{"schema":"vela.problem-packet.v0.1","problem":971,"statement":"Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c&#62;0$ such that, for all large $d$,\\[p(a,d) &#62; (1+c)\\phi(d)\\log d\\]for $\\gg \\phi(d)$ many values of $a$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_13abaf6ae36688cc","kind":"dead_end","claim":"Erdős #971 (does ∃c>0 s.t. for ALL large d the least prime p(a,d) > (1+c)φ(d)log d for ≫φ(d) reduced classes a?) remains OPEN — OBSTRUCTION MAP + conditional reduction, not a settlement; Erdős 1949 got infinitely many d. UNCONDITIONAL: exact occupancy model (N0 = #empty reduced classes; mean occupancy λ→1+c, verified d=1000003,c=1/2 → N0/φ=0.2138 vs e^(−3/2)=0.2231). The second-moment/Cauchy route is REFUTED — it lower-bounds OCCUPIED classes (wrong sign); an S2≪φ(d) bound is compatible with N0=0 (explicit {1,2}-box config). Right statistic = factorial moments: N0=φ−M+E. CONDITIONAL REDUCTION (Bonferroni, P3(3/2)=1/16): if F2(d;X)/φ→(3/2)^2 and F3/φ→(3/2)^3 at X=(3/2)φ(d)log d for all large d, then N0≥(1/16+o(1))φ(d), giving c=1/2. OBSTRUCTION: at X≈φ(d)log d one has d=x^(1−o(1)), beyond Bombieri-Vinogradov (level 1/2), Elliott-Halberstam (x^(1−ε)), and GRH (per-class error swamps the O(1) mean); Erdős selects totient-deficient moduli (φ(m)≤4δm) by averaging, and no individual-d lower bounds for the prime-pair/triple correlation sums F2,F3 along shifts ≡0 (mod d) are known.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_c16c08ba44d145cb","kind":"partial_proof","claim":"#971 (least prime ≡ a mod d exceeding (1+c)φ(d)log d for ≫φ(d) classes, for ALL large d) is open; Erdős got infinitely many d. I give a rigorous reduction: it reformulates as a prime-occupancy problem","grade":"partial_proof","gateStatus":"superseded","superseded":true}],"velaLean":[],"oeis":[{"id":"A226521","name":"Triangle read by rows: T(n,k) = smallest prime == k (mod n) if gcd(k,n)=1, otherwise 0, for n >= 2, 1 <= k < n.","terms":"3,7,2,5,0,3,11,2,3,19,7,0,0,0,5,29,2,3,11,5,13,17,0,3,0,5,0,7,19,2,0,13,5,0,7,17,11,0,3,0,0,0,7,0,19,23,2,3,37,5,17,7,19","url":"https://oeis.org/A226521"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}