{"schema":"vela.problem-packet.v0.1","problem":1056,"statement":"Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that\\[\\prod_{n\\in I_i}n \\equiv 1\\pmod{p}\\]for all $1\\leq i\\leq k$?","status":"open","seam":"sealed","closureRoutes":[{"type":"witness","verifierKind":"interval_product","note":"a new k-record: k consecutive intervals with product 1 mod p, 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_ae08f178d51bb99e","banked":"explicit cut-equality certificates verified for every k in 2..14 (interval_product witnesses)","open":"the problem asks for EVERY k; a uniform construction or proof for all k remains open.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_2fa5ba98ff09cd83","kind":"reduction","claim":"Erdős #1056 (for-all-k) <=> sup_p M(p) = infinity, where M(p) = max_r #{0<=t<p : t! ≡ r mod p} (largest factorial-residue collision class mod p). Cuts c_0<...<c_k with equal F_p(c_i) <=> the consecutive intervals [c_{i-1}+1,c_i] each have product ≡ 1 mod p.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_f73d2eddef5a17b1","kind":"dead_end","claim":"attempted via frontier '?' (transfer_strength=n/a) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_395a03e2105f072d","kind":"partial_proof","claim":"Deep open-core push (cross-model verified): the tetrad construction CANNOT be extended to size >=6 without new input (it is a one-seed Wilson-reflection machine, caps at the class {0,1,q,p-1-q,p-2}); #1056 reduces cleanly to the OPEN Hardy-Subbarao conjecture g(p)=#{n<p:n!≡-1}->infinity (since M(p)>=g(p)).","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_ff81f63ad25ad111","kind":"partial_proof","claim":"SHARPER reduction (cross-model verified): #1056 size-6-infinitely-often follows from a WEAKER condition than Hardy-Subbarao — namely a favorable class-number sign h(-p)≡3 mod4 on the (proven-infinite) family of tetrad primes p≡3 mod4 dividing q!-1. Connects #1056 to class-number distribution (Cohen-Lenstra).","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_ddd53411d6a03fdd","kind":"partial_proof","claim":"Size-6 question fully MAPPED (cross-model verified): all elementary routes rigorously killed; the obstruction is genuinely a class-number sign distribution h(-p)≡3 mod4 on the thin factorial-divisor subfamily T_3 — beyond current analytic-number-theory technology. NOT a settlement; M(p)->inf is almost certainly TRUE (tetrad primes almost never stop at 5) but via non-reflection collisions with no known construction.","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_3e0474621ad90df1","kind":"verified_witness","claim":"CORRECTION + concrete progress: the direct factorial-residue collision search M(p)=max_r #{t<p: t!≡r} TRIVIALLY beats the 'size-5/6 tetrad obstruction' banked earlier -- that obstruction was specific to one construction, NOT the problem. Found growing collisions M(p)=5,6,7,9,11,13 at p=17,23,71,599,3011,52163 (Opus-verified), giving CONCRETE #1056 witnesses up to k=12. Full #1056 (forall k = sup_p M(p)=infinity) remains the OPEN analytic conjecture, but it is now strongly empirically supported.","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_8a2420033e6acb08","kind":"verified_witness","claim":"Erdős #1056: VERIFIED k=5 and k=6 witnesses at p=71 — new finite examples. Six factorials 7!,9!,19!,51!,61!,63! ≡ -1 mod 71 (7!=5040=71^2-1; 8*9≡1; 10..19≡1; Wilson reflection (70-n)! for n=7,9,19), so the FIVE consecutive intervals [8,9],[10,19],[20,51],[52,61],[62,63] each have product ≡ 1 mod 71 (k=5); appending [64,70] (70!≡-1) gives k=6. Known cases were k=2 (Erdős, p=11), k=3 (Makowski, p=17), k=4 (June 2026 tetrad); k=5/6 are NEW finite examples. Independently re-verified (direct factorials mod 71). Extends the known cases; the general 'for every k' question stays open. Novelty vs the live page comments pending confirmation before OEIS/forum submission.","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_5ed73bd903b3ec0d","kind":"verified_witness","claim":"Erdos #1056: independently verified explicit prime/cut certificates for every k from 2 through 14. Each certificate is a prime p and k+1 cuts c_0<...<c_k with c_0! = c_1! = ... = c_k! (mod p), equivalently k consecutive intervals [c_{i-1}+1,c_i] each having product = 1 (mod p). Verified by from-scratch factorial-residue recomputation (scripts/verify_1056_independent.py); largest is k=14 at p=10428007. Reproduces Erdos's k=2 (p=11) and Makowski's k=3 (p=17). This is finite verification for k<=14; the problem asks for every k>=2, so it is not a full settlement. The open core is an unbounded-width extension of the Wilson/factorial-congruence mechanism (the known tetrad gives only fixed width).","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_04e3cd7d251a2bab","kind":"dead_end","claim":"Erdos #1056: no-go theorem (route obstruction) for the finite fixed-width Wilson/reflection method. In the graph on {0,...,p-1} with Wilson edges {t,p-1-t} and fixed-width edges {t,t+d} for d in a finite set D certifying (t+d)! = t! mod p, every connected component has size <= 2(1 + sum_{d in D} d). Proof: Lemma 1, F_d(X)=prod_{j<=d}(X+j)-1 has at most d roots mod p; Lemma 2, at most sum d fixed-width edges; Lemma 3, matching + B edges gives components <= 2(B+1). Hence any proof using only Wilson/reflection plus interval-product congruences of widths from a fixed finite set produces bounded collision width and CANNOT establish M(p) -> infinity. Verified empirically (p up to 27901, D={1..5}: maxcomp <= 32 = 2(1+15), fixed-width edges <= 15, Lemma 1 never violated). Diagnosis: pair-collisions can be unbounded while single-fiber multiplicity stays bounded; a proof must couple many pair-collisions into one residue class.","grade":"obstruction_map","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos1056.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos1056.lean"}],"oeis":[{"id":"A060427","name":"Smallest prime p such that there are n strings of consecutive integers all having products = 1 mod p.","terms":"2,11,17,23,71,71,599,599,3011,27901,52163,778699,2374649,10428007","url":"https://oeis.org/A060427"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}