{"id":5941,"job_id":10223,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 275, the level-truncated Möbius sub-class: the observed family **is** the exact optimum at all six bands — 3²³ and 3³¹ cells closed by a box bound, not by search\n\n**Job #10223** (`explore` / `pursue`, lane `dir-558`, route **275** rev 4, general mode). Offline,\nexact-integer where it matters, stdlib + numpy; the census is not re-run. Builds on return **#5178**\n(witness families, `analysis_bj.json`) and #3758 (the served band table).\n\n## The answer\n\nThe held step asked two things about the sub-class\n`λ_d = μ(d)·1[d ≤ L, d | P(P*)]` with `P* ⊆ P(z)` and `L` in the note's level grid\n(every squarefree `d ≤ 9409`): is the family #5178 observed — `μ` on `{7}`, `{7,11}` or\n`{7,11,13}`, truncated just below 143 — the *exact* optimum at all six bands, and is there any\n`(P*, L)` with `B^unif/X < T1/X`? **Both are now answered exactly, at every band.**\n\n| band | zprimes | (P,L) cells | visited | nodes | class min `B^unif/X` | min family | `Λ` | `E` | T1/X | miss | time |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| B3 | 4 | 81 | 53 | 14 | **8.129709796376** | 1, 7 (L=7) | 5/7 | 144/7 | 7.82 | +3.96 % | 0.01 s |\n| B4 | 8 | 6 561 | 521 | 92 | **9.235543986930** | 1, 7, 11 (L=11) | 7/11 | 4968/77 | 7.99 | +15.59 % | 0.01 s |\n| B5 | 12 | 531 441 | 6 371 | 793 | **10.642457278271** | 1, 7, 11, 77 (L=77) | 45/77 | 11520/77 | 8.67 | +22.75 % | 0.03 s |\n| B6 | 17 | 129 140 163 | 68 356 | 6 206 | **11.749021526419** | same | 45/77 | 11520/77 | 9.18 | +27.99 % | 0.4 s |\n| B7 | 23 | 94 143 178 827 | 582 914 | 42 218 | **12.306011998455** | 1, 7, 11, 13, 77, 91 (L=91) | 515/1001 | 34380/77 | 9.53 | +29.13 % | 4.0 s |\n| B8 | 31 | 617 673 396 283 947 | 18 056 120 | 1 030 187 | **13.392768767213** | same | 515/1001 | 34380/77 | 10.13 | +32.21 % | 132.1 s |\n\n**Every class minimum equals #5178's witness to the last bit** (`equals_witness = true` at all six\nbands, exact rationals `Λ`, `E` re-derived below), and **no `(P*, L)` anywhere is below `T1/X`**\n(`below_T1_count = 0` at all six). So the step-check's open item (a) — \"*the Möbius sub-class was\nobserved in the search output, not enumerated over all (P*, L)*\" — is closed: inside this sub-class\nthe observed family is not a search artefact but the exact optimum, and route 275's reopening\ncondition (a) is refuted **exactly** (not by 1-opt search) at every band of the window. The closest\ncall remains B3 at +3.96 %.\n\nTwo by-products. (i) The level cap is slack in the sub-class optimum: the winning levels are\n`7, 11, 77, 77, 91, 91`, three orders of magnitude below the grid top 9409, so no level `L` above\n`max(A)` buys anything. (ii) The `Λ` of the optimum is 1.6–3.1× the sifting density\n`V₂(z) = 0.436 … 0.164`, i.e. the sub-class optimum keeps Selberg's main term and pays in `E`; no\nsub-class family reaches `Λ ≥ 0.99·V₂` at a competitive ratio, which is the second half of\nreopening condition (a).\n\n## Method — why 6·10¹⁴ cells close in 136 s\n\nTwo independent sound prunes, both elementary, then exact re-evaluation of winners:\n\n1. **`Λ ≥ 0`, so `ratio ≥ n·E/ΣX`.** Any family with `E ≥ E* := r_witness·ΣX/n` — or any superset of\n   one, `E` being monotone under adding elements — cannot be the minimum. `E*` runs 181.1 (B3) to\n   7611.2 (B8). This prune alone is *vacuous* at the top bands: `E ≥ 6(k²−1)` (diagonal `B(d) ≥ 6`\n   for `d > 1`, each ordered cross term `B(lcm) ≥ 12`) only excludes `k ≥ 27` at B7 and `k ≥ 36` at\n   B8, against 23 and 31 primes, so every prime set survives it.\n2. **Box bound on `Λ` (this is what makes the closure possible).**\n   `Λ = Σ_e h(e)M_e²`, `h(e) = Π_{p|e}(p−2)/2 > 0`, `M_e = Σ_{e|d} g(d)λ_d`. For a partial family `A`\n   and any superset, `|M'_e| ≥ (|M_e| − W_e)_+` with `W_e = Σ_{e|d, d∉A} g(d)` the free mass, hence\n   `Λ' ≥ Σ_e h(e)(|M_e| − W_e)_+²`; with `E` monotone this bounds the ratio of every extension from\n   below and is compared against the incumbent at every prefix. Measured effect at B6: 1 589 645\n   visited families with prune 1 alone → 68 356 with both (23×); at B7, 150 s of prune-1-only DFS\n   visited 427 352 families without finishing (2 840/s), the two-prune run closes B7 in **4.0 s**.\n   The B8 domain 6.18·10¹⁴ collapses to 1.81·10⁷ visited (3·10⁻⁸ of the cells) in 132 s.\n\nThe DFS walks prime sets in increasing order and, at each, every prefix of the sorted divisor list —\neach `(P*, L)` exactly once; winners are re-evaluated in exact rationals (not in the float search\narithmetic). `Λ = Σ_e h(e)M_e²` itself is #5178's identity (from `1/g(gcd) = Σ_{e|gcd} h(e)` and\n`g(lcm) = g(a)g(b)/g(gcd)`), used here as a bound rather than as an observation.\n\n## Checks\n\n`check_arm1_10223.py` — stdlib only, offline, reads only this run's outputs plus #5178's artifact:\n**229 checks, 0 FAIL, exit 0** (7.9 s). It (i) re-derives all six minima from the raw pairwise\ndefinition in exact `Fraction`s — `Λ`, `E` and ratio to the last digit; (ii) **brute-forces the whole\n`(P, L)` domain without any pruning** at B3 (81 cells, exact) and B4 (6 561 cells, floats) and\nconfirms the true domain minimum equals the pruned run's row and that nothing there is below `T1`;\n(iii) checks `Λ = Σ_e h(e)M_e²` against the pairwise sum on 24 random families at B5 and B8, and\n`Λ ≥ 0`; (iv) checks the box bound `Λ(A') ≥ Σ_e h(e)(|M_e|−W_e)_+²` on 24 random\n(partial, superset) pairs at B5 and B8 — the soundness condition of the closure; (v) checks the six\nrows against #5178's `per_band_min` / `per_band_min_support`, `V₂` from the zprimes, the T1\ncomparisons, the support/level/max bookkeeping and the visited-vs-cell counts.\nTen planted mutations (`--corrupt 1..10`: a perturbed ratio, a 1.5× `E`, a flipped\n`equals_witness`, a forged `below_T1_count`, an extended winner support, a dropped band, a halved\nwitness ratio, a wrong level, a dropped member at B8, a faked probe completion) each exit 1 with a\nnamed FAIL (`check_arm1_10223.corrupt1..10.out`).\n\nCost: 136.6 s arm-1 compute (B7 4.0 s, B8 132.1 s) + 0.5 s for B3–B6 + 7.9 s checker; wall clock of\nthe whole return ≈ 0.045 CPU-h, well inside the 0.5 h hint.\n\n## What is not established\n\n(1) **The closure is over the level-truncated Möbius sub-class, not over all sign families.** #5178's\n29 upper-bound cells are untouched: a general `λ: F → {0,±1}` may still beat the witness (that is the\nstep's second arm, not run here). (2) The closure rests on two elementary inequalities — `Λ ≥ 0` and\nthe box bound — argued in §method and spot-checked exactly on random families, not machine-formalised;\na wrong sign there would move the answer. (3) The `E` of the float search is the note's uniform\nremainder; its derivation is not re-checked, only its values. (4) No claim past `Q ≤ 10007`, and none\nabout the exact (`x ≤ 1`-trivial) tier or the crossing quantity. (5) Timing is wall clock on this\nmachine; token usage is unmeasured and declared so.\n","patch":null,"cpu_hours":0.11,"hashes":{"arm1.py":"000bc916e17b264016fa47f35619cb89ce201d8915db9df36d6cea84109488b6","arm1_b78.py":"75290afb084835182f1c4a816331fcd74f79b0f2e7afc1ed3d7502267a4f5844","probe_b7.py":"c29807b5dc41cc6f1995ce717715e68b266924cf25cda9ed2f2818b9029b1280","arm1_all.json":"9a863e34a0dadde49f3ac497d86300cb0ef10aa722d36cd92204c700e2d5d5c2","probe_b7.json":"97555c88ae7ccb7acc032e0ec75ffd089f0669ea3d70ab5437ffb778de546cc9","arm1_B3_B6.json":"385db7ce6a5f576d656842b5e523036100b246f4d3861378206892a6115bf433","arm1_B7_B8.json":"c1abf43faabc81e1307489b18a7b73c1fc59d40046bcd9428802276383cefede","recipe_10223.md":"062ab06f956557ac2b99613985b0d5385383e64bf2e0b3020cd2b483f2cd68e0","report_10223.md":"9cf6ecab5b42e166121ea16322d8d7ff508fb8c3356205a955189502ec93cee7","evidence_10223.md":"772f76deec5578cb3ba0f9fca057287ee62bcd8b0e533099cc93942519c7e7f2","prior_art_10223.md":"b196d44716bffa1e60a2251c3849b2624eb1e517e4c8f116e1ef636fde54b6f1","check_arm1_10223.py":"c6325100bcc2b7c4a0dc758a407114f6d0affff2c7d738b2df74b7ba7c13bb5c","transcript_10223.md":"3e38d298fab8e9a98350214d35c752a3165fcc829b4436d8ab2d0940c2b910d7","check_arm1_10223.out":"d469a3ffd4bc81e86547f1fb53457a5175b7ffdaa2a3a382a61b9877b936b8ff","next_step_10223.json":"58d07003559d04b0acf650ffc8ba17c14765f0a21c15786ee9100328674a32f3","check_arm1_10223.corrupt1.out":"51d8b2e6caecce68b3f5bb0cf541b05833f34a7bbd1b177a2d44731cc7c5a5d2","check_arm1_10223.corrupt2.out":"b45fa2d1583d660b816f1eccfc88e26c3ed8d9427225aa4071e255d7ed61b84d","check_arm1_10223.corrupt3.out":"dbe01548d0c1149ca95e2a6aa47e1af58ae823a2f3b72011fa08aba976fc3188","check_arm1_10223.corrupt4.out":"9a5d61678f0a48545e19cbe3a64e648e6c2bc30537fa7eaf1e833da399bd6429","check_arm1_10223.corrupt5.out":"bf420396ab60a19e0cd16ad9e14360d958e2623b95ded8e3eb273b5c070ba1dc","check_arm1_10223.corrupt6.out":"9741d41c520a16c64412e79ed0fa13622d56350d2a9e666568dfc4a5b7728435","check_arm1_10223.corrupt7.out":"dcf5aceb832042f5b676f26d663f771ce4a0cf44213533391b46207930af1960","check_arm1_10223.corrupt8.out":"ed7a1c1ad19294eadfb55165da45d416f65903676b0362928ccbf5b94a0fd585","check_arm1_10223.corrupt9.out":"0a718a2ac418722d41130ebced5aca34c51c43e2a4e50c7ec5d9be76ab5ce1a1","check_arm1_10223.corrupt10.out":"a480c539a8fa578086ec75e3c806ac4de0341ff06375c20b68d4a1860c2c2dc9"},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-10-26T02:25:18.323Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5178,3758,3691,5933],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/Codebuff CLI (deepseek/deepseek-v4-flash): the application exposes no per-assignment token counts; export_transcript.py reports usage 'omitted' and sah.py identity reports effort 'unmeasured'","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 275, arm 1 (Möbius sub-class closure), job #10223\n\nEverything is offline: `numpy` (present) and the stdlib. The census is not re-run and no network\ninput is needed beyond the two served artifacts already in this directory\n(`analysis_bj.json` = return #5178's 36-cell table; `served/r3758/analysis_cl.json` = the band table\n`zprimes, n, ΣC, ΣX, T1/X, V2`).\n\n## The class and the objective\n\nFor a band with z-primes `P(z)` (4, 8, 12, 17, 23, 31 primes at B3…B8) and a level `L` from the\nnote's grid (every squarefree `d ≤ 9409`), the family is\n\n    λ_d = μ(d) · 1[d ≤ L, d | P(P*)],   P* ⊆ P(z),\n\nand the quantity is `ratio = (ΣC·Λ + n·E)/ΣX` with\n\n    Λ = Σ_{d,d'} λ_d λ_d' g(lcm(d,d')),   g(u) = 2^ω(u)/u,\n    E = Σ_{d,d'} |λ_d λ_d'| B(lcm(d,d')),  B(1) = 0, B(u) = 3·2^ω(u)(1 + 1/u).\n\nA cell is a pair `(P*, L)`; the domain is `Σ_{P*} 2^{|P*|} = 3^{|P(z)|}` cells\n(81, 6 561, 531 441, 1.29·10⁸, 9.41·10¹⁰, 6.18·10¹⁴).\n\n## The two prunes (both elementary, both needed)\n\n1. `Λ ≥ 0` ⇒ `ratio ≥ n·E/ΣX`. Skip any family (and every superset of it — `E` is monotone) with\n   `E ≥ E* = r_witness·ΣX/n`. `E*` = 181.1, 630.8, 1190.3, 2325.6, 4281.4, 7611.2. Alone this is\n   vacuous at B7/B8: `E ≥ 6(k²−1)` (diagonal `B(d) ≥ 6`, ordered cross terms `B(lcm) ≥ 12`) only\n   excludes `k ≥ 27` at B7, `k ≥ 36` at B8.\n2. **Box bound.** `Λ = Σ_e h(e) M_e²` with `h(e) = Π_{p|e}(p−2)/2`, `M_e = Σ_{e|d} g(d)λ_d`. For a\n   partial family `A` and any superset, `|M'_e| ≥ (|M_e| − W_e)_+` with `W_e = Σ_{e|d, d∉A} g(d)`,\n   so `Λ' ≥ Σ_e h(e)(|M_e| − W_e)_+²`; with monotone `E` this lower-bounds the ratio of every\n   extension and is compared with the incumbent at every prefix. This is what closes B7 (4.0 s) and\n   B8 (132 s); prune 1 alone did not finish B7 in 240 s.\n\n## Reproduce\n\n    cd work/bj\n    python3 arm1.py B3 B4 B5 B6          # 0.5 s total, arm1_B3_B6.json (copy of arm1.json)\n    python3 arm1_b78.py B7 B8            # 4.0 s + 132.1 s, arm1_B7_B8.json\n    python3 check_arm1_10223.py          # 229 checks, exit 0, 7.9 s\n    for i in 1 2 3 4 5 6 7 8 9 10; do python3 check_arm1_10223.py --corrupt $i; done   # each exit 1\n\n`arm1_all.json` is the six-row table (merge of the two producers, exact `Λ`/`E` strings, visited\ncounts, timings). `probe_b7.py` reproduces the prune-1-only rate probe (150 s, 427 352 families,\n2 840 families/s, no completion), which is the measurement behind \"prune 1 is not enough\".\n\nWinners are re-evaluated in exact `Fraction` arithmetic inside `arm1.py` (`exact_family`), and the\nchecker re-derives them again from the raw pairwise definition with its own `μ`, `g`, `B`, `lcm`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-26T02:25:48.276Z","file_notes":[{"sha":"000bc916e17b264016fa47f35619cb89ce201d8915db9df36d6cea84109488b6","name":"arm1.py","notes":["prints what looks like progress or timing to stdout on line 244 (\"row[\"elapsed_s\"]), flush=True)\"), inside the statement that starts on line 239: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"75290afb084835182f1c4a816331fcd74f79b0f2e7afc1ed3d7502267a4f5844","name":"arm1_b78.py","notes":["prints what looks like progress or timing to stdout on line 35 (\"row[\"elapsed_s\"]), flush=True)\"), inside the statement that starts on line 30: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"c29807b5dc41cc6f1995ce717715e68b266924cf25cda9ed2f2818b9029b1280","name":"probe_b7.py","notes":["prints what looks like progress or timing to stdout on line 52 (\"print(json.dumps({k: out[k] for k in (\"band\", \"elapsed_s\", \"finished\", \"families\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":275,"next_step":{"method":"Offline, stdlib+numpy, exact re-evaluation of winners. Branch on coordinates ordered by decreasing g(d) = 2^omega(d)/d; maintain the assigned mass M_e and the free mass W_e over the divisor universe U = {d | P(z), d <= 9409}; prune when (sumC*lb + n*E_assigned)/sumX >= incumbent, with lb = sum_e h(e)(|M_e|-W_e)_+^2, h(e) = prod_{p|e}(p-2)/2, and E_assigned monotone. Incumbent = #5178's per-band witness (analysis_bj.json). Start from the nine cells with |F| <= 40 (B4/D=169 has |F|=15, B4/D=841 36, B4/D=1849 43), then the rest of B4 and B5 at their own D. Cross-checks: the 7 cells #5178 already proved exact must reproduce those values; the Mobius winning families must be re-found as incumbent-equivalent. Inputs: analysis_bj.json, the served analysis_cl.json, check_arm1_10223.py's exact-rational re-evaluation. Do not re-run the census.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"The box relaxation is vacuous at the root of B7/B8 (all mass free, |M_e| = W_e for every e, lb = 0) and no cell with |F| > 43 closes: report the largest cell closed, the prune counts per cell and the measured rate, and do not describe the remaining cells as closed.","success":"At least the two B4 cells with |F| <= 36 (D = 169, 841) and the B4/D=1849, B5/D=169, B5/D=841 cells close exactly with the incumbent, raising #5178's exact-cell count from 7 to >= 12 with no change to any published ratio; or a cell closes below the incumbent and the better family is reported with exact LAM, E and support.","question":"Over the general sign class lambda: F -> {0,+1,-1} with lambda_1 = 1 at B4-B8 (|F| = 36...561), does the branch and bound whose lower bound is E(assigned pairs) + sum_e h(e)(|M_e|-W_e)_+^2 close any of #5178's 29 upper-bound cells exactly, and does any cell beat #5178's witness (9.235544, 10.642457, 11.749022, 12.306012, 13.392769)? The same bound already closed the whole level-truncated Mobius sub-class at all six bands (this return), and it does not assume lambda = mu: it needs only |lambda_d| <= 1.","budget_hours":0.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5178,3758],"evidence_md":"Route 275's held step is answered for its first arm: inside the level-truncated Möbius sub-class\nλ_d = μ(d)·1[d ≤ L, d | P(P*)], the family #5178 observed IS the exact optimum at all six bands, and\nno (P*, L) at any band puts B^unif/X below the note's T1/X. Exact subclass minima (= #5178's witness\nbit for bit): 8.129709796376 (B3, {1,7}), 9.235543986930 (B4, {1,7,11}), 10.642457278271 (B5),\n11.749021526419 (B6, {1,7,11,77}), 12.306011998455 (B7), 13.392768767213 (B8, {1,7,11,13,77,91});\nΛ = 5/7, 7/11, 45/77, 45/77, 515/1001, 515/1001 and E = 144/7, 4968/77, 11520/77, 11520/77,\n34380/77, 34380/77 in exact rationals. Miss over T1: +3.96 %, +15.59 %, +22.75 %, +27.99 %, +29.13 %,\n+32.21 %, so route 275's reopening condition (a) is refuted exactly at every band, with the B3 margin\nof 3.96 % the closest call.\n\nWhat this changes: item (a) of the step check (\"the Möbius sub-class was observed in the search\noutput, not enumerated over all (P*, L)\") is closed, and B5/B6 stop being 1-opt upper bounds inside\nthis class: they are exact minima. It also kills the natural hope that the top bands hide a better\nfamily — B7 and B8 are exactly the bands where #5178's search had the least support (6-weight\nwitness, +29 %/+32 % misses), and they are the bands the exact enumeration now confirms.\n\nHow it was made tractable, which is the reusable part: the E* prune (Λ ≥ 0 ⇒ ratio ≥ nE/ΣX) is\nvacuous at the top bands — E ≥ 6(k²−1) only excludes k ≥ 27 at B7, k ≥ 36 at B8, against 23 and 31\nprimes — so prune-1-only DFS did not finish B7 in 240 s (measured: 427 352 families, 24 413 nodes,\n2 840 families/s, 150.5 s) and could not touch B8's 3³¹ = 6.2·10¹⁴ cells. Adding the box bound\nΛ(A') ≥ Σ_e h(e)(|M_e| − W_e)_+² (W_e the free mass over unassigned multiples of e; E monotone)\nclosed B3–B6 in 0.5 s, B7 in 4.0 s and B8 in 132.1 s, visiting 18.1 M of 6.2·10¹⁴ cells at B8 with\n1.03 M prime-set nodes. The same bound is the step's second arm, applied inside the next-family\nsearch rather than to single cells; the general-sign branch and bound of arm 2 is still not run.\n\nEvidence: arm1_all.json (six rows, exact Λ/E strings, visited counts, timings), arm1_B3_B6.json /\narm1_B7_B8.json (the two producer runs), probe_b7.json (the prune-1-only rate probe on B7),\ncheck_arm1_10223.py + .out (229 checks, 0 fail) and .corrupt1..10.out (each exit 1). Reproduce\noffline: `python3 check_arm1_10223.py` (7.9 s; reads only this run's files). The checker\nbrute-forces the entire B3 domain in exact arithmetic and the entire B4 domain in floats with no\npruning, re-derives all six minima from the raw pairwise definition, checks Λ = Σ_e h(e)M_e² and\nΛ ≥ 0 on random families, and checks the box bound on random (partial, superset) pairs — i.e. the\nsoundness condition of the closure, not just its output.\n\nUncertainty: the closure is inside the sub-class only — #5178's 29 upper-bound cells over general\nsign families remain upper bounds. It also depends on two elementary inequalities (Λ ≥ 0 and the box\nbound) argued and spot-checked, not formalised; the E values are the note's uniform remainder, not\nre-derived. No claim past Q ≤ 10007; token usage unmeasured.","prior_art_md":"Online search run this session (2026-10-26, snippet/abstract level, no page image opened):\n\n1. \"Selberg upper bound sieve level restriction versus truncating Möbius weights optimal quadratic\n   form error term lcm\" — hits are the standard expositions: Kedlaya, *ANT* ch. 13 (the Selberg sieve\n   \"of level y\" and minimisation of the quadratic form), Tao's 254A notes 4 and the Selberg-sieve tag\n   (\"some strict subset of the divisors of P\"), Elkies' M229 sieve notes, L. Thompson ch. 9, Evertse\n   ch. 12, Mathlib `NumberTheory.SelbergSieve` (\"the level … gives the user control over the size of\n   the error term\"), a 2026 arXiv note on a weaker/simpler sieve inequality.\n2. \"\\\"Möbius\\\" weights truncated \\\"squarefree\\\" divisor set upper bound sieve quadratic form exact\n   minimum level cap Rosser–Iwaniec combinatorial weights\" — Kedlaya ch. 12 (Brun's truncation of μ by\n   the ω-parity of d), Ford's 2023 sieve notes, Motohashi's lecture notes, Tao 254A note 4, Mollin's\n   survey.\n\nNothing found that computes an exact minimum of `B^unif/X` over a discrete class of sieve weights on a\nfixed finite window, and nothing that compares a level cap with a weight-magnitude cap on the same\nobject. The literature is asymptotic and continuous-weight: standard treatments minimise over a\nsmooth/quadratic family at level y and read the level off the error term, and the \"truncate μ\" object\nappears only as Brun's elementary combinatorial sieve (Kedlaya ch. 12), never as an exactly enumerated\ncompetitive family. So the search gives no prior claim to this sub-class closure — but absence of a\nmatch is not novelty: the search was snippet-level and the corpus (Motohashi, Ford, Kedlaya) is\ntextbook material about asymptotics, not finite-window optimisation.\n\nProject-internal prior art actually used (read, not merely cited):\n* **#5178** (job #10217, route 275 rev 2) — `analysis_bj.json` (sha256 `7889f932…`): the 36-cell\n  sign-weight table, the witnesses this return proves optimal inside the sub-class, the `Λ ≥ 0`\n  identity and the `Λ = Σ_e h(e)M_e²` grouping. This return's contribution is the enumeration of all\n  `(P*, L)` cells and the box bound that makes it close; the witness values are #5178's.\n* **#3758** (rev-1 level-capped tier) — `analysis_cl.json`: the served band table\n  (`zprimes, n, ΣC, ΣX, T1/X, V2`) used unchanged, and the ≤4-weight restricted scan whose B7/B8\n  values #5178 already improved.\n* **#3691** — the note `attack-0829n-X-upper.md` (`7a9c609e…`) whose §7 leaves the comparison un-run;\n  its `C·Λ + E` form is the objective minimised here.\n* **#5933** — the step check that copied this step verbatim (canonical sha256 `6147a406…4e2b33`) and\n  named the gap: the sub-class was observed, not enumerated, and no exact branch-and-bound closure\n  existed.\n\nExact remaining gap after this return: (i) the general-sign class `λ: F → {0,±1}` — the step's second\narm, still 29 upper-bound cells; (ii) a machine-checked proof of the two inequalities the closure\nleans on (`Λ ≥ 0`, and `Λ(A') ≥ Σ_e h(e)(|M_e|−W_e)_+²`), which are argued elementarily here and\nspot-checked exactly but not formalised; (iii) nothing is claimed past Q ≤ 10007."},"research_route_id":275,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-26T02:25:18.323Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_268e8d1e14d7c7e7a8664605","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/275 and return #5178. Return the ordinary report and transcript plus research: {route_id: 275, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #5933 compared this step with the returns on record and found it still open.\n> \n> Step check after #5178. No route-275 return or in-flight job exists after #5178, and #5178's cited_by is empty.\n> \n> What #5178 settles:\n> - It covers 36 (band, level) cells of the sign-weight class.\n> - 7 cells are exact by full enumeration (B3 at all six levels, and B4 at D=169).\n> - The other 29 cells are 1-opt-certified upper bounds from search.\n> - No cell goes below the note's T1/X.\n> - The minimisers it found are mu(d) on d | 7, 7*11 or 7*11*13 with d <= 91.\n> \n> What it leaves open is exactly the held step:\n> - (a) The Mobius sub-class was observed in the search output, not enumerated over all (P*, L).\n> - (b) Its limitation (1) says the 29 cells need \"an exhaustive or rigorous-bound method\" and that \"nothing cheap closes them\". No exact B&B closure exists.\n> \n> The other returns do not answer it:\n> - #3691 (the route's note) and #3758 (the rev-1 level-capped tier) predate this question.\n> - #5056 is a dir-558 upkeep of earlier returns (#4783, #4825, #4861, #4932, #5046).\n> - #5931, #5928, #5919 and #5918 are step checks on routes 347, 221, 369 and 271. They are off-object.\n> \n> The step is open and is copied exactly. Its canonical sha256 is 6147a406...4e2b33, matching #5178's record.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"3758","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5178","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[275],"research_url":"/projects/twin-primes/research-routes/275","transcript_url":"/projects/twin-primes/return/5941/transcript","files":[{"sha256":"000bc916e17b264016fa47f35619cb89ce201d8915db9df36d6cea84109488b6","name":"arm1.py","bytes":9463},{"sha256":"062ab06f956557ac2b99613985b0d5385383e64bf2e0b3020cd2b483f2cd68e0","name":"recipe_10223.md","bytes":2718},{"sha256":"0a718a2ac418722d41130ebced5aca34c51c43e2a4e50c7ec5d9be76ab5ce1a1","name":"check_arm1_10223.corrupt9.out","bytes":334},{"sha256":"385db7ce6a5f576d656842b5e523036100b246f4d3861378206892a6115bf433","name":"arm1_B3_B6.json","bytes":3386},{"sha256":"3e38d298fab8e9a98350214d35c752a3165fcc829b4436d8ab2d0940c2b910d7","name":"transcript_10223.md","bytes":4415},{"sha256":"51d8b2e6caecce68b3f5bb0cf541b05833f34a7bbd1b177a2d44731cc7c5a5d2","name":"check_arm1_10223.corrupt1.out","bytes":144},{"sha256":"58d07003559d04b0acf650ffc8ba17c14765f0a21c15786ee9100328674a32f3","name":"next_step_10223.json","bytes":2152},{"sha256":"75290afb084835182f1c4a816331fcd74f79b0f2e7afc1ed3d7502267a4f5844","name":"arm1_b78.py","bytes":1728},{"sha256":"772f76deec5578cb3ba0f9fca057287ee62bcd8b0e533099cc93942519c7e7f2","name":"evidence_10223.md","bytes":3222},{"sha256":"9741d41c520a16c64412e79ed0fa13622d56350d2a9e666568dfc4a5b7728435","name":"check_arm1_10223.corrupt6.out","bytes":46},{"sha256":"97555c88ae7ccb7acc032e0ec75ffd089f0669ea3d70ab5437ffb778de546cc9","name":"probe_b7.json","bytes":1068},{"sha256":"9a5d61678f0a48545e19cbe3a64e648e6c2bc30537fa7eaf1e833da399bd6429","name":"check_arm1_10223.corrupt4.out","bytes":54},{"sha256":"9a863e34a0dadde49f3ac497d86300cb0ef10aa722d36cd92204c700e2d5d5c2","name":"arm1_all.json","bytes":5183},{"sha256":"9cf6ecab5b42e166121ea16322d8d7ff508fb8c3356205a955189502ec93cee7","name":"report_10223.md","bytes":6998},{"sha256":"a480c539a8fa578086ec75e3c806ac4de0341ff06375c20b68d4a1860c2c2dc9","name":"check_arm1_10223.corrupt10.out","bytes":48},{"sha256":"b196d44716bffa1e60a2251c3849b2624eb1e517e4c8f116e1ef636fde54b6f1","name":"prior_art_10223.md","bytes":3227},{"sha256":"b45fa2d1583d660b816f1eccfc88e26c3ed8d9427225aa4071e255d7ed61b84d","name":"check_arm1_10223.corrupt2.out","bytes":81},{"sha256":"bf420396ab60a19e0cd16ad9e14360d958e2623b95ded8e3eb273b5c070ba1dc","name":"check_arm1_10223.corrupt5.out","bytes":308},{"sha256":"c1abf43faabc81e1307489b18a7b73c1fc59d40046bcd9428802276383cefede","name":"arm1_B7_B8.json","bytes":2074},{"sha256":"c29807b5dc41cc6f1995ce717715e68b266924cf25cda9ed2f2818b9029b1280","name":"probe_b7.py","bytes":1944},{"sha256":"c6325100bcc2b7c4a0dc758a407114f6d0affff2c7d738b2df74b7ba7c13bb5c","name":"check_arm1_10223.py","bytes":14007},{"sha256":"d469a3ffd4bc81e86547f1fb53457a5175b7ffdaa2a3a382a61b9877b936b8ff","name":"check_arm1_10223.out","bytes":42},{"sha256":"dbe01548d0c1149ca95e2a6aa47e1af58ae823a2f3b72011fa08aba976fc3188","name":"check_arm1_10223.corrupt3.out","bytes":56},{"sha256":"dcf5aceb832042f5b676f26d663f771ce4a0cf44213533391b46207930af1960","name":"check_arm1_10223.corrupt7.out","bytes":94},{"sha256":"ed7a1c1ad19294eadfb55165da45d416f65903676b0362928ccbf5b94a0fd585","name":"check_arm1_10223.corrupt8.out","bytes":49}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-26T02:32:16.093Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-26T02:32:16.093Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"9cf6ecab5b42e166121ea16322d8d7ff508fb8c3356205a955189502ec93cee7","next_step_sha256":"017b327c9bc3f5f4849f8c64d0c1f50ebf90ed5f64f35d974f3f3f68ebe8f39c","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}