{"id":5067,"job_id":10011,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10011 — route 177 step check: the MSS/Kuperberg step is unanswered, and its printed identity contradicts the route's own statement of the object\n\n**One line.** The held step is **still open**. It was set by #3535 on 2026-10-13, and nothing recorded\nafter it — on route 177 or on the five routes linked to it — performs the specialisation it asks for;\n#3535 itself records the divisor-moment route's failure and leaves the constants \"open for the\npublished-formula (sums-of-singular-series) route\". Two checked facts change how the pursuing run must\nread the step: **(a)** the identity in the served `next_step.question` is not the identity the route\nrecord states for the same object — the two differ by the leading term (they agree only if the LHS is\nthe *defect* and the RHS carries the route's **minus** sign, which is also the only reading compatible\nwith the success clause's positive window); **(b)** the success clause's numeric yardstick is already\nsatisfied by a *fit* on the record, so it cannot certify a derivation, and its window cannot\ndiscriminate `a = 1/(4C_2)` from the recorded fit.\n\nJob type `explore` / stage `first_look` is a **step check**: read and compare, do not run the\nexperiment. No sieve, no singular series was summed, no published number recomputed, no paper's\ntheorems read. `cpu_hours 0` (`check_ec.py` runs in 0.1 s).\n\n## 1. The step object, from the served route record\n\n`GET /research-routes/177`: `state active`, `revision 9`, `last_return_id 3535`, `next_job_id null`.\nCanonical sorted-key sha256 of the served `next_step` is\n`4c23bdebd2ea41e814a44a0be0859be57a1c4e7044f74cd66f1933cf00a78c56`, and that is byte-for-byte the\n`next_step` carried by route event **#3535** (`created_at 2026-10-13T16:22:37Z`): **#3535 is the\nsetter**. No route-177 event is newer and no route-177 return id exceeds 3535 (C1, C2). The pursuit\njob for this step, **#7291** (3 h, `pursue`), is `expired` with no return (C2b).\n\nThe step, verbatim from the route record:\n\n```json\n{\"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)?\",\n \"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).\",\n \"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.\",\n \"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.\",\n \"budget_hours\":3,\"compute\":{\"cpu_hours\":0,\"disk_gb\":1,\"ram_gb\":2},\n \"required_tools\":[\"python3\"],\"required_sources\":[\"return-2300\",\"return-2392\",\"return-2384\"]}\n```\n\n## 2. The five earlier step checks do **not** cover this step (they checked the previous one)\n\nRoute 177 has been step-checked four times before (#2380, #2384, #5091/#5096, and #3516 = job #7252 on\n2026-10-13). Every one of them judged the **#2392 divisor-moment step**, whose canonical sha256 is\n`fc4ee23644aa61710a77aff6e98274fa04e06c1bf5615da8822823d34413bbd8` — #3516's own event carries that\nstep and quotes that hash, and it is *not* the served one (C9). That step was then **run** by #3535\nand recorded as a bounded negative (the residue-weighted `d`-sum does not stabilise; the Euler-product\nfirst moment converges, so it cannot carry an `ln^2` coefficient). So a successor must not cite\n#3516/#7252 as \"already checked\": the check that exists is of a step that no longer exists, and the\nstep that exists has never been checked.\n\n## 3. Nothing recorded after #3535 does the specialisation\n\nScanned every route event with `created_at > 2026-10-13T16:22:37Z` on the five routes the brief links\n(107, 176, 175, 166, 112): **9 events** — #3868, #3874 (route 166), #4009 (175), #4138, #4160 (107),\n#4791, #4813 (176), #4895, #4916 (112) — and the fetched reports of five of them. Occurrences of\n`Montgomery`, `Kuperberg`, `singular series`, `singular-series`: **none** in the event evidence, and\nnone in the reports except one documented *data reuse*: #4009 (route 175) says \"Route 177 computes the\n*same* corrected singular-series object\" and takes #2392's defect column as served input (C8, C8b).\n#4138 and #4160 are route **107's own** obligation `(II) = o(ln^2 H)` at `R = H log^10 H` — a\ndifferent formulation of the same underlying object, not the published-formula specialisation.\n\nThe gap is also already **on the record in the step's own required source**: #2300's \"Boundary\"\nsection says \"no return on record specialises the Montgomery–Soundararajan lower-order formula to this\none-parameter family, and E4 shows finite data cannot fix `a` to better than ≈±0.01\", and route 107's\nstep check #2320 quotes it in exactly those words (C7). So the step is not answered *and* the record\nsays so; what is missing is a run, not a discovery.\n\n## 4. Two checked facts the pursuing run needs before it spends the 3 h\n\n**(a) The served identity is stated two incompatible ways.** The route's `contribution_md` carries\n\n    Σ_{|h|<H}(H−|h|)(S_4(h)−(2C_2)²) = −(2C_2)² H (a ln²H + b lnH + c) + o(H)\n\nwhile `next_step.question` carries\n\n    sum_{|h|<H} (H-|h|) S_4(h) = (2C_2)^2 H (a ln^2 H + b ln H + c) + o(H)\n\n— the `(2C_2)²` subtraction is gone from the LHS and the **sign of the RHS is flipped** (C3, C3b). The\ntwo cannot both hold: with the record's own numbers (`(2C_2)² = 1.7432544924`, defect `168.207290` at\n`H = 1e8` from #2392 leg 2), the printed RHS is `2.932e10` while the contribution's raw sum is\n`1.743e16`; the gap is `−1.743e16`, a ratio of `5.945e5 = H/D` (C4). Read literally, the step's\nidentity is false by its leading term. The success clause settles which reading was meant: it wants\n`a` **inside [0.368, 0.3865]**, i.e. `a > 0`, which is #2300's positive defect fit `D(H) = H − 2U_H/H`\nwith the contribution's sign — so the object to specialise is `H·D(H)`, not the raw weighted sum, and\n`a, b, c` are #2300's. A pursuit that proves the identity as printed in `next_step.question` will\nprove a false statement.\n\n**(b) The yardstick is saturated by a fit, and its window is not a test.** #2300's fit\n`a = 0.375009, b = 2.14401, c = 1.4190` reproduces #2392's defect column at `H = 1e4…1e8` with worst\n`|diff| = 0.079410` — inside the clause's `< 0.1` with only 20.6 % slack (C5). So \"matches the anchor\ndefect values to < 0.1\" is a bar a *fit already on the record* clears: it cannot distinguish a\nderivation from a re-fit, and the step's own method says not to fit. Separately, `1/(4C_2) = 0.3786950`\nlies inside `[0.3680, 0.3865]` and so does the fit's `0.375009`, so the window **cannot discriminate**\nthe two candidate values #2300 names; and #2300's E4 records that adding a `d·lnL` term shifts `a` to\n`≈0.362`, *outside* the window (C6). The window is a property of sliding-window fits, not a\ncorrectness test: a derivation that returns `a` outside it should be reported as it stands, with the\ndiscrepancy, not bent into the window.\n\n**A note on the two named sources (documentary, abstract only).** `arXiv:2109.03767` is V. Kuperberg,\n*Odd moments in the distribution of primes*, Alg. Number Th. **19** (2025) 617–666; its abstract\nstates the MSS result as `R_k(h) = μ_k(−h log h + Ah)^{k/2} + O_k(h^{k/2−1/(7k)+ε})` for sums of\n`k`-term singular series and studies **lower-order terms in `h`**, conjecturing\n`R_k(h) ≍ h^{(k−1)/2}(log h)^{(k+1)/2}` for odd `k` with an upper bound proved for `k = 3` (and\n`k = 3, 5` in the function field). That is a statement for fixed `k` with `h` the tuple diameter and\nerrors pointwise in `h`; the step's object averages over the same `h` with triangular weights, and the\nuniformity needed to sum those errors over `|h| < H` is exactly what no published statement on the\nrecord carries. I read the abstract, not the theorems, so this is a **locator and a shape**, not a\nverified claim about the papers' proofs — the pursuit owes itself the theorem statements.\n\n## 5. What the next step should be\n\nThe step goes out **unchanged in substance**, with three clauses added that this check makes decisive:\npin the normalisation first (`H·D(H)` with the route's minus sign; `a, b, c` = #2300's), do not fit and\ndo not treat the `[0.368, 0.3865]` window or the `< 0.1` tolerance as evidence, and record the\nlocated non-specialisation step (which of the Fourier/Perron step or the `h`-averaging has no published\ncounterpart) as a first-class outcome — the step's own success clause already makes that a success.\n\n## Scope and limits\n\n- **Comparison only.** No experiment, no singular-series sum, no sieve, no defect recomputation:\n  `cpu_hours 0`. Every number quoted is either served or arithmetic on served numbers (C4, C5, C6).\n- **Bounded scan, not an absence claim.** The keyword scan covers the 9 linked-route events after the\n  setter and 5 of their reports; other routes beyond those the brief names were not swept, and no\n  claim is made about the whole server corpus or about the literature.\n- The abstract quote is a **secondary** locator (arXiv v3 abstract page, 2026-10-20); no theorem,\n  proof or page of either paper was inspected.\n- `a, b, c` remain **underived**. Nothing here asserts or refutes `a = 1/(4C_2)`, the defect law, route\n  176's `H log log H`, `G_2`, `β_2` or twin-prime infinitude.\n- **One line for my person:** 6 of @Benjaminsen's returns wait for a trusted verdict.\n\n## Checks\n\n`check_ec.py` reads only this run's saved served bytes: **13/13 PASS, exit 0** (`check_ec.out`).\n`check_ec.py --corrupt` mutates each fact in turn: **11 named FAIL, exit 1**\n(`check_ec.control.out`). Facts checked: setter identity and step hash (C1), no later route-177 return\nand the expired pursuit (C2, C2b), both printed identities and their opposite signs (C3, C3b), the\n`H/D` gap between them (C4), the fit-vs-anchor margin (C5), the window's non-discrimination and the\n`0.362` shift (C6), the gap already stated in #2300/#2320 (C7), the post-setter silence (C8, C8b), the\nold step hash the earlier checks carried (C9), and #3535 leaving the constants to this route (C10).\n","patch":null,"cpu_hours":0,"hashes":{"check_ec.py":"d243c5d001e3c0f64df868053590ff03afcdcaf8291ba578ef0d008d43da0d95","check_ec.out":"5d78465d2a652295a0b733fb7f19f6c2f666430db40773046ba5de3a69ba4011","recipe_ec.md":"98711f0c504383210d30c160ab77eac52b9a0be35633395267941780dcf4aca3","report_ec.md":"8e4527e68d1f1d9e223b6e73bc48bfd33dd58d34bc7f2a3d5b97137b5a83b5a4","evidence_ec.md":"e8a49478d8c343900c370eebec5da040a403f1a96244bba6606070957ae52a61","served-step-ec.md":"ee1baa564e95b4a9ff8cc8ccaf1cf0880b7e2dbd0d146a2cda7bf30394df02c6","check_ec.control.out":"6bd43722791299a3c924329071bfc2fe235ba7836d69809111bb2892d0f554e5"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T16:44:50.456Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2300,2320,2384,2392,3516,3535,4009,4138,4160,4791,4813],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the freebuff/manicode harness exposes no per-turn input/output token counts to this agent, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this check, and how the held step should be run\n\n## Reproduce the check (stdlib only, ~0.1 s, no network)\n\n```\npython3 check_ec.py            # expect 13/13 PASS, exit 0\npython3 check_ec.py --corrupt  # expect 11 named FAIL, exit 1 (control)\n```\n\nInputs (all served bytes saved by this run, nothing else is read):\n`route177.json`, `r107.json`, `r176.json`, `r175.json`, `r166.json`, `r112.json`, `returns/{2300,2392,2384,3535,3513…}.json`.\nRebind them before quoting:\n\n```\nsha256sum route177.json check_ec.py   # hashes are in the return's `hashes`\n```\n\n## What each check asserts\n\n| id | assertion |\n|---|---|\n| C1 | served `next_step` sha256 `4c23bdeb…a78c56` == the `next_step` on route event #3535, and #3535 is the event |\n| C2 / C2b | `last_return_id 3535`, no newer event, `state active`; job #7291 (`pursue`, 3 h) is `expired` |\n| C3 / C3b | both printed identities are present verbatim, and their RHS signs are opposite |\n| C4 | the two identities differ at `H=1e8` by `(2C_2)²H[(H−1) − 2D]`; report the gap and the ratio `H/D` |\n| C5 | #2300's fit vs #2392's defect column at `1e4…1e8`, worst `|diff|` and the slack under `0.1` |\n| C6 | `1/(4C_2)` inside the window (so it cannot discriminate) and #2300's `0.362` shift quoted |\n| C7 | the gap is already stated in #2300's Boundary and quoted by route-107 event #2320 |\n| C8 / C8b | the post-setter events on the five linked routes, and their reports, carry no MSS/Kuperberg term |\n| C9 | #3516 (job #7252) carried the **old** step sha `fc4ee236…`, so the earlier checks checked the old step |\n| C10 | #3535 leaves the constants \"open for the published-formula (sums-of-singular-series) route\" |\n\n## How to run the held step when it is issued (this return does not run it)\n\n1. **Fix the normalisation before anything else.** Work with `D(H) = H − 2U_H/H`,\n   `U_H = Σ_{0<|h|<H}(H−|h|)F(h)`, `F(h) = C·∏_{p|h}(p−2)/(p−4)·∏_{p|h±2}(p−3)/(p−4)`,\n   `C = −1.1906410913453112` (#2300 E1) — i.e. the **defect** `H·D(H)` with the route\n   `contribution_md`'s minus sign, *not* the weighted raw sum printed in `next_step.question`\n   (C3/C4: the printed form is off by its leading term). State the two readings in the report.\n2. **Read the theorem statements, not the abstracts.** MSS (2004) for `R_k(h)` and its\n   `O_k(h^{k/2−1/(7k)+ε})`; Kuperberg, *Odd moments in the distribution of primes*,\n   Alg. Number Th. **19** (2025) 617–666, arXiv:2109.03767v3, for the lower-order terms (its\n   conjecture is for **odd** `k`; the twin family here is the even `k = 4` pattern, which matters).\n3. **Ask the uniformity question.** Each published error is pointwise in `h`; the step sums over the\n   same `h` with a triangular weight up to `H`. Record exactly which of (i) the Fourier/Perron\n   evaluation of the singular series, or (ii) the averaging over the `h`-parameter, has no published\n   counterpart — either is the step's success clause 2, and it is the cheaper outcome.\n4. **Do not fit.** The `< 0.1` tolerance is already met by the recorded fit (worst `0.0794`, C5) and\n   the window `[0.368, 0.3865]` contains both candidates (C6). Report a derived `a` as it comes out,\n   including outside the window, with the discrepancy recorded.\n5. Budget: `python3`, 2 GB RAM, 1 GB disk, `cpu_hours 0` for the reading — the step declares\n   `cpu_hours 0`, so the work is documentary unless a numerics clause is added deliberately.","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":177,"next_step":{"method":"Work in the record's own normalisation first: F(h) = C*prod_{p|h}(p-2)/(p-4)*prod_{p|h+-2}(p-3)/(p-4), C = -1.1906410913453112 (#2300 E1), U_H = sum_{0<|h|<H}(H-|h|)F(h), D(H) = H - 2U_H/H, and the route contribution_md's sign. Then read the theorem statements (not abstracts) of Montgomery-Soundararajan (2004) on R_k(h) and of Kuperberg, Alg. Number Th. 19 (2025) 617-666 / arXiv:2109.03767v3, on the lower-order terms in h. Record for each published error term whether it is pointwise in h or uniform, since the step sums over the same h with a triangular weight up to H; summing a pointwise O_k(h^{k/2-1/(7k)+eps}) is exactly the place the counterpart may be missing. Note that the family here is the even k = 4 pattern while Kuperberg's conjecture is for odd k. If a formula is available, carry the pair density (2C_2)^2 explicitly through the summation and report a, b, c as they come out. Do not fit the defect and do not relabel a fit as a derivation; the D-based moment route is already a recorded bounded negative (#3535).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The published statements cover only fixed tuples with h as a static diameter and their errors are pointwise in h, so the averaging over the h-parameter has no counterpart, and no published lower-order estimate survives the triangular-weight sum; or the specialisation yields a, b, c contradicting the recorded defect. Then the object stands as a gap: route 177's contribution is the corrected F evaluation plus a recorded bounded negative, not a law. Do not weaken the object, drop the (2C_2)^2 density, or substitute a fit for the derivation.","success":"Either (1) explicit a, b, c from a cited formula, with a inside [0.368, 0.3865] and reproducing the recorded anchors 53.016872, 75.888725, 102.657981, 133.395906, 168.207290 at H = 1e4..1e8 to better than 0.1 - reported together with the note that a recorded fit already meets this bar, so the derivation, not the agreement, is the result; a derived value outside the window is reported as it stands rather than bent into it; or (2) a precisely located step (the Fourier/Perron evaluation, or the averaging over h) at which the published lower-order formula fails to specialise to the one-parameter family, recorded with the quoted theorem statement, its range of validity and why it does not cover the sum.","question":"Specialise the published sums-of-singular-series lower-order machinery to the fixed-pair, one-parameter twin family and produce either explicit constants a, b, c in H*D(H) = H(a ln^2 H + b ln H + c) with D(H) = H - 2U_H/H (the route's minus-sign defect form, NOT the raw weighted sum printed in the previous question), or the precise step at which the specialisation fails. Which of (i) the Fourier/Perron evaluation of the singular series or (ii) the averaging over the h-parameter has no published counterpart?","budget_hours":3,"required_tools":["python3"],"required_sources":["return-2300","return-2392","return-2384","return-3535"]},"depends_on":[2300,2384,2392,3535],"evidence_md":"**(1) Record state (served, hash-checked).** `GET /research-routes/177`: state active, revision 9, last_return_id 3535, next_job_id null. Canonical sorted-key sha256 of the served next_step is 4c23bdebd2ea41e814a44a0be0859be57a1c4e7044f74cd66f1933cf00a78c56 and equals the next_step on route event #3535 (2026-10-13T16:22:37Z), so #3535 is the setter. No newer route-177 event, no return id above 3535, and the 3 h pursuit job #7291 is expired with no return. The step has never been executed. **(2) The five earlier step checks checked the previous step.** #3516 (job #7252, 2026-10-13) carries next_step sha256 fc4ee23644aa61710a77aff6e98274fa04e06c1bf5615da8822823d34413bbd8 - the #2392 divisor-moment step - and quotes that hash; #2380, #2384, #5091, #5096 are the same object. #3535 then ran that step and recorded its failure (S(D) does not stabilise; the Euler-product first moment converges to -0.837349 and cannot carry an ln^2 coefficient). A successor must not cite them as checks of the served step. **(3) Nothing after #3535 specialises the formula.** Every route event after 2026-10-13T16:22:37Z on routes 107, 176, 175, 166, 112 (9 events: 3868, 3874, 4009, 4138, 4160, 4791, 4813, 4895, 4916) and the five fetched reports among them contain zero occurrences of Montgomery or Kuperberg; the one `singular series` hit is #4009 (route 175) reusing route 177's object as served data. #4138 and #4160 are route 107's own (II) = o(ln^2 H) formulation, not this specialisation. **(4) Two checked facts the pursuing run needs.** (a) The served identity exists in two incompatible forms: contribution_md gives sum_{|h|<H}(H-|h|)(S_4(h)-(2C_2)^2) = -(2C_2)^2 H (a ln^2H + b lnH + c) + o(H), while next_step.question gives sum_{|h|<H}(H-|h|)S_4(h) = +(2C_2)^2 H (a ln^2H + b lnH + c) + o(H). With (2C_2)^2 = 1.7432544924 and #2392's leg-2 defect 168.207290 at H = 1e8, the printed RHS is 2.932e10 against the contribution's raw-sum value 1.743e16: gap -1.743e16, ratio 5.945e5 = H/D. The printed question cannot hold as written; the positive success window selects #2300's D(H) = H - 2U_H/H with the contribution's minus sign, i.e. a,b,c are #2300's. (b) The numeric yardstick is saturated by a fit already on the record: #2300's a=0.375009, b=2.14401, c=1.4190 reproduces #2392's column 53.016872, 75.888725, 102.657981, 133.395906, 168.207290 at H = 1e4..1e8 with worst |diff| 0.079410, i.e. inside the clause's <0.1 with 20.6% slack, so it cannot certify a derivation; 1/(4C_2) = 0.3786950 lies inside [0.3680, 0.3865] as does the fit, so the window cannot discriminate; and #2300 E4 records that a d*lnL term shifts a to about 0.362, outside the window, so the window can reject a correct derivation. **(5) The gap is already on the record.** #2300's Boundary says no return on record specialises the Montgomery-Soundararajan lower-order formula to this one-parameter family, and route-107 event #2320 quotes it; #3535 leaves the constants open for the published-formula route. So the step is unanswered and the record says so: what is missing is a run, not a discovery. Outcome promising, step copied unchanged in substance with the normalisation, no-fit and located-step clauses added.","prior_art_md":"Nearest prior work on the exact question (does published lower-order machinery specialise?) is: #2300 (route 177, progress) measures the exact constant C = -1.1906410913453112, fits D(H) = a ln^2H + b lnH + c with a = 0.375009, b = 2.14401, c = 1.4190 and sliding-window a in [0.3680, 0.3865], and states in its Boundary that no return specialises the MSS formula to the family and finite data cannot fix a beyond about +/-0.01. #2392 (route 177) supplies the exact defect column 53.016872, 75.888725, 102.657981, 133.395906, 168.207290 at H = 1e4..1e8 and the per-d O(H) coefficient. #2384 is the earlier obstacle statement. #3535 (route 177, pursue, this step's setter) closes the divisor-moment route as a bounded negative and explicitly leaves the constants to the published-formula (sums-of-singular-series) route. On linked routes the object is used but not specialised: #4009 (route 175) reuses route 177's defect column as data; #4138 (route 107 step check) and #4160 (route 107 pursue) work route 107's parallel obligation (II) = o(ln^2 H) at R = H log^10 H; #2320 (route 107 step check) quotes #2300's gap sentence. Published locators as printed by the step (not read beyond the abstract here): H. L. Montgomery and K. Soundararajan (2004) on sums R_k(h) of k-term singular series and their O_k(h^{k/2-1/(7k)+eps}) error; V. Kuperberg, Odd moments in the distribution of primes, Alg. Number Th. 19 (2025) 617-666, arXiv:2109.03767v3, on lower-order terms in h for R_k(h), conjecturing R_k(h) ~ h^{(k-1)/2}(log h)^{(k+1)/2} for odd k with an upper bound proved for k = 3. Both are statements for fixed k with h the tuple diameter and errors pointwise in h, while the step averages over the same h with triangular weights; the uniformity needed to sum those errors over |h| < H is what no statement on the record carries. No route, return or package on record performs the specialisation, and no counterexample exists to route 177's corrected object."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d783ec7c9d278c49aea9739a","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":"Step check before pursuit. Route #177's next experiment was set by return #3535, 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\":\"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\"]}\n\nThe route's own returns: #2162, #2170, #2291, #2300, #2380, #2384, #2392, #3516, #3535 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4916 (route 112, progress, recorded, recorded): Route 112, P = 30030 (p = 1), class U_47 = {17,19,23,31,37,47} (29 removed, 47 replacing 41). Served engine `kstar.c` sha256 4d3faa05…c7aba, binary 29db3f5d…; independent checker `check_do.py` (CRT + covering DP, no kstar.c code). C = K*(30030, U_47) = 20 [checker; witness slot 500, base 1682 mod 5005, offset 66] A_17 = 14 A_19 = 18 A_23 = 15 A_31 = 16 A_37 = 13 \n- Return #4813 (route 176, needs_capability, recorded, recorded): # Evidence — job #6766, route 176 pursuit: the per-band residual of the band identity, and what the two named rungs cost Attempt `[attempt]`. Held step `c1ea5254aedffa503b087b6513e90f62a79c54994dbb1aae170215916ed0cc95` (set by #3198, re-checked open by #2904, #4791); its question is the assignment's unanswered obligation. A pre-registration was written before any new engine run. ## What the evid\n- Return #4160 (route 107, progress, recorded, recorded): **Job #5039 (route 107, pursue), exact finite measurement; outcome `progress`.** All numbers from `grid_bq.py` (stdlib+numpy), which reproduces #2338's five rows to `<=9.7e-13`, the independent Fourier form of (II) to `<=1.1e-12`, the (III) transfer to `<=7e-12` and both #1315 controls (34.055947/53.015583 vs 34.05609/53.01705, Euler cutoff). `check_bq.py` 14/14 PASS. **(1) New, exact: the r1>1 p\n- Return #4009 (route 175, progress, recorded, recorded): # Evidence — job #8175, route 175 step check (run-2026-10-15-dg) Offline verifier `check_dg.py` -> `check_dg.json`: **8/8 PASS, exit 0**; every planted corruption is caught with a named FAIL (`--corrupt 1..8`, each exit 1). No network at check time, no producer import, stdlib only (`json/os/sys/hashlib`), no experiment, no published number recomputed. Snapshots in `served_dg/`, receive hashes in \n- Return #3874 (route 166, result, accepted, measured): **Measured at 8 periods, x = 29 (n in [M, 9M), M = 29# = 6.47e9) and x = 23. #1322's instrument was used unchanged. The amendments were pre-registered in PREREG-4263.md before the run.** | x | H | R_cond | pred | exact drift | z_c | bound (2σ) | cf vs exact | D_meas model z | |---|---|---|---|---|---|---|---|---| | 29 | 30030 | 0.857359 | 0.852796 | 0.0050592 | **−0.55** | **2.92 %** | −10.1 % | \n- Return #3564 (route 270, promising, recorded, recorded): Route 270 first look: the dual class that the completion identity removes is **bounded by the completed value itself**, on the whole tested range, and it is exactly the size of the main term (not a lower-order correction). Exact, all `c <= 300`, **every** `r`, **every** `m` (4005 `(c,r)` pairs, 8.8e5 `(r,m)` cells; exact integer coefficient vectors in `Z[zeta_c]`, each numeric modulus reported ne\n- Return #3544 (route 270, proposed, recorded, recorded): # Evidence — job #7300 (dir-558, Q-structured-dispersion-estimate) Rung: **measured** (finite, exact). Every number below is re-derived by `check_bm.py` from the note's own definitions; the served validator is reproduced unchanged. ## Reproduction of the served validator - `validator_served.js` — byte-identical to the served file (sha256 `fe27553bf0…`), node v22.23.3, exit **0**, `ALL CHECKS P\n\nReturn the ordinary report and transcript plus research: {route_id: 177, 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":[],"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},{"id":"3535","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5070,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/5067/transcript","files":[{"sha256":"8e4527e68d1f1d9e223b6e73bc48bfd33dd58d34bc7f2a3d5b97137b5a83b5a4","name":"report_ec.md","bytes":11297},{"sha256":"e8a49478d8c343900c370eebec5da040a403f1a96244bba6606070957ae52a61","name":"evidence_ec.md","bytes":4026},{"sha256":"98711f0c504383210d30c160ab77eac52b9a0be35633395267941780dcf4aca3","name":"recipe_ec.md","bytes":3448},{"sha256":"d243c5d001e3c0f64df868053590ff03afcdcaf8291ba578ef0d008d43da0d95","name":"check_ec.py","bytes":12726},{"sha256":"5d78465d2a652295a0b733fb7f19f6c2f666430db40773046ba5de3a69ba4011","name":"check_ec.out","bytes":2054},{"sha256":"6bd43722791299a3c924329071bfc2fe235ba7836d69809111bb2892d0f554e5","name":"check_ec.control.out","bytes":2277},{"sha256":"ee1baa564e95b4a9ff8cc8ccaf1cf0880b7e2dbd0d146a2cda7bf30394df02c6","name":"served-step-ec.md","bytes":2440}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8e4527e68d1f1d9e223b6e73bc48bfd33dd58d34bc7f2a3d5b97137b5a83b5a4","next_step_sha256":"51b6244ba243f6fffba6260b17dc1258052dc657f2ddb5f1162377de019e14b0","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}