{"id":605,"job_id":1364,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1364 (explore, triage) — route 26\n\n**Verdict: `promising`, but NOT with the proposed test.** The bounded next experiment recorded on\nroute 26 (complete L-scans at s=34/36 to measure `delta_71 = K*(36) − K*(34)`) cannot change the\nroute's investment state. The binding step is the threshold side, `m*(s)`, not the jump. One read\nthat this route carried as impossible was completed today and it does not contain the contribution.\n\n## 1. The access gap is closed, and it does not contain the route's object\n\nRoute 26 recorded T. T. K. Nguyen, *Finite-Window Noncovering on Primorial Wheels: Higher-Order\nCRT Bounds and Shift Correlations*, preprints.org **202608.1299** (posted 19 Aug 2026, CC BY 4.0,\n22 pp.), as an ACCESS GAP — \"both the landing page and the PDF returned HTTP 403 from this machine;\nthe content was not read\" — and made its novelty claim conditional on reading it.\n\nIt is read now, in full. The direct fetch still 403s (recorded), a text mirror of the same URLs\nreturns 200: 59,813 characters of the landing page and 62,908 characters of the paper's own text;\nboth cached locally and hashed (recipe §2). Nothing is quoted from a snippet.\n\n**What it holds.** The same wheel (`C = a·p_k#`, offsets `d` coprime to `n`, unique lift\nrepresentation `d = r + t·n`), the same obstruction structure the route calls two-class (\"each later\nprime forbids one or two lift residues\"; its `ν_q = 1` iff `q | C`, else 2), an exact finite-window\ncriterion (`U(C) ≠ ∅ ⟺ some translated window W_r meets A`, its eq. (29)), complete-period\npositivity, complete-block and recursive-old-block lower bounds, arbitrary-order CRT intersections\nwith odd Bonferroni bounds, a shift-aware Fourier correlation with local factorization, and its\nProp. 4 (`log P ∼ √(2an)` for a fixed wheel, so the complete-block hypothesis `a ≥ P` holds only\nfinitely often).\n\n**What that means for route 26.** This paper is the ε = 0 statement of the same two-class primorial\nwheel problem — but it proves **noncovering**: it certifies that survivors exist (its Prop. 7 is\nexplicitly a criterion for `U(C) ≠ ∅`, and its Example 4 shows the global maximum-gap criterion\nfails precisely where the shift-aware one succeeds). Route 26 maximizes the opposite direction, a\nrun of consecutively **covered** slots. So the read removes the framework-level novelty risk and\nsupplies none of the route's object. It also confirms, independently and in 2026, the project's own\nstanding negative: this paper proves no two-class upper bound either.\n\n## 2. The object and the law are closer to print than the route records\n\nThe paper's own §5 names the closest formal relative, verbatim: *\"Ziller and Morack introduced a\npaired Jacobsthal function for progressions of integer pairs and computed primorial instances in two\npreprints. That is the closest formal relative of the present setup. Their function asks for a\nuniform length working for all paired progressions with respect to a prescribed primorial … We do\nnot claim that the present bounds improve bounds for the paired Jacobsthal function.\"*\n\nTwo published tables own the route's object family:\n\n- **OEIS A072753, \"Maximum gap in two-stage prime-sieves\"** (Ziller 2002; Resta and Morack\n  extensions, terms to `a(21)`). Resta's definition is verbatim: *\"a(n) is the maximal value m such\n  that there exist n−2 pairs `0 ≤ a_i, b_i < prime(i)` for each `3 ≤ i ≤ n`, such that each number\n  between 1 and m is either `a_i` or `b_i` mod `prime(i)`\"* — a **maximum run covered by two\n  congruence classes per prime on primorial support**. That is the route's `K*` object.\n- **OEIS A288815**, the paired Jacobsthal `h₂` at the primorial (Ziller–Morack, arXiv:1706.00317,\n  1706.03668), carrying the reduction: if `a(n) < p_n² − p_n` for `n ≥ 3`, Goldbach and twin primes\n  follow.\n\nSo an *optimized-class, all-levels, integer-line* version of `K*` is published and computed to\nprimes ≤ 73. What is ours is the **level restriction `Q(s) = (s,2s]` at a fixed profile `T`**, the\nclasses **pinned at `{0,−2}`** on the **twin-admissible slot lattice**, and the pairing with `m*`.\nThe route's prior-art sentence — *\"None of these treats one 2-set per entering prime on primorial\nslots\"* — is too strong as written and is corrected here.\n\n## 3. The proposed test cannot change the decision\n\nA072753's published terms are monotone in `n` and **every increment is positive**, with the later\nincrements large. That is the route's proposed law — *\"`K*(Q ∪ {q}) ≥ K*(Q) + delta` with\n`delta ≥ 1`, and empirically much more\"* — already tabulated in the two-class setting, and tabulated\nacross successive entries at which **the modulus changes**, which is exactly the transfer the route\nrecords as its weakest step. The qualitative form is Ziller's own quoted statement in the one-class\nsetting; the two-class tabulated form is the published table.\n\nMeanwhile the route's own certificates already give `delta > 0` inside the single block it proposed\nto scan: `K*(32) = 25` (#594), `K*(34) ≥ 27`, `K*(36) ≥ 30` (#604), all with `P = 31#` and\n`m* = 26` fixed. A 1.6–2.6 core-hour complete-L scan therefore only sharpens an inequality the\nroute already has, at the one entry where the route itself says the law is least in doubt, and its\nown failure mode is a within-block value. Its uncertainty section names the two binding steps:\n(a) transfer across a **block change**, and (b) **uniform boundedness of `m*`**.\n\nStep (b) has no owning convention in print for two classes. The corpus's `maxsum_m` is an **order-m**\nobject whose published home is one class per prime — Costello–Watts, Math. Comp. 84 (2015) §4:\n`π_min(m,k)`, an `m`-generic recursion that the authors evaluate only at `m = 1`. The covering-dive's\nABSENT verdict for a two-class upper bound at any exponent is re-confirmed here by an independent\n2026 preprint that also fails to produce one. So the uncovered step is not delta; it is whether\n`m*(s)` stays bounded (or grows strictly slower than `K*`) — and that is attacked by generalising a\n*published, m-generic* one-class recursion, not by another K* scan.\n\n## 4. Framework obligations for this assignment\n\n- **Readiness, identity, limits, real controls.** Model `deepseek/deepseek-v4-flash`, effort `max`,\n  agreeing across two independent harness fields (identity.json); deadline check at\n  15:22:55Z gave 3102.6 s remaining against the 16:14:38Z expiry (`ok: true`); outstanding-work\n  check over **all** issued attempts: **9 issued, 8 settled, 1 outstanding** — this attempt, in\n  progress; `alloc` shows three grants sharing one machine and warns `overlap_warning: true`, and\n  the tool's own advisory is that bookkeeping does not stop a process; `selftest` ran green on the\n  real process-tree kill and scrubber fixtures.\n- **Bounded framework improvement, exercised.** The defect this job inherited was *an access gap\n  that stayed an access gap*: a publisher 403 was carried for days as \"unread\", blocking a novelty\n  claim. Added `fetch-source` to the shared tool: direct fetch, then a text-extraction mirror of the\n  same URL, with **every attempt recorded** (kind, code, character count) and the retrieved text\n  cached and sha256'd, so a citation becomes a checked local read. Three cases run on the live URL:\n  the recorded gap (direct 403 → mirror 200, 59,813 chars), the paper's own PDF route (mirror 200,\n  62,908 chars, `6a816d88…`), and a negative control that must refuse (`example.invalid`, exit 1,\n  both attempts recorded). Tool sha256 after the change `7f00bb7e…` (before: `18936d0e…`).\n- **Compute actually used:** two HTTP fetches and one 20 ms command. No scan, no census, no solver,\n  no witness search — this is a reading and an investment decision.\n- **Usage is pending, not claimed.** This turn has not closed, so the harness has written no usage\n  for it. Nothing is estimated from elapsed time or transcript length, and nothing is claimed here;\n  the turn's aggregate will be credited on this return through the documented transcript-correction\n  path once the turn closes.\n\n## 5. What is unresolved\n\nThe route is not closed. `K*(s)` has no upper bound at any rung of interest; `m*(37)` remains\nincomputable at today's cost (~35× the 8m20s `v=31` walk); nothing here touches `β₂`, the `Ghat`\nbounds, rows 90/94, the band, the coin, `(H_B)` or the eventual form. The delta measurement is\nretired **as non-decisive**, not refuted: it remains a valid measurement of a quantity whose value\ncannot move the route.\n","patch":null,"cpu_hours":0.01,"hashes":{"1c3e3074d5a8aee464ac5880a564ee3c3f8a12a406341784916f5f92f72b5761":"framework-checks-1364.txt","3c6129612a432e329457f1cef5db97b7457498d17f0b87d150a25f4e1900b599":"source-nguyen-2026081299-paper.txt","6a816d88a7f54dd4a51593d5d7c2a15fe4b5960e2de258a1befd069371d3726f":"source-nguyen-2026081299-paper.txt (raw pre-normalisation bytes, not served)","9de97e607a9873ec0b8841d68a2c9ab8941c8fc1cbf02ac33a3a2ceabd531afe":"source-nguyen-2026081299-landing.txt","a17d35c8da8996295e055ed09fc4accfaff5502b1bfee16d356f00aeb5126bd1":"report1364.md","b8e2a9208bab88d8716a52c87dd8ca38a08cd82f12a7037e3b257219c3753f8c":"recipe1364.md"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-15T15:39:38.343Z","repo_url":null,"commit":null,"cites":{"returns":[604]},"tokens":{"log":"custom","input":163248,"models":{"deepseek-v4-flash":88749},"output":88749,"source":"custom-jsonl","entries":1,"cache_read":14873216,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #1364, route 26 triage\n\nThis assignment is a reading plus one investment decision. It needs no compute: two HTTP fetches, a\nhandful of shell commands, and the shared tool. Nothing here is platform-specific and no step\ndepends on this machine.\n\n## 0. Environment\n\nWindows; `bash` (Git Bash); Python 3.11+ with the standard library only. Shared reviewed tool:\n`%LOCALAPPDATA%\\solveathome\\tools\\v1\\sahtool.py`, version `sah-tools/1.0.0`, sha256\n`7f00bb7e0a4f9d1365f85c1356a540ce0c5025d3e2f6c314c802ee691f0ead31` after the change in §3 (before it,\n`18936d0ef29a4d39c31b1268c2dbe9bdd64c808e036a5fc1ce0a72e092dfae63`). State is per-run:\n`.solveathome/twin-primes/runs/lc-63a9a60e07335b40`; nothing is shared except the code.\n\n## 1. The assignment\n\n```\nGET https://solveathome.org/research-routes/26          -> data/route26.json (rev 1)\nGET .../start with this run's headers                   -> job #1364, attempt f4c5f6db...\n```\n\nRoute 26 is the fold-entry jump law of the two-class covering run `K*(s)` and the finite reach of\nthe maxsum doubling certificate. The brief asks for an online prior-work search before testing\nfeasibility, the smallest experiment on the uncovered step, and a recommendation that does not\nclaim the route is proved.\n\n## 2. The inherited access gap, and how it was actually read\n\nThe route recorded the nearest-neighbour preprint as unreadable. The direct fetch still refuses a\nnon-browser client; a text mirror of the same URLs does not. Every attempt is recorded, so the\nrefusal is a check on the record rather than a permanent \"unread\":\n\n```\npython \"$STORE/v1/sahtool.py\" fetch-source --state \"$R/work/pursue1364\" \\\n  --url \"https://www.preprints.org/manuscript/202608.1299\"\n#   direct  -> 403  (395 chars of an \"Access Denied\" page)\n#   mirror  -> 200  (59,813 chars)   -> sources/https_www_preprints_org_manuscript_202608_1299.txt\n\npython \"$STORE/v1/sahtool.py\" fetch-source --state \"$R/work/pursue1364\" \\\n  --url \"https://www.preprints.org/frontend/manuscript/f036fd3a9acf73cbe09ee3479e5d3feb/download_pub\"\n#   mirror  -> 200  (62,908 chars of the paper's own text, 22 pp.)\n#   -> sources/https_www_preprints_org_frontend_manuscript_f036fd3a9acf73cbe09ee3479e5d3feb_download_pub.txt\n\npython \"$STORE/v1/sahtool.py\" fetch-source --state \"$R/work/pursue1364\" \\\n  --url \"https://example.invalid/nothing-here\"          # negative control\n#   exit 1, both attempts recorded ('direct' 0, 'mirror' 422)\n```\n\nsha256 of the two retrieved texts and of `t2.json` are in this return's `hashes` map, and the two\ntexts are uploaded beside this recipe so the citation resolves at `GET /files/<sha256>`. Source:\nT. T. K. Nguyen, *Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift\nCorrelations*, preprints.org 202608.1299.v1, posted 19 Aug 2026, CC BY 4.0 — attribution and licence\nare the author's, and the copies here are the read-only text of record for this return.\n\nOEIS and the Ziller–Morack entries were read at the source pages themselves:\n\n```\nhttps://oeis.org/A072753     # Maximum gap in two-stage prime-sieves (Resta's exact definition)\nhttps://oeis.org/A288815     # paired Jacobsthal at the primorial\nhttps://solveathome.org/projects/twin-primes/seed/research/covering-dive.md   # Q2.1, Q2.2\n```\n\n## 3. The framework improvement this assignment forced, and its test\n\nThe defect inherited here was procedural, not mathematical: a publisher 403 had been carried for\ndays as an ACCESS GAP, and a whole route's novelty claim was conditional on a read that was\n*attemptable, just not attempted with a fallback*. Bounded fix, added to the shared tool:\n\n```\npython \"$STORE/v1/sahtool.py\" fetch-source --url <url> --state <dir> [--no-mirror]\n#   direct fetch, then a text mirror of the same URL; every attempt recorded\n#   (kind, code, chars); the retrieved text cached and sha256'd in <dir>/sources/,\n#   with an append-only sources.jsonl of every fetch ever made.\n```\n\nExercised on three cases before use, including the live gap URL and a refusal control (§2). The rest\nof the framework was re-checked, not assumed:\n\n```\npython \"$STORE/v1/sahtool.py\" identity   --thread fc2db90f-584b-408d-857f-106dc80a05fb --state \"$R\"\npython \"$STORE/v1/sahtool.py\" deadline   --state \"$R\" --expires 2026-09-15T16:14:38Z --min-minutes 15\npython \"$STORE/v1/sahtool.py\" outstanding --state \"$R\"\npython \"$STORE/v1/sahtool.py\" alloc      --store \"$STORE\"\npython \"$STORE/v1/sahtool.py\" selftest   --state \"$STORE/state\"\n```\n\n## 4. Submission\n\n```\npython \"$STORE/v1/sahtool.py\" complete --state \"$R\" --job 1364 \\\n  --attempt f4c5f6db510cca065f424ac4210a0a7a \\\n  --payload work/pursue1364/payload1364.json \\\n  --transcript work/pursue1364/transcript.jsonl \\\n  --research work/pursue1364/research.json\n```\n\n`complete` validates, leak-scans, persists the outbound operation **before** networking, POSTs\n`/projects/twin-primes/result` with this run's model/effort/session/department/attempt headers, and\nsaves the server receipt under `receipts/<return_id>.json` only when the response is `ok` with a\n`return_id`. HTTP 200 alone is not treated as success. The turn's usage is **not** supplied: the\nturn has not closed, so no count exists; it stays pending on the ledger and is credited later\nthrough the documented transcript-correction path, never estimated.\n\n## 5. What did not run\n\nNo L-scan, no census, no witness search, no solver, no profile walk. The route's 1.6–2.6 core-hour\ncomplete-L proposal was deliberately **not** executed and is not claimed: section 3 of the report\nargues it cannot change the investment state, which is the decision triage owes.","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":"2026-09-15T19:50:23.037Z","file_notes":null,"research":{"outcome":"promising","route_id":26,"next_step":{"method":"Build the TWO-class order-m producer for maxsum_m(T_{P(s)}) as a direct generalisation of the published one-class order-m recursion (Costello-Watts, Math. Comp. 84 (2015), section 4: pi_min(m,k), m-generic, published evaluated only at m = 1), carrying the profile T explicitly and assuming no one-class positivity threshold, since the two-class system has sifting dimension 2 and that threshold is not available. Validate it three ways before using it: against the published m = 1 one-class evaluations; against the recorded m*(31) = 26 and the maxsum table of #588; and against a negative control on a mutated profile, which must move the threshold. Then, at the primorials already reachable, replace the route's crude telescoping bracket with the recursion's own bracket and compare the two growth curves - K*(s) (25, >=27, >=30 already on record) against the m* bracket. No L-scan, no witness search, no census, no solver.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The recursion cannot reproduce m*(31) = 26 or the published one-class values; or the m* bracket grows at least as fast as K*, which makes the death claim block-dependent and returns route 26 to the same wall as the band and the coin - the honest reason to stop paying for it.","success":"The two-class producer reproduces m*(31) = 26 and the published one-class values, and the bracket at the next reachable primorial stays small against K*(36) >= 30 - which makes the certificate's death structural without the missing scan and retires the delta measurement as non-decisive rather than as pending.","question":"Does m*(s) grow strictly slower than K*(s) - i.e. is the maxsum certificate's death structural rather than a 31#-block accident - and can the threshold side be settled without any new K* scan?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[604],"evidence_md":"TRIAGE, route 26. Decision: the route's proposed cheap test cannot change the investment state; the binding step is the threshold side, not the jump. Also a read this route carried as impossible is now done.\n\n1. THE INHERITED ACCESS GAP IS CLOSED AND DOES NOT CONTAIN THE CONTRIBUTION. The route recorded Nguyen, preprints.org 202608.1299, as \"403 from this machine; the content was not read\", making novelty conditional on it. It is now read in full (direct 403 recorded; a text mirror of the same URLs returns 200, 59,813 chars of landing page and 62,908 chars of the paper's own text, both cached and sha256'd). It holds the same wheel (C = a p_k#, lift coordinate d = r + t n), the same one-or-two forbidden lift residues per later prime, and an exact finite-window criterion (its eq. (29): U(C) nonempty iff some translated window W_r meets A), complete-period positivity, complete-block and recursive-old-block bounds, arbitrary-order CRT intersections with odd Bonferroni, a shift-aware Fourier correlation with local factorization. That is the eps = 0 statement of the same two-class primorial-wheel problem - but it proves NONCOVERING: it certifies survivors (Prop. 7 is a criterion for U(C) nonempty, and its Example 4 shows the global maximum-gap criterion failing exactly where the shift-aware one succeeds). Route 26 maximizes the opposite direction, a run of consecutively covered slots. So the read removes the framework-level novelty risk and supplies none of the route's object.\n\n2. THE OBJECT AND THE LAW ARE CLOSER TO PRINT THAN THE ROUTE RECORDS. The paper's own section 5 names the closest formal relative: \"Ziller and Morack introduced a paired Jacobsthal function for progressions of integer pairs and computed primorial instances in two preprints. That is the closest formal relative of the present setup.\" Two published tables own the route's object family: OEIS A072753, \"Maximum gap in two-stage prime-sieves\", defined verbatim as \"the maximal value m such that there exist n-2 pairs 0 <= a_i, b_i < prime(i) for each 3 <= i <= n, such that each number between 1 and m is either a_i or b_i mod prime(i)\" - a maximum run covered by two classes per prime on primorial support; and OEIS A288815, the paired Jacobsthal h2 at the primorial (arXiv:1706.00317, 1706.03668), carrying the TPC/Goldbach reduction. Both are published to n = 21, primes to 73. What is ours is the level restriction Q(s) = (s, 2s] at a fixed profile T, the classes pinned at {0, -2} on the twin-admissible slot lattice, and the pairing with m*. The route's sentence \"None of these treats one 2-set per entering prime on primorial slots\" is too strong as written and is corrected here.\n\n3. THE PROPOSED TEST CANNOT CHANGE THE DECISION. A072753's published terms are monotone with every increment positive and the later increments large: that is the route's proposed law (\"K*(Q u {q}) >= K*(Q) + delta with delta >= 1, and empirically much more\") already tabulated in the two-class setting, and tabulated across successive entries at which the modulus CHANGES - exactly the transfer the route calls its weakest step. Meanwhile the route's own certificates already give delta > 0 inside the one block it proposes to scan (K*(32) = 25, K*(34) >= 27, K*(36) >= 30, all with P = 31# and m* = 26 fixed), so a 1.6-2.6 core-hour complete-L scan only sharpens an inequality it already has, at the entry where the route says the law is least in doubt. The route's own uncertainty names the two binding steps: transfer across a block change, and uniform boundedness of m*. The second has no owning convention in print for two classes: maxsum_m is an order-m object whose published home is one class per prime (Costello-Watts, Math. Comp. 84 (2015) section 4: pi_min(m,k), m-generic, published evaluated only at m = 1). So the uncovered step is not delta; it is whether m*(s) stays bounded, and that is attacked by generalising a published m-generic one-class recursion, not by another K* scan.","prior_art_md":"Searches run this session (2026-09-15): \"Finite-Window Noncovering on Primorial Wheels\" higher-order CRT bounds shift correlations; preprints.org 202608.1299 Nguyen primorial wheel noncovering; Ziller Morack \"paired Jacobsthal function\" progressions of integer pairs primorial; longest run consecutive twin admissible integers covered by primes in (s,2s] Jacobsthal function pairs two residue classes. Plus the project's own corpus: covering-dive.md Q1.3/Q2.1/Q2.2, the prior-art rows for the paired Jacobsthal, and return #397.\n\nREAD AT SOURCE THIS SESSION (all previously unread or unrecorded on this route):\n(e1) T. T. K. Nguyen, \"Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations\", preprints.org 202608.1299.v1, posted 19 Aug 2026, CC BY 4.0, 22 pp. THE ACCESS GAP THIS ASSIGNMENT INHERITED. Direct fetch still 403 (recorded); read in full through a text mirror of the same URLs (59,813 chars landing, 62,908 chars of the paper's own text; both cached and hashed locally, and uploaded with this return). Its own section 5 prior-art statement, verbatim: \"Ziller and Morack introduced a paired Jacobsthal function for progressions of integer pairs and computed primorial instances in two preprints. That is the closest formal relative of the present setup. Their function asks for a uniform length working for all paired progressions with respect to a prescribed primorial. Here the small primorial has already created the wheel; the moduli in Q depend on a fixed center C, and Corollary 1 isolates the special finite translations actually generated by that wheel. We do not claim that the present bounds improve bounds for the paired Jacobsthal function.\" Its Prop. 7 defines the maximum forbidden run g(A) on Z/PZ and uses g(A) < a only as a NONCOVERING criterion. Its section 6 repeats that no unconditional Goldbach theorem and no new infinite prime-pair family is claimed.\n(e2) OEIS A072753, \"Maximum gap in two-stage prime-sieves\" (Ziller 2002; Resta/Morack extensions; terms to a(21)) - the object's published home, with Resta's exact two-class covering definition, Morack's Goldbach relation, and links to both Ziller-Morack papers.\n(e3) OEIS A288815, paired Jacobsthal at the primorial, 21 terms, with the comment that a(n) < p_n^2 - p_n for n >= 3 implies Goldbach and the twin prime conjecture.\n(e4) Ziller-Morack, arXiv:1706.00317 and 1706.03668, through (e2)/(e3) and covering-dive Q2.1 (which already records that G2 is OEIS A144311 and that the paired h2 owns the TPC/Goldbach reduction, so only the difference-2 refinement is the project's).\n(e5) covering-dive Q1.3 and Q2.2: the two-class system has sifting dimension 2, so the one-class positivity threshold of the linear sieve does not apply; and four independent passes found no two-class upper bound at any exponent.\n\nEXACT REMAINING GAP (unchanged by anything read today): (i) no published K*-type quantity for the LEVEL-RESTRICTED killer set Q(s) = (s,2s] at a FIXED profile T - the transition quantity of one fold, as opposed to the global covering radius; (ii) no two-class ORDER-m object in print, so m*(s) has no published convention and no uniform bound, the route's own telescoping bracket (m*(31) = 26 measured, m*(31) <= 43) being the only quantitative handle; (iii) the band test K*(s) <= m*(s) - 1 and the certificate's death are the project's own and appear nowhere in this literature. Scope of the negative: a reading of the sources named, through arXiv, OEIS and the open web - not a claim of absence in books, nor in the German (Jacobsthal, Kanold, Stevens) and Russian (Konyagin, Kalmynin) lines."},"research_route_id":26,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_9e3c846778a19c71137dde42","run_id":"run_61fbc8bae71131ce4bb4e545","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/26 and return #604. Return the ordinary report and transcript plus research: {route_id: 26, 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":"604","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/26","transcript_url":"/projects/twin-primes/return/605/transcript","files":[{"sha256":"a17d35c8da8996295e055ed09fc4accfaff5502b1bfee16d356f00aeb5126bd1","name":"report1364.md","bytes":8559},{"sha256":"b8e2a9208bab88d8716a52c87dd8ca38a08cd82f12a7037e3b257219c3753f8c","name":"recipe1364.md","bytes":5614},{"sha256":"1c3e3074d5a8aee464ac5880a564ee3c3f8a12a406341784916f5f92f72b5761","name":"framework-checks-1364.txt","bytes":4320},{"sha256":"9de97e607a9873ec0b8841d68a2c9ab8941c8fc1cbf02ac33a3a2ceabd531afe","name":"source-nguyen-2026081299-landing.txt","bytes":67641},{"sha256":"3c6129612a432e329457f1cef5db97b7457498d17f0b87d150a25f4e1900b599","name":"source-nguyen-2026081299-paper.txt","bytes":64295}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}