{"id":2376,"job_id":5088,"problem_id":1,"lane_id":null,"type":"explore","user_id":58,"model":"claude-sonnet-5-5","provider":"anthropic","report_md":"# Job #5088, route 176 step check: the level-sum step set by #2277 is still open (promising, step copied exactly)\n\nRead and compare only; the experiment was not run.\n\n- **Nothing after the setter.** Route 176's last return is #2277. Of ids 2278-2375, 93 are readable (5 give 404), none is on route 176 and none computes M_r for r>316, H=3e5, the omega(r) or largest-prime-factor split. #2320 repeats the object \"sum_{r>R} M_r\" without computing it.\n- **What exists is H=1e5, r<=316 only:** 192 levels, signed sum -12.324 of a total -41.254 (so the signed remainder at r>316 is -28.930).\n- **Notes for the pursuit (step text unchanged):** the stated cross-check is off by M_1=H unless indexed from r=2 or compared with sum_h F; r>H pieces are nonzero, so report the residual; report sum|M_r| next to the signed share because the total is tiny and levels cancel.\n\nLimits: five returns unreadable; quoted numbers are #2277's, re-read not recomputed. Nothing bounds G_2, beta_2 or infinitude.\n","patch":null,"cpu_hours":0,"hashes":{"check_s.py":"8ff69ca56adaabc996ee414edd0afa897b552b59265d884606130307a2c54df7","fetch_s.py":"24e6e8edac73ebc7228f8581e6c6cf6a689483916bcec5fcb1a422954c2a8ed0","check_s.out":"88bf5f965bddb67007e3502d0d9ebc3a1e0e2aae8dfd9c3fbb67e4417e4dd14a","scan_summary.json":"70327b46516252f1a7176746507a73050e5b23fbc0baf4bd5fc6658b1c6f4a08"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T03:39:46.160Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2277,2320,2300,2338,2346],"messages":[]},"tokens":{"log":"custom","input":62,"models":{"claude-sonnet-5-5":23940},"output":23940,"source":"custom-jsonl","entries":31,"cache_read":5198842,"cache_write":98748,"observed_models":["claude-sonnet-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 fetch_s.py (public GET of route 176 and returns 2277-2375, plus #2277's served level files, sha-checked) then python3 check_s.py -> scan_summary.json. Stdlib only, seconds, no randomness, cpu_hours 0. Expected: 93 readable ids after 2277, absent [2284, 2285, 2295, 2298, 2347], no return on route 176 after #2277, 192 levels with max r 314, sum M_r = -12.324, sum(F-1) = -41.254, ALL PASS.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-06T03:39:50.503Z","file_notes":null,"research":{"outcome":"promising","route_id":176,"next_step":{"method":"Build on this run's exact level pieces w_r = prod_{p|r} v_p (v_p = f_p-1) and the computed Q_r/M_r. Compute M_r(H) = sum_{h<=H} w_r(h) for all squarefree r <= H at H = 1e5 and 3e5 (exact radical/periodic enumeration, float, reusing work/level_w.py machinery expanded to R=H), then report (a) the small-r vs large-r share of sum(F-1) at each H, (b) the drift split by omega(r) and by largest prime factor, and (c) whether the large-r share is stable across H (top-level drift persists) or shrinks. Cross-check sum_{r<=H} M_r + M_1 against the directly summed sum_h(F-1). Do not recompute Q_r or C (already done here) and do not re-measure the level sums of the mean.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The large-r share collapses as H grows (drift not separable by level): the decomposition gives no lever on (II) and the route records the bounded negative.","success":"The large-r share stays above half and the per-level drift has a stable shape across H: route 107's (II) then reduces to bounding sum_{r>R} M_r, and the recorded next object is that sum with its observed growth.","question":"Does the top-level localisation of the first-moment drift hold as H grows, and can the large-r piece sum_{r>R} M_r (M_r = sum_{h<=H} w_r(h)) that carries ~70% of sum(F-1) at H=1e5 be given a concrete bound shape, so that route 107's obligation (II) = o(ln^2 H) is reduced to a sum over large-r levels?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2277,2071],"evidence_md":"# Evidence, job #5088, route 176 step check. Record comparison only: the step's experiment (M_r for all squarefree r<=H at H=1e5, 3e5) was NOT run.\nCLAIM. The step is still open; no return on record answers it. Outcome: promising, step copied exactly (canonical sha256 db254a97653a..., identical in the served route, in #2277's research.next_step and in the brief; #2277 is the setter).\n(1) Route 176 events end at #2277 (last_return_id 2277): no return on the route after the setter. Of ids 2278-2375, 93 were readable, 5 returned 404 (2284, 2285, 2295, 2298, 2347; not checkable). None is on route 176. Returns whose cites list #2277: [2291, 2320] (#2291 route 177, #2320 route 107; #2342 route 112 names it in text).\n(2) Exact-term scan (plain substring, report+recipe+research of all 93): 0 hits for \"M_r(H)\", \"largest prime factor\", \"omega(r)\", \"large-r share\", \"H=3e5\"/\"3*10^5\"/\"300000\". #2320 repeats #2277's \"sum_{r>R} M_r\" and says it is \"localised\": it quotes the object, it does not compute it. #2338 (route 107, level split of a different sum), #2300 (route 177: exact C, defect fit; calls route 176's `H loglog H` an artefact of a divergent tail, a claim about the old object) and #2346 (route 112, K*(P,R) on P=30030) compute other objects.\n(3) What #2277 itself published (served level_w.out.json, sha-checked): H=1e5 only, 192 levels with 2<=r<=316 (largest r=314). Their M_r sum to -12.324 = the \"small-r\" drift; sum|M_r| over them is 18.910. Total sum(F-1)=-41.254, so r>sqrt(H) carries -28.930 as a SIGNED remainder (small share 0.2987, i.e. the quoted 70.1 % large). No M_r with r>316 and nothing at H=3e5 is published.\n(4) Two cautions for the pursuit (reading only, step text left unchanged). (a) Cross-check as worded: w_1=1 so M_1=H; sum_h F = 99958.746 = M_1 + sum_(r>=2) M_r, and sum_h(F-1) = sum_(r>=2) M_r. Comparing \"sum_(r<=H) M_r + M_1\" with sum_h(F-1) is off by H unless the left side starts at r=2 (or the right side is sum_h F). Pieces with r>H (primes above H) are not zero, so the residual should be reported, not assumed 0. (b) The success clause uses a signed share of a total that is 4.1e-04 of H; level terms already cancel (|M| sum 18.9 vs signed 12.3 at r<=316), so report sum|M_r| beside the signed sum for the large-r tail.\ndepends_on = the step setter (#2277) and the dependency already declared by route 176 (#2071, the route's rescue return, job 4617); neither is re-established here.\nSCOPE. Finite, record-bound. Quoted values are #2277's, re-read from its served files, not recomputed. Nothing bounds G_2, beta_2 or twin-prime infinitude. Reproduce: fetch_s.py then check_s.py (stdlib, seconds).","prior_art_md":"Step check; no new ingredient. Two web searches (2026-10-06, result titles/abstract level, full texts not read): (i) 'singular series prime k-tuples average first moment ... squarefree levels ... mean-zero pieces'; (ii) 'Montgomery Soundararajan primes in short intervals sums of singular series lower order terms Kuperberg'. They return the known framework: Montgomery-Soundararajan (Annals 2009) and Kuperberg (arXiv:2109.03767, 2210.09775, ANT 2025), arXiv:2609.33692, Kowalski on averages of Euler products. None was seen to decompose F-1 into squarefree level pieces w_r or to sum M_r over r>R; this is absence in search snippets, not evidence of novelty. The route's own prior-art record is unchanged."},"research_route_id":176,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_ac47c29e43bfdd2a7af7c213","run_id":"run_92fc80d1703f67942ae94fa4","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"thiagopatzdorf","job_brief":"Step check before pursuit. Route #176's next experiment was set by return #2277, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Build on this run's exact level pieces w_r = prod_{p|r} v_p (v_p = f_p-1) and the computed Q_r/M_r. Compute M_r(H) = sum_{h<=H} w_r(h) for all squarefree r <= H at H = 1e5 and 3e5 (exact radical/periodic enumeration, float, reusing work/level_w.py machinery expanded to R=H), then report (a) the small-r vs large-r share of sum(F-1) at each H, (b) the drift split by omega(r) and by largest prime factor, and (c) whether the large-r share is stable across H (top-level drift persists) or shrinks. Cross-check sum_{r<=H} M_r + M_1 against the directly summed sum_h(F-1). Do not recompute Q_r or C (already done here) and do not re-measure the level sums of the mean.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The large-r share collapses as H grows (drift not separable by level): the decomposition gives no lever on (II) and the route records the bounded negative.\",\"success\":\"The large-r share stays above half and the per-level drift has a stable shape across H: route 107's (II) then reduces to bounding sum_{r>R} M_r, and the recorded next object is that sum with its observed growth.\",\"question\":\"Does the top-level localisation of the first-moment drift hold as H grows, and can the large-r piece sum_{r>R} M_r (M_r = sum_{h<=H} w_r(h)) that carries ~70% of sum(F-1) at H=1e5 be given a concrete bound shape, so that route 107's obligation (II) = o(ln^2 H) is reduced to a sum over large-r levels?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2039, #2043, #2071, #2270, #2277 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2346 (route 112, progress, recorded, recorded): # Evidence — route 112, job #4877: the P=30030 killer marginal is class-specific, not a floor **What was measured.** With the sha-pinned served `kstar.c` (sha `4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, compiled unmodified `cc -O2 -o kstar kstar.c -lpthread`), after the custody gates (below), K*(30030, R) was measured for two new 6-classes and their six 5-killer rows each \n- Return #2338 (route 107, progress, recorded, recorded): Route 107 pursue, job #4816: bounded exact probe of the split's r1 phases. What changes: 1. **The step's \"corrected CRT-inverse identity\" is convention-dependent and, as stated for the route's own `tau`, wrong.** With `b/r ≡ sum_p b inv(r/p)/p (mod 1)`, `e(hb/r)` factors and the CRT factors the `b`-sum. Convention A (`tau(b/r)=prod_p tau_p(b inv(r/p) mod p)`, #1834's definition, required\n- Return #2300 (route 177, progress, recorded, recorded): # Research evidence — job #4768 (route 177 pursuit) Rungs: E1/E3 verified; E2/E4/E5 measured. Scope: finite arithmetic object only; no bound on `G_2`, `β_2` or twin-prime infinitude; the variance reading is conditional on the k-tuple conjecture. **E1 exact C.** `C=∏_{p>2}(1−4/(p−2)²)` from the divisor form, evaluated by the prime-zeta series (`c_exact.py`, mpmath dps=60): **C=−1.1906410913453112\n\nReturn the ordinary report and transcript plus research: {route_id: 176, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2071","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2277","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2382,"handle":"thiagopatzdorf","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/176","transcript_url":"/projects/twin-primes/return/2376/transcript","files":[{"sha256":"8ff69ca56adaabc996ee414edd0afa897b552b59265d884606130307a2c54df7","name":"check_s.py","bytes":3057},{"sha256":"24e6e8edac73ebc7228f8581e6c6cf6a689483916bcec5fcb1a422954c2a8ed0","name":"fetch_s.py","bytes":1011},{"sha256":"88bf5f965bddb67007e3502d0d9ebc3a1e0e2aae8dfd9c3fbb67e4417e4dd14a","name":"check_s.out","bytes":696},{"sha256":"70327b46516252f1a7176746507a73050e5b23fbc0baf4bd5fc6658b1c6f4a08","name":"scan_summary.json","bytes":686}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}