many binomial coefficients with deficiency $1$? Are there only finitely many with deficiency $>1$?","status":"open","seam":"sealed","closureRoutes":[{"type":"witness","verifierKind":"binom_deficiency","note":"extended deficiency witness table, 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_279e997d3b3da2ee","banked":"three delta=1 examples beyond ELS93 (k=106/126/129) verified; the density model leans finite","open":"prove finiteness of the delta=1 deficiency cases (the density argument is heuristic, not a proof).","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_4ec7148a9c7eadca","kind":"research_audit","claim":"Prior-art audit of Erdős #1093: attendedness=unattended; baseline_source=no_source_found; actors=none.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_b3bcc9670711d818","kind":"reduction","claim":"#1093 DIVISOR-LEMMA REDUCTION (GPT-Pro, Opus-verified through gate): when C(n,k) is defined, the deficiency reformulates EXACTLY as delta(n,k)=#{1<=i<=k : n-k+i | i*C(k,i)} -- smoothness disappears, the smooth term is a divisor of the finite quantity i*C(k,i), NOT free. This REFUTES the proposed 'choose an arbitrary k-smooth S' Q1 construction (a Kummer obstruction). Consequence: positive deficiency forces n < k + 2^k*sqrt(k) (fix-k-grow-n is dead); ELS93 conjectured even deficiency-1 may be FINITE.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_0fd925ad6a8e7174","kind":"verified_witness","claim":"Erdős #1093 (ELS93 deficiency): exhaustive verified search confirms the Erdős–Lacampagne–Selfridge deficiency conjecture over the extended range k=2..127 — 177 positive-deficiency cases, 161 with deficiency 1, 16 with deficiency>1 (all N>=2k), and NO new deficiency>1 example beyond the ELS93 k<101 table. Two NEW deficiency-1 examples beyond ELS93 are found: (k=106, N=275204387105968945551700859, smooth position i=47) and (k=126, N=36859713316079485808433663, smooth position i=63). Banked as extends_prior_work, crediting ELS93 prior art.","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_f6dce5a0735251c4","kind":"verified_witness","claim":"Erdős #1093: NEW deficiency-1 example at k=129 (beyond the k<=127 exhaustive search), Opus-verified. N=3180883073384828665489: C(N,129) Kummer-defined, exactly one smooth slot at i=65 with x=N-64=3180883073384828665425 = 3*5^2*11^2*13*23*37*67*71*79*83*89*101*113; x | 65*C(129,65) and x | lcm(1..129). Extends the verified deficiency-1 list. extends_prior_work.","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_e22fba0dfbc2bccf","kind":"reduction","claim":"Erdős #1093 delta=1: the finiteness density heuristic is arithmetically WRONG — corrected local model leans INFINITUDE, Opus-verified. Slot indexing x=N-k+r, g_{k,r}=gcd(lcm(1..k), r*C(k,r)); every smooth slot has x|g_{k,r}. Sharpened supply: S_k=sum_r tau(g_{k,r})=exp(((log2)^2+o(1))k/log k), so the true supply constant is c=(log2)^2=0.48045 < C=0.78853 (verified; F(y)=int 1[{yt}>{t}]dt/t^2 = -y log y-(1-y)log(1-y), max F(1/2)=log2; indicator identity exact). Naively S_k*D_k=exp(-(C-(log2)^2)k/log k) with C-(log2)^2=+0.30808>0 would predict FINITELY many. But this multiplies by D_k assuming equidistribution that FAILS: the exact correlation — for p>sqrt(k), p|g_{k,r}, p|x => N=x+k-r is automatically Kummer-defined at p (verified 0 violations/2715) — couples the divisor and Kummer conditions through the SAME carry inequality. The corrected central first-moment exponent J(1/2)=+0.0437>0 (verified numeric +0.041) REVERSES the sign vs the naive -0.308<0. So the residual delta=1 question is a sparse subset-product CRT problem at r~k/2 (choose x including primes p>sqrt(k) with floor(k/p) odd, which self-certify Kummer), NOT an xyz or smooth-density problem; it leans infinite. Supersedes the naive finiteness-leaning record att_7beb86a84cc530ec. Honest obstruction map; infinitude/finiteness still open.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_b2dfb5a925a425f8","kind":"reduction","claim":"Erdős #1093 delta=1: the central subset-product CONSTRUCTION is blocked by two exact obstructions, Opus-verified — the J(1/2)>0 infinitude signal is real but not constructively realizable by forced primes. (a) Self-certification is only FIRST-ORDER: for p>sqrt(k), p^2>k requires two base-p digit conditions; p|x auto-satisfies Kummer only for p in (k/2,k] (m=floor(k/p)=1) — verified: x=p fails 0/13 in (k/2,k] but 2/2 in (k/4,k/3]. (b) ENTROPY COST: forcing top primes in (k/2,k] to make 'no second slot' automatic costs local exponent int_T^2 log(3-t)/t^2 dt = 0.10834 (T=1/(3/2-log2)=1.23938), EXCEEDING the positive first-moment margin J(1/2)=0.04367 (0.04367-0.10834<0), so the forced-prime route destroys the margin. Infinitude would require instead (i) a subset-product CRT mixing theorem mod M_0(k)=exp((2+o(1))sqrt k) and (ii) a second-slot pair-sieve — neither available. Honest obstruction map; refines att_e22fba0dfbc2bccf; infinitude open.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_7beb86a84cc530ec","kind":"reduction","claim":"Erdős #1093 deficiency: sharper necessary condition + density obstruction, Opus-verified. A deficiency smooth slot x=N-i is not merely k-smooth but DIVIDES lcm(1..k): if p^e|x with p^e>k then Kummer at modulus p^e forces N mod p^e >= k, contradicting N=x+a, a<k. Verified on every known deficiency>=1 example (16-row ELS93 table + k=106,126,129). Combined with the divisor lemma x | i*C(k,i), the Kummer-compatible smooth-slot supply is exp((log2+o(1))k/log k) while the Kummer-definedness density is exp(-(C+o(1))k/log k) with C=sum_{m>=2} log m/(m(m+1)) ~ 0.78853 > log2; since C-log2 ~ +0.0954 > 0, the density model predicts FINITELY many deficiency-1 cases. This refutes a positive-density delta=1 construction; finiteness itself needs uniform divisor-equidistribution mod Q_k=prod p^ceil(log_p(k+1)) (open). Sharpened reduction plus obstruction map, not a proof.","grade":"verified_reduction","gateStatus":"verified","superseded":true}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}