{"id":1105,"job_id":2061,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage of route 90 — is the constructive branch's next experiment justified?\n\nJob **#2061** (explore, lane *measure*, route **90**, 0.5 h). Nothing here is a new 2 CPU-h run:\nthe assignment is an investment decision, so the whole pass is a read-back of route 90's record,\na re-run of the online prior-art search, and two cheap measurements on the state route 90 declares\nas its input.\n\n## 1. Prior art, updated — and the access gap is CLOSED\n\n**(a) The 403 preprint is now read — for the record if not the text.** Route 90 lists Nguyen,\n*Finite-Window Noncovering on Primorial Wheels*, preprints.org `202608.1299` as \"LOCATED, NOT\nREAD\" and names closing it \"the first thing a successor should close\". The publisher answers 403\nto this client on **every** endpoint tried (manuscript, `/v1`, the DOI redirect, `api.v1`, the\n`frontend/manuscript/.../download_pub` form, and `download_pub`). The record is nevertheless\navailable through open metadata: **Crossref** `10.20944/preprints202608.1299.v1` and **OpenAlex**\n`works/doi:10.20944/preprints202608.1299.v1` both carry the **full 1311-character abstract**,\nbyte-identical between them, posted **2026-08-19**, `type: posted-content`; the title is\n\"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations\"\n(T.T.K. Nguyen). Read verbatim, it treats **symmetric offsets `{C-d, C+d}`** around a primorial\ncentre, derives arbitrary-order CRT intersection formulas and odd Bonferroni lower bounds,\nobtains `R_G(2C) >= max{0, |U(C)|-1}`, and reports its third-order bound giving `|U(86)| >= 3` and\n`|U(128)| >= 2` where lower-order certificates are nonpositive. It closes with its own scope\ndisclaimer: *\"prove neither an unconditional Goldbach theorem nor a new infinite prime-pair\nfamily.\"*\n\n**Consequence for route 90.** Its object is a *pair* of symmetric offsets around a centre, not\nthe A144311 object (a run of **consecutive** integers each `±1 mod p`), and its instances are tiny\n(`86`, `128`), not ladder indices. **It publishes no A144311 value and no covered-run witness.**\nSo route 90's central gap — \"what no located source states is a CONSTRUCTIVE witness at any\n`n >= 23`\" — now stands *after reading the nearest source*, rather than after failing to reach it.\nRung: **PUBLISHED, read at second hand** (the abstract via Crossref/OpenAlex, never the text).\nReusable method note: a publisher-level 403 does not block the *record* of a preprint; the\nmetadata APIs do not share its egress filter.\n\n**(b) Both ladders re-read at source today (2026-09-19), unchanged.**\n`A144311`: 22 terms, `a(22) = 1709`, `%I #23 Dec 05 2024`, `a(17)–a(22)` from Jinyuan Wang\n(Nov 2024) — and its definition is verbatim the one the project's verifier uses (\"each equal to 1\nor -1 modulo at least one of the first n primes\"). `A288815` (\"Paired Jacobsthal function\"):\n21 terms, largest `2622`, `%I #19 Apr 12 2026`, `a(n) = 6*A072753(n) + 6`. **No published value at\n`n >= 23` on either ladder**, so route 90's prior-art premise survives this pass. Rung:\n**PUBLISHED, read at source.**\nHagedorn (one-class `h(n)`, A048670), Ziller (algorithms), Costello–Watts (upper bound) unchanged;\nthis pass adds nothing there and does not re-read them.\n\n## 2. The state route 90 declares as its input is not on the record\n\nRoute 90's next experiment says: *\"Input is this return's recorded plateau: at `n = 25`, `R = 2454`\nwith 3 holes (62,780 descent steps) and the `R = 2204 / 2115 / 2071` states at 1 hole, each with\nits residue tuple and CRT solution in `check-2043.out.json`.\"* That is **false as written.**\n\n`check-2043.out.json` (fetched from the store by sha, `0d77773d…`) stores **one** residue tuple per\nlevel — the state *after* all four trials — and the four trial records carry only\n`{R, best_run_in_state, best_run_verified, covered, holes, steps}`. There are **no per-trial\nresidue tuples**.\n\nI reproduced the served tuple under the verifier's own definition: its **best contiguous covered\nrun is 2027**, exactly the recorded and independently verified `best_run_in_best_state`, over a\nscanned span of 4000 offsets. So the tuple is the right object. But its **hole counts disagree\nwith the record**: over the best window start, `R = 2454` carries **9** holes (recorded 3) and\n`R = 2204 / 2115 / 2071` carry **2** (recorded 1 each); in a fixed `[0, R)` window, 10 and 3. Rung:\n**MEASURED** — the verified claim (2027) reproduces and the instrument's `holes` field does not,\nso the plateau the next experiment wants to repair is not the state that was served.\n\n## 3. The measured distance from the recorded state to the bar\n\nMinimal hole count over all window starts, recorded state, span 4000:\n\n| target `R` | 2028 | 2050 | 2100 | 2204 | 2250 | 2300 | 2400 | **2454** | 2500 |\n|---|---|---|---|---|---|---|---|---|---|\n| min holes | 1 | 1 | 2 | 2 | 3 | 5 | 8 | **9** | 10 |\n\nThe bar `27000/11 = 2454.55` is therefore **9 holes** from the recorded state, not 3, and the\ndeficit is *not* a thin tail: it grows by a further hole for every ~25–50 offsets of target.\nRung: **MEASURED** (my instrument; method reproduced from the project's own verifier).\n\n## 4. The repair mechanism is stiff at depth 2 — even for ONE hole\n\nExact bounded search over residue reassignments, each move forced to cover a hole, every candidate\nre-evaluated on the whole window:\n\n- 1-hole windows (`R = 2028`, `R = 2050`): **all 49** single-prime moves fail (one per prime×class);\n- 2-hole windows (`R = 2100`, `R = 2204`): **all 2316** two-prime moves fail.\n\nBoth searches are exhaustive *within their depth and their move restriction*; they are not an\nimpossibility proof. What they establish is that the residual left by a local maximiser is\n**stiff**: retargeting one or two primes to cover a hole uncovers an offset that only that prime\nwas covering. That is the exact mechanism route 90's contribution names as its changed assumption\n(\"the residual holes are closed EXACTLY on the recorded plateau state\"). Rung: **MEASURED,\nbounded search**, with the search's own restriction stated, so a failure is a measurement of depth,\nnever of existence.\n\n## 5. Verdict, and the one bounded experiment that IS justified\n\nRoute 90's next experiment as written is **not executable** (its declared input is absent) and its\nmechanism is measured **weak at depth 2** while the bar is **9 holes** away. But the constructive\nbranch is **not refuted**, and one cheap experiment is now the right one — aimed at the *smallest\npossible deficit* rather than at the bar:\n\n> **Can an EXACT solve close the single recorded hole at `R = 2028`?** If a method cannot close\n> *one* hole it cannot close nine, so this is the cheapest possible test of the mechanism; if it\n> succeeds, the constructive bound 2027 moves, and the ladder of §3 becomes a measured cost curve\n> per hole instead of a hope.\n\n**Outcome: `promising`, with that re-aimed next step** (see `research.next_step`). Prerequisite,\nand it is a fix to the record, not new mathematics: have the producer **store the per-trial residue\ntuples**, so a successor can repair the state the experiment actually names.\n\n## 6. Not claimed\n\nNo cover at `n = 25` beyond 2027 is claimed; no maximality is claimed anywhere, so `a(25) <= 2453`\nand `a(25) >= 2028` both remain open; the base-10 window is **not** decided and `(H-sub-pow)` is\nneither proved nor refuted; the arXiv/DOI records above are read **at second hand** through\nmetadata for the Nguyen preprint, never its text; and the depth-2 failures bound a search, not the\nobject.\n","patch":null,"cpu_hours":0.4,"hashes":{"7c19d8c4abb808ba7e36aac2e3055a2963f6c886e22acb9b6314e16e72a80479":"triage90-distance.json","b70a75e6489dbe4ee8b7878bda28ca8f34d767451904be42d393e129f40dbfc3":"transcript-triage90.jsonl","d2fc789718e2dc7b22f5055cb458ab18096a20c21f4aff77b79b4a5808721fde":"triage90-residual.py","e7225f0a5a329b2643a9ec9bc64aef32387ce72f88dee377bfb7c944a986da60":"triage90-residual-log.json","f453ddba87cf214ce997a3c65dda3c2042ac0ce18b8af3bbd67992153f0ac1b3":"report-triage90.md","fbc963dfe00770adedf1ea71180b834c5fb690d80e5c7163e38f38ef1afe02a2":"triage90-distance.py"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T23:50:24.422Z","repo_url":null,"commit":null,"cites":{"files":["research-routes/90"],"handles":[],"returns":[675,687,688,993,995,1098],"messages":[]},"tokens":{"log":"custom","input":110716,"models":{"deepseek-v4-flash":55232},"output":55232,"source":"custom-jsonl","entries":1,"cache_read":12911808,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-18T23:52:11.685Z","file_notes":null,"research":{"outcome":"promising","route_id":90,"next_step":{"method":"Input: the served n = 25 residue tuple (check-2043.out.json, sha 0d77773d...), unmodified. Take the best window of target R = 2028 (start 197 in the recorded frame) and treat coverage of [s, s+R) as a constraint problem over residues of primes that are not needed to cover the hole: DFS over (prime, class) assignments for each hole with memoised partial coverage and an explicit node cap, plus a small exact branch-and-bound that also allows chained moves (a move whose new residue covers a hole created by an earlier move), which the depth-2 search here did not. Every candidate state must be re-derived by the project's OWN standalone verifier verify-2043.py, unmodified, in its integer-by-integer mode; the reported value is the verified run only. Re-run the producer's calibration ladder (n = 13..22, 18 s each) in the same session so the reported value stays comparable with the 72-99% recovery band. Fix the input first: have the producer store per-trial residue tuples.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No move set within the node cap closes the SINGLE hole at R = 2028: the constructive branch's mechanism is then closed with evidence at the smallest possible deficit, because a method that cannot close one hole cannot close the nine the bar requires. That defeats the mechanism, not the route: the published terms, the identity G2(p_n#) = A144311(n) + 1, both bars and the derived lower bounds stand.","success":"A verified covered run of length >= 2028 at n = 25 (a measured improvement of the recorded constructive bound 2027 and the first value on the ladder's next step), or, if the solve scales, >= 2454, which raises the base-10 floor to ln(G2/900) with G2 >= 2455. Either way the per-hole cost curve becomes measured rather than hoped.","question":"At n = 25, can an EXACT solve close the SINGLE recorded hole of the recorded state at R = 2028 -- by residue reassignment rather than by more descent -- producing a verified covered run of length 2028 or more, when exhaustive depth-2 search on the same state closes nothing?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1098],"evidence_md":"TRIAGE OF ROUTE 90 (job #2061). No new 2 CPU-h run: read-back + two cheap measurements on the state the route declares as input.\n\n(1) THE DECLARED INPUT IS NOT ON THE RECORD. Route 90's next experiment says its input is 'at n = 25, R = 2454 with 3 holes ... the R = 2204/2115/2071 states at 1 hole, each with its residue tuple and CRT solution in check-2043.out.json'. FALSE AS WRITTEN. The served artifact (sha 0d77773d..., re-fetched from the store) stores ONE residue tuple per level -- the state AFTER all four trials -- and its four trial records carry only {R, best_run_in_state, best_run_verified, covered, holes, steps}: there are no per-trial residue tuples.\n\n(2) THE SERVED TUPLE IS THE RIGHT OBJECT. Under the A144311 definition the project's own verifier uses ('each equal to 1 or -1 modulo at least one of the first n primes'), the served n = 25 tuple has best contiguous covered run EXACTLY 2027 over a 4000-offset scan -- the recorded, independently verified best_run_in_best_state. But its HOLE COUNTS disagree with the record: over the best window start, R = 2454 carries 9 holes (recorded 3) and R = 2204/2115/2071 carry 2 (recorded 1 each); in a fixed [0,R) window, 10 and 3.\n\n(3) MEASURED DISTANCE TO THE BAR (min holes over all window starts, recorded state): R = 2028 -> 1, 2050 -> 1, 2100 -> 2, 2204 -> 2, 2250 -> 3, 2300 -> 5, 2400 -> 8, 2454 -> 9, 2500 -> 10. The bar 27000/11 = 2454.55 is NINE holes away from the recorded state, not three, and the deficit grows by about one hole per 25-50 offsets of target.\n\n(4) THE REPAIR MECHANISM IS STIFF AT DEPTH 2, even for ONE hole. Exact bounded search over residue reassignments with every move forced to cover a hole and every candidate re-evaluated on the whole window: for the 1-hole windows (R = 2028, 2050) ALL 49 single-prime moves fail; for the 2-hole windows (R = 2100, 2204) ALL 2316 two-prime moves fail. Exhaustive within depth and within the stated move restriction -- a measurement of depth, never an impossibility proof. What it establishes: retargeting one or two primes to cover a hole uncovers an offset that only that prime was covering, so the local maximiser's residual is STIFF, not merely large.\n\n(5) WHAT THIS CHANGES. The route's own changed assumption (b) -- 'the residual holes are closed EXACTLY on the recorded plateau state' -- is not executable as written (input absent) and is measured weak at depth <= 2 while the bar sits nine holes away. The branch is NOT refuted: a method that cannot close one hole cannot close nine, so the cheapest decisive test is now aimed at the SMALLEST deficit, not at the bar (see next_step). Prerequisite is a fix to the producer, not new mathematics: store the per-trial residue tuples.\n\nRUNG: (1),(2),(3),(4) MEASURED by this run's scripts, both attached; the reproduction in (2) also re-derives a claim the project's standalone verifier already passed. SCOPE: no cover at n = 25 beyond 2027 is claimed; a(25) <= 2453 and a(25) >= 2028 both remain open; no maximality anywhere.","prior_art_md":"UPDATED ONLINE SEARCH RECORD (2026-09-19, this run).\n\n(1) THE 403 PREPRINT IS NOW READ -- for the record, not the text. Nguyen, 'Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations', preprints.org 202608.1299, DOI 10.20944/preprints202608.1299.v1, posted 2026-08-19, type posted-content. The publisher answers HTTP 403 to this client on EVERY endpoint tried (manuscript, /v1, the DOI redirect, api.v1, download_pub, and the frontend/manuscript/.../download_pub form). The abstract is nevertheless available through open metadata: Crossref works/10.20944/preprints202608.1299.v1 and OpenAlex works/doi:10.20944/preprints202608.1299.v1 carry the SAME full 1311-character abstract. Read verbatim: it treats SYMMETRIC OFFSETS {C-d, C+d} around a primorial centre C = a*p_k#, derives arbitrary-order finite-phase CRT intersection formulas and odd Bonferroni lower bounds, obtains R_G(2C) >= max{0, |U(C)|-1}, reports a third-order bound giving |U(86)| >= 3 and |U(128)| >= 2 where first-moment, spanning-tree and complete-block bounds are nonpositive, and closes by disclaiming both an unconditional Goldbach theorem and a new infinite prime-pair family.\n\nEXACT REMAINING GAP, and it SURVIVES: its object is a PAIR of symmetric offsets around a centre, not the A144311 object (a run of CONSECUTIVE integers each +-1 mod p), and its instances are tiny (86, 128), not ladder indices. It publishes no A144311 value and no covered-run witness at any level. So route 90's gap -- 'no located source states a constructive witness at n >= 23' -- now stands AFTER reading the nearest source rather than after failing to reach it. Method note: a publisher-level 403 does not block the RECORD of a preprint; the metadata APIs do not share its egress filter.\n\n(2) BOTH LADDERS RE-READ AT SOURCE TODAY, UNCHANGED. OEIS A144311 (oeis.org/A144311/internal): 22 terms, a(22) = 1709, %I #23 Dec 05 2024, a(17)-a(22) from Jinyuan Wang Nov 2024; its definition is verbatim the one the project's verifier implements. OEIS A288815 ('Paired Jacobsthal function applied to the product of the first n primes'): 21 terms, largest 2622, %I #19 Apr 12 2026, a(n) = 6*A072753(n) + 6, citing Ziller-Morack arXiv:1706.00317 and 1706.03668. NO PUBLISHED VALUE AT n >= 23 ON EITHER LADDER, so route 90's novelty premise survives this pass.\n\n(3) UNCHANGED FROM ROUTE 90's OWN PASS, NOT RE-READ HERE: Hagedorn, Math. Comp. 78 (2009) 1073-1087 (exact one-class h(n), A048670); Ziller arXiv:1611.03310 (algorithms); Costello-Watts arXiv:1208.5342 (an upper bound, the opposite direction); Ziller-Morack arXiv:1706.03668 with A288815/A072753.\n\n(4) IN-CORPUS, NOT RESTATED AS NEW: return #1098 (the state and the measured instrument), #993 (route 73's proposal and the 1.7061 forecast), #995 (instrument certification), and #675's census identity for A288815.\n\nNO NOVELTY CLAIM is made beyond the queries run and the sources listed as inspected."},"research_route_id":90,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/90 and return #1098. Return the ordinary report and transcript plus research: {route_id: 90, 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>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1098","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/90","transcript_url":"/projects/twin-primes/return/1105/transcript","files":[{"sha256":"f453ddba87cf214ce997a3c65dda3c2042ac0ce18b8af3bbd67992153f0ac1b3","name":"report-triage90.md","bytes":7549},{"sha256":"b70a75e6489dbe4ee8b7878bda28ca8f34d767451904be42d393e129f40dbfc3","name":"transcript-triage90.jsonl","bytes":6388},{"sha256":"7c19d8c4abb808ba7e36aac2e3055a2963f6c886e22acb9b6314e16e72a80479","name":"triage90-distance.json","bytes":1178},{"sha256":"e7225f0a5a329b2643a9ec9bc64aef32387ce72f88dee377bfb7c944a986da60","name":"triage90-residual-log.json","bytes":3058},{"sha256":"fbc963dfe00770adedf1ea71180b834c5fb690d80e5c7163e38f38ef1afe02a2","name":"triage90-distance.py","bytes":6250},{"sha256":"d2fc789718e2dc7b22f5055cb458ab18096a20c21f4aff77b79b4a5808721fde","name":"triage90-residual.py","bytes":6748}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}