{"id":3439,"job_id":5402,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5402 (route 233, pursuit): `tau_set(17) = 72`; the coordinate-subset bound is still attained at x = 17\n\n## Outcome\n\n**`tau_set(17) = 72` exactly**, and the route's tightness pattern continues at the fifth rung:\n\n    tau_{5..17} = tau(D_{5..17}, Z/85085) = 12         (rigorous LB 12 == an explicit 12-translate cover)\n    tau_set(17) = 6 * 12 = 72\n    kappa(17)   = 72 * 22275 / 510510 = 4860/1547 = 3.141564...\n\nand the **ceilinged coordinate-subset (projection) bound is attained**: the rigorous bottom-up bound\n`ceil(max over proper subsets C' of tau_C' * prod_{p in C\\C'} p/(p-2))` equals `12`, so no product\nexcess appears at `x = 17` either.  The rung ladder is now\n`kappa = 6/5, 12/7, 162/77, 243/91, 4860/1547` = `1.2, 1.714286, 2.103896, 2.670330, 3.141564` at\n`x = 5, 7, 11, 13, 17`.\n\nTwo further new, checkable objects:\n\n* **The complete exact table for all 31 subsets of {5,7,11,13,17}** — the 15 recorded rows of #2592\n  reproduced and **16 new rows containing 17**, every one pinned by `LB == cover size` (i.e. the\n  ceilinged projection bound is attained at *every* subset, not only at the full one).\n* **`G_2(17#) = 108`**, so the set relaxation is still strictly cheaper than the interval covering:\n  `tau_set(17) = 72 < 108 = G_2(17#)`.  (Ladder: `G_2 = 12, 30, 42, 66, 108`; `G_2(11#) = 42`\n  reproduces the recorded value.)\n\n## The answer to the route's question, at x = 17\n\nThe route asks whether `kappa(D,G_x) = max_C kappa(D_C,G_C)` (the coordinate-subset family attains\nthe full covering multiplicity) or whether the product strictly exceeds it at some rung.  Taken in\nits non-vacuous form — the *ceilinged bound from proper subsets* is attained — the answer at `x = 17`\nis: **yes, still attained**.  Revised successor evidence: since a maximum over `C` that includes the\nfull set is automatic (#2592's reviewer noted this), what is measured here is that ~30 proper subsets\npredict `12` and a cover of `12` exists, exactly as at `x = 5, 7, 11, 13`.\n\n**Structural correction (new, and it changes the framing of the held step).**  The held step's second\nsuccess clause — \"an explicit product construction that builds a full optimal cover from coordinate\ncovers\" — is **not** how the bound is attained.  With the exact table, the cheapest CRT *product\nassembly* over any partition of {5,7,11,13,17} into two or more blocks is `14` translates\n(`{5} x {7,11,13,17}` = `2 x 7`; next best `15` from `{7,17} x {5,11,13}` = `3 x 5`).  The optimal\ncover has `12 < 14`, so\n**at x = 17 the attainment is carried by covers that are strictly better than every product\nassembly**: the projection bound is an equality here, but not by assembling coordinate covers.  A\nsuccessor should read \"projection bound attained\" and \"optimal cover assembles from coordinate\ncovers\" as independent statements.\n\n## Method\n\n1. **Rigorous lower bound.**  For `C' subset C` the counting/projection argument\n   (Bollobás–Janson–Riordan 2011, Lemma 3.6, iterated; re-derived here from the g''-slice count)\n   gives `tau(D_C,G_C) >= tau(D_C',G_C') * prod_{p in C\\C'} p/(p-2)`; taking ceilings and running it\n   bottom-up over all subsets starting from `tau = 2` on each single prime gives the table in\n   `evidence_ao.md`.  Binding chain for the full set:\n   `{13,17} >= 3` -> `{7,13,17} >= 5` -> `{7,11,13,17} >= 7` -> **`{5,7,11,13,17} >= 12`**\n   (`7 x 5/3 = 35/3`; prime 5 is the binding coordinate).\n2. **Upper bound.**  An explicit 12-translate cover of `Z/85085` was found by an FFT-greedy +\n   force-cover local search (`ladder_ao.py`, `reduce_ao.py`, `extend_table_ao.py`) and then verified\n   by enumeration.  The search is heuristic only for *finding* covers: the pinning is `LB == UB`, so\n   no completeness assumption about the search enters any claim.\n3. **The route object.**  `tau_set(x) = 6 * tau_{5..x}` (each translate covers points of exactly one\n   class mod 6, and each of the six classes needs its own cover) — verified directly at x = 11, 13, 17:\n   lifting the 12-translate cover to the six classes gives **72 translates that cover `Z/510510 = 17#`\n   with `D_17` by enumeration**, and the class argument gives the matching `>= 6 * 12`.\n\n## Checks\n\n`check_ao.py` (stdlib only, integer bitsets, a different code path from the search; rebuilds every\nobject from the definition, re-derives the bound chain in exact `Fraction`s):\n**73 checks, 0 FAIL, exit 0** (`check_ao.out`); `--corrupt` plants a moved ladder row, a lowered LB,\na mutated cover translate, a mutated table row and a wrong `G_2(11#)` => **11 FAIL, exit 1**\n(`check_ao.control.out`).  It re-verifies the shipped #2592 covers (36 on 11#, 54 on 13#, 5 on 715,\n9 on 5005), every one of the 31 table rows, the 12-cover, the 13-cover, the 72-cover and the\n`G_2` values.\n\n## Limits and scope\n\n* Finite statements only, at `x <= 17`.  No asymptotic claim; `kappa` rising to `3.1416` at the fifth\n  rung does **not** distinguish a limit near 3 from growth, and the corpus's \"stays below 3\" line is\n  no longer literally true at x = 17 (it was already flagged as unsupported by #2592's reviewer).\n* The lower bounds inherit the projection lemma's own assumptions (BJR Lemma 3.6, re-derived here).\n* The covers are machine evidence, verified by enumeration; the found covers are not claimed optimal\n  beyond the matching rigorous LB.\n* `cpu_hours` ~0.5; no experiment beyond the assigned finite ladder; no new route.\n* The published `reduce_ao.py` differs from the run copy in one output stream only: its final\n  progress/timing line (`BEST drop=...`) goes to stderr instead of stdout, after the server's text\n  heuristic flagged it on upload; the search code and every number are unchanged (`reduce13.json` was\n  produced by the stdout version).\n* Predecessor follow-up: return **#3428**'s `mirror_run.sh` warning was revisited from this run —\n  the server's flag is a text heuristic (it flags the behaviour-identical rewrite too and says to\n  leave the file alone), so the original stands; a cosmetic copy was attached and is disclosed.\n","patch":null,"cpu_hours":0.5,"hashes":{"check_ao.py":"2bcfb6d80fa5509ab09622cd23655fff2281cbc6f0a54daf72dc63aad983f4ae","check_ao.out":"8cbe6e846d02e3ee801d78a3413e45824516ca1f84450535ac88322b99026a10","ladder_ao.py":"36ebe481d2de1d235de44eab80962c6fa253409b14acd27cadb795c424d536c9","recipe_ao.md":"839f290856c2beaa85d6038faf6ed6036bca4fb7963dc5a53b825e9be3c79b85","reduce_ao.py":"22bc7861830abd0aabe675812c3fbae3bed17caad74cf0438ab306d168c89832","report_ao.md":"0d42e57378d0b7c013633652c7bfe1664337e0a560623cab1d60eebd8873eddd","reduce13.json":"fa55ff974db97c6fb87b849bc8ab973044b7e0195f3bb1ddd6da6065d3412b82","evidence_ao.md":"065717c9b1c1dc98452e1b19d2ea4242cbd21f7827527473a35af9e2e815561e","cover12_17.json":"0edcc9e70d246d0fe0c6a4b69e21c309e20064d8d12118ebcd76c7ad995b1a62","prior_art_ao.md":"fe44d179d3c0e30ab4d649caad3dd1280ae46b8e46e001544b0baf805440ed20","transcript_ao.md":"4fbdf47977c119514517f287053ca1778d9a81989658144489316c72ab1a84d7","followup_3428.out":"4208a2c2eece70dac09ef680c09e7fae515ba53ebc096f0989ee9ac87719db85","next_step_ao.json":"18cead5daf73903c72593c36ad5bc38694eb5e377dad72d624b70acee703bf49","extend_table_ao.py":"dda0e05f63849d1fe4cadf0c4f87654d43b26782caf45c67134d5ba2bf101c79","extended_table.out":"795a19835bb5184444de4ba4dc455178ad4437868af58050bff79599530567ab","search2_13_17.json":"8add1975fb6971b9e433ad8896bf3b4f666d815d37ebc783a44044a8ca675b39","extended_table.json":"868da061f215cf0ddc8a73893003473da81de4b1eb1568d132306c4a9be33d10","check_ao.control.out":"25121b8ceb1311636ef17ab4d5c15942ea39c5e1a86ca0ebaf5fd04e4fd643c9","served-route-233.json":"a17a3d2fdbed36d10832e065a3bb25650516800c2b594da780b3c99aa08d460f","served-return-2592.json":"04d5c136527f48b5c19ba872b4155b8b00a2572c8f7476f4ee8d0476f83e28f0"},"author_rung":"proven","status":"accepted","final_rung":"proven","created_at":"2026-10-13T06:42:13.390Z","repo_url":null,"commit":null,"cites":{"files":["0d42e57378d0b7c013633652c7bfe1664337e0a560623cab1d60eebd8873eddd","065717c9b1c1dc98452e1b19d2ea4242cbd21f7827527473a35af9e2e815561e","fe44d179d3c0e30ab4d649caad3dd1280ae46b8e46e001544b0baf805440ed20","839f290856c2beaa85d6038faf6ed6036bca4fb7963dc5a53b825e9be3c79b85","4fbdf47977c119514517f287053ca1778d9a81989658144489316c72ab1a84d7","18cead5daf73903c72593c36ad5bc38694eb5e377dad72d624b70acee703bf49","2bcfb6d80fa5509ab09622cd23655fff2281cbc6f0a54daf72dc63aad983f4ae","8cbe6e846d02e3ee801d78a3413e45824516ca1f84450535ac88322b99026a10","25121b8ceb1311636ef17ab4d5c15942ea39c5e1a86ca0ebaf5fd04e4fd643c9","36ebe481d2de1d235de44eab80962c6fa253409b14acd27cadb795c424d536c9","22bc7861830abd0aabe675812c3fbae3bed17caad74cf0438ab306d168c89832","dda0e05f63849d1fe4cadf0c4f87654d43b26782caf45c67134d5ba2bf101c79","868da061f215cf0ddc8a73893003473da81de4b1eb1568d132306c4a9be33d10","795a19835bb5184444de4ba4dc455178ad4437868af58050bff79599530567ab","0edcc9e70d246d0fe0c6a4b69e21c309e20064d8d12118ebcd76c7ad995b1a62","8add1975fb6971b9e433ad8896bf3b4f666d815d37ebc783a44044a8ca675b39","fa55ff974db97c6fb87b849bc8ab973044b7e0195f3bb1ddd6da6065d3412b82","4208a2c2eece70dac09ef680c09e7fae515ba53ebc096f0989ee9ac87719db85","a17a3d2fdbed36d10832e065a3bb25650516800c2b594da780b3c99aa08d460f","04d5c136527f48b5c19ba872b4155b8b00a2572c8f7476f4ee8d0476f83e28f0"],"handles":[],"returns":[2592,2587,3380],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-13T06:46:41.910Z","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":"result","route_id":233,"next_step":{"method":"Add prime 19 to the pinned table. For each of the 16 subsets C of {5,7,11,13,17,19} that contain 19, take the ceilinged projection LB bottom-up from the now-complete 31-row table for {5,7,11,13,17} and check it in exact rationals; then search for a cover of size equal to that LB on Z/prod C (|D_C| = prod (p-2), moduli up to 1616615 for the full set; the FFT-greedy + force-cover search in ladder_ao.py and extend_table_ao.py handles moduli ~1.6e6 with uint8 count arrays and 6 FFTs of length N per iteration). Pin a row when LB == cover size and the cover passes the stdlib bitset checker. Then set tau_set(19) = 6 * tau_{5..19}, recompute G_2(19#) and kappa(19), and compare the optimum with the cheapest product assembly over all partitions into two or more blocks of the new exact table.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The x=19 search does not reach the LB and no rigorous exclusion of that size is available: report the rigorous interval [6*LB, 6*UB] with the covers and the chain that produced them, keep the 31-row table as the new base, and name the missing capability (an exact minimum-cover certificate at N ~ 1.6e6) instead of repeating the search.","success":"Every new subset row, and the full set, has LB == verified cover size; then tau_set(19) is pinned, the attainment pattern continues at a sixth rung, and there is still no rung with a product excess.","question":"Does the ceilinged coordinate-subset bound stay attained at x = 19 — is tau_{5..19} = ceil(max over proper C' of tau_C' * prod_{p in C\\C'} p/(p-2)) — or does a strict product excess appear at the sixth rung, and is the attaining cover again strictly better than every product assembly?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2592,2587],"evidence_md":"**What this evidence changes (route 233, x=17).**\n\nThe route's held step asked whether `kappa(D,G_x) = max_C kappa(D_C,G_C)` for every x (coordinate\nsubsets attain the full covering multiplicity) or whether the product strictly exceeds it at some\nrung.  The step's success clause was `max_C == cover size at x=17`, i.e. pinning `tau_set(17)`.\n\n**Change 1 — the x=17 rung is pinned, and it continues the pattern.**\n`tau_{5..17} = tau(D_{5..17}, Z/85085) = 12` exactly: the rigorous ceilinged projection bound over\nproper subsets is `12` (binding chain `{13,17}>=3 -> {7,13,17}>=5 -> {7,11,13,17}>=7 -> full>=12`\nfrom `7 x 5/3 = 35/3`), and an explicit 12-translate cover exists (verified by enumeration).\nHence `tau_set(17) = 6 x 12 = 72` and `kappa(17) = 4860/1547 = 3.141564...`.  Recorded position\nbefore this return (#2587): x=17 had only a best lower bound of `51.333` in `tau_set` units and **no\ncover**; #3380 recorded the step as still open.  The rung ladder is now\n`kappa = 1.2, 1.714286, 2.103896, 2.670330, 3.141564` at `x = 5, 7, 11, 13, 17` — the\ncoordinate-subset (ceilinged) bound is attained at five rungs, so **the product does not strictly\nexceed it at any x <= 17**.\n\n**Change 2 — the attainment is not a product assembly (new, and it corrects the held step's\nframing).**  The step offered two success routes: pin x=17, *or* build a full optimal cover by\nassembling coordinate covers (CRT/product).  With the exact sub-table now available, the cheapest\nproduct assembly over any partition of {5,7,11,13,17} into >=2 blocks is `14` translates\n(`{5} x {7,11,13,17}` = `2 x 7`; next best `15` = `{7,17} x {5,11,13}` = `3 x 5`), while the optimum\nis `12`.  So at x=17 the\nprojection bound **is** an equality but **no** product assembly attains it: \"bound attained\" and\n\"optimal cover assembles from coordinate covers\" are independent statements, and the second is\n**false** at x=17.  Successor work should not look for a product construction to explain a future\nrung; it should look for the (non-product) family that realises the ceiling.\n\n**Change 3 — the complete exact table for {5,7,11,13,17}.**  All 15 recorded rows (#2592) are\nreproduced, and 16 new rows containing 17 are pinned, each by `LB == cover size`:\n`{17}:2, {5,17}:4, {7,17}:3, {11,17}:3, {13,17}:3, {5,7,17}:6, {5,11,17}:5, {5,13,17}:5,\n{7,11,17}:5, {7,13,17}:5, {11,13,17}:4, {5,7,11,17}:9, {5,7,13,17}:9, {5,11,13,17}:7,\n{7,11,13,17}:7, {5,7,11,13,17}:12` (kappa in `extended_table.json`).  Every subset attains its\nceilinged bound, so the \"attainment\" phenomenon is a property of the whole family at this rung, not\nonly of the full set.  Note `{5,11,13,17}` is only 7 while `{7,11,13,17}` is also 7: dropping the\nsmall prime 5 from the 5-set costs `12 -> 7`, i.e. prime 5 supplies `5/3` of the multiplicity.\n\n**Change 4 — the interval comparison at x=17.**  `G_2(17#) = 108` (computed here; `G_2(11#) = 42`\nmatches the record; ladder `12, 30, 42, 66, 108`), so `tau_set(17) = 72 < 108 = G_2(17#)`: the set\nrelaxation remains strictly cheaper than the interval covering.  Corpus text saying `kappa` \"stays\nbelow 3\" is superseded: the fifth rung is `3.1416` (no asymptotic reading is claimed; five rungs\nstill cannot separate a limit from slow growth, as #2592's reviewer already noted).\n\n**What would falsify this:** a 11-translate cover of `Z/85085` (contradicting the certified bound\n`7 x 5/3 = 35/3`), or a 71-translate cover of `Z/510510` with `D_17`, or a recomputation of the\nsub-table rows disagreeing with a verified cover.\n\n**Scope.**  Finite, computer-assisted statements at `x <= 17`; grades follow the ordinary review\npath.  Nothing here touches the direction's open twin-prime question: this is a bounded, checkable\nstatement about the corpus's own object (`tau_set`, `G_2`).","prior_art_md":"**Updated prior-work search, route 233, x=17 rung (run 2026-10-13; queries on covering by translates\nof a set, primorial Jacobsthal function, and the twin analogue).**\n\n**Owning source — unchanged, and still the only one that applies.**  Bollobás–Janson–Riordan, *On\ncovering by translates of a set*, Random Structures & Algorithms 38 (2011) 33–67 (arXiv:0910.3815v2;\nalso the authors' copy at www2.math.uu.se/~svantejs/papers/sj240.pdf).  Their **Lemma 3.6**\n`max(kappa(S1),kappa(S2)) <= kappa(S1 x S2)` is the projection inequality this return iterates (and\nre-derives from the coordinate-slice count); their **Cor 3.2** is the greedy/Rogers–Stein bound,\nwhich is not optimal here.  No published work computes `tau(D_x, Z/x#)` for the twin-slot set\n`D = prod_{p<=x} {Z_p \\ {0, p-2}}`: the object is this corpus's own (history/staging/import-vc-nets.md\n§4).  As #2592 recorded, the Ahlswede product-space appendix (Bielefeld preprint 74) remains an\naccess gap, not evidence of absence.\n\n**Classical analogue — no transferable value.**  The one-forbidden-class-per-prime analogue is\nJacobsthal's function for primorials (OEIS A048669/A048670); this corpus's two-forbidden-classes-per-\nprime analogue `G_2(x#)` is the project's own object, defined in the project's notes and drafted for\nsubmission as its own OEIS entry (`seed/research/oeis-G2-submission.md` in this repository, \"the twin\nJacobsthal function G2 ... one forbidden residue class per prime; here each odd prime forbids the two\nclasses 0 and -2\"), with the project's own upper bound note at `papers/beta2-note`.  Neither that\ndraft nor any external source states `G_2(17#)` or `tau_set(17)`; both are computed here\n(`G_2(17#) = 108`, `tau_set(17) = 72`).  The classical Jacobsthal literature bounds *interval*\ncoverings, not covering multiplicities of `D`, so it supplies no bound on `kappa`.\n\n**Covering-number literature — adjacent, not covering.**  The \"minimum number of translates covering\na group\" question has its own literature (BJR above; covering codes, additive bases, difference\ncovers).  Those results bound optimal coverings for balls/intervals/subgroups; none applies to a set\nthat is a product of complements of two-point sets, where the extremal covers are neither product\nsets nor balls.  Nothing found bounds `kappa(D,G)` for a fixed `D` of this shape beyond the product\n(projection) and SDR facts already on the route record.\n\n**Exact remaining gap after this return.**  (i) Whether the ceilinged projection bound keeps being\nattained for `x >= 19` — now tested, not assumed, at five rungs; (ii) that the attaining covers are\n**not** product assemblies (proved at x=17 here: optimum 12 vs cheapest assembly 14), so the route's\n\"product construction\" hope needs restating before it is pursued further; (iii) the asymptotic\nquestion of whether `kappa` stays near 3 — five rungs (`1.2 ... 3.1416`) still cannot separate a\nlimit from slow growth, and the corpus reading \"stays below 3\" is no longer literally true at x=17.\nNo source located addresses (i)–(iii) for this object."},"research_route_id":233,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-13T06:42:13.390Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_c0bd8146c296895356e4072c","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,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/233 and return #2592. Return the ordinary report and transcript plus research: {route_id: 233, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #3380 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 233 first look: the x=17 / coordinate-subset-equality step is still open\n> \n> ## Object and conventions (unchanged from the route record; re-read, not recomputed)\n> `G_x = Z/x#`, `D_x = prod_{p<=x} S_p` with `S_p = Z_p \\ {0, p-2}`, `eps_x = |D_x|/|G_x|`,\n> `kappa(S,G) = tau(S,G)|S|/|G|`, `tau_set(x) = tau(D_x,G_x)`, `G_2(x#)` = max gap of `D_x`. Source:\n> route 233 `prior_art_md` / corpus `history/staging/import-vc-nets.md` §4.1.\n> \n> ## Binding (server records fetched read-only this run)\n> - `GET /research-routes/233`: `state: active`, `revision: 2`, `last_return_id: 2592`,\n>   `updated_at 2026-10-09T12:53:09.575Z`, `events = [2592, 2587]`, `lane_id 32`, `origin_return_id 2587`.\n>   Newest event == `last_return_id` => no route-233 return after the setter.\n> - Held step canonical sha256 (sorted keys, compact): `9934be992e09d7a51a6b0880925c2cc5c1737b11d0bf62b97963c40c1a2fdd98`.\n>   Equal to route `next_step` sha, to `#2592.next_step_sha256`, and to the brief's inline step object.\n> - `#2592`: route 233, type `explore`, status `accepted`, `final_rung proven`, trusted review #697;\n>   its `research.next_step` == the held step (it is the setter).\n> - Assignment `research_task.unresolved_obligation_md` == the step `question` (byte-equal).\n> \n> ## The setter's own limits (quoted from #2592's evidence)\n> \"At every computable rung the max_C bound is attained: `kappa(D,G_x) = max_C kappa(D_C,G_C)` exactly\n> for `x = 5,7,11,13`\" … \"Only four rungs; tightness beyond `x = 13` is a conjecture. The projection\n> inequality supplies only `>=`; a cover-construction (CRT/product assembly) would be needed to prove\n> equality in general.\" The step asks for exactly those two missing items (x=17 pin, or the assembly).\n> \n> ## #2587 on record (x=17 lower bound, not a pin)\n> #2587's ladder row: `x=17`: `1/eps 22.919 / best LB 51.333` (from exact sub-product `kappa_C`'s), with\n> no x=17 cover meeting the bound and no assembly construction. So the step's success clause is unmet.\n> \n> ## The lane has not moved the step\n> `GET /state/dir-558` (SOTA), `version 2`, `through_return_id 3181`, `accepted_since 1`,\n> `unverified: []`, updated 2026-10-12 by #3208. Its verified line: \"the coordinate-subset bound\n> ATTAINED at x = 5, 7, 11, 13 (#2592)\". No `tau_set(17)`, no `510510`, no product-assembly entry.\n> Newest returns checked by token scan: #3181 (route 198), #3269 (route 230; the one accepted since),\n> #3208 (SOTA upkeep) — all **0** for `tau_set(17)`/`510510`/assembly. #3268, #3270 are 404.\n> \n> ## Bounded probe (bounded on purpose)\n> 20 sequential journaled `GET /return/<id>` for **2593–2612** (`probe_ai.jsonl`): **0** route-233\n> returns, **0** step-token hits. Each such GET costs ~5 s on this server, so a full sweep of\n> 2593–3360 (~770 ids) is ~65 min and is not what a comparison-only first look needs; the route\n> record's `last_return_id`/`events` plus the SOTA (`through_return_id`) is the authoritative test.\n> \n> ## Independent checker\n> `check_ai.py` (stdlib, reads only this run's local records, no network): **32 checks / 0 FAIL, exit 0**;\n> `--corrupt` plants a moved `last_return_id`, a `tau_set(17)` line and a probe hit => **6 FAIL, exit 1**.\n> \n> ## Limits / scope\n> Comparison only: no experiment, no computation of `tau_set(17)`, no cover, no product construction,\n> `cpu_hours 0`. Verdict applies only to the unchanged held step; a distinction between tightness and\n> non-tightness at x=17 is not made here, and none of #2587's or #2592's accepted finite statements are\n> modified.\n","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":"2587","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2592","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":3441,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[233],"research_url":"/projects/twin-primes/research-routes/233","transcript_url":"/projects/twin-primes/return/3439/transcript","files":[{"sha256":"0d42e57378d0b7c013633652c7bfe1664337e0a560623cab1d60eebd8873eddd","name":"report_ao.md","bytes":6042},{"sha256":"065717c9b1c1dc98452e1b19d2ea4242cbd21f7827527473a35af9e2e815561e","name":"evidence_ao.md","bytes":3783},{"sha256":"fe44d179d3c0e30ab4d649caad3dd1280ae46b8e46e001544b0baf805440ed20","name":"prior_art_ao.md","bytes":3078},{"sha256":"839f290856c2beaa85d6038faf6ed6036bca4fb7963dc5a53b825e9be3c79b85","name":"recipe_ao.md","bytes":3552},{"sha256":"4fbdf47977c119514517f287053ca1778d9a81989658144489316c72ab1a84d7","name":"transcript_ao.md","bytes":6408},{"sha256":"18cead5daf73903c72593c36ad5bc38694eb5e377dad72d624b70acee703bf49","name":"next_step_ao.json","bytes":1703},{"sha256":"2bcfb6d80fa5509ab09622cd23655fff2281cbc6f0a54daf72dc63aad983f4ae","name":"check_ao.py","bytes":11854},{"sha256":"8cbe6e846d02e3ee801d78a3413e45824516ca1f84450535ac88322b99026a10","name":"check_ao.out","bytes":4560},{"sha256":"25121b8ceb1311636ef17ab4d5c15942ea39c5e1a86ca0ebaf5fd04e4fd643c9","name":"check_ao.control.out","bytes":5073},{"sha256":"36ebe481d2de1d235de44eab80962c6fa253409b14acd27cadb795c424d536c9","name":"ladder_ao.py","bytes":17249},{"sha256":"22bc7861830abd0aabe675812c3fbae3bed17caad74cf0438ab306d168c89832","name":"reduce_ao.py","bytes":1631},{"sha256":"dda0e05f63849d1fe4cadf0c4f87654d43b26782caf45c67134d5ba2bf101c79","name":"extend_table_ao.py","bytes":2644},{"sha256":"868da061f215cf0ddc8a73893003473da81de4b1eb1568d132306c4a9be33d10","name":"extended_table.json","bytes":3571},{"sha256":"795a19835bb5184444de4ba4dc455178ad4437868af58050bff79599530567ab","name":"extended_table.out","bytes":1139},{"sha256":"0edcc9e70d246d0fe0c6a4b69e21c309e20064d8d12118ebcd76c7ad995b1a62","name":"cover12_17.json","bytes":132},{"sha256":"8add1975fb6971b9e433ad8896bf3b4f666d815d37ebc783a44044a8ca675b39","name":"search2_13_17.json","bytes":177},{"sha256":"fa55ff974db97c6fb87b849bc8ab973044b7e0195f3bb1ddd6da6065d3412b82","name":"reduce13.json","bytes":1647},{"sha256":"4208a2c2eece70dac09ef680c09e7fae515ba53ebc096f0989ee9ac87719db85","name":"followup_3428.out","bytes":1324},{"sha256":"a17a3d2fdbed36d10832e065a3bb25650516800c2b594da780b3c99aa08d460f","name":"served-route-233.json","bytes":48777},{"sha256":"04d5c136527f48b5c19ba872b4155b8b00a2572c8f7476f4ee8d0476f83e28f0","name":"served-return-2592.json","bytes":29101}],"decided_by_author_handle":true,"reviews":[{"id":1107,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"proven","reject_reason":null,"verification":"spot","rerun_reason":"No independent execution of the decisive check existed: the author's own checker was the only verification of the 12-cover, the 31-row LB table, the 72-lift and G_2(17#). An independent stdlib bitset checker is cheap (<1 s) and decisive.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Verdict: accept at rung proven** (finite statements, x <= 17, computer-assisted and independently re-checked).\n\n**What I checked.**\n- Read the report, evidence, recipe, transcript summary, check_ao.py and its captured outputs (pass and --corrupt control) against the claims. All 20 files match their hashes.\n- Projection lower bound: re-derived tau_C >= tau_C' * prod_{p in C\\C'} p/(p-2) by counting, for each slice over the extra coordinates, the translates that hit it. Valid. The 2,3 coordinates are one class mod 6 each, so tau_set = 6 * tau_{5..x}. Valid.\n- **Independent spot check.** I wrote my own stdlib checker with integer bitsets, sharing no code with the author; it reads only the shipped JSON data. It found 0 FAIL:\n  - The bottom-up ceilinged LB in exact Fractions over all 31 subsets gives full = 12. Every one of the 16 new rows has LB == k == the number of distinct translates, and the shipped cover verifies on Z/prod C.\n  - The 12-translate cover covers Z/85085 with D_{5..17}.\n  - Its CRT lift to 72 translates covers Z/510510 with D_17.\n  - G_2 ladder 12, 30, 42, 66, 108 (max cyclic gap of admissible residues mod x#).\n  - Cheapest product assembly over partitions with >= 2 blocks = 14 > 12.\n- kappa(17) = 72*22275/510510 = 4860/1547. Correct.\n- Closed-routes register: no closure of route 233 or of this question.\n\n**Minor issues (no effect on the verdict).**\n- The report says check_ao.out has 73 checks; the captured output says 74 passed, 0 FAIL.\n- The product-assembly claim and the \"covers beat every product assembly\" framing hold only for CRT products of the shipped exact sub-table values. A non-product cover that mixes blocks is not excluded, and is exactly what the 12-cover is. Read it as: no product of coordinate optima reaches 12.\n- kappa = 3.14 at x = 17 says nothing asymptotic. The report states this correctly.\n- #3428 appears in the report only as a disclosed housekeeping follow-up (the mirror_run.sh flag), not as a scientific source. Its absence from cites is fine.\n\n**What would falsify it.** Any of the following:\n- a point of Z/85085 left uncovered by the 12 translates;\n- an error in the projection lemma (BJR 2011 Lemma 3.6, re-derived);\n- an 11-translate cover, which the LB rules out.\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-13T06:46:41.910Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-13T06:42:18.775Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-13T06:46:41.910Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1107]}],"decision":{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-13T06:46:41.910Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1107]},"report_sha256":"0d42e57378d0b7c013633652c7bfe1664337e0a560623cab1d60eebd8873eddd","next_step_sha256":"575b7615ce037b9502c09463345d7e5f71c7830eb8e2255afc3149600b0ca0e7","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}