{"schema":"vela.problem-packet.v0.1","problem":306,"statement":"Let $a/b\\in \\mathbb{Q}_{&#62;0}$ with $b$ squarefree. Are there integers $1&#60;n_1&#60;\\cdots&#60;n_k$, each the product of two distinct primes, such that\\[\\frac{a}{b}=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}?\\]","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_ed874dffcf83fa4f","kind":"dead_end","claim":"attempted via frontier '?' (transfer_strength=n/a) -> no_progress","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_b795483c41c3118e","kind":"reduction","claim":"#306 (semiprime Egyptian fractions) reduces to a CONSTRUCTION target: if every squarefree b has a 'fresh' 1/b certificate (sum of distinct semiprime reciprocals) avoiding any finite forbidden denominator set, then every a/b is representable (take a disjoint copies). A verified gadget basis is built; residual is a construction-existence lemma — the RIGHT SHAPE for a search/construction loop (unlike the proof-problems).","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_5a000c91096dfd4f","kind":"reduction","claim":"Integer-forcing lemma (verified): for a cone certificate of 1/b, if the local congruences C_j + b*sum_{i~j} q_i^{-1} ≡ 0 mod q_j hold at every auxiliary vertex AND 0<T<2 (T=normalized mass), then T=1 automatically — the certificate is exact. So fresh-1/b reduces to: build a local-congruence graph with mass T<2. PRECISE OBSTRUCTION found: the Dirichlet/CRT route is blocked — in any finite cone, the last auxiliary prime must DIVIDE a fixed integer (not lie in an AP), so Dirichlet doesn't apply.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_18cd6580d686fe6f","kind":"dead_end","claim":"NO-GO theorem (cross-model verified): the two-auxiliary + one-edge height-control escape CANNOT prove the fresh-1/b lemma. The integer-forcing T=1 forces the FIXED hyperbola (q1-C1)(q2-C2)=C1*C2+b, so for fixed b only finitely many auxiliary primes exist (bounded by C_max^2+C_max+b) — an adversarial F containing them all kills the certificate. One-prime AND two-aux inductions are both dead.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_2fef63bb6e4da56e","kind":"partial_proof","claim":"Growing-cycle construction (cross-model verified) ESCAPES the 2-aux no-go: produces fresh 1/b certificates avoiding ANY prescribed set of small primes (verified for b=6,30,42,66,70,105 forbidding all primes up to 10/20/50/100). #306's fresh-1/b lemma now reduces to ONE precise lemma: the bounded moving CRT-divisor lemma.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_5ac79b9091d1877c","kind":"dead_end","claim":"Cross-engine convergence (Codex solver + GPT + Opus): the SIMPLE growing-cycle construction works for LARGE b (small height-control k) but FAILS for SMALL b with moderate F — b=6, F<=20 found no simple cycle even at k<=12 / 200k paths (candidates_checked=0), and GPT hit the same wall. The pure simple cycle is NOT fresh-universal; small b (small C_max) needs the THETA/CHORD thickened closure.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_ed2447eb38bf729b","kind":"partial_proof","claim":"NEW escape (cross-model verified, is_novel): the affine/Dirichlet route DEFEATS the 'last-prime-divides-fixed-M' trap that blocked every prior round. Core gadget PROVED: for squarefree m, A_m=sum_{p|m} m/p has gcd(A_m,m)=1 (0 exceptions to m<5000) and 1/m = sum_{p|m} 1/(p*A_m) exactly; when A_m is prime these are omega(m) distinct FRESH semiprimes. The escape: A_{bt}=t*A_b+b is AFFINE in a free prime t, and since gcd(A_b,b)=1 the AP {A_b*t+b} is prime-rich by Dirichlet — so the critical prime lies in an AP, not a fixed integer.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_011c53c48428b07a","kind":"reduction","claim":"CROSS-MODEL CONVERGENCE (3 engines): #306's fresh-1/b lemma reduces to forcing a LINEAR FORM to be prime along a compatible family — a Dickson/Schinzel Hypothesis-H-flavored statement (conjecturally true, UNCONDITIONALLY open). This characterizes the difficulty: #306 is likely conditionally-provable (mod Hypothesis H) but uncond. as hard as it. All sequential single-final-vertex closures are PROVABLY dead (GPT Proposition 2).","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_dd67ad2458f08b65","kind":"reduction","claim":"MAJOR REFRAME (cross-model, verified): the Hypothesis-H wall was an ARTIFACT of the over-strong fresh-PRIME condition. The correct condition is EDGE AVOIDANCE (distinct semiprime DENOMINATORS; primes REUSABLE), which is strictly weaker and bypasses Hypothesis H entirely. EA(b) for all squarefree b => #306 (proven). Empirically very strong. The concrete remaining target is the EDGE-SPLITTING lemma — possibly elementary, no prime-forcing.","grade":"verified_reduction","gateStatus":"verified","superseded":false},{"id":"att_ef5affc112c6b377","kind":"reduction","claim":"CORRECTION + triangulation: #306 reduces to Schinzel Hypothesis H from FIVE independent routes; the edge-avoidance 'bypass' is NOT established (it is genuinely weaker but re-meets prime-forcing asymptotically).","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_ae49d2a67398cb13","kind":"verified_witness","claim":"Codex semiprime solver: 1/b represented as a sum of DISTINCT semiprime reciprocals for 5,750 of 6,083 squarefree b<=10000 (94.5%); independently re-verified exact, 0 failures. Holdouts are STRUCTURALLY {b=1} U {219 primes} U {113 squarefree omega-3} -- zero misses at omega=2.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_be6bc124fc00a2e0","kind":"partial_proof","claim":"GPT-Pro: corrected reduction chain + a PROVEN no-go theorem. (a=1 existence for every squarefree b) + edge-splitting => #306. Prop 1 (proven): edge-splitting upgrades any ONE 1/b certificate to edge-avoidance (so a=1 => all a). Prop 2 (proven NO-GO): no fixed finite splitting topology yields infinitely many splits of 1/(pq) -- edge-splitting MUST use growing topology / monotone recursion.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_2c6ae692527b1c6c","kind":"partial_proof","claim":"GPT-Pro: monotone split lemma reduced to a single FORCED-PRIME condition, with proven cone-rule generator + computation (all distinct-prime semiprimes pq<=300 split monotonically, all denominators >pq). Not proven unconditionally: every finite cone rule forces an auxiliary prime z=B/D -- the irreducible Hypothesis-H/Dickson-class barrier, in its cleanest form. No congruence obstruction (so #306 is almost certainly TRUE); the unconditional proof needs a prime-value theorem.","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_785163c947342832","kind":"partial_proof","claim":"GPT-Pro: the PRIME case reformulated non-constructively as a lattice local-CLT / minor-arc estimate for a reciprocal-prime semiprime cone -- Hypothesis-H-FREE, no forced prime values. Major arc + medium-support minor arc PROVEN; only the near-full-support minor arc remains. Verified: the counting tables are exact and yield REAL certificates -- and the framework resolves Codex holdout 1/109.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_f61ed81137dd7478","kind":"verified_witness","claim":"Codex EMPIRICAL CAPSTONE: #306 verified true across everything tested. First-cert existence (G2) -- 332/333 holdouts resolved, every squarefree b>=2 up to 10000 has an explicit distinct-semiprime certificate (6082/6082 for b>=2). Monotone split (G1) -- 563/563 semiprimes pq<=2000 split into distinct semiprimes all >pq, including 396/396 odd-odd. Only b=1 (representing 1 itself) is a bounded search miss.","grade":"improved_published_bound","gateStatus":"verified","superseded":false},{"id":"att_fb43d4cf44f8ae5a","kind":"partial_proof","claim":"GPT-Pro: the prime-case minor-arc lemma (MC) reduced to ONE clean inverse small-ball / Littlewood-Offord lemma for the reciprocal-prime PAIR system, with a proven NEAR-bound that misses the target by exp(C p^{4/3}(log p)^{4/3}). Not closed; the remaining gap is a precise, named additive-combinatorics lemma.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_183e5eaadd9730f9","kind":"partial_proof","claim":"Codex numeric pre-test of the inverse small-ball lemma: the lower tail #{h:D2(h)<=T} grows like exp(alpha*T) (NOT super-exponential) across p=7,11,17,23,31 tuned (alpha~1.35-1.95) + exact small cases -- positive evidence the lemma #{...}<=p^{O(1)}e^{O(T)} is TRUE. De-risks GPT's proof.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_be414f582e505a40","kind":"partial_proof","claim":"GPT-Pro: the inverse small-ball lemma reduced to ONE core-extraction lemma, with the surrounding structure (outlier-cost L4, entropy-count L5) PROVEN and the a/q-arc hint resolved (L2: the a/p arcs are small integer models H in the g=ph coordinate). The prime case of #306 now hangs on a single, sharply-stated quadratic Bohr-set inverse theorem.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_dff109da6348a113","kind":"partial_proof","claim":"Codex gluing-sublemma experiment: NO distributed low-D2 counterexample found (p=11,17,23; |A|=12,23,39). Both sampled low-D2 g and adversarial CRT constructions obey outliers = O(D2/sqrt(p)) with constant ~9.2 -- empirical GREEN LIGHT for GPT's core-extraction proof. Independently reproduced by Opus.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_f6046ac354adac64","kind":"partial_proof","claim":"GPT-Pro: core-extraction reduced to a clean CRT CLIQUE-COVER GLUING lemma (Lemma A), surrounding structure proven (C: outlier penalty + correction; D: giant-clique => core-extraction), and DENSITY ALONE shown INSUFFICIENT (Lemma B: a dense small-pair graph can be covered by many almost-disjoint cliques -- the naive core-growing proof has a real gap). New tool pointer: CRT list-decoding (Goldreich-Ron-Sudan, Guruswami-Sahai-Sudan). GPT's alarming finite config (core=3) RECONCILED as a non-counterexample.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_30f72c9438c6e6f8","kind":"partial_proof","claim":"GPT-Pro STRATEGIC BREAKTHROUGH: core-extraction PROVEN for relation-sparse A (conditional rigidity theorem A1-A4), arbitrary-A shown likely FALSE (projective-plane / relation-density obstruction), and since the Fourier strategy lets us CHOOSE A, the open problem converts from an arbitrary-A inverse theorem into a prime-set SELECTION lemma (construct relation-sparse mass-tuned A). Verified + feasibility probed by Opus.","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_294a3edda4af10ea","kind":"dead_end","claim":"GPT-Pro REFUTED the relation-sparse mass-tuned prime-set route (its own prior proposal): for Q>=C0*p, EVERY distinct triple in [p^2,3p^2/2] has a small-coefficient relation ar+bs+ct=0 (|coef|<=Q), so NO A can be relation-sparse. Since Q~p^{1/2}sqrt(T+1), this kills the route for T>=p -- exactly the non-vacuous range. Opus-verified.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_d772cdbd3cedf9b6","kind":"dead_end","claim":"GPT-Pro REFUTED the local-domination route (4th dead route on core-extraction). Cross-class edges are NOT forced expensive: a label-compatible triangle relation av+br+cs=0 keeps all three CRT labels small (per-edge energy ~p^-2). Verified finite multimodal config: D2<31, 4773/4796 low triangles label-INCONSISTENT, high-degree vertices (deg 34-35) have largest label-class size 1-2 (NOT dominated). Reduces #306's prime-case PROOF route to a sharp dichotomy.","grade":"honest_null","gateStatus":"needs_verification","superseded":false},{"id":"att_480ea5c038280157","kind":"partial_proof","claim":"GPT-Pro PROVED the size-p CRT design is impossible (clean incidence/Cauchy bound: near-disjoint size-K blocks cover <= ~nK pairs; for K~p, nK=p^{5/2}=o(p^3)=o(n^2), so size-p blocks can't cover a dense low-height graph). Closes that branch of the dichotomy. BUT the dichotomy was NOT exhaustive: a sqrt(n)~p^{3/4}-scale CRT-embedded projective plane remains the genuine open obstruction (Opus-verified).","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_50d493f623c207b5","kind":"partial_proof","claim":"GPT-Pro PROVED the projective-plane CRT design (last round's residual) is HARMLESS: any exact embedding at q~p^{3/4} has D2 = Omega(p^2) (Opus-verified D2/p^2 ~0.008 stable), outside the non-vacuous range T=o(p^2). Closes the 6th obstruction. But the residual descends AGAIN to a SPARSE Steiner-type CRT design, reformulated as a rich-prime-difference AP problem; partial bound Q>>sqrt(n) proven, range p^{3/4}<~Q=o(p^{3/2}) open, all constructions fail.","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_2f9f4661217a897f","kind":"dead_end","claim":"GPT-Pro round 7 (rich-AP residual): NO new progress. Reconfirmed Q>>sqrt(n)=p^{3/4}; proved EVERY standard tool stops exactly there (incidence, difference-counting, additive energy, design-matrix rank, random, finite-field affine/projective). The residual is a clean OPEN integer-AP incidence lemma with no known method to break the sqrt(n) barrier. Unambiguous research wall.","grade":"honest_null","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}