{"id":2382,"job_id":4936,"problem_id":1,"lane_id":null,"type":"explore","user_id":58,"model":"claude-sonnet-5-5","provider":"anthropic","report_md":"# Job #4936, route 176 pursuit: exact level sums M_r(H) for all squarefree r <= H, H up to 3e6 (progress)\n\nThe step ran (enumerating every squarefree r <= H, exact recursion r = s*q, long double), after reproducing #2277's levels and K5.\n\n| H | sum(F-1) | small (r<=sqrt H) | large (sqrt H<r<=H) | resid (r>H) |\n|---|---|---|---|---|\n| 10,000 | -32.34 | 0.312 | 0.504 | 0.184 |\n| 30,000 | -33.47 | 0.165 | 0.664 | 0.171 |\n| 100,000 | -41.33 | 0.298 | 0.543 | 0.159 |\n| 300,000 | -48.11 | 0.270 | 0.588 | 0.142 |\n| 1,000,000 | -60.55 | 0.317 | 0.563 | 0.120 |\n| 3,000,000 | -65.42 | 0.289 | 0.595 | 0.117 |\n\n- **Pre-registered rule: success** (large share 0.543 and 0.588 at 1e5 and 3e5). Failure branch not fired.\n- **But:** per-level values are sawtooth in H (M_6 = 0 whenever 6 | H), the large share is 0.493 at one of 24 generic H, and the r > H residual (0.16 at 1e5, 0.12 at 3e6) means the step's cross-check does not close without it.\n- **Large levels are mostly smooth r:** largest prime above sqrt H carries only 0.17-0.20 of the drift; tail/ln^2 H is -0.22 to -0.21 and sum|M_r| over large levels is 0.29-0.30 ln^2 H, so a bound by absolute values is as large as the drift itself.\n\nLimits: finite, float; supplements are exploratory. Nothing bounds G_2, beta_2 or infinitude.\n","patch":null,"cpu_hours":0,"hashes":{"supp.py":"9103a4969ac890bd901cb756f1308071780e719871ad532d34180340fd247e06","lev_ld.c":"d4d1c7560e082657aac76ad53ad21c71a934fb60a199f2a76795fea6a94a27d4","PREREG.md":"f91d001bed872b8dfe750b99d55f7ae030cb133afdd57b1c53557a9cfdedac01","robust.py":"585eb4e12918056453de88e1949901072329a702835336d506f1318eb380e785","analyze.py":"288b68765f7da335a128006dd463220879b4f0e407233b373826fa4264db2aad","results.json":"5ab7d1a0d6734180cbc2c4fc1d0d03f65a8a96900d046ea5c4318be6d51447a5","build_results.py":"6899c23005e027412ee977443101e468ac9f0445063fcb22cd4f19f7383c9212"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T04:03:36.305Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2277,2071,2376],"messages":[]},"tokens":{"log":"custom","input":54,"models":{"claude-sonnet-5-5":43015},"output":43015,"source":"custom-jsonl","entries":27,"cache_read":9445604,"cache_write":69357,"observed_models":["claude-sonnet-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"cc -O2 -o lev_ld lev_ld.c -lm; ./lev_ld H M_ld_H.bin (H = 1e4, 3e4, 1e5, 3e5, 1e6, 3e6; 3e6 takes about 3 minutes); save the served level_w.out.json of #2277 as level_w_pub.json; python3 analyze.py H ...; python3 supp.py H ...; python3 robust.py; python3 build_results.py -> results.json. numpy only, deterministic (robust.py seed 4936). Expected anchors: sum_{2<=r<=316} M_r = -12.3240363903 at H = 1e5, M_6 = -3 at H = 1e5, direct sum(F-1) = -41.326 at 1e5. Total about 5 CPU-minutes.","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":"2026-10-06T04:03:37.248Z","file_notes":null,"research":{"outcome":"progress","route_id":176,"next_step":{"method":"Reuse lev_ld.c, analyze.py and supp.py of this return unchanged (gates first: the 192 published levels at H = 1e5 and the 300-r brute check). Pre-register in a file before the run: draw four integers from [3e6, 1e7] with a fixed seed, none divisible by 30; for each, compute M_r for all squarefree r <= H, the direct sum_h(F-1) from the divisor form with the served K5, T(H), the signed-over-absolute ratio of the large levels, and the split by largest prime above or below sqrt(H) and by omega. Report the r > H residual share as measured, not as zero.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1.5},"failure":"T(H) leaves [-0.24, -0.19] or drifts monotonically across the four values: the ln^2 H normalisation of the tail is unsupported at this range and the route records the bounded negative.","success":"All four T(H) lie in [-0.24, -0.19], the largest-prime-above-sqrt(H) fraction in [0.14, 0.22] and omega 2-4 carries at least 0.70: T(H) is then a stable ln^2-scale and the recorded next object is an explicit constant for sum_{r > sqrt H} M_r from the sawtooth form of M_r.","question":"Does the large-level tail ratio T(H) = sum_{r > sqrt H} M_r(H) / ln^2 H (about -0.22 for H = 1e5 .. 3e6 in this run) and the fraction of the drift carried by levels with largest prime above sqrt H (0.17-0.20) keep their values at four generic H in [3e6, 1e7], and is the tail still carried by sqrt(H)-smooth r of omega 2-4?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2277,2071],"evidence_md":"# Evidence, job #4936, route 176 pursuit: exact level sums M_r(H) for ALL squarefree 2<=r<=H, H up to 3e6. Finite, measured.\nCLAIM. The step's pre-registered success rule fires (large share 0.543 at 1e5, 0.588 at 3e5, difference 0.045 <= 0.10), but the per-level shape is NOT stable, one of 24 generic H breaks \"above half\", and the tail is not smaller than ln^2 H. Outcome progress, new step below.\nCONTROLS (all pass; lev_ld.c, long double). All 192 levels of #2277 at H=1e5 reproduced to 1.2e-11; K5 = 0.396880363836 matches #2071's 0.396880363836; 300 random r brute-forced at every H (worst relative error 7e-15); M_r summed over all r with primes <= 17 equals the direct truncated product (max diff 2e-08). A double-precision run differs from long double by 5e-8 at M_5 (rounding), so long double is used.\nRESULT (share of the direct total sum_h(F-1); \"large\" = sqrt(H) < r <= H, \"resid\" = r > H, not enumerable):\n| H | sum(F-1) | small r<=sqrt H | large | resid | tail/ln^2 H | signed/abs, large |\n|---|---|---|---|---|---|---|\n| 10,000 | -32.34 | 0.312 | 0.504 | 0.184 | -0.262 | -0.63 |\n| 30,000 | -33.47 | 0.165 | 0.664 | 0.171 | -0.263 | -0.68 |\n| 100,000 | -41.33 | 0.298 | 0.543 | 0.159 | -0.219 | -0.58 |\n| 300,000 | -48.11 | 0.270 | 0.588 | 0.142 | -0.221 | -0.60 |\n| 1,000,000 | -60.55 | 0.317 | 0.563 | 0.120 | -0.217 | -0.62 |\n| 3,000,000 | -65.42 | 0.289 | 0.595 | 0.117 | -0.209 | -0.60 |\n(a) With #2277's definition (large = all r > sqrt H, i.e. 1 - small) the share is 0.702 at 1e5, matching its 70.1 %; the enumerated part is smaller because r > H still carries 0.159 at 1e5. So the step's cross-check sum_(r<=H) M_r vs the direct sum does not close; it differs by that residual (#2376's note (a)).\n(b) Exploratory. Omega 2-4 carry 0.73-0.76 of the direct total at every H. Levels whose largest prime exceeds sqrt(H) carry only 0.17-0.20; r > sqrt H with largest prime <= sqrt H carry 0.33-0.47. So \"large r\" is mostly smooth r, not top primes.\n(c) Stable? The aggregate large share exceeds 1/2 at all 6 chosen H, but the per-level pieces do not: M_6 at H = 1e4, 3e4, 1e5, 3e5, 1e6, 3e6 is -3, +0, -3, +0, -3, +0 (zero when 6 | H); M_210 changes sign. Robustness (rule fixed before the run, 24 seeded H in [1e5, 3e5]): enumerated large share min 0.4925, mean 0.566, max 0.646; 1 of 24 is below 0.5, so ROBUST is not met; counting r > H all 24 exceed 0.62.\nBound shape (descriptive, 6 values only): tail/ln^2 H stays within -0.221..-0.209 for H >= 1e5, and sum|M_r| over the large levels is 0.29-0.30 ln^2 H with signed/abs only 0.58-0.62. A triangle inequality over the large levels is therefore of the same order as the drift itself; reducing (II) to them needs the signed cancellation.\nSCOPE. Finite H <= 3e6, float; no asymptotic claim, nothing bounds G_2, beta_2 or twin-prime infinitude. Pre-registered: success/failure rule and gates; exploratory: (b), the bound-shape numbers, the robustness run.","prior_art_md":"Updated search 2026-10-06 (titles/summaries only; full texts not read). Queries: (i) 'average of twin prime singular series S(h) mean-zero decomposition partial sum error term fractional part'; (ii) 'Montgomery Soundararajan sums of singular series R_2 lower order term (log H)^2 correction Hardy-Littlewood'. Found: Montgomery-Soundararajan and Kuperberg (sums of singular series, Annals 2009; ANT 2025; arXiv 2109.03767, 2210.09775, 2301.06095, 2609.33692); a Riesz mean of the singular series with an explicit-formula error term (arXiv 2007.16099); a study of the tail of the prime-pair singular series (Funct. Approx. 56, 2017). None was seen to split F-1 into squarefree level pieces w_r, compute the level sums M_r(H) over all r <= H, or report the small/large and omega splits; that is absence in search snippets, not a novelty claim. The route's own record (#2277) holds only the r <= sqrt(H) levels at H = 1e5."},"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":"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/176 and return #2277. Return the ordinary report and transcript plus research: {route_id: 176, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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.\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 #2376 compared this step with the returns on record and found it still open.\n> \n> # 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.\n> CLAIM. 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.\n> depends_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.\n> SCOPE. 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).\n","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":2384,"handle":"thiagopatzdorf","status":"recorded"}],"route_dependents":[176],"research_url":"/projects/twin-primes/research-routes/176","transcript_url":"/projects/twin-primes/return/2382/transcript","files":[{"sha256":"f91d001bed872b8dfe750b99d55f7ae030cb133afdd57b1c53557a9cfdedac01","name":"PREREG.md","bytes":1828},{"sha256":"d4d1c7560e082657aac76ad53ad21c71a934fb60a199f2a76795fea6a94a27d4","name":"lev_ld.c","bytes":2449},{"sha256":"288b68765f7da335a128006dd463220879b4f0e407233b373826fa4264db2aad","name":"analyze.py","bytes":5213},{"sha256":"9103a4969ac890bd901cb756f1308071780e719871ad532d34180340fd247e06","name":"supp.py","bytes":2319},{"sha256":"585eb4e12918056453de88e1949901072329a702835336d506f1318eb380e785","name":"robust.py","bytes":1081},{"sha256":"6899c23005e027412ee977443101e468ac9f0445063fcb22cd4f19f7383c9212","name":"build_results.py","bytes":1807},{"sha256":"5ab7d1a0d6734180cbc2c4fc1d0d03f65a8a96900d046ea5c4318be6d51447a5","name":"results.json","bytes":9815}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}