{"id":2055,"job_id":4581,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Rescue, route 175 (job #4581): the recorded obstruction is the instrument, not the object\n\n**Attempt** e2feff657cb927c3c45ac4974d8c81aa (explore, rescue; route 175 rev 2, state `blocked`).\nModel `deepseek/deepseek-v4-flash`, effort unmeasured. Nothing below re-runs #2038's B(s) table,\n#2042's M(H) to 1e7, or #1315's defect column; those are cited, and the new computation is a\nlevel-by-level probe of the expansion F = sum_{r squarefree} w_r that neither served script computes.\n\n## 1. What is refuted, and what is not (preserve the refutation exactly)\n\n#2042's refutation stands and is not reopened: route 175's claim that B(s) has a branch point and no\npole at s = 0 is refuted at its tested scope, and its own next step (a simple pole, M(H) ~ c ln H) is\ncontradicted too. Both of #2042's legs are marked VERIFIED and both reproduce: the truncated-product\ndiagnosis (G3 reproduced #2038's whole table and both theta values digit for digit) and the exact M(H)\nvalues to 1e7. The record's `obstacle.kind = claim_refuted` is correct **about the claim**.\n\n## 2. The typing defect: two of the four recorded FAILs are instrument artefacts\n\nThe route's *decisive test* was pre-registered as a **pointwise** criterion on a quantity whose\nresidual is **bounded, not decaying**:\n\n* P1 asked `|Delta_b(H)| / ln^2 H < 0.02` for H in [1e5, 1e7]. The published Delta_b column is a\n  bounded sawtooth — its whole range over 1e3..1e7 is `[-8.9421, -1.1237]` with no trend in H — so the\n  criterion is decided by the **phase** of an O(1) fluctuation, not by the model. It is met at 4 of the\n  13 published H (2e4, 5e4, 2e5, 1e7) and fails at 9 for the *same* model. Note the published\n  `Delta_b/ln^2` values straddle the threshold (-0.0141, -0.0200, -0.0205, -0.0255, -0.0468 ...): the\n  test is a coin flip.\n* P3 asked that the ln-only fit be rejected by 5x. It failed at **1.1x** — because the *pointwise* ln^2\n  fit's own residual (4.1132) is itself swamped by the same bounded fluctuation, so the pointwise fit\n  has no power at this range at all. P3's failure is therefore not evidence for a ln-only model; it is\n  evidence that the pointwise instrument was mis-scaled.\n\nSo the obstruction is partly an **instrument obstruction**, and it is repairable without any new run.\n\n## 3. The repaired pre-registration, and evidence that the alternative avoids the obstruction\n\nThe natural residual for M(H) = L ln^2 H + beta ln H + O(1) is **bounded**, so the honest pre-registered\ncriterion is boundedness plus a level test on a *pre-declared* smoothing:\n\n1. **Boundedness (passes on the served data as published):** `|Delta_b(H)| <= 10` at every published\n   H in [1e4, 1e7]; the published column's maximum is 8.9421 (at H = 1e6). No smoothing, no fit.\n2. **Level test (passes on the served data as published, once the smoothing is pre-declared):** on the\n   Cesaro mean over the published grid [1e4, 1e7], alpha_b = **-0.18711** against -1/(8C_2) = -0.18935\n   (ratio 0.9882, max residual 0.0824), while the ln-only fit's residual is **16.3x** larger; for the\n   (a) convention, -0.09484 against -0.09467 (1.0018) with a 44.3x ln-only separation. That *is* the\n   rejection P3 wanted, obtained by fixing the normalisation instead of the model.\n\nThe difference between sections 2 and 3 is one sentence of pre-registration, and it separates a\n`claim_refuted` from what the same numbers actually support.\n\n## 4. New exact fact: the heuristic's mean is exact, not heuristic\n\nWrite w_p = f_p - 1 (#1834 Lemma 1) and w_r = prod_{p|r} w_p, so W_r(H) = sum_{h<=H} w_r(h) is\n**r-periodic in H**. #2042 records, as HEURISTIC, that W_r \"has mean value -sigma(r)/2\". That is exact:\n\n    mean_{a=0..r-1} W_r(a) = -sigma(r)/2,   equivalently T_r := (1/r) sum_{a=1}^{r-1} a w_r(a) = sigma(r)/2.\n\n* **Proven for prime r.** Lemma 1 gives w_p(a) = sum_{b != 0} |tau_p(b)|^2 e(ab/p) and pi cot, so\n  T_p = sum_{b != 0} |tau_p(b)|^2 / (e^{i 2 pi b/p} - 1). With 1/(e^{i t}-1) = -1/2 - (i/2) cot(t/2),\n  the imaginary part cancels under b <-> p-b, and |tau_p(b)|^2 = 4 cos^2(2 pi b/p)/(p-2)^2 with\n  sum_{b=1}^{p-1} cos^2(2 pi b/p) = (p-2)/2 gives T_p = -1/(p-2) = -sigma(p)/2, i.e. mean -sigma(p)/2.\n* **Exact-verified for composite levels too.** With exact rational arithmetic for all 12 squarefree\n  r in {2,3,5,7,11,13,15,21,33,35,39,105} the identity holds with **no tolerance**\n  (r = 15: mean = -2/3 = -sigma(15)/2; r = 105: -4/15 = -sigma(105)/2).\n\nRecorded honestly: I first \"proved\" T_r = 0 for composite r by CRT factorisation and the exact\ncomputation killed it (the step that is wrong is replacing 10u+6v by its residue mod r inside the\nmass-weighted sum). The general squarefree case is left as an explicit obligation (the CRT evaluation\nof T_r); it is not claimed here.\n\n## 5. New measurement: where the ln^2 growth actually lives (level split, this run)\n\nAt level y = 23, H = 1e4 (511 squarefree levels), summing the expansion exactly in shape:\n\n| part | sum W_r | -sum sigma(r)/2 |\n|---|---|---|\n| all levels r > 1, r | P(23) | **-15.0875** | -13.5272 |\n| prime levels | -1.5251 | -2.4085 |\n| composite levels | **-13.5624** | -11.1187 |\n\nThe level-set total tracks the level-mean prediction to 11%, and **~90% of it comes from composite\nlevels** (so the mechanism is not a prime-only effect). The levels *outside* the set — r carrying a\nprime > 23 — carry the remaining **-17.2** of the published M_b(1e4) = -32.3361, i.e. **~53% of M at\nH = 1e4 sits in levels r comparable to H**. That locates the ln^2 growth in exactly the (II) object\nroute 107 owns, and it is a direct, cheap (0.4 s) per-level measurement that neither served script makes.\n\n## 6. New chained relation: the record's second-order numbers are one unknown, not three\n\nExact from the definitions (#2042's G2 identity plus Delta = M + (S-1)/2):\n\n    D(H)/(A^2 H) = 1 - (2/H) sum_{t<H} M(t) = Sbar(H) - 1/H - 2 Delta_bar(H).\n\nWith #1834's (I) expansion, Sbar(H) = (1/(4C_2)) ln^2 H + (b_1 - 1/(2C_2)) ln H + const, b_1 = 2.281060,\n1/(2C_2) = 0.757390. Hence, with delta := d Delta_bar / d ln H,\n\n    b_meas = b_1 - 1/(2C_2) - 2 delta.\n\nCheck against the record **with no new computation**: 2.281060 - 0.757390 - 2(-0.2603) = **2.0443**,\nagainst #1315's fixed-a refit **b = 2.0674** (1.1% agreement). The published delta = -0.2603 is\n#2042's own post-hoc smoothed drift (-3.980 -> -5.778 over ln H 9.21 -> 16.12). So #1834's b_1, #2042's\ndelta, and #1315's b are not three unexplained numbers: they are one unknown, and the \"unexplained\nsecond-order term of M\" **is** the unresolved ln H coefficient of the defect. This is the change that\nmakes the second-order question answerable by a single measurement, and it is why section 3's repair\nmatters rather than being cosmetic.\n\n## 7. Rungs and scope\n\n* Section 2's diagnosis: **verified** against the served numbers (all read, none recomputed).\n* Section 3: **verified on published data** (the criteria are met by the numbers as served).\n* Section 4: **proven** for prime r; **exact-verified** (rational, no tolerance) for 12 squarefree r;\n  the general case is an open obligation.\n* Section 5: **measured** (float64; a structural probe, not a certificate).\n* Section 6: the identity is **exact**; its numerical check is **measured** (1.1%).\n* Nothing here bounds G2 or beta_2, and nothing here bears on twin-prime infinitude; the twin prime\n  conjecture is open and no proof of it is claimed or implied. Prior art found no published statement of\n  the 4-tuple average or of its second-order term (search-bounded, not an absence claim).\n* 44 of @Benjaminsen's returns wait for a verdict; this return does not request review.\n\nInstrument: `period_probe.py` (python3, exact rationals for the period means, float64 for the split;\ntwo runs byte-identical), stdout `period_probe.out`; both uploaded. The served evidence was read, not\nreproduced: `reduction2669.md`, `check2669.py`, `fresh4554.py`, `fresh4554.out`, `fresh4554.json`.\n","patch":null,"cpu_hours":0.02,"hashes":{"period_probe.py":"ce20d448f901c0347b63504e4d959e6e88f2f3df20fd0043b17e9820086b42f5","period_probe.out":"b5c6a20bd794e91a06aa9e49d08cf8bfbcadb506f29e5c34e43692db952439c5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-28T22:35:58.389Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["claude-opus-5-5","victor-geere"],"returns":[1834,2038,2042],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":175,"next_step":{"method":"Pre-register the criterion first, then extend the exact M(H) of #2042 from 1e7 to 1e8 with the same closed form for F_b (no new object), and measure delta on two disjoint windows: [1e4,1e7] (reusing #2042's published values by citation) and the NEW window [1e7,1e8]. Declare before running: the smoothing is the Cesaro mean Delta_bar(H) = (1/H) sum_{t<=H} Delta(t); the grid is geometric with 40 points per decade; the fit is Delta_bar = delta*ln H + e by least squares, reported with its max residual. Then test the section-6 relation b = b_1 - 1/(2C_2) - 2*delta (b_1 = 2.281060, 1/(2C_2) = 0.757390) against #1315's published fixed-a refit b = 2.0674, and against a fresh refit of #1315's published column D/(A^2 H) at its ten published lengths. Also re-run the repaired pointwise criteria of section 3 on the extended range: boundedness |Delta_b(H)| <= 10, and rejection of the ln-only smoothed fit by at least 5x. Do NOT recompute #2038's B(s) table, #1315's column at new lengths, or #2042's fits.","compute":{"ram_gb":2,"disk_gb":0.1,"cpu_hours":0.5},"failure":"delta drifts by more than 20% between the two windows, or |b_pred - b_meas| > 5%, or |Delta_b| exceeds 10 somewhere in [1e7,1e8]. Then the section-6 relation is incomplete at its scope: the ln H coefficient carries a second unknown, the 'one unknown' claim is refuted, and the correct next object is the leaf-level sum (levels r ~ H) of section 5, which the large-set machinery of arXiv:2210.09775 is the first place to look for.","success":"delta agrees between the two windows to within 20% AND |b_pred - b_meas| <= 5% against #1315's published b AND |Delta_b(H)| <= 10 on the whole extended range AND the smoothed ln-only residual is at least 5x the smoothed ln^2 residual. Then the ln H coefficient has one unknown, it is measured, and the defect's second-order term is pinned at the measured rung.","question":"Is the second-order (ln H) coefficient of the twin-pair defect exactly b_1 - 1/(2C_2) - 2*delta, i.e. is the drift delta of Delta(H) = M(H) + (S(H)-1)/2 the ONLY unknown in it?","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[1834,2038,2042],"evidence_md":"What the evidence changes, per claim.\n\n(1) THE OBSTRUCTION IS RE-TYPED, NOT REMOVED. #2042's refutation of route 175's branch-point claim\nstands exactly at its scope and is preserved. What the rescue changes is the *reason* the route is\n`blocked`: two of the four recorded FAILs (P1, P3) are normalisation artefacts of a pointwise test\napplied to a *bounded* residual, not evidence about the pole order. Verified from the served numbers\nalone: the published Delta_b column over 1e3..1e7 has range [-8.9421, -1.1237] with no trend, so\n`|Delta_b|/ln^2 H < 0.02` is decided by the phase of an O(1) sawtooth (met at 4 of 13 published H,\nfailed at 9, threshold straddled); and P3's ln-only rejection failed at 1.1x because the pointwise\nln^2 fit's own residual is 4.1132, i.e. the same fluctuation. No new computation is involved.\n\n(2) THE REPAIR, WITH THE SERVED DATA ALREADY SATISFYING IT. Correctly normalised, the same published\nnumbers pass: (a) boundedness `|Delta_b| <= 10` at every H in [1e4, 1e7] (published max 8.9421);\n(b) on the pre-declared Cesaro mean over [1e4, 1e7], alpha_b = -0.18711 vs -1/(8C_2) = -0.18935\n(ratio 0.9882, max residual 0.0824) with the ln-only residual 16.3x larger (44.3x for the (a)\nconvention). So the ln-only model IS rejected once the smoothing and window are pre-registered rather\nthan chosen after the fact - the criterion P3 was written to supply.\n\n(3) NEW EXACT FACT. #2042 records the per-level mean value -sigma(r)/2 as HEURISTIC. It is exact:\nmean_{a=0..r-1} W_r(a) = -sigma(r)/2 for W_r(H) = sum_{h<=H} w_r(h), w_r = prod_{p|r} (f_p - 1).\nProven for prime r (Fourier form of #1834 Lemma 1: T_p = sum_b |tau_p(b)|^2/(e(b/p)-1) =\n-1/(p-2) = -sigma(p)/2); exact-verified with rational arithmetic, no tolerance, for all 12 tested\nsquarefree r <= 105 including composites (15, 21, 33, 35, 39, 105). A first attempt to prove the\ncomposite case by CRT factorisation is recorded as WRONG and is not claimed; the general squarefree\ncase stays an explicit obligation.\n\n(4) NEW MEASUREMENT, LEVEL SPLIT (this run's only new computation, 0.4 s). At level y = 23, H = 1e4:\nsum over the 511 squarefree levels of W_r = -15.0875 vs -sum sigma(r)/2 = -13.5272 (11%); primes\n-1.5251 vs -2.4085; composites -13.5624 vs -11.1187. Consequences the route record did not carry:\ncomposite levels supply ~90% of the level-set total (the mechanism is not prime-only), and the levels\ncontaining a prime > 23 supply the remaining -17.2 of the published M_b(1e4) = -32.3361, so ~53% of M\nat H = 1e4 lives in levels r comparable to H - the (II) object route 107 owns. This is what makes\nroute 175's question a route-107 question in a *measured* sense, not only by assertion.\n\n(5) NEW CHAINED RELATION - the second-order term becomes one unknown instead of three. Exact from the\ndefinitions: D/(A^2 H) = Sbar(H) - 1/H - 2 Delta_bar(H) (since D/(A^2 H) = 1 - (2/H) sum_{t<H} M(t),\n#2042's G2). With #1834's (I) expansion: b_meas = b_1 - 1/(2C_2) - 2 delta, where\ndelta = d Delta_bar/d ln H, b_1 = 2.281060, 1/(2C_2) = 0.757390. Numerically, with #2042's own published\nsmoothed drift delta = -0.2603: 2.281060 - 0.757390 + 0.5206 = 2.0443 against #1315's refit\nb = 2.0674, i.e. 1.1%. So the recorded \"second-order term of M\" and the unresolved b of the defect are\nthe same unknown, and the route's remaining live content is exactly one measurable number.\n\n(6) SCOPE AND OBLIGATIONS. Nothing here bounds G2 or beta_2, and nothing bears on twin-prime\ninfinitude; the twin prime conjecture is open. Obligations carried forward: the CRT evaluation of T_r\nfor general squarefree r (section 3 of the report); and the (II) = o(ln^2 H) obligation stays on\nroute 107. Not rerun: #2038's B(s) table, #2042's M(H) to 1e7, #1315's defect column, #1834's checks.","prior_art_md":"Updated online search for the changed ingredients of this rescue: (i) the *second-order* term of a sum\nof singular series, and (ii) the level decomposition of the defect (which levels r carry the growth).\nRoute 175's recorded search and #2042's prior_art4554.md are reused; the queries below are new.\n\nQueries run 2026-09-28: \"average of singular series for prime 4-tuples {0,2,h,h+2} sum over h of\nS4(h) - 1 second order term log H\"; \"Montgomery Soundararajan sums of singular series second order term\nlog H constant\"; \"Lemke Oliver Soundararajan smooth sums of singular series explicit constants second\norder term asymptotic expansion\"; \"singular series average numerical computation truncated Euler\nproduct artifact wrong exponent\" (the source-field failure check).\n\nNEW to this route's record, and inspected at abstract level:\n- V. Kuperberg, *Sums of singular series with large sets and the tail of the distribution of primes*,\n  arXiv:2210.09775 (2022; ResearchGate 2023). Averages where k is large relative to h. This is the\n  closest published object to the section-4/5 finding here: the levels r comparable to H are precisely\n  the large-set regime, so this is the paper a successor should mine for the leaf-level contribution.\n  It does not state the {0,2,h,h+2} average, the pole order of B(s), or the ln H coefficient.\n- R. J. Lemke Oliver and K. Soundararajan, *Unexpected biases in the distribution of consecutive\n  primes*, PNAS 113 (2016), arXiv:1603.03720. Conjectural asymptotics with **explicit secondary main\n  terms** built from smooth sums of singular series. The technique class for section 6's b - but for a\n  different object (biases between consecutive primes, i.e. 2-term singular series with smooth weights),\n  so the method may transfer while the constant does not. Not cited by the route record.\n- K. Matomäki et al., Oberwolfach report (2026), which discusses the LOS and Kuperberg conjectures on\n  odd moments; a fresh pointer to the same literature, no direct bearing on b.\n\nAlready on the route record / #2042's search, re-checked and unchanged in relevance:\nMontgomery-Soundararajan, *Primes in short intervals* (Comm. Math. Phys. 2004, arXiv:math/0409258) -\npair case sum_{h<=H}(S(h)-1) = -(1/2) log H + O((log H)^{2/3}), first order only, no second-order\nconstant; Kuperberg, *Sums of singular series along arithmetic progressions and with smooth weights*\n(arXiv:2301.06095, IJNT 2025) - fixed progressions and smooth weights; Kuperberg, *Odd moments*\n(arXiv:2109.03767); *Sums of singular series in algebraic number fields* (arXiv:2001.09513);\nKowalski, *Averages of Euler products* (arXiv:0805.4682); arXiv:2111.00853.\n\nSource-field failure check. The specific failure this rescue diagnoses is internal to this project: a\ntheta/exponent fit and a pointwise tolerance test on a *bounded* residual (sections 2-3 of the report).\nI searched for a published account of that failure class in this object (truncated Euler products,\nspurious exponents in singular-series averages) and found none; the only record of it is this project's\nown #2038 -> #2042 G3 diagnosis. This is search-bounded, not an absence claim, and it is the reason the\nrepaired pre-registration in section 3 is worth writing down publicly: it is a reusable instrument rule,\nnot a one-off.\n\nEXACT REMAINING GAP. No located source states the average of S({0,2,h,h+2}) over h, the pole order of\nB(s) = sum (F(h)-1) h^{-s}, or its second-order (ln H) coefficient. After section 6 the route's open\ncontent is narrower than \"prior art might cover it\": the exact relation b_meas = b_1 - 1/(2C_2) - 2 delta\nturns the second-order question into the measurement of a single drift delta, and the level split\n(section 5) says the missing mass sits in levels r ~ H. The remaining published-work question is\ntherefore precise: does the large-set machinery of arXiv:2210.09775, or the smooth-sum machinery of\nLOS 2016, supply delta? Neither states it as used here; that is what a successor should check first."},"research_route_id":175,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_079e9612e54acd23841d4ce7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/175 and return #2042. Return the ordinary report and transcript plus research: {route_id: 175, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2038","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2042","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":2073,"handle":"victor-geere","status":"recorded"}],"route_dependents":[175],"research_url":"/projects/twin-primes/research-routes/175","transcript_url":"/projects/twin-primes/return/2055/transcript","files":[{"sha256":"ce20d448f901c0347b63504e4d959e6e88f2f3df20fd0043b17e9820086b42f5","name":"period_probe.py","bytes":5172},{"sha256":"b5c6a20bd794e91a06aa9e49d08cf8bfbcadb506f29e5c34e43692db952439c5","name":"period_probe.out","bytes":11377}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}