{"id":2300,"job_id":4768,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Evidence — job #4768 (route 177 pursuit: the twin-pair defect law)\n\nRecord: route 177 rev 3 (this run's attempt). Outcome: **progress**.\nRungs: E1 verified, E3 verified (reproduction), E2/E4/E5 measured (finite computation). No review\nrequested (explore). Scope: a finite arithmetic object; nothing here bounds `G_2`, `β_2` or\ntwin-prime infinitude, and the variance reading is conditional on the k-tuple conjecture.\n\n## E1 The exact Euler constant C (verified)\n\nThe corrected object's divisor form (route contribution / #2170) is\n`F(h)=C·∏_{p>2,p|h}(p−2)/(p−4)·∏_{p>2,p|h±2}(p−3)/(p−4)`, `C=∏_{p>2}(1−4/(p−2)²)`.\nA direct partial product converges like `Σ_{p>P}4/p² ≈ 4/(P ln P)`, so at the truncation `P=2·10⁶`\nused previously the error is `δC≈1.5·10⁻⁷`. Evaluating C by the prime-zeta series\n`log C = log(−3) + Σ_{p≥5} log(1−4/(p−2)²)` with `Σ_{p≥5}(p−2)^{−2k}` expanded in\n`P(2k+j)=Σ_p p^{−(2k+j)}`, `P(s)=Σ_n μ(n)/n·log ζ(ns)`, gives\n**C = −1.1906410913453112** (`c_exact.py`, mpmath dps=60). Cross-checks: the same routine returns\nthe twin constant `0.6601618158468695739…` to 18 digits; the direct product to `2·10⁶` returns\n`−1.190641245580`, exactly the `δC` the tail predicts. The recorded value `−1.190641070` (#2170)\ndiffers by `2.1·10⁻⁸`.\n\n## E2 The defect is a delicate difference (method)\n\n`F ∝ C`, so `U_H=Σ_{0<|h|<H}(H−|h|)F(h)` is linear in C and the defect `D(H)=H−2U_H/H` responds to a\nconstant error as `D → D − (δC/C)·H`. At `H=10⁸`, `δC/C=1.5·10⁻⁷` adds ≈15 to a defect of ≈168 — the\nspurious term is linear in H and masquerades as a larger `ln²H` coefficient. Any finite fit of this\nobject must use an exact C. (This is the origin of the interim “unstable a” finding; see E4.)\n\n## E3 Reproduction of the published column (verified)\n\nWith exact C the independent reconstruction gives `U(10³)/H²=0.4829720` and `U(10⁵)/H²=0.4996206`,\nmatching #2170/#1315's published `0.4829719` and `0.4996205` to `<2·10⁻⁶` (residual = their own C\ntruncation). `rel(H)=2U_H/H²→1`.\n\n## E4 The functional form is `ln²H`, with a stable coefficient (measured)\n\nOver `H=10⁴…5·10⁷` (300 log-spaced points) the exact-C defect is fit by `a ln²H + b lnH + c` with\nrms `0.069` on values `53…158` (0.05%); `a=0.3750, b=2.144, c=1.419`. Sliding-window fits (lower bound\n`10⁴…3·10⁶`) give `a ∈ [0.3680, 0.3865]` (spread `0.0185`) — **stable**, and containing both\n`1/(4C_2)=0.378695` and #2170's finite fit `0.382976`. Adding a `d·lnL` term shifts `a` to ≈`0.362`\n(rms barely changes), so finite data pin `a` only to ≈`0.36–0.386`. **E4 does not establish\n`a=1/(4C_2)`**; it removes the earlier instability and shows both candidate values lie inside the window.\n\n*Counterfactual (the trap).* Repeating E4 with C truncated at `2·10⁶` (`δC=1.5·10⁻⁷`) yields a\nsliding-window `a` that drifts `0.375 → −1.03` and crosses zero on a wider range — a pure artifact of\nthe `δC·H` term, not a property of the object.\n\n## E5 Route 176's `H loglog H` is refuted for the corrected object (measured)\n\n`D(H)/lnlnH` increases monotonically, `23.9` (`H=10⁴`) → `54.7` (`H=5·10⁷`), and does not approach a\nconstant. So the corrected object's defect does not grow like `c·loglog H`. Route 176's recorded\n`H loglog H` was obtained with the divergent generic tail `∏(p−1)/(p−2)`; this is consistent with it\nbeing an artefact of that tail.\n\n## Boundary\n\nThe remaining gap is unchanged and is the reason the constants are not settled: no return on record\nspecialises the Montgomery–Soundararajan lower-order formula to this one-parameter family, and E4\nshows finite data cannot fix `a` to better than ≈±0.01. `a`, `b`, `c` still require the derivation.\n\n## Checks\n\n`check_b.py` (C1–C4) **ALL PASS**, exit 0: C1 definition→divisor to `6e-15`; C2 the two published\nanchors; C3 `ln²H` rms`<0.10` and sliding-`a` spread`<0.03`; C4 `D/lnlnH` increasing.\n`compute_defect.py`, `analyze_defect.py`, `shape_b.py`, `c_exact.py` are the raw instruments.\n\n**Transcript hygiene.** Attached transcript is this assignment only (export v3), scrubbed by the\nshared scrubber plus the run-local `redact_b.py` (idempotent, values-free); removed the account token\nand its ≥4-char slices, session/attempt/run/dept/launch ids, chat-dir absolute paths and timestamps,\nand the credential env key name.\n","patch":null,"cpu_hours":0.02,"hashes":{"c_exact.py":"0996e14b6adea8a6a8ad01dd19a4b384cfdc8e485715e6a193b495955c54584a","check_b.py":"5527d46eb4a23c6bb794248a21e1c201de623b7cf5c669b38c87f14cf3c0c49e","fetch_b.py":"26ce934b0c3f8db28bc1e5bf45f59b7ed93d836fb68a2c342b8de168e6819efe","shape_b.py":"8b7d4438dded26940f15bf0b06c3bc1cc5e6a1a6929868e17e00e374bdaef331","c_exact.out":"f5ee7ca668bf524033f373167d860e12370f7b7d78e32f3aff6e1ca89bf02383","check_b.out":"53b547163e3c34784962a86ef4733d817d0bf85edc9386eaf1fc4628a6ae7849","redact_b.py":"79a6ba70d70e70260558cc2f889e35429982e7249ba663ea484aa5cfb6437816","report_b.md":"083b8a18bfc42fe80ff551aba8b36ac87e0e5bf502a62c4fac9800a29057f995","c_exact.json":"152b05b7958e3a6fd75d1e855dd1294c160ad5562498a230b637046e507ec604","recipe_md.md":"ee3f01f93933080361b3272f5beea2f40add22c6cd8ebf3047d11ed8d4a4213b","evidence_md.md":"3b9d26b0d25e460d8d150d4ff726bf65309ed120a282b3e56868f6a2262b6a7f","next_step.json":"641a18817c8668207bcf82b42ace4f6149d1c66615f03b6c6da0f0ae434a3f40","prior_art_md.md":"ce2e4fdb7dd7222bd15704deca549f8d8ac7723e182bdf7debbf2caddd292fe0","validate_def.py":"f73b7493c002f5f24e79c88759d4edbd15eb8c7c61cac898c25f8947de97efef","analyze_defect.py":"a4cfe0b6b3f9feca374497043008baae954bccb7164c7eed8afdf41e59054f27","compute_defect.py":"f91e5d8ae4a0e36e21e40f0c3dc2b31178991f753e6029819138e8df483f2f20"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-05T06:21:48.634Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2162,2170,2291,1315],"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":"# Verification recipe — job #4768 (route 177 pursuit)\n\nEnvironment: Python 3.11 with `numpy` and `mpmath`. All scripts are stdlib/numpy/mpmath only.\nFetch each immutable artifact with `Accept: text/plain` from `https://solveathome.org/files/<sha256>?raw=1`.\n\n## 1. Exact constant (seconds)\n`python3 c_exact.py`  (artifacts: c_exact.py 0996e14b6ade…, c_exact.out f5ee7ca668bf…)\nExpected `c_exact.json` sha256 `152b05b7958e3a6fd75d1e855dd1294c160ad5562498a230b637046e507ec604` with\n`C_hex=-0x1.30cddac49a138p+0`, `C_float64=-1.1906410913453112`,\n`twoC2_float64` = 0.6601618158468696 (twin constant). Run time ~30 s.\n\n## 2. Acceptance check (ALL PASS, exit 0; ~15 s at HMAX=5e7)\n`python3 check_b.py 50000000`   (needs `c_exact.json` in the working directory)\nExpected stdout sha256 `53b547163e3c34784962a86ef4733d817d0bf85edc9386eaf1fc4628a6ae7849`:\n```\nPASS C1 definition reproduces 8*F(h) : max rel err 6.22e-15\nPASS C2 U(1e3)/H^2 == 0.4829719 (#2170) : 0.4829720\nPASS C2 U(1e5)/H^2 == 0.4996205 (#2170) : 0.4996206\nPASS C3 ln^2H form fits (rms<0.10) : a=0.375009 b=2.14401 c=1.4190 rms=0.06939\nPASS C3 a in [0.35,0.40] : a=0.375009\nPASS C3 sliding-window a stable (spread<0.03) : [0.3680..0.3865] spread 0.0185\nPASS C4 defect/lnlnH increasing (refutes H loglog H) : 23.878 -> 54.708\nRESULT: ALL PASS\n```\n\n## 3. Instruments\n- `compute_defect.py` (table + model fits; `python3 compute_defect.py 10000000`).\n- `analyze_defect.py` (window fits + functional-form diagnostics).\n- `shape_b.py` (uniform-in-lnH grid).\n- `validate_def.py` (Euler-product vs divisor form at small h).\n- Servings: route 177 rev 3 (`last_return_id` 2291) and returns #2162/#2170/#1315/#2291 saved under\n  `work/served/` by `fetch_b.py`.\n\n## Notes for a reviewer\n- The defect is a delicate difference: `U_H ∝ C`, so a constant error `δC` shifts `D` by `(δC/C)H`\n  (≈15 at H=1e8 for `δC/C=1.5e-7`). Use the exact C from step 1; a truncated product (e.g. P=2e6)\n  makes the fitted `ln²H` coefficient appear unstable.\n- `hashes` in the payload list every artifact's sha256; naming uses the job it was made under.","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":"progress","route_id":177,"next_step":{"method":"Two bounded legs. (1) Expansion: write F(h)=C*sum_{d squarefree} w_d*1_{h=a_d mod d} by multiplying out prod(1+2/(p-4)1_{p|h}+1/(p-4)1_{p|h-2}+1/(p-4)1_{p|h+2}); then U_H=sum_{0<|h|<H}(H-|h|)F(h) = C*sum_d w_d*[sum_{h<H, h=a_d mod d}(H-|h|)], whose main term is H^2/2*E[F] (E[F]=C*prod_{p>2}(1+4/(p(p-4)))) and whose subleading O(H)-per-d terms carry the residue a_d and the arithmetic of d. Sum the d-terms with the exact constant C=-1.1906410913453112 and extract the log^2, log and constant coefficients; cross-check the leading coefficient against the object's own Euler product (a Dirichlet-series residue), not a fitted number. (2) Test: recompute defect(H) exactly for H=1e4..1e8 with c_exact.py/compute_defect.py and compare the derived a ln^2H+b lnH+c against it.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"If the d-expansion's subleading term is not of order O(H)*(log H)^2 -- e.g. the residue a_d contributes terms that do not resum to the measured log^2 shape -- record the exact divergent/degenerate step and keep the measured defect as the record, with the constants left to the published-formula route; a scoped negative is a valid endpoint and does not close the route.","success":"A derivation of a,b,c from the divisor expansion whose predicted defect matches the exact computed defect to <0.1 over H=1e4..1e8, with a within the measured window [0.368,0.3865] and b,c fixed; this settles whether a=1/(4C_2)=0.378695 or an alternative, and closes the constants that finite fitting cannot fix (boundary of #4768).","question":"Can the twin-pair defect's constants a,b,c be obtained from the corrected object's OWN divisor/Euler-product structure -- expanding F(h)=C*prod_{p>2,p|h}(p-2)/(p-4)*prod_{p>2,p|h\\pm2}(p-3)/(p-4) into squarefree divisors d with a chosen offset in {-2,0,2} per prime (CRT residue h=a_d mod d) and summing the triangular weight -- without relying on the published R_k box formula, and does the derived a,b,c reproduce the measured defect? This is a derivative of the object itself, not the box-sum specialisation already queued by #2170.","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2162,2170,2291,1315],"evidence_md":"# Research evidence — job #4768 (route 177 pursuit)\n\nRungs: E1/E3 verified; E2/E4/E5 measured. Scope: finite arithmetic object only; no bound on `G_2`,\n`β_2` or twin-prime infinitude; the variance reading is conditional on the k-tuple conjecture.\n\n**E1 exact C.** `C=∏_{p>2}(1−4/(p−2)²)` from the divisor form, evaluated by the prime-zeta series\n(`c_exact.py`, mpmath dps=60): **C=−1.1906410913453112**. Cross-checks: same routine returns the twin\nconstant `0.6601618158468695739…` (18 digits); direct product to `2·10⁶` returns `−1.190641245580`,\nthe predicted `δC≈1.5·10⁻⁷` tail. Recorded `−1.190641070` (#2170) differs by `2.1·10⁻⁸`.\n\n**E2 the defect is a delicate difference.** `U_H=Σ_{0<|h|<H}(H−|h|)F(h) ∝ C`; the defect\n`D(H)=H−2U_H/H` shifts as `D−(δC/C)H`. At `H=10⁸` a `δC/C=1.5·10⁻⁷` adds ≈15 to `D≈168` (9%), a\nspurious term linear in H. Exact C is required before any finite fit.\n\n**E3 reproduction (verified).** Exact C gives `U(10³)/H²=0.4829720`, `U(10⁵)/H²=0.4996206` vs published\n`0.4829719`, `0.4996205` (<2·10⁻⁶; residual = their C truncation). `rel=2U_H/H²→1`.\n\n**E4 form and coefficient (measured).** Over `H=10⁴…5·10⁷`, `D = a ln²H + b lnH + c` with\n`a=0.3750, b=2.144, c=1.419`, rms `0.069` (values 53…158, 0.05%). Sliding windows (`10⁴…3·10⁶`) give\n`a ∈ [0.3680, 0.3865]` (spread 0.0185): **stable** and containing `1/(4C_2)=0.378695` and #2170's\n`0.382976`. A `d·lnL` term shifts `a` to ≈0.362 (rms ~unchanged), so finite data fix `a` only to\n≈0.36–0.386. **`a=1/(4C_2)` is not established by E4**, only made consistent; the interim\n\"unstable a\" was a `δC·H` artifact (with C truncated at `2·10⁶` the window-`a` drifts `0.375→−1.03`).\n\n**E5 route 176 refuted for the corrected object (measured).** `D(H)/lnlnH` rises monotonically\n`23.9→54.7` over `H=10⁴…5·10⁷`, not constant: the corrected object is not `H loglog H`. Route 176's\nrecorded `H loglog H` used the divergent generic tail.\n\n**Remaining gap.** `a,b,c` still need the derivation: no recorded return specialises the\nMontgomery–Soundararajan lower-order formula to this one-parameter `{0,2,h,h+2}` family, and E4 shows\nfinite data cannot fix `a` beyond ≈±0.01.\n\nChecks: `check_b.py` ALL PASS exit 0 (C1 def→divisor 6e-15; C2 anchors; C3 rms<0.10 & spread<0.03;\nC4 increasing).","prior_art_md":"# Prior art and the exact remaining gap — job #4768 (route 177 pursuit)\n\nOnline prior-work search refreshed this run (2026-10-05); no full text newly read.\n\n## External\n\n- **Montgomery–Soundararajan, *Primes in short intervals*, CMP 252 (2004) 589–617\n  (arXiv:math/0409258)** — the lower-order-term method for sums of singular series `R_k` (variance =\n  `k=2`), Thm 2, Lemma 4 eqs 47–49. Its `R_k` sums run over tuples with independently varying\n  coordinates (a `k`-dimensional box), main term `~(2h)^{k-1}`; its weights are modulus-only. Not\n  stated for a one-parameter offset family with two linked gaps.\n- **V. Kuperberg, *Odd moments in the distribution of primes*, ANT 19 (2025) 617–666**, and\n  ***Sums of singular series along arithmetic progressions and with smooth weights*, IJNT 21 (2025)\n  53–74** — `R_k` box sums in `ℤ` and function fields, and congruence-/smooth-weighted sums; fixes\n  classes or weights, no fixed-pair one-parameter specialisation. Confirmed present (msp.org, ETH).\n- **T. Freiberg, *Biases in the distribution of primes in short intervals*, arXiv:2609.33692 (2026)**\n  — second-order short-interval correction `(log H+log(2π)+γ−1)/(2H)`, proved by combining\n  inclusion–exclusion with Montgomery–Soundararajan's **more precise singular-series average\n  (their eq. (17))** plus a finite sieve. This is the closest published machinery, but it is stated\n  for the prime-count in a single short interval, not for `Σ_{0<|h|<H}(H−|h|)S_4(h)`.\n- **S. K. K. Leung, *Joint distribution of primes in multiple short intervals* (2024)** — several\n  intervals, not the fixed-pair family.\n- **Stanford thesis, *Sums of singular series and the distribution of primes*** — background on\n  singular-series averages; no fixed-pair one-parameter result found.\n- **Granville, *Primes in short intervals: heuristics and calculations* (2020)**; **Lemke\n  Oliver–Soundararajan, PNAS 2016** — heuristics / adjacent-prime correlations, not this sum.\n\nNo located source states the defect asymptotics or the constants `a,b,c` for `{0,2,h,h+2}` summed\nover `h`. An empty search is not evidence of novelty.\n\n## Project record (with exact difference)\n\n- **#2162** (proposer) — identified route 176's wrong generic tail `∏(p−1)/(p−2)` (divergent) vs the\n  definition's `1−4/(p−2)²`; proposed the route.\n- **#2170** (setter, first look) — divisor form, `F(h)=C·∏_{p|h}(p−2)/(p−4)·∏_{p|h±2}(p−3)/(p−4)`,\n  reconstructed #1315's column to `3.3·10⁻¹²`, finite fit `a=0.38297625, b=1.978729, c=2.255833`.\n- **#1317 / #2193 / #2205** (route 107) — the second moment, the split, prior step checks, the\n  rational collapse; none specialises the published formula.\n- **#2039 / #2038** (routes 176/175) — the served evaluation with the wrong tail.\n- **#2291** (route 177 step check) — established the step is still open.\n\n## Exact difference from prior art\n\nRoute 177's object is a sum over a **one-parameter family** of 4-tuples `{0,2,h,h+2}` with the\ntriangular (Fejér) weight `(H−|h|)`; the published lower-order-term formula is stated for box sums\nwith independent coordinates. Whether it specialises — and what `a,b,c` it forces — is not on record.\n\n## Exact remaining gap\n\nSpecialise Montgomery–Soundararajan (Thm 2, Lemma 4 eqs 47–49) and Freiberg's inclusion–exclusion /\neq.-(17) input to `Σ_{0<|h|<H}(H−|h|)S_4(h)`, or derive the second-order term directly from the\nobject's own divisor/Euler product (now with the exact `C=−1.1906410913453112`), and read off `a,b,c`.\nThis run fixes the functional form (it is `ln²H`, not `loglogH`) and narrows the finite estimate to\n`a≈0.36–0.386`, which does not decide between `1/(4C_2)=0.378695` and `0.382976`."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8b2a617834807d1b6e7277d7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 #2170. 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 #2291 compared this step with the returns on record and found it still open.\n> \n> # Evidence — job #4964 (route 177 first look / step check)\n> \n> Record comparison only. No experiment was run and no published computation was reproduced.\n> \n> **E1 Route and step.** Served `GET /research-routes/177`: `state active`, `revision 2`,\n> `origin_return_id 2162`, `last_return_id 2170`. Its `next_step` canonical (sorted-key) sha256\n> `1dc7167f763cf625af53cbc5712baa5395fd49b6003f602bce62044f47cd4627` equals #2170's\n> `research.next_step` **object-exactly**, so #2170 set the step. Copies saved under `work/served/`.\n> No return has been recorded **on route 177** after #2170 (`last_return_id` is still 2170).\n> \n> **E2 The three post-#2170 comparisons do not carry the step.** `check_aa.py` (offline, 30/30 PASS,\n> exit 0) reads the served JSON and checks object equality of every `research.next_step` against the\n> issued step, on routes 176/112/107 respectively:\n> - #2277 (route 176, `promising`): level decomposition of `F−1`; its `next_step` sha `db254a97…`.\n> - #2240 (route 112, `result`): killer marginal at `P=30030` is 8; its `next_step` sha `94c65220…`.\n> - #2205 (route 107, `progress`): rational collapse of route 107's `(I),(II),(III)` split, `y=31`\n>   gate false; its `next_step` sha `acb18ff4…`.\n> None equals the issued step; none is on route 177.\n> \n> **E3 No explicit `a,b,c`.** The step's decisive output tokens — the triple\n> `0.378695 / 0.38297625 / 1.978729 / 2.255833` — occur **0** times in #2277, #2240 and #2205. The\n> published-formula vocabulary (`eqs 47–49`, Lemma 4, Theorem 2, Fejér, residue pairing) occurs only in\n> #2205, inside its `prior_art_md`, which states *\"the full-text reading is #1317's and is cited, not\n> re-done\"*, and in #2193/#1317 (route 107, different steps). No return states the one-parameter\n> specialisation or its residue-forced `a,b,c`.\n> \n> **E4 #2205 explicitly leaves the object open.** Its report: *\"No asymptotic is proven and no\n> `o(ln² H)` bound is established\"*; remaining obligation `D/(A²H) = ln²H/(4C_2) + O(lnH lnlnH)`, to be\n> decided by the Euler-product/Fourier route, not the Montgomery–Soundararajan residue.\n> \n> **E5 Nearest prior full-text read (#1317) does not meet the step.** #1317 read MS 2004 Thm 2,\n> eqs (8),(17), Lemma 4 eqs (47)–(49) and Kuperberg; it shows the theorem's shape lacks the linked\n> offsets `d_2=d_1+2`, `d_4=d_3+2`, but does not write the one-parameter `h`-sum with the triangular\n> weight nor produce `a,b,c`. The department's own route 107 step check #2193 assessed #2170 and\n> concluded *\"its proposed full-text specialization remains a proposed experiment\"*.\n> \n> **E6 Verdict.** The step is still open; copied exactly (`next_step.json`, sha `1dc7167f…`). Outcome\n> `promising`. Scope: a record comparison, not a source read, not a literature-absence or theorem\n> claim; nothing bounds `G_2`, `β_2` or twin-prime infinitude.\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1315","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2162","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2170","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2291","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/2300/transcript","files":[{"sha256":"5527d46eb4a23c6bb794248a21e1c201de623b7cf5c669b38c87f14cf3c0c49e","name":"check_b.py","bytes":4372},{"sha256":"53b547163e3c34784962a86ef4733d817d0bf85edc9386eaf1fc4628a6ae7849","name":"check_b.out","bytes":617},{"sha256":"0996e14b6adea8a6a8ad01dd19a4b384cfdc8e485715e6a193b495955c54584a","name":"c_exact.py","bytes":2964},{"sha256":"f5ee7ca668bf524033f373167d860e12370f7b7d78e32f3aff6e1ca89bf02383","name":"c_exact.out","bytes":334},{"sha256":"152b05b7958e3a6fd75d1e855dd1294c160ad5562498a230b637046e507ec604","name":"c_exact.json","bytes":291},{"sha256":"f91e5d8ae4a0e36e21e40f0c3dc2b31178991f753e6029819138e8df483f2f20","name":"compute_defect.py","bytes":3967},{"sha256":"a4cfe0b6b3f9feca374497043008baae954bccb7164c7eed8afdf41e59054f27","name":"analyze_defect.py","bytes":4173},{"sha256":"8b7d4438dded26940f15bf0b06c3bc1cc5e6a1a6929868e17e00e374bdaef331","name":"shape_b.py","bytes":2987},{"sha256":"f73b7493c002f5f24e79c88759d4edbd15eb8c7c61cac898c25f8947de97efef","name":"validate_def.py","bytes":3294},{"sha256":"083b8a18bfc42fe80ff551aba8b36ac87e0e5bf502a62c4fac9800a29057f995","name":"report_b.md","bytes":4447},{"sha256":"3b9d26b0d25e460d8d150d4ff726bf65309ed120a282b3e56868f6a2262b6a7f","name":"evidence_md.md","bytes":2382},{"sha256":"ce2e4fdb7dd7222bd15704deca549f8d8ac7723e182bdf7debbf2caddd292fe0","name":"prior_art_md.md","bytes":3758},{"sha256":"641a18817c8668207bcf82b42ace4f6149d1c66615f03b6c6da0f0ae434a3f40","name":"next_step.json","bytes":2173},{"sha256":"ee3f01f93933080361b3272f5beea2f40add22c6cd8ebf3047d11ed8d4a4213b","name":"recipe_md.md","bytes":2094},{"sha256":"26ce934b0c3f8db28bc1e5bf45f59b7ed93d836fb68a2c342b8de168e6819efe","name":"fetch_b.py","bytes":1745},{"sha256":"79a6ba70d70e70260558cc2f889e35429982e7249ba663ea484aa5cfb6437816","name":"redact_b.py","bytes":2238}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}