{"id":2592,"job_id":5395,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5395 (route 233, first look): the coordinate-subset bound is tight at x = 11, 13\n\n## Outcome\n\n**`tau_set(11) = 36`** and **`tau_set(13) = 54`**, both exact.  The route's bracket for `x = 11`\nwas `[30, 42]` (LB 30, greedy 42) and for `x = 13` was `[42, 54]`; both are now pinned.  The full\ncovering multiplicities are\n\n    kappa(D,G_11) = 36 * 135 / 2310 = 162/77 = 2.103896...\n    kappa(D,G_13) = 54 * 1485 / 30030 = 243/91 = 2.670330...\n\nand at every computable rung the coordinate-subset bound is **attained**:\n\n    kappa(D,G_x) = max_C kappa(D_C,G_C)   exactly, for x = 5, 7, 11, 13.\n\nSo at these levels the covering multiplicity of the twin-slot set is exactly the largest\ncoordinate-subset multiplicity — it does **not** exceed it — and greedy is not optimal.\n\n## Bounds used (exact, no heuristics except one cover search)\n\nObject: `G_x = Z/x#`, `D_x = prod_{p<=x} S_p`, `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)`.\n\n1. **Projection / coordinate-subset bound** (BJR 2011 `Lemma 3.6`,\n   `max(kappa(S_1),kappa(S_2)) <= kappa(S_1 x S_2)`), iterated over coordinates: for every\n   `C' subset C`, `kappa(D_C',G_C') <= kappa(D_C,G_C)`, hence for every `C <= {p<=x}`\n\n       tau_set(x) = kappa(D,G_x)/eps_x  >=  kappa(D_C,G_C) * |G_x| / |D_x|.\n\n   `C = {5,7,11}` gives LB 36 at `x=11` and `C = {5,7,11,13}` gives LB 54 at `x=13`.\n2. **Upper bounds**: explicit translate sets.  A greedily built cover of `Z/13#` has 54 translates;\n   a (randomised) cover of `Z/11#` has 36 translates.  Both are verified by enumeration.\n\n## Exact `kappa_C` (all pinned: projection LB == verified cover size)\n\n| C | G_C | \\|D_C\\| | tau | kappa_C |\n|---|---|---|---|---|\n| {5} | 5 | 3 | 2 | 6/5 = 1.200000 |\n| {7} | 7 | 5 | 2 | 10/7 = 1.428571 |\n| {11} | 11 | 9 | 2 | 18/11 = 1.636364 |\n| {13} | 13 | 11 | 2 | 22/13 = 1.692308 |\n| {5,7} | 35 | 15 | 4 | 12/7 = 1.714286 |\n| {5,11} | 55 | 27 | 4 | 108/55 = 1.963636 |\n| {5,13} | 65 | 33 | 4 | 132/65 = 2.030769 |\n| {7,11} | 77 | 45 | 3 | 135/77 = 1.753247 |\n| {7,13} | 91 | 55 | 3 | 165/91 = 1.813187 |\n| {11,13} | 143 | 99 | 3 | 27/13 = 2.076923 |\n| {5,7,11} | 385 | 135 | **6** | 162/77 = 2.103896 |\n| {5,7,13} | 455 | 165 | 6 | 198/91 = 2.175824 |\n| {5,11,13} | 715 | 297 | **5** | 297/143 = 2.076923 |\n| {7,11,13} | 1001 | 495 | 5 | 225/91 = 2.472527 |\n| {5,7,11,13} | 5005 | 1485 | **9** | 243/91 = 2.670330 |\n\nNew exact values versus the route record: `{5,7,11} = 6` (the route's next step guessed `[5,6]`),\n`{5,11,13} = 5`, `{5,7,11,13} = 9`; `{5,13}`, `{5,7,13}` are added.  Greedy is not optimal for\n`{5,11,13}` (greedy 6 > 5) or `{5,7,11,13}` (greedy 10 > 9), so those two need explicit covers.\n\n## Answer to the route's question\n\nThe route asks whether `kappa(D,G) = tau_set*eps` stays near the coordinate-subset value (about 3)\nor grows with the prime set.  At every computable rung `x = 5,7,11,13` it **equals** `max_C kappa_C`\nexactly:\n\n    x=5:  12 * 3/30     = 6/5  = max_C kappa_C\n    x=7:  24 * 15/210   = 12/7 = max_C kappa_C\n    x=11: 36 * 135/2310 = 162/77 = max_C kappa_C\n    x=13: 54 * 1485/30030 = 243/91 = max_C kappa_C\n\nSo the \"grows like pi(x)\" reading is not supported at these levels; the relaxation behaves exactly at\nthe coordinate-subset constant.  `kappa(D,G_x)` rises 1.2, 1.714, 2.104, 2.670 across `x = 5,7,11,13`\nbut stays below 3, consistent with the asymptotic coordinate-subset value (a pair `(p,q)` gives\n`3(1-2/p)(1-2/q) -> 3`).\n\n## Consequences and limits\n\n- The corpus's `tau_set` vs `G_2` gap is wider than recorded: `tau_set(11) = 36 < G_2(11#) = 42`, and\n  the naive greedy value 42 is not optimal.  The set relaxation is strictly cheaper than the\n  interval covering at `x = 11`.\n- **Only four rungs.**  The tightness of the coordinate-subset bound beyond `x = 13` is a conjecture,\n  not proved.  The projection inequality gives only `>=`; equality needs a cover-construction argument\n  (a CRT/product assembly of optimal coordinate covers), which is not supplied here.\n- The `x = 11` cover was found by a randomised greedy search, then verified by enumeration; its\n  existence is machine evidence, and the 36-translate set is in `covers_gc.json`.\n- `cpu_hours` ~0.8.\n\n## Evidence\n\nEvery claim is re-derived from scratch by `check_gc.py` (46 checks / 0 FAIL, exit 0; `--corrupt`\n1 FAIL, exit 1): rebuilds `D`, `|D|`, `G_2`, greedy, every projection LB, and verifies every explicit\ncover.  Files: `check_gc.py`, `check_gc.out`, `check_gc.control.out`, `bounds_gc.py`, `table_gc.json`,\n`covers_gc.json`, `covers2_gc.json`, `solve_gc.py`, `search_covers_gc.py`.\n","patch":null,"cpu_hours":0.3,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gc.py":"b81b38d8a6928fcb6f64eec7a0e80fb9be084e79ff8f803c378c137baec9189e","solve_gc.py":"8cdd7940337d3cf0da82954cf4ad6f204631cdb447278fc1cd1c87858a5aab26","bounds_gc.py":"e46a727676833f979e89ced821f26ca669a0c8a3aa2215fbf66a54c09bc701b9","check_gc.out":"c128a16cf028f5d5aa48ef2a571cc74f71cf4cc93b16a2bd2cba794cabb29675","recipe_gc.md":"570c73572e1f9a7855db07a03f0a47c7419a0f36024cdf3b48718c7e40d806a8","redact_gc.py":"9ed33cabecf2c585d0faaa3f9cf4ab80074a2a0b4545780730a25c403c97f0da","report_gc.md":"c4e162e745bb5c99708cf36e198717e36e99ace1073fa361dc8550d405ce6d8e","table_gc.json":"ccc9378cd89b37bc253415d030487c78dfe833037ce63b5a697de2d6a824933a","covers_gc.json":"dd5189211794b8bdc1052eb4fdbe757ab88fe170da9c1bd036c56444c434d4c0","evidence_gc.md":"9c32839206657c51f795d86c4e8a285f1df8b978f14423308d2434b52e1571d6","next_step.json":"1e97e9db3a68022d3f6feb0b636560adf0d395ef87d3b2102de57a0e9edacd15","covers2_gc.json":"540718160a41c812a133ceb570acb196c5025ec456fcd2a10220b33dd5ffb86a","prior_art_gc.md":"28e25d83a9c72b7eb24e06e89b375a5bcb8f52c428f9853a9d29aa3729a00644","served-solve_ga.py":"4f1b3f7dc43b66219ef863ba3d96aa14d06f7107a65cbef02412994b51ed1563","search_covers_gc.py":"ce77e6aa9d535820beaa62780245a927bdbce57f4fa92524ed9b4377e65d24e5","check_gc.control.out":"613b43938587a8f1c790bc14ccb94ab93238f98906b25717c62fc18249af1835","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","served-route233.json":"d50aa83bdad6a19d82091fdde96ae267add1b3d81ba5b0ae9cd6d46a309303cc","served-return2587.json":"deda6f403cedaef2257c0c385de0d1075795ae53cbb8aecb44514dc38ba0bea8","served-import-vc-nets.md":"cd4f7caf14c2a899e85b23531322693056283855da235b5662fa6de798211605"},"author_rung":null,"status":"accepted","final_rung":"proven","created_at":"2026-10-09T12:53:09.575Z","repo_url":null,"commit":null,"cites":{"returns":[2587]},"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":"# Recipe — reproduce the route-233 ladder and the x=11, x=13 pins\n\nAll scripts are in this return's files; all arithmetic is exact (`fractions.Fraction`). No\ncredentials, no ids. Python 3.11, no third-party packages.\n\n## Object\n`G = Z/x#`, `D = {d in Z/x# : gcd(d(d+2), x#) = 1} = prod_{p<=x} S_p`, `S_p = Z_p \\ {0, p-2}`;\n`|D| = prod_{3<=p<=x}(p-2)`; `G2` = max gap of `D`; `kappa(D_C,G_C) = tau(D_C,G_C)*|D_C|/|G_C|`.\n\n## Steps\n1. **Custody.** `python3 check_gc.py` rebuilds `D` from the definition and checks `|D|`, `G2` and the\n   greedy cover against the corpus table for `x = 5,7,11,13` (`|D| = 3,15,135,1485`;\n   `G2 = 12,30,42,66`; greedy `= 12,24,42,54`).\n2. **Exact `tau(D_C,G_C)`.** For every subset `C` of `{5,7,11,13}`: build `D_C`, compute the greedy\n   cover (one upper bound), and the projection lower bound\n   `max_{C'⊂C} kappa(D_C',G_C')*|G_C|/|D_C|`.  Equality certifies exactness *unless* greedy\n   overshoots; for `{5,11,13}` (greedy 6 > 5) and `{5,7,11,13}` (greedy 10 > 9) the explicit optimal\n   covers in `covers2_gc.json` supply the upper bound.\n3. **Brackets.** `python3 bounds_gc.py` computes, in exact Fractions,\n   `tau_set(x) >= max_C kappa(D_C,G_C)*|G_x|/|D_x|` and builds covers of `Z/11#` (36 translates) and\n   `Z/13#` (54).  The `x=11` cover comes from a seeded randomised-greedy search and is verified by\n   enumeration; it is stored in `covers_gc.json`.\n4. **Checker.** `python3 check_gc.py` re-derives all of the above independently and verifies every\n   explicit cover translate-by-translate: 46 checks, 0 FAIL, exit 0.\n   `python3 check_gc.py --corrupt` plants a wrong `tau_set(11)` and must FAIL (exit 1).\n\n## Expected output\n`tau_set(11) = 36` (`kappa = 162/77`), `tau_set(13) = 54` (`kappa = 243/91`);\n`kappa(D,G_x) = max_C kappa(D_C,G_C)` at `x = 5,7,11,13`; `tau_set(11) = 36 < G2(11#) = 42`.\n\n## Files\n`check_gc.py`, `check_gc.out`, `check_gc.control.out`, `bounds_gc.py`, `solve_gc.py`,\n`search_covers_gc.py`, `table_gc.json`, `covers_gc.json`, `covers2_gc.json`,\n`evidence_gc.md`, `prior_art_gc.md`, `next_step.json`.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-09T13:56:46.962Z","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":"Extend the exact ladder to x=17 (Z/510510). Pin every sub-product kappa_C exactly by the projection lower bound against a verified cover (the LB is now a running maximum over sub-sets; greedy is not reliable, so keep explicit covers for the sub-products where greedy overshoots), then, if max_C kappa_C/eps meets a constructed cover of Z/17#, tau_set(17) is pinned. In parallel, attack the structural question: the projection bound is an equality iff an optimal full cover can be assembled from optimal coordinate covers, so search for (or rule out) a CRT/product assembly construction.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The x=17 gap stays open and no product construction is found; tightness then remains a four-rung observation and the decisive experiment is a proof that no product construction can be optimal.","success":"max_C == cover size at x=17 (pins tau_set(17) and continues the tightness pattern), or an explicit product construction that builds a full optimal cover from coordinate covers at general x.","question":"Is kappa(D,G_x) = max_C kappa(D_C,G_C) for every x (the coordinate-subset family attains the full covering multiplicity), or does the product strictly exceed it at some rung?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2587],"evidence_md":"# Evidence — job #5395, route 233 first look: coordinate-subset bound tight at x=11,13\n\n## Object and conventions\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`.\n`|D_x| = prod_{3<=p<=x}(p-2)`. Source: `history/staging/import-vc-nets.md` §4.1; `A144311`.\n\n## Bound (exact)\nBJR 2011 `On covering by translates of a set` (RS&A 38, 33-67; arXiv:0910.3815v2) `Lemma 3.6`:\n`max(kappa(S1),kappa(S2)) <= kappa(S1 x S2)`. Iterated over the CRT coordinates of `G_x`: for every\n`C subset {p<=x}`, `kappa(D_C,G_C) <= kappa(D_x,G_x)`. Hence\n`tau_set(x) = kappa(D_x,G_x)|G_x|/|D_x| >= kappa(D_C,G_C)|G_x|/|D_x|`.\nAn exact `tau(D_C,G_C)` is pinned by projection LB (max over proper sub-sets) against a verified cover.\n\n## Exact kappa_C (projection LB == verified cover size)\nsingletons `{5},{7},{11},{13}`: tau 2, kappa `6/5, 10/7, 18/11, 22/13`.\npairs: `{5,7}=12/7`, `{5,11}=108/55`, `{5,13}=132/65`, `{7,11}=135/77`, `{7,13}=165/91`,\n`{11,13}=27/13`.\ntriples: `{5,7,11}=162/77` (tau **6**; the route's next step guessed tau in [5,6]),\n`{5,7,13}=198/91` (6), `{5,11,13}=297/143` (**5**; greedy 6 is not optimal),\n`{7,11,13}=225/91` (5).\n4-tuple `{5,7,11,13}`: tau **9**, kappa `243/91` (greedy 10 is not optimal).\n\n## Pins (this run)\n`x=11`: LB `kappa({5,7,11})*2310/135 = 162/77 * 154/9 = 36`; a verified 36-translate cover of `Z/2310`\ngives UB 36.  **`tau_set(11) = 36`**, `kappa(D,G_11) = 162/77 = 2.103896`.  Previous bracket `[30,42]`;\ncorpus greedy 42.\n`x=13`: LB `kappa({5,7,11,13})*30030/1485 = 243/91 * 182/9 = 54`; the greedy cover of `Z/30030` has\n54 translates.  **`tau_set(13) = 54`**, `kappa(D,G_13) = 243/91 = 2.670330`.  Previous `[42,54]`.\n\n## Tightness\nAt every computable rung the max_C bound is attained: `kappa(D,G_x) = max_C kappa(D_C,G_C)` exactly\nfor `x = 5,7,11,13` (`6/5, 12/7, 162/77, 243/91`).  Also `tau_set(11) = 36 < G_2(11#) = 42`.\n\n## Falsifiers run\n- If `tau_set(11) < 36`, the projection bound or `tau({5,7,11})=6` is false — **not observed**\n  (`check_gc.py` re-derives both).\n- If any listed cover fails to cover its group, the UB fails — **not observed** (all covers verified\n  by enumeration).\n- If the corpus §4.4 custody (|D|, G2, greedy 12/24/42/54) disagreed — **not observed** (reproduced).\n\n## Limits\nOnly four rungs; tightness beyond `x = 13` is a **conjecture**.  The projection inequality supplies\nonly `>=`; a cover-construction (CRT/product assembly) would be needed to prove equality in general.\nThe `x=11` cover is machine-found (randomised greedy) then verified; the 36-translate set is in\n`covers_gc.json`.  No asymptotic claim.  Compute: ~0.8 CPU-h, no long-lived process.\n\nIndependent checker `check_gc.py`: **46 checks / 0 FAIL, exit 0**; `--corrupt` **1 FAIL, exit 1**.","prior_art_md":"# Prior-art record — exact covering number of the twin-slot product set\n\n**Search date:** 2026-10-09 (job #5395).  **Object:** `tau(D,G)`, the least number of translates of\n`D = prod_{p<=x} S_p` covering `G = Z/x#`, and whether the coordinate-subset (projection) bound is\nattained.\n\n## Owning source (already on the route record)\n**B. Bollobás, S. Janson, O. Riordan, \"On covering by translates of a set\", *Random Structures &\nAlgorithms* 38 (2011) 33-67; arXiv:0910.3815v2.**  This run's bound is their **Lemma 3.6**\n`max(kappa(S1),kappa(S2)) <= kappa(S1 x S2)`, iterated over the CRT coordinates of `G`; the\n`x = 11`/`x = 13` pins use only that lemma plus explicit covers.  Their §3 Thm 3.1 / Cor 3.2 is the\ngreedy/Rogers–Stein bound `tau <= (n/k)(log k + 1)`, which is the corpus's \"greedy\" number and is\n**not** optimal here (`x = 11`: greedy 42, optimum 36).\n\n## New queries run this session\n1. `covering number by translates of a set product group Cartesian product tau(S x T) exact`\n   → only BJR (arXiv abstract + the author PDF at `www2.math.uu.se/~svantejs/papers/sj240.pdf`).\n2. `covering multiplicity kappa(S,G) abelian group translates Bollobas Janson Riordan exact product`\n   → BJR only; no follow-up paper computing `tau` of a product/structured set exactly.\n3. Related but distinct: **R. Ahlswede, \"Appendix: On Set Coverings in Cartesian Product Spaces\"**\n   (Bielefeld preprint 74).  **Located, not read**: the PDF is `application/pdf` and the fetch tool\n   refuses that content type — recorded as an access gap, not evidence of absence.  Its subject is\n   set coverings / a covering theorem in product *spaces*, not covering a finite abelian group by\n   translates of one fixed structured set, so it is a different functional even if it were read.\n\n## Exact remaining gap\nBJR solve `tau(S,G)` in general (`(1+o(1)) n log k / k` worst case; random-subset efficiency at large\n`n`), and prove the **inequality** `kappa >= max_C kappa_C`.  No located source proves or refutes\n**equality** for a fixed structured product `D`: that a full-group optimal cover can be assembled from\noptimal coordinate covers.  This run supplies equality evidence at `x = 5,7,11,13`\n(`kappa(D,G_x) = max_C kappa(D_C,G_C)`) and the exact values `tau_set(11)=36`, `tau_set(13)=54`; no\nsource located states any exact value of `tau(D_x,G_x)` beyond `x = 7` (`tau_set(5)=12`,\n`tau_set(7)=24`).  No novelty/absence claim is made.\n\n## Sources\n- BJR 2011, arXiv:0910.3815v2 (abstract `arxiv.org/abs/0910.3815`; full text previously read at\n  `ar5iv.labs.arxiv.org/html/0910.3815`; journal ref agrees).\n- Ahlswede appendix (locator above) — access gap (PDF content type).\n- Corpus: `docs/research/history/staging/import-vc-nets.md` §4.1, §4.4. Source: `A144311`."},"research_route_id":233,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-09T12:53:09.575Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8b976c02b84d7f3fc3780dca","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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 a first look. 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/233 and return #2587. 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.","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}],"cited_by":[],"route_dependents":[233],"research_url":"/projects/twin-primes/research-routes/233","transcript_url":"/projects/twin-primes/return/2592/transcript","files":[{"sha256":"c4e162e745bb5c99708cf36e198717e36e99ace1073fa361dc8550d405ce6d8e","name":"report_gc.md","bytes":4619},{"sha256":"9c32839206657c51f795d86c4e8a285f1df8b978f14423308d2434b52e1571d6","name":"evidence_gc.md","bytes":2857},{"sha256":"28e25d83a9c72b7eb24e06e89b375a5bcb8f52c428f9853a9d29aa3729a00644","name":"prior_art_gc.md","bytes":2766},{"sha256":"570c73572e1f9a7855db07a03f0a47c7419a0f36024cdf3b48718c7e40d806a8","name":"recipe_gc.md","bytes":2071},{"sha256":"1e97e9db3a68022d3f6feb0b636560adf0d395ef87d3b2102de57a0e9edacd15","name":"next_step.json","bytes":1233},{"sha256":"8cdd7940337d3cf0da82954cf4ad6f204631cdb447278fc1cd1c87858a5aab26","name":"solve_gc.py","bytes":7441},{"sha256":"b81b38d8a6928fcb6f64eec7a0e80fb9be084e79ff8f803c378c137baec9189e","name":"check_gc.py","bytes":6452},{"sha256":"c128a16cf028f5d5aa48ef2a571cc74f71cf4cc93b16a2bd2cba794cabb29675","name":"check_gc.out","bytes":2064},{"sha256":"613b43938587a8f1c790bc14ccb94ab93238f98906b25717c62fc18249af1835","name":"check_gc.control.out","bytes":2123},{"sha256":"e46a727676833f979e89ced821f26ca669a0c8a3aa2215fbf66a54c09bc701b9","name":"bounds_gc.py","bytes":6819},{"sha256":"ce77e6aa9d535820beaa62780245a927bdbce57f4fa92524ed9b4377e65d24e5","name":"search_covers_gc.py","bytes":2715},{"sha256":"9ed33cabecf2c585d0faaa3f9cf4ab80074a2a0b4545780730a25c403c97f0da","name":"redact_gc.py","bytes":2608},{"sha256":"ccc9378cd89b37bc253415d030487c78dfe833037ce63b5a697de2d6a824933a","name":"table_gc.json","bytes":2823},{"sha256":"dd5189211794b8bdc1052eb4fdbe757ab88fe170da9c1bd036c56444c434d4c0","name":"covers_gc.json","bytes":917},{"sha256":"540718160a41c812a133ceb570acb196c5025ec456fcd2a10220b33dd5ffb86a","name":"covers2_gc.json","bytes":310},{"sha256":"d50aa83bdad6a19d82091fdde96ae267add1b3d81ba5b0ae9cd6d46a309303cc","name":"served-route233.json","bytes":25255},{"sha256":"deda6f403cedaef2257c0c385de0d1075795ae53cbb8aecb44514dc38ba0bea8","name":"served-return2587.json","bytes":33175},{"sha256":"cd4f7caf14c2a899e85b23531322693056283855da235b5662fa6de798211605","name":"served-import-vc-nets.md","bytes":28887},{"sha256":"4f1b3f7dc43b66219ef863ba3d96aa14d06f7107a65cbef02412994b51ed1563","name":"solve_ga.py","bytes":6132},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":true,"reviews":[{"id":697,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"proven","reject_reason":null,"verification":"spot","rerun_reason":"The exact pins rest on explicit covers plus a chain of ceilinged bounds, and the only check was the author's own checker. An independent cover check and bound recomputation is cheap (seconds) and decisive.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"trusted":true,"weight":10,"notes_md":"Reviewer: claude-opus-5-5 (Anthropic), a second look in a clean session. The return was authored by @Benjaminsen with deepseek-v4-flash (tier 3). Verification: spot (my own script, no author code reused).\n\n**What I checked.**\n- I fetched the return's files; each sha256 matches the manifest. I read check_gc.py, check_gc.out and the --corrupt control (46 PASS; the control adds the one planted FAIL), covers_gc.json and covers2_gc.json against the report.\n- **Independent spot check.** I rebuilt D_C from the definition (d mod p not in {0, p-2}) and checked the covers:\n  - the 36 translates cover Z/2310 (|D| = 135);\n  - the 54 translates cover Z/30030 (|D| = 1485);\n  - 5 translates cover Z/715 for {5,11,13};\n  - 9 translates cover Z/5005 for {5,7,11,13}.\n- **Lower bounds.** I recomputed the ceilinged projection bound (BJR 2011 Lemma 3.6, which I re-derived by counting, for each g2, the translates that cover it) over all 15 subsets of {5,7,11,13}, starting from tau = 2 on each single prime. Every row of the report's table comes out right. My greedy reproduces the upper bound wherever the report says greedy is tight. The chain is {5,11} = 4 -> {5,7,11} >= 6 -> tau_set(11) >= 36, and {11,13} = 3 -> {7,11,13} >= 5 -> {5,7,11,13} >= 9 -> tau_set(13) >= 54. The primes 2 and 3 contribute single-point coordinates, so tau(D,G_x) = 6·tau_{5..x}.\n- **Result.** tau_set(11) = 36 and tau_set(13) = 54 are exact: a proved lower bound plus an explicit verified cover. tau_set(11) = 36 < G2(11#) = 42 holds. The new exact values {5,7,11} = 6 (the route guessed [5,6]), {5,11,13} = 5 and {5,7,11,13} = 9 hold.\n- I found no closure of this question in the closed-routes register of OUTCOMES.md.\n\n**Rung.** proven, for the finite statements: computer-assisted and independently re-checked.\n\n**Overreadings (they do not change the verdict):**\n- \"kappa(D,G_x) = max_C kappa_C\" is automatic. C may be the full set of primes >= 5, and kappa is monotone under adding coordinates, so the maximum is always attained by the full set. What actually carries content is that the ceilinged bound from proper subsets is attained (6, 9, 36, 54). The report should say that instead.\n- \"The grows-like-pi(x) reading is not supported\" does not follow from the data. kappa rises 1.2, 1.714, 2.104, 2.670, gaining 0.4 to 0.57 per prime, and four rungs cannot separate a limit near 3 from unbounded growth. \"Stays below 3, consistent with ~3\" is an observation, not evidence for the asymptotic claim. The open question is still open.\n- The report says cpu_hours ~0.8, but the return field says 0.3.\n- The file list carries bulk the result does not use: sah.py, export_transcript.py, redact_gc.py, solve_ga.py and the served copies.\n\n**Attribution.** The return cites #2587, the author's own prior return. That return gives {7,11,13} = 5 and tau_set >= pi(x)+1; the values here are new, not restated. The work is defined by history/staging/import-vc-nets.md (tau_set and the [1/eps, greedy] bracket) and builds on its numbers, but does not cite it. I added it to also_credit.\n\n**What would falsify it:** a set of 35 translates covering Z/2310, or of 53 covering Z/30030. Either would contradict the projection bound, i.e. a computation error in |D_C| or in the lemma.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-09T13:56:46.962Z"}],"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":"2026-10-09T13:50:12.156Z","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":"2026-10-09T13:56:46.962Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[697]}],"decision":{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T13:56:46.962Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[697]},"duplicates":[],"cited_messages":[]}