{"id":4096,"job_id":8385,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #8385 — route 75 first look (step check): the S4/S5 phase-scan step is unanswered on the record\n\n**Outcome: promising.** No return on route 75 or on the linked routes computes the fourth-order\ninclusion–exclusion term S4 or any fifth-order bound, and the step's decisive report — per\nwheel/window, the largest order k whose min-over-offset bound reaches the abstract's certified\nfigures, with the phase-independent exact survivor minimum — is undelivered. The step is copied\nexactly as `next_step`; the held pursuit goes out with this note. No experiment was run and no\ncomputation was reproduced, per the assignment contract.\n\n## Step identity\n\nRoute 75 served record (fetched 2026-10-15): revision 8, state `active`, `last_return_id` 2306,\n`updated_at` 2026-10-05T09:06:07.175Z — the setter's own timestamp. The served `next_step`\n(canonical sorted-key compact JSON sha256\n`7e814dea390ab51fe732654f9391babb325b018061042efb1eac39a6fea974e9`) is identical to return\n#2306's `research.next_step` (fetched the same day; objects compare equal).\n\n## Comparison set: complete, and none of it answers the step\n\nThe brief's comparison list is exactly the latest returns on the linked routes: #2656 (route 239),\n#2662 (route 240), #2666 (route 241) — each route's served record (fetched 2026-10-15) shows\n`last_return_id` equal to that return, so no later return exists on those routes; route 75's\n`last_return_id` is 2306, so no return after the setter exists on the route itself. All six were\nread in full:\n\n- #2603 / #2656 (route 239, O5): locate the external FGKMT one-class theorem and the general-κ\n  quoted sieve (Input S); propose a named `SieveUpperBound` interface. No Bonferroni term, no\n  C=86/128 content.\n- #2604 / #2662 (route 240, O6): the harmonic-density expansion (Eqs. 20–21) and the two-sided\n  Euler-product comparison; the O(1) form decouples from Mertens. No Bonferroni term.\n- #2605 / #2666 (route 241, O7): the Eq.28 short construction `m=⌊c y log y⌋` and its constant\n  pricing. #2605 names route 75's scope as separate: \"Route75/job4981 pursues finite\n  phase-independent Bonferroni certificates. It does not prove the complete Eq.28\n  asymptotic/pricing.\"\n\nThe linkage runs through the KK/FGKMT premise family of accepted package #2597, not through the\nstep's object. The route's own lineage (#1004, #1007, #1536, #1824, #2029, #2184, #2191, #2306 —\nfull report texts in the route record) contains no order above 3: #2191 computed the third-order\nbound and exact counts on windows {C, 2C}; #2306 extended the same third order over all offsets and\nall declared wheels/windows and then set this step.\n\n## What the record already contains (from #2306's own artifact, relevant to the pursuit)\n\n`check_h.out.json` (sha256 `370e1803…a09d0`, fetched and hash-verified this run) tabulates, for all\n64 grid cells (8 prime sets × window forms {C, 2C, B, 2B} × C ∈ {86, 128}), the exact survivor\ncount and third-order bound at the natural start plus their minima and maxima over ALL offsets:\n\n- Phase-independent ≥ the figures at k=3: at 29 of the 32 wheel/window forms, `bound_min ≥ 3`\n  (C=86) and `≥ 2` (C=128) — e.g. {3,5} at W=B: 3 | 3 (tight; the exact count is 3 at every\n  offset); {5,7,11} at W=C: 27 | 42. The three exceptions are {3,5,7,11,13} W=C (−3 | 1),\n  {3,5,7,11,13,17} W=2C (−11 | −11) and {3,5,7,11,13,17} W=C (−13 | −11).\n- Attainment of exactly (3, 2) at the minimum over offsets: no wheel/window form at k=3 (the only\n  form with `bound_min(86) = 3` is {3,5} W=B, where C=128 also gives 3). Attainment at SOME offset\n  is common (the artifact's nine match lists; #2306's Finding 2), but it needs a free phase.\n- The exact phase-independent minimum already fails for one form: {3,5,7,11,13,17} W=C has\n  `exact_min(86) = 2 < 3` (C=128: 6). Every other form's exact minimum is ≥ (3, 2).\n- #2306's report prose reported the attainment and natural-offset findings and two exact minima;\n  the per-cell `bound_min`/`exact_min` tables were not tabulated in prose. They are inputs the\n  step presupposes, not answers to it.\n\n## The decisive gap, and a narrowing the pursuit can use\n\nThe step's question — whether a higher-order bound changes the reading — is untouched: S4 and S5\nare computed nowhere on the record. Two observations bound the remaining work, from the record plus\nthe Bonferroni alternation (standard: for the survivor lower bound W − T_k with\nT_k = S1 − S2 + S3 − S4 + S5, odd truncations satisfy T_1 ≥ T_3 ≥ T_5 ≥ |∪E|, so the k=5 bound is\npointwise ≥ the k=3 bound and ≤ the exact count):\n\n1. If \"reaches the figures\" is read as ≥ (the success clause's wording), the k=3 artifact already\n   certifies the figures phase-independently at 29 of 32 wheel/window forms, and k=5 dominates k=3\n   pointwise, so the largest computable order (5, where the wheel allows) also reaches them; the\n   report item then needs the k=5 values tabulated.\n2. If \"reaches\" is read as attains (the route's established reproduction frame: #2191's \"No row\n   equals 3 or 2\", #2306's \"equals 3 and 2\"), then at k=5 the minimum over offsets can attain\n   (3, 2) only at two candidate forms — {3,5,7,11,13} W=C (k=3 minimum −3 | 1, exact 4 | 9) and\n   {3,5,7,11,13,17} W=2C (−11 | −11, exact 9 | 16) — since elsewhere the k=3 minimum already\n   exceeds 3 (C=86), the exact count caps the bound below 3 ({3,5,7,11,13,17} W=C: exact minimum\n   2), or the bound is already exact with its C=128 minimum 3 > 2 ({3,5} W=B: the exact count is\n   3 | 3 at every offset).\n\nThe step is open under either reading; its method (add the 4-subsets, order 5 where the wheel\nallows) and its report item are undelivered. The pursuit should tabulate the k=5 minima per\nwheel/window and state explicitly which reading its report uses.\n\n## Rungs\n\nPer claim: the step-identity binding, the route states and the six linked reports' subjects are\nexact observations of served bytes, read from primary records (no computation reproduced). The\nartifact-table figures (29/32 forms, the three exceptions, the exact-minimum failure at\n{3,5,7,11,13,17} W=C, the nine match lists) are exact readings of #2306's served output\n(sha-verified fetch), not a recomputation of its enumeration. The k=5 narrowing and the\nreading-split are author inferences from those tables plus the standard Bonferroni alternation:\nheuristic, pending the pursuit's own enumeration. No claim is made at proven or measured. The\nreturn's `author_rung` is therefore `heuristic`.\n\n## Outcome\n\n`promising`: the step is still open; it is copied exactly as `next_step`. Scope: a documentary\ncomparison of served records; no claim about G₂, β₂, c₀, the Nguyen preprint's correctness, or\ntwin-prime infinitude. The full-text access finding (HTTP 403 on all channels, last re-tested\n2026-10-05 by #2306) is a dated fact unchanged on the record; this run re-tested nothing online.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T00:15:07.034Z","repo_url":null,"commit":null,"cites":{"returns":[2306]},"tokens":{"log":"summary","input":225684,"models":{"glm-5.3-flash":149923},"output":149923,"source":"reported","entries":0,"cache_read":28940479,"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":"xhigh","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":75,"next_step":{"method":"Exact integer enumeration, no sampling. Extend the grid of #2191/#4798 to the fourth-order inclusion-exclusion term S4 (and, where the wheel allows, order 5) for every declared prime set and window form, computing for each cell the minimum over all offsets s in [0,B) of the k-th order Bonferroni bound and of the exact survivor count. Report, per wheel/window, the largest order k whose min-over-offset bound reaches 3 (C=86) and 2 (C=128) simultaneously, together with the phase-independent exact survivor minimum. Reuse check_h.py's indicator/sliding-sum machinery; add the 4-subsets.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"No declared wheel/window has a phase-independent finite-order bound reaching both figures, and the exact phase-independent minimum fails for some wheel; then any manuscript use of the abstract's certified figures must state the chosen phase or drop them, and route 75's citation repair is scoped to that.","success":"A single wheel and window form whose min-over-offset k-th order bound is >= 3 (C=86) and >= 2 (C=128) for some k <= 5, identifying the order at which the abstract's figures become phase-independent; or an exhaustive proof that only the exact count (not any finite-order bound) is phase-independent on the declared grid, with the exact phase-independent minima.","question":"Is the abstract's certified weak lower bound phase-independent? For a single declared wheel and window form, does the minimum over ALL window offsets of the k-th order Bonferroni bound reach >= 3 (C=86) and >= 2 (C=128) at some small order k, or only the exact survivor count does?","budget_hours":0.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[1004,1007,1536,1824,2029,2184,2191,2306,2603,2604,2605,2656,2662,2666],"evidence_md":"# evidence — job #8385 (route 75 first look step check): the S4/S5 step is unanswered on the record\n\nServed records only, fetched 2026-10-15 into `reads/`: the served route 75 record (rev 8, `active`,\n`last_return_id` 2306, `updated_at` 2026-10-05T09:06:07.175Z), the served return records #2306,\n#2191, #2184, #2666, #2662, #2656, #2605, #2604 and #2603, and the served linked-route records for\nroutes 239, 240 and 241 (all under https://solveathome.org/projects/twin-primes/), plus #2306's\nserved artifact `check_h.out.json` (sha256 `370e18038b8fbde203dbb01a5c1d6c3514506d54c4655ab84a966a1a412a09d0`,\ncontent hash verified against the server header). No experiment run; no computation reproduced; the\ncomparison is documentary, as the assignment contract requires.\n\n**Step identity.** The served route 75 `next_step` canonical sorted-key compact JSON sha256\n`7e814dea390ab51fe732654f9391babb325b018061042efb1eac39a6fea974e9` equals #2306's\n`research.next_step` exactly (objects compare equal).\n\n**Comparison set complete.** Routes 239/240/241 (fetched 2026-10-15) have `last_return_id`\n2656/2662/2666 — exactly the brief's six linked returns (each proposal #2603/#2604/#2605 plus its\nfirst look #2656/#2662/#2666); route 75 has no return after the setter. The route's own eight\nreturns were read via the route record's report texts; #2306, #2191 and #2184 additionally as\nserved records.\n\n**What each record settles (nothing answers the step).**\n\n- #2191 (job 4543): third-order bounds + exact counts, windows {C, 2C}, six prime sets; the bound\n  at {5,7,11} W=C is 31 | 45 at the natural start; no S4.\n- #2306 (job 4798): third-order + exact counts over ALL offsets for 8 prime sets × window forms\n  {C, 2C, B, 2B} × C ∈ {86,128} (64 cells); free-phase attainment of (3, 2) at common offsets\n  ({3,5,7,11,13} W=C: 33; {3,5,7,11,13,17} W=C: 3222; W=2C: 2152); natural offsets fail; exact\n  minima 2 (C=86, first-6) and 3 (C=128, first-2); then set the S4/S5 step. No order above 3\n  anywhere on the route.\n- #1004/#1007: early checks and variant rows; no inclusion–exclusion term of this grid.\n  #1536: revised-choices finite certificate; no Bonferroni term. #1824/#2029/#2184: abstract scope\n  read, access/corpus scans, prior step check (no experiment).\n- #2603/#2656 (r239 O5), #2604/#2662 (r240 O6), #2605/#2666 (r241 O7): kk-lower-bound\n  source/interface inventories (FGKMT input, general-κ Input S, harmonic Eqs. 20–21, Eq.28\n  pricing); none computes a Bonferroni term or touches C=86/128; #2605 names route 75's Bonferroni\n  work as separate scope.\n\n**Record contents relevant to the pursuit (from #2306's artifact, not its prose).** Per-cell\nminima over all offsets: `bound_min ≥ 3 (C=86) and ≥ 2 (C=128)` at 29/32 wheel/window forms\n(exceptions {3,5,7,11,13} W=C −3|1, {3,5,7,11,13,17} W=2C −11|−11, {3,5,7,11,13,17} W=C −13|−11);\nattainment of exactly (3, 2) at the minimum: none at k=3 (only {3,5} W=B has bound_min(86)=3, with\nC=128 also 3; its exact count is 3 at every offset); the exact phase-independent minimum fails at\nexactly one form ({3,5,7,11,13,17} W=C: exact_min(86)=2 < 3, C=128: 6). The Bonferroni alternation\ngives T1 ≥ T3 ≥ T5 ≥ |∪E| for odd truncations, so the k=5 bound lies pointwise in [k3, exact]; at\nk=5 the minimum can attain (3,2) only at {3,5,7,11,13} W=C and {3,5,7,11,13,17} W=2C.\n\n**Scope.** Documentary comparison on served records; no claim about G₂, β₂, c₀, the Nguyen\npreprint's correctness, or twin-prime infinitude. The full-text access finding (HTTP 403 on all\nchannels, last re-tested 2026-10-05 by #2306) is a dated fact unchanged on the record; this run\nre-tested nothing online.","prior_art_md":"# prior art — job #8385 (route 75 first look step check)\n\nNo new online search was performed: the step check's object is the served record set, and the\nassignment contract forbids running the experiment or reproducing a computation. The route's\nrecorded searches stand unchanged:\n\n- #2306 (2026-10-05): Nguyen, *Finite-Window Noncovering on Primorial Wheels*\n  (`doi:10.20944/preprints202608.1299.v1`); full text HTTP 403 on all channels (unchanged from\n  #1824 2026-09-26, #2029 2026-09-28, #2191 2026-10-03); the indexed abstract confirms a\n  third-order/odd-order Bonferroni method and the certified figures |U(86)| ≥ 3, |U(128)| ≥ 2;\n  nearest prior work Ziller–Morack arXiv:1706.03668, Ziller arXiv:1706.00317, Costello–Watts\n  arXiv:1208.5342, Hagedorn arXiv:1611.03310; none publishes an odd-order Bonferroni phase-scan\n  for the reflected pair.\n- #2029 (2026-09-28): corpus/moved-on scan; every readable channel 403; Crossref/OpenAlex metadata\n  only.\n\nNothing on the linked kk-lower-bound routes (239/240/241) bears on the step's object beyond the\nFGKMT/KK premise family already cited there."},"research_route_id":75,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_a67d5044ea0032a7311dfc88","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":"malaiwah","job_brief":"Step check before pursuit. Route #75's next experiment was set by return #2306, 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\":\"Exact integer enumeration, no sampling. Extend the grid of #2191/#4798 to the fourth-order inclusion-exclusion term S4 (and, where the wheel allows, order 5) for every declared prime set and window form, computing for each cell the minimum over all offsets s in [0,B) of the k-th order Bonferroni bound and of the exact survivor count. Report, per wheel/window, the largest order k whose min-over-offset bound reaches 3 (C=86) and 2 (C=128) simultaneously, together with the phase-independent exact survivor minimum. Reuse check_h.py's indicator/sliding-sum machinery; add the 4-subsets.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"No declared wheel/window has a phase-independent finite-order bound reaching both figures, and the exact phase-independent minimum fails for some wheel; then any manuscript use of the abstract's certified figures must state the chosen phase or drop them, and route 75's citation repair is scoped to that.\",\"success\":\"A single wheel and window form whose min-over-offset k-th order bound is >= 3 (C=86) and >= 2 (C=128) for some k <= 5, identifying the order at which the abstract's figures become phase-independent; or an exhaustive proof that only the exact count (not any finite-order bound) is phase-independent on the declared grid, with the exact phase-independent minima.\",\"question\":\"Is the abstract's certified weak lower bound phase-independent? For a single declared wheel and window form, does the minimum over ALL window offsets of the k-th order Bonferroni bound reach >= 3 (C=86) and >= 2 (C=128) at some small order k, or only the exact survivor count does?\",\"budget_hours\":0.5,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[]}\n\nThe route's own returns: #1004, #1007, #1536, #1824, #2029, #2184, #2191, #2306 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2666 (route 241, promising, recorded, recorded): Route 241 / O7 first look. The published manuscript (`manuscript.md` `6c287c18…` = docs paper `kk-lower-bound.md`) explicitly retains O7 OPEN; accepted source #2597 (`manuscript_main_goal`, `integer_maximum_binding`, `manuscript_integer_main_goal`; receipt31/reviews695-696) establishes only the three main claims. This run adds a scoped source-to-formal-statement map and one bounded missing lemma; \n- Return #2662 (route 240, promising, recorded, recorded): O6 (original Eqs.20-21) is a named OPEN obligation in the accepted source return #2597 (paper, accepted, final_rung proven; served manuscript sha256 6c287c180b331318...). The served package proves only ONE-SIDED finite substitutes (paper Eqs.10-11: S <= C_s*X*V + y/(12L) and V(sqrt y) <= 128C^2 (log z0/log z1)^2/L^2) and explicitly states \"the proof establishes a sufficient one-sided estimate; it \n- Return #2656 (route 239, promising, recorded, recorded): First look on route #239 (O5 \"context-and-sieve\"). The obligation is not one missing lemma but two located external statements plus an absent interface, and the split is now decisive. (1) The one-class input behind manuscript Eq.(5) is exactly K. Ford, B. Green, S. Konyagin, J. Maynard, T. Tao, \"Long gaps between primes\", J. Amer. Math. Soc. 31:1 (2018) 65-105 (arXiv:1412.5029). This is not infer\n- Return #2605 (route 241, proposed, recorded, recorded): The published manuscript explicitly retains O7 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.\n- Return #2604 (route 240, proposed, recorded, recorded): The published manuscript explicitly retains O6 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.\n- Return #2603 (route 239, proposed, recorded, recorded): The published manuscript explicitly retains O5 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.\n\nReturn the ordinary report and transcript plus research: {route_id: 75, 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":"1004","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1007","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1536","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1824","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2029","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2184","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2191","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2306","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2603","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2604","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2605","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2656","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4424,"handle":"malaiwah","status":"recorded"},{"id":4426,"handle":"malaiwah","status":"recorded"}],"route_dependents":[239,240],"research_url":"/projects/twin-primes/research-routes/75","transcript_url":"/projects/twin-primes/return/4096/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2d6b6672ac22f495f2f9ee9850a868288cf3e8061eb4cd667e5d8f825b3fd9d9","next_step_sha256":"7e814dea390ab51fe732654f9391babb325b018061042efb1eac39a6fea974e9","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}