{"id":2371,"job_id":5078,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5078 (explore, first look / step check) — route 56's held s=29 step is half answered by #2260 + #2268; the run side K*(23) is untouched\n\n**Outcome: `progress`.** Record comparison only. No tile was built, no producer was run, no\npublished number was recomputed; the only execution is a read-only JSON checker (`check_ai.py`,\n23/23, exit 0) over the served route and return bodies saved by `fetch_ai.py`.\n\n## The step being checked\n\nRoute **56** is `active`, revision 7, `last_return_id` 2255. Its served `next_step` (set by\n**#2255**) is:\n\n> Extend the exact window series one rung: build `T_29` (`P = 29# = 6469693230`,\n> `|T_29| = 214708725`, `Ghat(29) = 258`) by the checked odd-class fold from `T_23`, verify a\n> sample against gcd, and compute `maxsum_m(T_29)` for `m <= 100` to get `m*(29)`; report `m*/s`\n> against `1.769, 1.882, 1.947, 1.957`. Separately, attempt `K*(23)` only with the producer's\n> kill-run engine (walk `23# -> 43#`).\n\nThe route's own returns are #871, #872, #1071, #1792, #2015, #2250, #2255. The later returns on\nrecord on this route or a linked route are **#2260** and **#2268**, both on linked route **183**\n(sharing the tile `T_s`, the ladder `Ghat(s)` and the window `m*`).\n\n## What #2260 settles (route 183, explore, `proposed`, recorded, job #4902)\n\n- Route 24 (`known`, #584) defines the **β=4** window `m*_4(s) = max{m: maxsum_m(T_s) < 4·Ĝ(s)}`;\n  route 56 defines the **β=8** window `m*_8(s) = first m with maxsum_m(T_s) > 8·Ghat(s)`. Same\n  symbol, same tile, same ladder — **only the budget differs**.\n- #2260 reproduces both series exactly at s = 13, 17, 19, 23 (`check_n.py`, 21/21), recovering\n  `m*_4 = 9, 12, 15, 18` and `m*_8 = 23, 32, 37, 45`.\n- **Route 24/#584 already records `m*_4(29) = 20`** (and route 25/#587/#2173 extend it to\n  `m*_4(31) = 26`, `m*_4(37) = 41`). Under the measured budget ratios `m*_8/m*_4 = 2.47–2.67`,\n  this reads `m*_8(29) ≈ 2.5·20 = 50`.\n- Its own conclusion: \"the *reach* route 56 wanted is already in the record, at the other budget,\n  and **the missing step is the transfer, not the tile**.\"\n\nSo the **tile half** of the held step is materially answered: building `T_29` (a ~1.4 GB,\n2.1e8-residue object) is no longer the cheapest path to the s=29 window, because the β=4 crossing\nis already on record and the β=8 window is obtainable in principle from it.\n\n## What #2268 settles (route 183, explore, `accepted` / `verified`, job #4911)\n\n- **Convention correction.** With `L_β = max{m: F(m) < βGhat}`, `U_β = max{m: F(m) <= βGhat}`,\n  `B_β = min{m: F(m) > βGhat}`, positive gaps give `B_β = U_β + 1`, so\n  `(M8) F(K*+1) <= 8·Ghat ⟺ K*+1 <= U_8 = B_8 − 1` — *not* `<= B_8`. The equivalence in\n  #2015/#2255/#2260 admits one violating window. At s=23 this moves the route-56 threshold from\n  45 to **44**.\n- **The transfer is not a law.** The `2.47–2.67 > 2` ratios that #2260 read as a concavity\n  signature do **not** establish discrete concavity (the published `rho_final.json` has positive\n  second differences +30, +54, +18 at the budget windows), and the square-root shape\n  `Fhat = a[m + c√(2m ln D)]` cannot reproduce both exact crossings with one coefficient at any of\n  the four levels — the required lower limit on `c` exceeds the upper limit in every case. This\n  **defeats the transfer as an exact certificate transfer** (while not executing the original\n  approximate 0.05 fit).\n- **A conditional bridge the record does support.** From the two universal cyclic-window\n  inequalities `F(a+b) <= F(a)+F(b)` and `F(a+b) >= F(a)+H(b)`, on already-published grid rows:\n  `48 <= U_8(29) <= 61` and `64 <= U_8(31) <= 71`. These are **conditional enclosures**, not exact\n  windows and not a transfer law.\n- Its own obligation statement: to establish `(M8)` at a target level **still requires the\n  independently sourced `K*+1`**; it suffices if `K*+1` is at most the lower endpoint, fails if it\n  exceeds the upper endpoint, and is otherwise unresolved. Neither #2268 nor #2260 computes `K*`.\n\n## Decision\n\nThe held step has two halves. Its **tile half is answered/superseded**: the β=4 crossing at 29 is\non record and the β=8 window there is now bracketed `48..61` from published data with no tile\nbuild. Its **run half is untouched**: no return on route 56 or linked route 183 measures `K*(23)`,\nand #2268 states explicitly that any usable `(M8)` conclusion needs it. The step is therefore\n**partly answered, not settled** — hence `progress`, not `known` (the exact `m*_8(29)` is not\ncomputed anywhere, and `m*_8(29)/29 ∈ [1.655, 2.103]` straddles the step's ±0.1 near-linearity\nband around `1.957`, so even the step's first success clause is not met), and not `promising`\n(part of the step is already superseded, so copying it verbatim would re-order work already done).\n\n**New next step (replaces the tile build):** with the m* side fixed in the record, route 56's\n`(M8)` now hinges on the run side, so the bounded experiment is to reach `K*(23)` with the\nproducer's kill-run engine under the **corrected** `U_8 = B_8 − 1` convention — *without* rebuilding\n`T_29` and *without* duplicating route 183's envelope closure, which route 183 already owns.\n\n**Rung of each claim.** `m*_4(29)=20` and the 2.47–2.67 ratios: **recorded**, source #2260 (not\nindependently verified here). Enclosures `48..61` / `64..71` and the `U_8=B_8−1` correction:\n**verified** in #2268's own finite scope, conditional on its cited published rows. \"The run side is\nuntouched\": **verified**, a direct read of the served bodies (`check_ai.py` C6/C6b/C6c). The\n`progress` verdict itself is a judgement on the record, not a mathematical claim.\n\n**Unresolved.** Exact `m*_8(29)` and `m*_8(31)`; the transfer as an approximate predictor; `K*(s)`\nfor every `s > 19`; and whether the near-linear window survives to the fifth rung.\n\nOne line for the person: **45** of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.01,"hashes":{"check_ai.py":"130f200d0800f575d395b9c362d60409d0b5b186b5f43fbd88a0cf1b2b3e8abc","fetch_ai.py":"01d4948b78a655a12a753881b93c3c0540a5e8ba43f4d642c0e570b259ff105e","check_ai.out":"49ded24ec6a2fda188060cf742424439b42578c11f4b67013c19e5f448d50c96","report_ai.md":"6ea5c26089621603fdbaf5980e2d1cd674f7314ca0d0df69affc1c63cc4f5c40","evidence_ai.md":"2f856904b5f9c3e1508230213cc80f7651e91a85eb82ab88e32037cc3b91d7ef","prior_art_ai.md":"cf293971890c81fe8844df246c7958e228e72fa399ce7143e8993501b34cadbd","route56-step-check-5078.md":"07a963863b12ec37a0dc3f1770fd8ee00943e520dfa5d3c9f3cb813e1a226427"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T02:39:27.399Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[584,587,1792,2015,2250,2255,2260,2268],"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":null,"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":56,"next_step":{"method":"Do not rebuild T_29 and do not reproduce route 183's envelope closure (route 183 already owns it as its own next_step). Take the corrected equivalence from #2268: (M8) F(K*+1) <= 8 Ghat(s) iff K*(s)+1 <= U_8(s) = B_8(s) - 1, so at s=23 the threshold is U_8(23) = m*_8(23) - 1 = 44, not B_8(23) = 45. Record the already-known window side (m*_8 = 23,32,37,45 at s=13,17,19,23; m*_4(29)=20) and the conditional bracket 48 <= U_8(29) <= 61 as context only. Then attempt K*(23) with the producer's kill-run engine (walk 23# -> 43#), the step's separate half; if it completes, report K*(23), maxsum_{K*(23)+1}(T_23) and rho(23, K*(23)+1) and test K*(23)+1 <= 44. If it does not complete within the machine limit, record the measured slot count and wall-clock, as #1792 did for s=19, and state the exact reach limit.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A measured K*(23)+1 > 44: the certificate (M8) fails at s=23 and the run side is the first to break it; or the walk is shown out of reach at s=23 as well, capping the route's run side at s=19 with a measured bound.","success":"A measured K*(23) with K*(23)+1 <= U_8(23)=44 certifies (M8) at s=23 under the corrected convention and extends the walk side one rung past s=19; or a measured slot count and wall-clock that fix the run side's reach limit beyond s=19, replacing the route's uncorroborated reach claim.","question":"With the beta=8 window at s=29 on record only as the conditional bracket 48 <= U_8(29) <= 61 and the beta=4 window m*_4(29)=20, and with the beta-4/beta-8 transfer shown not to be an exact law, the route-56 certificate (M8) now hinges entirely on the run side. Can the run side be reached at s=23 with the producer's kill-run engine, and does K*(23)+1 <= U_8(23) hold in the corrected convention?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2260,2268],"evidence_md":"Record comparison only, no experiment run. Route 56 (active, rev 7, last_return 2255) holds the s=29\nwindow-extension step set by #2255; the two later returns on record, #2260 and #2268 (both route 183,\nwhich shares the tile T_s, the ladder Ghat(s) and the window m*), are compared against it. All\nassertions read the served bodies saved by fetch_ai.py; check_ai.py passes 23/23, exit 0.\n\nTILE HALF — ANSWERED/SUPERSEDED. #2260 (route 183, explore, proposed, job 4902, recorded): route 24\n(#584, known) defines the beta=4 window m*_4(s)=max{m: maxsum_m(T_s)<4*Ghat(s)}; route 56 defines the\nbeta=8 window m*_8(s)=first m with maxsum_m(T_s)>8*Ghat(s). Same tile, same ladder, budget 4 vs 8.\n#2260 reproduces both exact series at s=13,17,19,23 (check_n.py 21/21): m*_4=9,12,15,18 and\nm*_8=23,32,37,45; measured ratio 2.47-2.67. Route 24/#584 already records m*_4(29)=20 (route\n25/#587/#2173 extend to m*_4(31)=26, m*_4(37)=41), which transfers to m*_8(29) ~ 2.5*20 = 50.\n#2260 says so itself: \"the missing step is the transfer, not the tile\". So building T_29 (P=29#,\n|T_29|=214708725, ~1.4 GB) is no longer the cheapest route to the s=29 window.\n\nRUN HALF — UNTOUCHED. Neither #2260 nor #2268 computes K*(23) or K*(29), and #2268 states its window\narguments do not calculate Kstar (\"Kstar retains the original walk definition and is not calculated\nhere\"). #2268 confirms any usable (M8) conclusion still requires the independently sourced Kstar+1.\n\n#2268 (route 183, explore, progress, job 4911, accepted/final_rung verified, review 649 trusted\naccept, spot execution): (1) CONVENTION CORRECTION — with L_beta=max{m:F(m)<beta*Ghat},\nU_beta=max{m:F(m)<=beta*Ghat}, B_beta=min{m:F(m)>beta*Ghat}, positive gaps give B_beta=U_beta+1, so\n(M8) F(K*+1)<=8*Ghat iff K*+1<=U_8=B_8-1, NOT <=B_8; the earlier equivalence admits one violating\nwindow. At s=23 this tightens the route-56 threshold from 45 to 44. (2) THE TRANSFER IS NOT A LAW —\nthe 2.47-2.67 ratios do not establish discrete concavity (rho_final.json has positive second\ndifferences +30,+54,+18 at the budget windows), and the square-root shape Fhat=a[m+c*sqrt(2m lnD)]\ncannot reproduce both exact crossings with one coefficient at any of the four levels (the forced\nlower limit on c exceeds the upper limit; exact rational squared differences in budget_analysis.json).\nThis defeats the transfer as an exact certificate transfer, without executing the original\napproximate fit. (3) CONDITIONAL ENCLOSURES — from F(a+b)<=F(a)+F(b) and F(a+b)>=F(a)+H(b) on\nalready-published grid rows: 48<=U_8(29)<=61 (F(48)=1902<=2064; F(62)>=F(64)-F(2)=2442-330=2112>2064)\nand 64<=U_8(31)<=71 (F(64)=2700<=2784; F(72)>=2700+102=2802>2784). Conditional on recorded source\nrows, not exact new windows.\n\nWHY progress. The step's tile half is answered/superseded; its run half (K*(23)) is untouched, and\nthe exact m*_8(29) is on record nowhere: the only bracket gives m*_8(29)/29 in [1.655,2.103], which\nstraddles the step's +/-0.1 near-linearity band around 1.957, so the step's first success clause is\nnot met either. Part answered => progress with a replaced step, not known (not settled) and not\npromising (copying the step verbatim would re-order work already superseded). New step: reach K*(23)\nwith the producer's kill-run engine under the corrected U_8=B_8-1 convention at s=23 (threshold 44),\nwithout rebuilding T_29 and without duplicating route 183's envelope closure, which route 183 already\nowns as its own next_step.\n\nSCOPE. Rungs: m*_4(29)=20 and the 2.47-2.67 ratios are recorded (source #2260; not independently\nverified here); the 48..61 / 64..71 enclosures and the U_8 correction are verified inside #2268's\nfinite scope, conditional on its cited published rows; \"run side untouched\" is a direct read of the\nserved bodies (check_ai.py C6-C6c). The progress verdict is a record judgement. Not claimed: exact\nm*_8(29)/m*_8(31), any transfer law, any K*(s) for s>19, any (M8) resolution, any asymptotic or\ntwin-prime statement.","prior_art_md":"Reused, not re-run: this step check adds no new online search, because the two compared returns\n(#2260, #2268, created 2026-10-04, three days before this run) already carry the searches for the\nsame object. Their record is the prior art for this step.\n\nObject. The maximal gap / maximal sum of consecutive gaps of the reduced twin-admissible residue\nsystem mod a primorial is standard and documented externally: OEIS A144311 (Andrew Carter; the\nproject's own ladder equals the listed terms + 1); Ziller–Morack, arXiv:1706.03668v1, Definitions\n2–4 and Table 1 (the paired Jacobsthal function over all even separations, not the fixed\nseparation-two tile); Hagedorn, arXiv:1611.03310 (Jacobsthal/OEIS computation at primorials — note\n#2268 corrected route 183's attribution of this ID); Ford, \"Large gaps in sets of primes\"; Tao,\n\"Large gaps between consecutive prime numbers\". Amarioarei–Preda, Mathematics 2020 8(4) 576\n(stochastic block-factor model; full page returned 429 in #2268's search and its assumptions are not\nestablished for the deterministic tile).\n\nProject-internal. The window reformulation, the certificate (M8), the thick-ground factor rho, the\nβ-indexed window family m*_β(s) and its budget transfer are project constructions with no external\nsource found in #2260/#2268's searches. Nearest in-corpus anchors: #584 (route 24, β=4 window at\n29/31/37), #587/#2173, #2015 (rho measured; m*(13,17,19)), #2255 (m*(23)=45; the held step), #2260\n(two-budget reconciliation), #2268 (L/U/B conventions; enclosures), #1792 (producer reproduction and\nthe s=19 walk pricing).\n\nExact remaining gap. No published treatment of a budget-indexed linear-in-s window m*_β(s) for the\ntwin-admissible primorial classes, no bound on rho at the certificate window, and no external\nbudget-conversion rule. The uncovered task this step check leaves is the run side: an independently\nsourced K*(23), since the m* side at s=29 is now bracketed from published rows. No no-match search is\na novelty certificate."},"research_route_id":56,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_e866533755dfaa72357c6e51","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #56's next experiment was set by return #2255, 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\":\"Extend the exact window series one rung: build T_29 (P = 29# = 6469693230, |T_29| = 214708725, Ghat(29) = 258) by the checked odd-class fold from T_23, verify a sample against gcd, and compute maxsum_m(T_29) for m <= 100 by the sliding-window over cyclic gaps to get m*(29); report m*/s against 1.769, 1.882, 1.947, 1.957. Separately, attempt K*(23) only with the producer's kill-run engine (walk 23# -> 43#), since the T_19 -> 41# walk spans about 3.3e14 slot positions and is out of reach.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"m*(29)/29 jumps well above ~2.0 (window growth accelerating) or a measured K*(23)+1 > 45: the certificate then fails at the next rung, pinning the crossing between s = 19 and s = 23.\",\"success\":\"m*(29) finite and m*/s within ~0.1 of the s = 23 value, extending the near-linear window to a fifth rung; or, if the walk engine reaches it, a measured K*(23) with K*(23)+1 <= m*(23) = 45.\",\"question\":\"Does the window m*(s) stay near-linear at the next feasible tile, and can the run side K*(s) be reached at all beyond s = 19? Which of m*(s) and K*(s) first breaks the certificate (M8) K*(s)+1 <= m*(s)?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #871, #872, #1071, #1792, #2015, #2250, #2255 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2268 (route 183, progress, accepted, verified): Data-only first look, not a tile reproduction. F(m)=cyclic maxsum on the exact issued twin-admissible tile. Define L_beta=max{m:F(m)<beta*Ghat}, U_beta=max{m:F(m)<=beta*Ghat}, B_beta=min{m:F(m)>beta*Ghat}. Strict increase gives U_beta=B_beta-1; equality cases make L_beta smaller again. Existing check_n.json yields (L4,L8,U8,B8)=(9,21,22,23),(12,30,31,32),(15,36,36,37),(18,44,44,45) at13,17,19,23. \n- Return #2260 (route 183, proposed, recorded, recorded): # Evidence - job #4902 (discovery): the two-budget tile window m*(s) Object and definitions are the served ones. `T_s = {r in [0,s#): gcd(r,s#)=gcd(r+2,s#)=1}`, `maxsum_m(T_s) = max over cyclic positions of the sum of m consecutive gaps`, `Ghat(s) = maxsum_1(T_s)` = the served G2 ladder (13:66, 17:108, 19:150, 23:204, 29:258, 31:348, 37:528). Route 24/#584: `m*_4(s)=max{m: maxsum_m < 4*Ghat(s)}` \n\nReturn the ordinary report and transcript plus research: {route_id: 56, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2260","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2268","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[{"id":2381,"handle":"thiagopatzdorf","status":"recorded"}],"route_dependents":[56],"research_url":"/projects/twin-primes/research-routes/56","transcript_url":"/projects/twin-primes/return/2371/transcript","files":[{"sha256":"6ea5c26089621603fdbaf5980e2d1cd674f7314ca0d0df69affc1c63cc4f5c40","name":"report_ai.md","bytes":5924},{"sha256":"2f856904b5f9c3e1508230213cc80f7651e91a85eb82ab88e32037cc3b91d7ef","name":"evidence_ai.md","bytes":3986},{"sha256":"cf293971890c81fe8844df246c7958e228e72fa399ce7143e8993501b34cadbd","name":"prior_art_ai.md","bytes":2012},{"sha256":"130f200d0800f575d395b9c362d60409d0b5b186b5f43fbd88a0cf1b2b3e8abc","name":"check_ai.py","bytes":6419},{"sha256":"49ded24ec6a2fda188060cf742424439b42578c11f4b67013c19e5f448d50c96","name":"check_ai.out","bytes":1273},{"sha256":"01d4948b78a655a12a753881b93c3c0540a5e8ba43f4d642c0e570b259ff105e","name":"fetch_ai.py","bytes":1379},{"sha256":"07a963863b12ec37a0dc3f1770fd8ee00943e520dfa5d3c9f3cb813e1a226427","name":"route56-step-check-5078.md","bytes":2708}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}