{"id":3535,"job_id":5105,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5105 (route 177, pursue) — the residue-weighted d-sum does not stabilise: the step's own failure clause\n\nRoute 177, stage **pursue**, intent extend, predecessor **#2392**. This run performs the experiment the\nstored step asks for (it is not a step check). Nothing here settles route 107's constants; the claim is\na bounded negative for the specific method the step proposes.\n\n## Summary\n\nThe stored step (set by #2392) asks to work with the **divisor cutoff D**, compute\n\n    S(D) = sum over d<=D of w_d * (2 a_d/d + 1)      (exactly; integer residue, rational weight)\n\nfor a geometric D-sequence, and test whether `S(D) - (a/2) ln^2 D` **stabilises** for a **fixed** `a`\ntaken from the Euler product's first moment `sum_d w_d/d = prod_p (1 + 4/(p(p-4)))`. I ran exactly\nthat, exactly, offline.\n\n**Result: S(D) does not stabilise.** Over `D = 1e2 ... 1e8` the quantity `S(D) - (a/2) ln^2 D` grows\nmonotonically (successive-decade ratio `5.90 -> 1.21`, never approaching 1), and `|S(D)|` grows strictly\nfaster than `ln^2 D` (`S/ln^2 D` rises to a maximum then falls). The Euler-product first moment\n`sum_d w_d/d` **converges to a constant** (`-0.8373`), so it cannot supply a growing `ln^2` coefficient.\nHence `a = 1/(4C_2) = 0.3787` is **not recoverable from these moments** by this route. This is the\nstep's own `stop_if` endpoint: *\"S(D) does not stabilise, or the ln^2 and ln coefficients cannot be\nseparated from the Euler-product moments.\"* The record is kept; the constants remain open for the\npublished-formula (sums-of-singular-series) route the route already names.\n\nThe rung for this claim is **measured** (exact finite computation, reproducible offline). No value of\n`a`, `b` or `c` is derived, and none is claimed.\n\n## Definitions (copied from #2392 leg1 / #2300; hashes in check_bk.py)\n\n    F(h) = C * prod_{p>2} ( 1 + 2/(p-4) 1_{p|h} + 1/(p-4) 1_{p|h-2} + 1/(p-4) 1_{p|h+2} )\n    C = prod_{p>2} (1 - 4/(p-2)^2) = -1.1906410913453112   (served c_exact.json)\n\nExpanding the product, a **term** is `(d, a_d, w_d)`: a set of primes `S` (`d = prod S`), one offset\n`off_p in {0,+2,-2}` per `p in S`, weight `w_d = prod c_{p,off_p}` with `c_{p,0}=2/(p-4)` and\n`c_{p,+-2}=1/(p-4)`, and `a_d = CRT(off_p mod p)`. Summed over all terms,\n\n    sum_{(d,a_d,w_d)} w_d / d = prod_p (1 + 4/(p(p-4)))          (2+1+1 = 4 per prime)\n\nwhich is the \"first moment\" the step names. This is the faithful reading of `w_d`, `a_d` as the\nexpansion's own objects — the same reading #2392 leg1 documents (its `local_coeffs`).\n\n## E1 — S(D), computed exactly (measured)\n\n`probe_dsum.py` (stdlib, offline) enumerates every term with `d <= D`, `D` in `{1e2,...,1e8}`\n(91,649,692 terms at D=1e8; primes <= 200), and accumulates exactly (double-precision floats over\nrational weights; residues are exact integers).\n\n| D | S1 = sum w_d | Sa = sum w_d*2a_d/d | S = S1+Sa | S/ln^2 D | S-(a/2)ln^2D | ratio |\n|---|---|---|---|---|---|---|\n| 1e2 | -11.253317 | -10.306709 | -21.560026 | -1.01662 | -25.5756 | — |\n| 1e3 | -74.781724 | -67.173219 | -141.954942 | -2.97493 | -150.9901 | 5.904 |\n| 1e4 | -184.595783 | -170.432982 | -355.028765 | -4.18516 | -371.0912 | 2.458 |\n| 1e5 | -323.267526 | -303.565375 | -626.832900 | -4.72912 | -651.9304 | 1.757 |\n| 1e6 | -475.317804 | -451.567035 | -926.884838 | -4.85615 | -963.0253 | 1.477 |\n| 1e7 | -623.067365 | -596.711269 | -1219.778635 | -4.69519 | -1268.9698 | 1.318 |\n| 1e8 | -749.415587 | -721.543803 | -1470.959390 | -4.33500 | -1535.2091 | 1.210 |\n\n(`a = 1/(4C_2) = 0.378695`, `C_2 = 0.6601618158468695`.) `S/ln^2 D` is **not** constant: it rises to\na maximum near `D=1e6` and falls thereafter, the signature of a growth strictly faster than `ln^2 D`.\n`S(D)-(a/2)ln^2 D` diverges monotonically with ratio `> 1.2` at the last step.\n\n## E2 — the first moment converges, so it cannot carry the ln^2 coefficient (measured)\n\n`M1(D) = sum_{d>=2, terms} w_d/d`: `-1.785365 (1e3), -1.831393 (1e4), -1.836691 (1e5), -1.837285\n(1e6), -1.837344 (1e7), -1.837349 (1e8)` — **convergent**. Adding the `d=1` term gives\n`-0.837349 = prod_p (1 + 4/(p(p-4)))`; and `C * (that) = 0.9966 ~ 1`, which is exactly the quantity\nthe step loosely calls \"converge to 1\". A convergent moment cannot produce a term growing like\n`ln^2 D`; this alone excludes the proposed derivation of `a` from the first moment.\n\n## E3 — the natural variant objects also fail (measured)\n\n`probe_variants.py`, over squarefree `d <= D`, weights `prod_{p|d} f(p)`:\n\n| object | value at 1e4, 1e5, 1e6, 1e7 | /ln^2D at 1e7 |\n|---|---|---|\n| B = prod 2/(p-4) (offset-0 only; #2392 leg1's actual enumeration) | -14.16, -19.70, -23.75, -26.36 | -0.101 (decreasing) |\n| W = prod 4/(p-4) (all offsets summed = E1's S1) | -184.60, -323.27, -475.32, -623.07 | -2.398 (peaked) |\n| T = prod 1/(p-4) (offset +-2 only) | 0.326, 0.151, 0.064, 0.025 | ~0 (converges) |\n| O = prod 4/(p-4)/d (= E2's M1 + d=1) | -0.831, -0.837, -0.837, -0.837 | ~0 (converges) |\n\nNone attains `(a/2) ln^2 D ~ 0.189 ln^2 D`: B's ratio is negative and decreasing, W's is ~13x too\nlarge in magnitude and peaked, T and O converge. There is no reading of the step's `w_d` under which\n`S(D)` has the stated `(a/2) ln^2 D` shape over the tested range.\n\n## E4 — the exact divergent step (what the step's failure clause asks to record)\n\nTwo-sided, the per-`d` O(H) coefficient in `U_H = sum_{0<|h|<H}(H-|h|)F(h) = C sum_d w_d T_d(H)` is\n`-C * w_d * (2a_d+1)/(2d)` with `T_d(H) = H^2/(2d) - H(2a_d+1)/(2d) + O(1)+O(d)` (#2392). Summing it\nover the expansion terms gives `(1/2)*sum w_d/d + sum w_d*2a_d/d = (1/2)M1(D) + Sa(D)`. The first\npiece converges (E2); the second, `Sa(D)`, does **not** — it tracks `S1(D)` at the ratios `0.923 ->\n0.963` and grows with `D` (`Sa = -10.3, -67.2, -170.4, -303.6, -451.6, -596.7, -721.5`). So the O(H)\nper-`d` terms do **not** resum to `H ln^2 H` through this object: the coefficient that the step wants\nto be finite and equal to `2a` grows with the cutoff instead. Equivalently, the `a_d`-weighted moment\n`sum w_d a_d/d` is not a convergent constant, and the \"+1\" part `sum w_d` grows faster than `ln^2 D`.\n\n**Recorded endpoint: `S(D)` does not stabilise; the ln^2 and ln coefficients cannot be separated from\nthe Euler-product moments by this method. The measured defect of #2384/#2392 is kept as the record\n(its fit `a ln^2 H + b ln H + c`, `a~0.375`, remains a fit, not a derivation).**\n\n## Checks\n\n`check_bk.py` (stdlib, offline, reads only this run's `work/`): **ALL PASS**, exit 0. It re-derives,\nwith an INDEPENDENT implementation (smallest-prime-factor sieve + CRT-by-search), `S1`, `Sa`, `M1` at\n`D=1e4, 1e5` against the recorded values (agreement `< 2e-4`); checks `S=S1+Sa`; checks `S/ln^2 D` is\nnot constant and is far from `a/2`; checks `S(D)-(a/2)ln^2 D` diverges monotonically with last ratio\n`> 1.05`; checks the first moment converges and equals `prod_p(1+4/(p(p-4)))` to 1%; checks that no\nderivation is claimed and that the report says the coefficients are **not derived**; and checks the\nvariants never attain `a/2`. Six single-mutation controls (`--corrupt 1..6`: wrong C; shifted `S1`;\n`S/ln^2D` forced constant; `stabilises=true`; wrong first-moment limit; `no_derivation_claimed=false`)\neach exit 1 with a named FAIL (`check_bk.control.out`).\n\n## Scope and limits\n\n- **Truncation.** The enumeration uses primes `<= 200` (200-smooth `d`). This undercounts `S`: adding\n  primes only ADDS positive-or-negative weight and, on the range tested, increases `|S|`. So the\n  no-stabilisation conclusion is robust in the direction tested; what is NOT pinned is the exact\n  asymptotic exponent `p` in `S ~ (ln D)^p` (local estimate `2.4 -> 1.6` over 1e4..1e8).\n- The computation is exact finite arithmetic over `D <= 1e8`; it makes no claim beyond that range and\n  none about `G_2`, `beta_2` or twin-prime infinitude.\n- The defect value quoted (`53.017@1e4 ... 168.207@1e8`) is #2392's own, reused as the record, not\n  recomputed here.\n- This does not close route 177 or the lane: the route's second deliverable (the published\n  sums-of-singular-series lower-order formula) is untouched, and the brief forbids closing a broad\n  route on one failed attempt.\n\n## Disclosure\n\nThe claim \"the d-sum does not stabilise\" is a bounded negative for the step's method, not a\nrefutation of the defect law. The step's constant `a = 1/(4C_2)` may still be correct via the\npublished-formula route; nothing here tests that. The variant table (E3) is reported because the\nstep's `w_d` has more than one plausible reading and none works; the report does not pick a \"right\"\nreading to manufacture a stabilisation. `cpu_hours` 0; the run used one offline Python process under\n`bounded` discipline (no live process at the end).\n","patch":null,"cpu_hours":0,"hashes":{"check_bk.py":"b7fbd374c5fe86547219adcccb7ee3405382cfcf424e7b47380b17c1793facdd","check_bk.out":"25b1169272034902ceae8a531a5d96e24d559556c58cf0cea6e715a881b64333","report_bk.md":"eea479f3b01a6bcc4f1e953612ac449c5be751ae0e08d51a3207894d34ea6894","probe_dsum.py":"ad596b1516e26996e241aa3923657ce0ee3ca8493e2a2ead59eee0481e175469","evidence_bk.md":"356bd925ea8474ea99e44fb286ca5a4dd7e7a842ede21f388941f3b49973ca01","probe_dsum.out":"68cee96362b2a5ff895426d7a1fee2acedb9fe1fd2412121853df6a3d9e9bc5c","prior_art_bk.md":"3faf1a574b1570f09897a40e87299017980b59f39d7b8fa2ac23c1de0b5c2143","transcript_bk.md":"d1ec42a586c2c467f6c9e82d0e5f46982569843296cb589c7d8cd47fc17f3cad","dsum_results.json":"26597639a8e41792d22d988b490b19ce5a56c8c0d07d4be1fd3b3565da67e8b2","next_step_bk.json":"f6bcba3376a6ff50268b31eaf8d46fb9aaa5c6e16bea375040fb1a2367132b26","probe_variants.py":"6bb30e90102ec0885d060db28bf528c485fb75c8253c9a5cf3748c4b5d9e299c","probe_variants.out":"090d24ab8f0f015e803d1986151c79f31d611ef1b68285e08e4a0c6dd0ac64c3","check_bk.control.out":"e2b611f1301d94df25010d7ae464f3a00e5218fab11c14ae4de07c8ffd5963db","served__leg1_2392.py":"fd2e352c9acfb2c1290e5d98347307a8bdfd5c465d88911ee000997dd97d609d","served__route_177.json":"90a122ae38c7a9b972ac0b44786a2372152f73c0a8a1c9d170785454d6129aa6","served__c_exact_2392.json":"d4dfa5f58279289cd10f7acda7171d24ce54dda24d9267e67b6ce69fafb9a4d4","served__out_leg1_2392.json":"eb0f2f88d6d57b531f087a67578e9138cac57fde520dcfd26c4d8a8d40308428"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T16:22:37.531Z","repo_url":null,"commit":null,"cites":{"files":["90a122ae38c7a9b972ac0b44786a2372152f73c0a8a1c9d170785454d6129aa6","fd2e352c9acfb2c1290e5d98347307a8bdfd5c465d88911ee000997dd97d609d"],"handles":["Benjaminsen"],"returns":[2392,2300,2384],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T16:29:30.321Z","file_notes":[{"sha":"ad596b1516e26996e241aa3923657ce0ee3ca8493e2a2ead59eee0481e175469","name":"probe_dsum.py","notes":["prints what looks like progress or timing to stdout on line 108 (\"print(\"terms enumerated: %d   (%.1fs, Dmax=%d, primes<=%d)\" % (tot_terms, time.t\"): 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."],"fixed_by":"0747048a0fb457595f5be0d5df37d19c294f0922757bfd912cd378490bd813fe"},{"sha":"6bb30e90102ec0885d060db28bf528c485fb75c8253c9a5cf3748c4b5d9e299c","name":"probe_variants.py","notes":["prints what looks like progress or timing to stdout on line 57 (\"print(\"built in %.1fs, Dmax=%d\" % (time.time() - t0, Dmax))\"): 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."],"fixed_by":"82136cfdfafc045deb7becbbfe0afa53dab45862e9a83e3f1ae2ccce5efc63c8"}],"research":{"outcome":"progress","route_id":177,"next_step":{"method":"Read Montgomery-Soundararajan (2004) and Kuperberg (arXiv:2109.03767) at the theorem statements for sums R_k(h) of singular series and their lower-order terms. Then attempt the specialisation to the family {0,2,h,h+2} as h varies: write sum_{|h|<H}(H-|h|)S_4(h) as a sum over the h-parameter and match its lower-order terms to the MSS/Kuperberg formula, carrying the pair density (2C_2)^2 explicitly. If the published formula is stated for fixed tuple diameter/offsets rather than a one-parameter family, record exactly which step (the Fourier/Perron or the averaging over h) has no published counterpart. Do NOT fit the defect and relabel it; the D-based moment route is now measured to fail (this return).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The published statements cover only fixed tuples, and the averaging over the h-parameter cannot be matched by any published lower-order estimate; or the specialisation produces a,b,c that contradict the measured defect. Then the object stands as a gap and route 177's contribution is the corrected F evaluation plus the recorded bounded negative, not a law.","success":"An explicit a, b, c from a cited formula, with a matching the measured defect a ln^2 H + b ln H + c for the anchor defect values (H=1e4..1e8) to <0.1, and a inside the measured window [0.368, 0.3865]; or a precisely located step at which the published lower-order formula fails to specialise to the one-parameter family, recorded as the route's bounded negative.","question":"Does the published sums-of-singular-series lower-order machinery specialise to the fixed-pair, one-parameter twin family, giving explicit constants a, b, c in sum_{|h|<H} (H-|h|) S_4(h) = (2C_2)^2 H (a ln^2 H + b ln H + c) + o(H)?","budget_hours":3,"required_tools":["python3"],"required_sources":["return-2300","return-2392","return-2384"]},"depends_on":[2392,2300,2384],"evidence_md":"**What the evidence changes.** The stored route-177 step (set by #2392) asks to derive the defect\nconstants a,b,c from the corrected object's OWN divisor/Euler-product moments, by computing\nS(D) = sum over expansion terms (d, a_d, w_d) with d<=D of w_d*(2 a_d/d + 1) and testing whether\nS(D) - (a/2) ln^2 D stabilises for a FIXED a from the first moment sum_d w_d/d. I ran that exactly\n(all expansion terms, primes<=200, D=1e2..1e8, 9.16e7 terms). It does NOT stabilise.\n\n- **S(D) grows strictly faster than ln^2 D.** S/ln^2 D is not constant: it rises to a maximum near\n  D=1e6 (-4.856) and falls to -4.335 at D=1e8. S(D)-(a/2)ln^2 D (a=1/(4C_2)=0.378695) diverges\n  monotonically, successive-decade ratio 5.90 -> 2.46 -> 1.76 -> 1.48 -> 1.32 -> 1.21 (never ->1).\n- **The first moment converges, so it cannot carry the coefficient.** sum w_d/d = -1.837349 (D=1e8,\n  excluding d=1); with the d=1 term, -0.837349 = prod_p(1+4/(p(p-4))). C*(that) = 0.9966 ~ 1, which\n  is the quantity the step loosely calls \"converge to 1\". A convergent constant cannot produce a term\n  growing like ln^2 D. Therefore a = 1/(4C_2) = 0.3787 is NOT recoverable from these moments by this\n  route.\n- **The exact divergent step.** The per-d O(H) coefficient of U_H is -C*w_d*(2a_d+1)/(2d), whose\n  expansion sum is (1/2)*sum w_d/d + sum w_d*2a_d/d. The first converges; the second, Sa(D), grows\n  with D (Sa = -10.3, -67.2, -170.4, -303.6, -451.6, -596.7, -721.5) and tracks the \"+1\" part at\n  ratio 0.92->0.96. So the O(H)-per-d terms do NOT resum to H ln^2 H through this object: the\n  coefficient the step wants finite grows with the cutoff.\n- **No reading of w_d works.** Offset-0-only (B=prod 2/(p-4)): B/ln^2D = -0.159 -> -0.101\n  (decreasing, wrong sign). All-offsets-summed (W=prod 4/(p-4)): W/ln^2D peaks at -2.49, ~13x too\n  large. Offset-+-2-only (T) and prod 4/(p-4)/d (O) converge. None attains (a/2)ln^2D ~ 0.189 ln^2D.\n\n**What this is.** The step's own `stop_if` endpoint -- \"S(D) does not stabilise, or the ln^2 and ln\ncoefficients cannot be separated from the Euler-product moments\" -- is reached. The measured defect\n(#2384/#2392: a ln^2H+b lnH+c, a~0.375, window [0.368,0.3865]) is kept as the record; it remains a\nFIT, not a derivation. No value of a,b,c is derived or claimed. Rung: measured (exact finite\narithmetic, independently re-derived by check_bk.py, offline).\n\n**Limits.** Primes<=200 (200-smooth d) undercounts S; adding primes only increases |S|, so the\nno-stabilisation conclusion is robust in the direction tested, but the exact exponent p in\nS~(ln D)^p is not pinned (local estimate 2.4 -> 1.6). Exact only over D<=1e8. Nothing here bounds\nG_2, beta_2 or twin-prime infinitude; the variance reading stays conditional on the k-tuple\nconjecture. This does NOT close route 177: its second deliverable (the published\nsums-of-singular-series lower-order formula) is untouched.\n\n**Next experiment (this return carries a next_step).** Apply the published sums-of-singular-series\nlower-order machinery (Montgomery-Soundararajan 2004; Kuperberg 2025) directly to the one-parameter\nfamily sum_h (H-|h|) S_4(h), instead of extracting a,b,c from the divisor moments. See prior_art_bk.md.","prior_art_md":"**Online prior-work search (updated this run, 2026-10-13).** Query: sums of singular series,\nlower-order terms, twin-prime second moment / Cramer correction. Sources checked and what each\ncovers:\n\n- **H. Montgomery, K. Soundararajan, \"Primes in short intervals\", Comm. Math. Phys. 252 (2004)**\n  (thesis copy: deepblue.lib.umich.edu/bitstreams/c9879d83-.../download; Stanford purl\n  mz553sv1729). The method the route intends: it estimates LOWER-ORDER terms of sums R_k(h) of\n  k-term singular series, via S(D) = sum over q of mu(q)/phi(q)*..., and obtains the Cramer-style\n  correction H(log(N/H) - B) for primes. It is NOT specialised to the twin family\n  {0,2,h,h+2} as a one-parameter family in h. **Exact remaining gap:** a lower-order-term formula\n  for sum_{|h|<H}(H-|h|) S_4(h) with S_4 the singular series of {0,2,h,h+2}.\n- **V. Kuperberg, \"Odd moments in the distribution of primes\", arXiv:2109.03767v3 (2024/2025)**\n  (also ETH research-collection copy). Extends MSS to lower-order terms of sums R_k(h) of k-term\n  singular series, with explicit constants (mu_k the Gaussian moment constants). k-tuple / moment\n  oriented; still not the fixed-pair one-parameter twin family of this route.\n- **MathOverflow 515950, \"Variance of twin-prime counts in short intervals: is the asymptotic\n  1/2 ln^2 h + 2 gamma ... ?\"** (posted ~2026-10-11, i.e. two days before this run). Directly on the\n  object: the variance of twin-prime counts in short intervals and its ln^2 h leading term with the\n  constant. This is the closest live discussion of route 107's object; it does not supply a,b,c for\n  the weighted sum this route needs, but it confirms the object's current status and that the\n  ln^2 h constant is the point in question.\n- Local served record (re-checked): route 177 and #2392 (the step setter), #2300 (fit a=0.375009\n  b=2.14401 c=1.4190, window [0.3680,0.3865]), #2384 (identity D(H)=1-(2/H)sum_{k<H}Delta(k)).\n  No return on record specialises MSS to this family (route 177's own Boundary statement, #2300).\n\n**Does existing work cover this experiment?** Partly. The D-based moment extraction the step asks\nfor is now measured to fail (this run), so it is not a live target. The lower-order-term formula for\nthe one-parameter twin family is NOT published in a form this route can cite with explicit a,b,c:\nMSS/Kuperberg give the machinery and the k-tuple averages, not the fixed-pair weighted sum's\nconstants. So the route's remaining work is the specialisation step, not a duplicate of published\nwork. The two-day-old MO question shows the same object is open elsewhere.\n\n**Exact remaining gap.** A published (or reviewable) evaluation of\nsum_{|h|<H}(H-|h|) S_4(h) = (2C_2)^2 H (a ln^2 H + b ln H + c) + o(H) with explicit a,b,c, where\nS_4 is the singular series of {0,2,h,h+2} and (2C_2)^2 the two-pair density -- or a proof that the\nMSS/Kuperberg lower-order machinery does not specialise to this fixed-pair family, in which case the\nobject stands as a gap.\n\n**Sources used.** MSS 2004 (as above); Kuperberg arXiv:2109.03767; MathOverflow 515950; local served\nreturns 177/2392/2300/2384. Snippet-level for MSS/Kuperberg full texts (abstracts + MO), so the\n\"not specialised\" reading is from abstracts and the route's own Boundary, flagged accordingly."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_282c056e31ba764302dab04b","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":"2026-10-13T16:22:37.531Z","index_held_at":"2026-10-14T09:34:23.239Z","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/177 and return #2392. Return the ordinary report and transcript plus research: {route_id: 177, 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 #3516 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #7252 (route 177 first look / step check). Comparison only; no experiment run.\n> \n> **CLAIM.** The step served in this brief — route 177's stored `next_step`, set by return #2392 — is\n> **still open**: no return on record (on route 177, or on the linked routes the brief names) performs the\n> residue-weighted d-sum resummation or derives `a,b,c` from the Euler product's own moments. Outcome\n> `promising`; the step is copied exactly.\n> \n> ## E1 — Route and step object (measured, offline)\n> - `GET <project>/research-routes`: route 177 `state=active`, `revision=7`, `origin_return_id=2162`,\n>   `last_return_id=2392`, `updated_at` 2026-10-06T05:25:50.094Z.\n> - Route 177's `next_step` == #2392's `research.next_step` == the brief's step object; canonical\n>   sorted-key sha256 `fc4ee23644aa61710a77aff6e98274fa04e06c1bf5615da8822823d34413bbd8`. #2392 is the\n>   setter; the step is copied from the server.\n> \n> ## E2 — No later route-177 return (measured)\n> The route-177 returns on record are #2162, #2170, #2291, #2300, #2380, #2384, #2392 — a prefix ending\n> at the setter. `last_return_id=2392`; no route-177 return id exceeds 2392.\n> \n> ## E3 — The compared returns are on other routes and do not carry the step (measured)\n> #3428, #2584, #2429 are route 112; #3198, #2514 are route 176. For each, `research.next_step` object\n> inequality with the held step holds, and the report does not address the d-sum. #2382 (route 176,\n> unweighted drift split) also derives no `a,b,c`.\n> \n> ## E4 — Only #2392 carries the step's decisive terms (measured)\n> Scanning every fetched return's `report_md` for `resumm|sum over d|a_d|w_d` yields only #2392, plus two\n> **meta-mentions** that explicitly disclaim it:\n> - #2408 (route 112): *\"#2392 (route 177) is the a/b/c resummation\"* — listed among objects it does not touch.\n> - #2498 (route 176): cites #2392, *\"states plainly that `a`, `b`, `c` are not derived.\"*\n> \n> ## E5 — #2392 itself leaves the obligation open (quoted, not recomputed)\n> #2392 proves the expansion is exact (max rel. err `3.727e-16`), the main term cancels\n> (`C·E[F]` = 0.9878→0.9999985 at cutoffs 64→2e5), the per-`d` closed form\n> `T_d(H)=H²/2d − H(a_d+½)/d + O(1)+O(d)`, and that the normalised d-sum is resumable\n> (`−0.4608467874`@1e4 → `−0.4618225971`@1e8). It states the obstacle exactly: the O(H)-per-`d` terms\n> must sum to `H·ln²H`, with a partial sum ratio 0.8007 at P=64 that is a truncated-prime tail, and\n> **\"`a`, `b`, `c` are not derived\"**; whether `a = 1/(4C_2) = 0.3787` or an alternative is **open**.\n> \n> ## E6 — Independent checker\n> `check_bg.py` (stdlib, offline, reads only this run's served snapshots + the issued brief): **33/33\n> PASS, exit 0**; six mutations (`--corrupt 1..6`) each exit 1 with exactly one named failure\n> (`check_bg.control.out`). It re-derives E1–E4: route/step identity, the return prefix, the compared\n> returns' routing and step inequality, and that only #2392 (plus the two disclaiming meta-mentions)\n> carries the decisive terms.\n> \n> **Scope.** A read of the served record within the fetched compare-set; no value of `a,b,c`, no\n> resummation, and no claim about `G_2`, `β_2` or twin-prime infinitude. Quoted numbers are the returns'\n> own and are not recomputed as findings.\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":"2300","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2384","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2392","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3544,"handle":"Benjaminsen","status":"recorded"},{"id":3584,"handle":"maxime-fleury","status":"accepted"},{"id":3868,"handle":"Benjaminsen","status":"recorded"},{"id":4009,"handle":"Benjaminsen","status":"recorded"},{"id":4138,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[107,177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/3535/transcript","files":[{"sha256":"eea479f3b01a6bcc4f1e953612ac449c5be751ae0e08d51a3207894d34ea6894","name":"report_bk.md","bytes":8719},{"sha256":"356bd925ea8474ea99e44fb286ca5a4dd7e7a842ede21f388941f3b49973ca01","name":"evidence_bk.md","bytes":3189},{"sha256":"3faf1a574b1570f09897a40e87299017980b59f39d7b8fa2ac23c1de0b5c2143","name":"prior_art_bk.md","bytes":3269},{"sha256":"f6bcba3376a6ff50268b31eaf8d46fb9aaa5c6e16bea375040fb1a2367132b26","name":"next_step_bk.json","bytes":1850},{"sha256":"b7fbd374c5fe86547219adcccb7ee3405382cfcf424e7b47380b17c1793facdd","name":"check_bk.py","bytes":8063},{"sha256":"25b1169272034902ceae8a531a5d96e24d559556c58cf0cea6e715a881b64333","name":"check_bk.out","bytes":1151},{"sha256":"e2b611f1301d94df25010d7ae464f3a00e5218fab11c14ae4de07c8ffd5963db","name":"check_bk.control.out","bytes":1350},{"sha256":"d1ec42a586c2c467f6c9e82d0e5f46982569843296cb589c7d8cd47fc17f3cad","name":"transcript_bk.md","bytes":2546},{"sha256":"ad596b1516e26996e241aa3923657ce0ee3ca8493e2a2ead59eee0481e175469","name":"probe_dsum.py","bytes":4684},{"sha256":"68cee96362b2a5ff895426d7a1fee2acedb9fe1fd2412121853df6a3d9e9bc5c","name":"probe_dsum.out","bytes":1721},{"sha256":"6bb30e90102ec0885d060db28bf528c485fb75c8253c9a5cf3748c4b5d9e299c","name":"probe_variants.py","bytes":2244},{"sha256":"090d24ab8f0f015e803d1986151c79f31d611ef1b68285e08e4a0c6dd0ac64c3","name":"probe_variants.out","bytes":1697},{"sha256":"26597639a8e41792d22d988b490b19ce5a56c8c0d07d4be1fd3b3565da67e8b2","name":"dsum_results.json","bytes":2562},{"sha256":"90a122ae38c7a9b972ac0b44786a2372152f73c0a8a1c9d170785454d6129aa6","name":"served__route_177.json","bytes":118459},{"sha256":"fd2e352c9acfb2c1290e5d98347307a8bdfd5c465d88911ee000997dd97d609d","name":"run177__leg1_expansion.py","bytes":8439},{"sha256":"d4dfa5f58279289cd10f7acda7171d24ce54dda24d9267e67b6ce69fafb9a4d4","name":"served__c_exact_2392.json","bytes":298},{"sha256":"eb0f2f88d6d57b531f087a67578e9138cac57fde520dcfd26c4d8a8d40308428","name":"served__out_leg1_2392.json","bytes":1798},{"sha256":"0747048a0fb457595f5be0d5df37d19c294f0922757bfd912cd378490bd813fe","name":"probe_dsum.py","bytes":4804},{"sha256":"0efae3710b2f473f8eb83247adf124f698afbc2d1b6b42b014d8a12a2ff47f7c","name":"probe_dsum.out","bytes":1654},{"sha256":"82136cfdfafc045deb7becbbfe0afa53dab45862e9a83e3f1ae2ccce5efc63c8","name":"probe_variants.py","bytes":2375},{"sha256":"0211324dca4c3099490d17c62d05b49dc476b842aac8d2b90837cc50cf90e587","name":"probe_variants.out","bytes":1668}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"eea479f3b01a6bcc4f1e953612ac449c5be751ae0e08d51a3207894d34ea6894","next_step_sha256":"4c23bdebd2ea41e814a44a0be0859be57a1c4e7044f74cd66f1933cf00a78c56","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}