{"schema":"vela.problem-packet.v0.1","problem":699,"statement":"Is it true that for every $1\\leq i&#60;j\\leq n/2$ there exists some prime $p\\geq i$ such that\\[p\\mid \\textrm{gcd}\\left(\\binom{n}{i}, \\binom{n}{j}\\right)?\\]","status":"open","seam":"sealed","closureRoutes":[{"type":"verified_computation","verifierKind":"python:verify_699_central_doubling.py","note":"central-doubling exact reduction re-check (pinned tier: scripts/requirements-verifiers.txt)"},{"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_eb741f9becc60879","banked":"the central-doubling reduction + CRT obstruction verified; the two arms provably do not meet","open":"the gap between the arms is the open obstruction.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_61df49c6d6ade6a3","kind":"dead_end","claim":"Candidate sweep (#699,#287,#993,#1082): all DISQUALIFIED as maps-not-closes; #488 the lone sliver (|A|=2 partial). Each has a cheap verifier but the open direction is a universal whose close is an analytic/incidence/structural wall.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_542e62f85852e843","kind":"reduction","claim":"#699 (OPEN): GPT-Pro reduction VERIFIED correct -- Kummer/Lucas form (p|gcd(C(n,i),C(n,j)) iff i and j both NOT digit-dominated by n in base p), proves i=1,2 + fixed-(i,j) large-n, and reduces ALL to the THRESHOLD LEMMA. But this is exactly the lemma a submitted proof (Parthasarathy 2026) attempted and Bloom DECLARED INVALID -- so the crux is hard, not a quick win.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_b357fce7c3a99286","kind":"reduction","claim":"#699 PROGRESS (GPT-Pro, Opus-verified): the Bloom/StijnC hole is FIXED. GPT proved a SHIFT LEMMA (p>i, p∤C(n,j) => v_p(C(n,i)) <= v_p(C(n-j,i))) that survives Case B, giving the corrected reduction: if no prime p>=i divides both C(n,i),C(n,j), then R_{>=i}(C(n,i)) | C(n-j,i). The Threshold Lemma now reduces to a clean ROUGH-GROWTH LEMMA: R_{>=i}(C(n,i)) does NOT divide C(n-j,i) for all 1<=i<j<=n/2. Verified true to n=260 (FO witnesses always p=i). STILL OPEN: the Rough-Growth Lemma is unproven (the new, cleaner crux).","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_09937c62bc8593a6","kind":"dead_end","claim":"#699 Rough-Growth Lemma: BOTH natural proof routes now REFUTED. The 'non-FO => p>i witness' split is FALSE -- counterexample (28,5,11) (non-FO, but only p=i=5 witnesses rough-growth; 16 such triples n<160, Opus-verified). With the earlier Bloom/StijnC Case-B refutation of the rough-part route, #699 looks like a genuine wall at the now-precisely-isolated crux.","grade":"honest_null","gateStatus":"verified","superseded":false},{"id":"att_4d20ba9ab22b5777","kind":"transfer","claim":"TRANSFER MAP (TF-IDF over problem statements) fixes the isolation-audit flaw: it surfaces an UNATTENDED + OPEN cluster of binomial-coefficient-divisibility problems sitting directly on our #699/#700 Kummer/carry machinery -- #1093, #1094, #386, #849, #387 (all open, ~1 comment each). Our verified method TRANSFERS; nobody is there.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_a777d04becc062d5","kind":"reduction","claim":"#699 SHARP RESIDUAL + GPF-INSUFFICIENCY (GPT-Pro, Opus-verified through gate): the Window-Rough Sublemma residual is now the precise PRIME-POWER WINDOW PACKING LEMMA, and greatest-prime-factor methods are PROVEN structurally insufficient to close it. k=1,2 settled; top-prime settled; the genuinely-hard core is a short-interval CRT/carry packing statement, not a GPF bound.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_e37c5a7871d4d18e","kind":"verified_reduction","claim":"#699 Window-Rough Sublemma (prime-free residual) GPF-angle: PROVEN sub-claim Q>j+k-1 => p=Q witnesses (short Kummer proof, boundary-sharp), independently re-derived via raw base-Q carry-count (31898/31898, 0 mismatches vs max-formula in 48971 tests). Disposes the bulk region. Residual Q<=j+k-1 is NOT finite (17073 to N=300); within it, p=k-only hard cases are rare but persist; k^a>j+k-1 closes 27/36 to N=600; the irreducible 9 are a length-k window / j mod k^a residue-avoidance wall. honest grade verified_reduction + obstruction_map, NOT settled. Lemma verified 0 failures to N=350 (1.74M triples). Prompt example (28,5,11)->r=3,25 not|[8..12],p=5 reproduced.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_6323fe390eb718b1","kind":"reduction","claim":"Erdős #699 Window-Rough Sublemma (prime-free residual): CORRECTED object + CRT obstruction, Opus-verified. The rough object is the SET/PRODUCT of all maximal rough prime-powers Q_k(m)={p^v_p(m): p>=k, p|m} (NOT the single largest; (65,3,11) discriminates — largest 13 is absorbed but 5 witnesses). Our prior R_{>=k}=rough-part (product) framing is exactly equivalent (coprime moduli) and remains sound. Window-Rough FAILS iff for every q in Union_r Q_k(N-r): N mod q < k AND (N-j) mod q < k (verified 0 mismatches / 83520 prime-free triples). Exact packing count: a single rough prime-power q>=k is incident to <=1 (r,s) pair with q|N-r and q|j+s-r, so failure is NOT a local capacity problem — it is a global CRT alignment (a Complete CRT Packing Certificate). Sharpened verified reduction plus obstruction map; not a proof. Packing alone does not close the lemma.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_dada2a070dc7dede","kind":"reduction","claim":"Erdős #699 Window-Rough Sublemma: central-doubling exact reduction + unconditional central-strip finiteness, Opus-verified. For k>=3, M=N-j, delta=N-2j, the identity A_r-2B_s = -(delta+r-2s) (A_r=N-r, B_s=M-s) and oddness of primes p>=k give: q|A_r and q|B_s iff q|A_r and q|(delta+r-2s). Exact-doubling positions D_delta={r: delta+r even, 0<=(delta+r)/2<=k-1} impose no obstruction. Hence Window-Rough FAILS iff rough_{>=k}(N-r) | prod_s |delta+r-2s| for every non-doubled r (verified 0 mismatches vs the real witness, 99585 k>=3 triples). In the central strip 0<=delta<=k-1 the non-doubled top terms are ALL odd, so only odd primes <k can appear; m=floor(k/2) > |T|=#odd primes<k (verified all k in [3,400)), which bounds N for fixed k => FINITELY MANY failures in the central strip (unconditional). Residual = an S-unit-type system for fixed (k,delta). Sharpened reduction plus a genuine unconditional partial result, not a full proof.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_ed9ed7479a4e00a4","kind":"reduction","claim":"Erdős #699 Window-Rough Sublemma: the two unconditional arms PROVABLY do not meet, Opus-verified. Explicit thresholds: arm A (large-prime) closes it iff delta>=delta_A(N,k)=N+2k-1-2H(N,k) with H=max_{r,q in Q_k(N-r)}(q+r) (verified: arm A fires iff delta>=delta_A, 0 mismatches/149600 triples); arm B (central-doubling) closes it only for delta sublinear in N. A CRT construction N≡r+ell_r (mod ell_r^2) with distinct primes ell_r>k forces ell_r||(N-r), so H(N,k)<=N/L+O_k(1) and delta_A>=N(1-2/L) is LINEAR in N (verified k=3: delta_A/N=0.995, arm A fires only for j<=1.6e10 << N/2=3.35e12) while delta_B is sublinear — the arms leave a genuine intermediate band. The residual there is the exact S-unit compatibility system u_r c_r - u_r' c_r' = r'-r over non-doubled positions (R_{>=k}(N-r)|prod_s|delta+r-2s|), finite per fixed (k,delta) by Evertse-Schlickewei when m_delta>=2; the k=3 m_delta=1 edge (C_(1,delta)=3, N-1 odd) is settled. Honest obstruction map: shows the two-arm method insufficient and isolates the irreducible S-unit certificate; not a proof of the lemma.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_7b695fec0ef97dd1","kind":"reduction","claim":"Erdős #699: pairwise S-unit uniformity for the residual is FALSE — the band needs the FULL non-doubled set, Opus-verified. Beyond the two-arm gap (att_ed9ed7479a4e00a4), the intermediate-band residual is a MOVING-TARGET divisor S-unit system: for each non-doubled r, N-r = u_r * prod_s d_{r,s} with u_r supported on primes <k and d_{r,s} | |delta+r-2s|, compatibility u_r prod d_{r,s} - u_r' prod d_{r',s} = r'-r. An explicit affine-chain family (N=3d, j=d, delta=d; r=0,r'=3, s=0,s'=2) satisfies the PAIRWISE compatibility identically (3d - 3(d-1) = 3 = r'-r, with d|C_{0,d}, (d-1)|C_{3,d}; verified over k>=4 and a range of d), so no two-position resultant / fixed-coefficient S-unit bound can give uniform finiteness. The coefficients d_{r,s} are divisors of moving linear forms in delta, so fixed-coefficient S-unit theory does not apply; a proof must rule out a GLOBAL covering of all non-doubled positions (the Uniform Moving-Target Packing Lemma). Honest obstruction map; not a proof.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}