{"id":5423,"job_id":10868,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #10868: route 107 step check after #4160. Outcome `promising`: the step is still open\n\nRoute **107** (twin-pair singular-series defect), revision 12. I read and compared the record only and ran nothing.\n\n**The held step.** Its source is #4160. The served route-107 next_step, canonical sorted-key sha256 `cf1a2751…240e`, is copied verbatim below as next_step. It asks how the centred residual delta(R,H) = Σ_{0<|h|<H}(1−|h|/H)A_R(h) grows in R at fixed H, at R up to 10^6 (H = 1000, 3000) and 2·10^5 (H = 10^4), against ln²H = 47.7 / 64.1 / 84.8.\n\n**What #4160 already settles.** These facts carry forward unchanged:\n- The exact r1 split (II) = diagonal + phase holds, with phase = M + delta.\n- rms_h|A_R(h)| is flat in R: 0.51 / 0.53 / 0.53 at H = 1e3 / 3e3 / 1e4.\n- max_h|A_R| = 4.43 against the sufficient uniform bound 0.048 at H = 1000, so that uniform form is false at the tested rungs.\n- |delta| ≤ 0.594 on its grid.\n\n**What #4160 leaves open, which is the step itself.** delta's R-scaling is undetermined. At H = 1000 it goes from 0.0013 to 0.336 for R = 10^3 to 2·10^4, which is slow but not fitted. No R beyond 2·10^4 at H = 1000 has been measured.\n\n**No later return answers it.** No route-107 return or route event exists after #4160. The comparison returns, read in full and screened by token for A_R / delta(R / secondmoment / Fejér / grid_bq:\n- #5371 (route 175) fits Delta_bar = delta ln H + e on [1e4,1e8] windows. It is a different object, and it states that nothing in it bears on route 107's (II).\n- #5243 (route 166) measures bootstrap Var R_cond.\n- #5067 (route 177) is an S_4-defect formula step check. It names #4138/#4160 as route 107's own (II) formulation and computes no delta.\n- #4916 (route 112) computes K*(30030, U_47).\n- #4813 (route 176) is a band residual, outcome needs_capability.\n- #4565 (route 192) measures the t-curve's x-dependence.\n- The citers of #4160: #4519 (route 36), #4555 (route 192, comparison only), #4791 (route 176, which says #4160 \"computes no M_r level sums… reaches no H above 1e4\") and #4895 (route 112, comparison only). None extends R.\n- The brief's locators #4499 (G2(89#)), #4511 (route 317 F_slot) and #5384 (g2 reruns) are unrelated objects.\n- Route 143, the failure fallback (G2 moment dial), is a different object.\n- The lane chat (last 60 messages) holds no route-107 claim. Mine is claim 8595.\n\n**Decision.** The step is still open, so the outcome is promising with the step copied exactly. Its cost (0.5 CPU-h, h-form R·H) is bounded, and its success and failure rules decide between the centred criterion and route 143.\n\n5 returns wait for a verdict.\n","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T16:45:44.494Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4160,2338,5371,5243,5067,4916,4813,4565,4519,4555,4791,4895],"messages":[8595]},"tokens":{"log":"summary","input":124,"models":{"claude-opus-5-5":20855},"output":20855,"source":"reported","entries":0,"cache_read":2982973,"cache_write":121086,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (job #10868, step check: reading and comparison only, nothing run)\n\n1. `GET <project>/research-routes/107` gives revision 12. I dumped `next_step` sorted-key with separators (',',':'); its sha256 is cf1a27519b28ce8516def15bb181601f024ed6d1fb8fe096479ce0cd7cad240e, and it equals the brief's step text.\n2. `GET <project>/return/<id>` for 4160, 5371, 5243, 5067, 4916, 4813, 4565, 4519, 4555, 4791, 4895, 4499, 4511 and 5384, saved one JSON per return in an isolated directory.\n3. A token screen (python -I) over report_md, recipe_md, research.evidence_md, prior_art and next_step for `A_R`, `secondmoment`, `delta(R`, `Fejer/Fejér`, `Lambda_r`, `rms_h`, `max_h`, `ln^10` and `grid_bq`. The only hit outside #4160 is #4791 (one `grid_bq` mention, as a comparison).\n4. Read #4160's evidence table: rms flat 0.51/0.53/0.53, |delta| 0.0013..0.336 / 0.074..0.395 / 0.107..0.594, max_h|A_R| 4.43 versus 0.048 needed.\n5. `GET <project>/chat/messages?limit=60` showed no route-107 claim. I posted claim 8595.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":107,"next_step":{"method":"Reuse grid_bq.py's `secondmoment` mode (the h-form, cost R*H; do not re-run the b-loop grid or the controls). At fixed H = 1000 and H = 3000 extend R to 10^6 and at H = 10^4 to 2*10^5, recording delta(R,H), max_h |A_R(h)|, rms_h |A_R(h)| and the signed sum's partial sums over R; fit delta's R-growth (ln R, ln R lnln R, or a power) and compare with ln^2 H = 47.7 / 64.1 / 84.8. Use sah.py bounded for every run and checkpoint every block. This builds on #2338 (the split's identity), #1834 (Theorem B and the Weil/Kloosterman diagnosis) and #2205 (the split) and needs no new source.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":0.5},"failure":"delta or max_h |A_R(h)| grows at or above ln^2 H in R at fixed H: then the centred criterion fails too and the route must bound the signed sum directly, route 143 / H(a,A) (served, active).","success":"delta bounds stay below c ln^2 H with a sub-ln^2 fitted growth at >= 2 values of H, so the centred object is the right one to attack and the Fejer-weighted linear input is the obligation to state.","question":"After the elementary marginal shift is removed, how does the centred residual delta(R,H) = sum_{0<|h|<H}(1-|h|/H) A_R(h) grow in R at fixed H -- does it stay o(ln^2 H) beyond R = H ln^10 H, and what is the R-scaling of A_R(h)?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[4160],"evidence_md":"Step check after #4160. No route-107 return or event exists after #4160.\n\nWhat #4160 settles: the exact r1 split (II) = diagonal + phase, with phase = M + delta. rms_h|A_R(h)| is flat in R (0.51 / 0.53 / 0.53 at H = 1e3 / 3e3 / 1e4). The uniform sufficient bound max_h|A_R| = o(ln²H/H) is false at the tested rungs (4.43 against 0.048 at H = 1000). |delta| ≤ 0.594 on its grid.\n\nWhat it leaves open: delta's R-scaling, which goes 0.0013 → 0.336 for R = 1e3 → 2e4 at H = 1000 and is not fitted. That is exactly the held step.\n\nThe comparison returns do not answer it:\n- #5371 (route 175) fits Delta_bar windows and states it does not bear on route 107's (II).\n- #5243 (route 166) measures bootstrap Var R_cond.\n- #5067 (route 177) is an S_4-defect formula check that names #4160 as route 107's own (II) and computes no delta.\n- #4916 (route 112) computes K*.\n- #4813 (route 176) is a band residual, outcome needs_capability.\n- #4565 (route 192) measures the t-curve.\n- The citers of #4160 (#4519, #4555, #4791, #4895) cite it as a comparison only. #4791 records that it \"reaches no H above 1e4\".\n\nNone computes delta(R,H) or A_R(h) at a larger R. The step is open and is copied exactly."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_60a17b2e8f6b38e3ba787c6d","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 #107's next experiment was set by return #4160, 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\":\"Reuse grid_bq.py's `secondmoment` mode (the h-form, cost R*H; do not re-run the b-loop grid or the controls). At fixed H = 1000 and H = 3000 extend R to 10^6 and at H = 10^4 to 2*10^5, recording delta(R,H), max_h |A_R(h)|, rms_h |A_R(h)| and the signed sum's partial sums over R; fit delta's R-growth (ln R, ln R lnln R, or a power) and compare with ln^2 H = 47.7 / 64.1 / 84.8. Use sah.py bounded for every run and checkpoint every block. This builds on #2338 (the split's identity), #1834 (Theorem B and the Weil/Kloosterman diagnosis) and #2205 (the split) and needs no new source.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.05,\"cpu_hours\":0.5},\"failure\":\"delta or max_h |A_R(h)| grows at or above ln^2 H in R at fixed H: then the centred criterion fails too and the route must bound the signed sum directly, route 143 / H(a,A) (served, active).\",\"success\":\"delta bounds stay below c ln^2 H with a sub-ln^2 fitted growth at >= 2 values of H, so the centred object is the right one to attack and the Fejer-weighted linear input is the obligation to state.\",\"question\":\"After the elementary marginal shift is removed, how does the centred residual delta(R,H) = sum_{0<|h|<H}(1-|h|/H) A_R(h) grow in R at fixed H -- does it stay o(ln^2 H) beyond R = H ln^10 H, and what is the R-scaling of A_R(h)?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1315, #1317, #1834, #2034, #2054, #2073, #2193, #2205, #2320, #2338, #4138, #4160 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5371 (route 175, progress, recorded, recorded): The held step (route 175's `next_step`, set by #2055 and repaired by #4009) is a delta measurement on two disjoint windows: W1 = [1e4,1e7], W2 = [1e7,1e8], 40 grid points per decade, fit Delta_bar = delta ln H + e, >= 5 admissible grid points per window. It was executed. Three findings. (1) THE CITATION ROUTE IS INADMISSIBLE, TWICE OVER. The served record supplies 4 smoothed Delta_bar values for \n- Return #5243 (route 166, progress, recorded, recorded): **Claim.** Route 166's held step is unrun but cheap (0.12 CPU-h with a streamed sieve, against a cited instrument needing 10 GB on a 6 GiB container). Reproducing its cells exactly shows: (i) the iid bootstrap does understate Var R_cond, but only at lengths the step does not pre-register (1.50x at 0.36 % of a period at x = 29, versus 0.93x/1.07x at its 13/169 windows); (ii) at that scale the serve\n- Return #5067 (route 177, promising, recorded, recorded): **(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 353\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 #4565 (route 192, result, accepted, measured): **What changes.** Route 192's held step asked whether the residual x-dependence of the `t`-curve (`t = h*delta(x)`) survives an interpolation-free measurement across several rungs. Measured on matched phases (`|dt| <= 1e-3`) over **x = 53, 97, 101, 107, 113, 127, 131** — with three new rungs of the exact integer tables `C(m), A(m)` computed here — it **does**: the matched-phase residual `dln = ln \n\nReturn the ordinary report and transcript plus research: {route_id: 107, 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":"4160","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5436,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/5423/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"790c1ca9f4ca121d11ef75bc9661f4b87fef1992c8cf29c967310e0856ecbb00","next_step_sha256":"cf1a27519b28ce8516def15bb181601f024ed6d1fb8fe096479ce0cd7cad240e","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":8595,"channel_path":"","handle":"Benjaminsen","model":"claude-opus-5-5","kind":"claim","body_md":"Job 10868 (route 107, step check before pursuit, first_look): checking whether returns recorded after #4160 answer route 107's held step (grid_bq.py secondmoment: delta(R,H), max/rms_h |A_R(h)| to R=10^6 at H=1e3/3e3 and R=2e5 at H=1e4, R-growth fit vs ln^2 H). Read-and-compare only; no runs. Screening #5371, #5243, #5067, #4916, #4813, #4565 and the #4160 citers #4519/#4555/#4791/#4895.","created_at":"2026-10-22T16:43:03.899Z","url":"/projects/twin-primes/chat/messages/8595"}]}