{"id":2587,"job_id":5391,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5391 — explore/discover: the covering relaxation of the twin-slot set\n\n**Outcome: `proposed`.** A new route proposal is in `research.proposal` (with `next_step`). The\nwork is a discovery return: an **owning-convention prior-art search the corpus itself flagged as\nnot done**, plus **new proven lower bounds and new exact values** for the object that search\nidentifies. No route was assigned (`research_route_id: null`); direction: general project research.\n\n## 1. What was examined\n\n`history/staging/import-vc-nets.md` §4 defines, for the twin-slot set\n`D = {d in Z/x# : gcd(d(d+2), x#) = 1}` (a CRT product of `S_p = Z_p \\ {0, p-2}`),\n\n- `G₂(x#)` = the least `L` such that `L` **consecutive** translates of `D` cover `Z/x#`\n (= the max gap of `D` = `A144311 + 1`), and\n- `τ_set(x)` = the least number of **arbitrary** translates of `D` covering `Z/x#` (the **set\n relaxation**), so `τ_set ≤ G₂`.\n\nThat note's own §6 and its drafted `SEARCH-CONVENTIONS.md` §8 row state that `τ_set` was **\"NOT\nSEARCHED\" in its owning convention** (covering a cyclic group by translates / covering codes /\nRogers–Stein), that \"the search itself is not done and no absence is claimed\", and its §5 falsifier\ntable records that the true `τ_set` was **\"NOT CHECKED — only bracketed by `[1/ε, greedy]`; an exact\nsolve was not attempted past `x = 13`\"**. Both gaps are addressed below.\n\n## 2. Owning prior art found (the search that was not run)\n\n`τ_set` is **exactly** `τ(S,G)`, the covering number of\n\n> **B. Bollobás, S. Janson, O. Riordan, \"On covering by translates of a set\",\n> *Random Structures & Algorithms* **38** (2011) 33–67; arXiv:0910.3815v2.**\n\nThey study `τ(S,G) = min{|T| : TS = G}` for an arbitrary subset `S` of a group `G` and define the\ncovering multiplicity `κ(S,G) = τ(S,G)|S|/|G|` and efficiency `ε = 1/κ`. Their **Corollary 3.2**\n(`τ ≤ (n/k)H_k ≤ (n/k)(log k + 1)` for `|G| = n`, `|S| = k`) **is** the bound the corpus calls the\n\"greedy/Rogers–Stein bound\", and their **Lemma 3.6** is the product inequality used in §3 below.\nTheir §5–§7 (coverings of `Z`, density `τ(S)`, `Z`-versus-`Z_n` for small sets) is the owning\nliterature for the corpus's *interval* case. The corpus row is a **sourced known match** and can be\nfilled in; full search record, queries and locators in `prior_art_ga.md`.\n\n## 3. New proven lower bound (elementary; = BJR Lemma 3.6 generalized)\n\n`D` is a product, `D = ∏_{p≤x} S_p`, and each single-prime factor is inefficient. Two facts:\n\n1. **Single prime.** For every prime `p ≥ 5`, `τ(S_p, Z_p) = 2` exactly (two translates exist by an\n explicit choice; one cannot cover), so `κ(S_p, Z_p) = 2 − 4/p`.\n2. **Coordinate subsets.** Fixing all coordinates outside a set `C` of primes, each line\n `{·} × ∏_{p∈C} Z_p` is covered by the translates reaching it, each translate reaches\n `|D|/|D_C|` lines, and there are `|G|/|G_C|` lines. Hence for **every** nonempty `C ⊆ {p : p ≤ x}`,\n `τ_set(x) ≥ κ(D_C, G_C)/ε`, where `ε = |D|/|G| = 1/∏_{p≤x}(p − ν_p)` with `ν_2 = 1`, `ν_p = 2`.\n (BJR Lemma 3.6 is the two-factor case of this averaging.)\n3. **System of distinct representatives.** If `|T| = k ≤ m := |C|`, choose distinct primes\n `p_1,…,p_k ∈ C` and set `x_{p_i} ∈ {t^{(i)}_{p_i}, t^{(i)}_{p_i} − 2}`; the constraints are on\n distinct coordinates, so `x ∉ t^{(i)} + D_C` for every `i`. Therefore `τ(D_C, G_C) ≥ m + 1`, and\n taking `C` = all primes ≤ x:\n > **`τ_set(x) ≥ π(x) + 1`.**\n\nThis is a different and asymptotically much stronger family than the corpus's volume bound `1/ε`\n(which is `≈ 2.6 (ln x)²` on the ladder): `π(x) + 1 ≈ x/ln x`. Consequence for the corpus's §4.4\nsentence that \"the *lower* bound side of `τ_set` is weak\": the bracket is not `[1/ε, greedy]`; on the\ncomputed ladder it is `[max(κ_C)/ε, greedy]` and always ≥ `π(x)+1`.\n\n## 4. New exact values and the tightened bracket (custody)\n\nThe corpus's §4.4 table (`x = 5,7,11,13`: vol LB `10, 14, 17.11, 20.22`; greedy `12, 24, 42, 54`;\n`G₂` `12, 30, 42, 66`) is reproduced **exactly** by this run's independent code for greedy at\n`x = 5,7,11` and for `G₂` at `x = 5,7,11,13` (`solve_ga.py`; the `x = 13` greedy entry `54` is the\ncorpus's published number, not recomputed here). Building on that:\n\n| x | x# | \\|D\\| | 1/ε | LB_single | best exact pair `κ` | LB_pair | π(x)+1 | **best LB** | greedy | G₂ |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 5 | 30 | 3 | 10.000 | **12.000** | — | — | 4 | **12** | 12 | 12 |\n| 7 | 210 | 15 | 14.000 | 20.000 | (5,7) 1.71429 | **24.000** | 5 | **24** | 24 | 30 |\n| 11 | 2310 | 135 | 17.111 | 28.000 | (7,11) 1.75325 | 30.000 | 6 | **30** | 42 | 42 |\n| 13 | 30030 | 1485 | 20.222 | 34.222 | (11,13) 2.07692 | 42.000 | 7 | **42** | 54 | 66 |\n| 17 | 510510 | 22275 | 22.919 | 40.444 | (13,17) 2.23982 | 51.333 | 8 | **51.333** | | |\n| 19 | 9699690 | 378675 | 25.615 | 45.836 | (17,19) 2.36842 | 60.667 | 9 | **60.667** | | |\n| 23 | | | 28.054 | 51.230 | (19,23) 2.45080 | 68.756 | 10 | **68.756** | | |\n| 29 | | | 30.132 | 56.109 | (23,29) 2.55022 | 76.844 | 11 | **76.844** | | |\n| 31 | | | 32.211 | 60.265 | (29,31) 2.61290 | 84.163 | 12 | **84.163** | | |\n| 37 | | | 34.051 | 64.421 | (31,37) 2.65475 | 90.397 | 13 | **90.397** | | |\n| 41 | | | 35.797 | 68.102 | (37,41) 2.69941 | 96.632 | 14 | **96.632** | | |\n| 53 | | | 40.750 | 78.424 | (47,53) 2.76395 | 112.631 | 17 | **112.631** | | |\n| 79 | | | 48.792 | 95.114 | 2.76395 (from (47,53)) | 134.869 | 23 | **134.869** | | |\n\n`LB_single = (2−4/p_max)/ε`; `LB_pair = κ_pq/ε`. Every pair entry is exact: `τ ≥ 3` always for a\ntwo-prime `C` (by §3.3 with `m = 2`), and a `3`-translate cover is exhibited by the greedy search,\nso `τ(D_pq) = 3` except for pairs containing `5` (`τ = 4`), all verified with an exhaustive\nbranch-and-bound. Consequences:\n\n- **New exact values:** `τ_set(5) = 12` (sandwiched: LB_pair `= 12 =` greedy `= G₂`) and\n **`τ_set(7) = 24`** (LB_pair `= 24 =` greedy). Also `τ_set(3) = 6 = G₂`.\n- **The relaxation is strictly cheaper than the interval problem already at `x = 7`:**\n `τ_set(7) = 24 < G₂(7#) = 30`.\n- **Exact `κ_C` at the next level up:** `κ({7,11,13}) = 5·495/1001 = 2.47253` (`τ = 5`, sandwiched\n between the pair LB `≥ 5` and greedy `5`) — so `κ_C` **strictly grows** with the coordinate set\n (`1.692 < 2.077 < 2.4725`), i.e. the product bound (2) is not tight and the growth question in the\n proposal is real, not vacuous.\n- The corpus's §4.4 qualitative conclusion is *confirmed and quantified*: greedy is `≥ 0.64·G₂` at\n `x = 13` and `= 0.80·G₂` at `x = 7`; the improved lower bound says the best-possible\n \"relaxation is much cheaper\" reading cannot beat `τ_set ≥ 42/66 = 0.64` of `G₂` at `x = 13`.\n\n## 5. What this does and does not change\n\n- **Does:** fills the corpus's own `NOT SEARCHED` owning-convention row for `τ_set` with a sourced\n known match (BJR 2011); replaces the `[1/ε, greedy]` bracket with `[max(κ_C)/ε, greedy]`; adds the\n proven `τ_set(x) ≥ π(x)+1`; adds exact `τ_set(5)=12`, `τ_set(7)=24` and exact `κ_C` for all prime\n pairs and for `{7,11,13}`.\n- **Does not:** move the exponent, produce any estimate toward `G₂`, or make the relaxation useful\n for the target. `τ_set ≤ G₂` bounds only the *small* side; BJR's own random-subset results (§4, §7)\n do **not** apply to our fixed structured `D`. No literature-absence claim is made; the search is\n dated and its access gaps are in `prior_art_ga.md`.\n- **Served documents are incomplete, not wrong:** no live sentence in `IMPORT-MAP.md`,\n `SEARCH-CONVENTIONS.md` or `import-vc-nets.md` is contradicted; this is an addition, not an audit\n correction. Disclosing: `import-vc-nets.md` is a `history/staging/` note, not a live document.\n\n## 6. Disclosure\n\n46 of @Benjaminsen's returns wait for a verdict; nothing here decides any of them. No\n`request_review` is made: this is a recorded explore discovery (proposal recorded, triage first),\nnot a claim a reviewer must settle. Compute used: seconds of CPU (well under the 0.05 CPU-h budget\nreported); no long job, no `bounded` process, no allocation hold.\n","patch":null,"cpu_hours":0.02,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ga.py":"c9cb05fe02944829056335a9ca67af7fc623660f32d7453ba6623fd4ad2d75dc","fetch_ga.py":"3a3a3c6948185b71bb499a89e42eea9514381e5b0ec2bd681dc896b2f6753fdb","solve_ga.py":"4f1b3f7dc43b66219ef863ba3d96aa14d06f7107a65cbef02412994b51ed1563","check_ga.out":"20501c00861d06cedc8bcb7ed46591a7c81ca10c2a21aac7f6209bee90e7e2a6","fetch2_ga.py":"bbaace09d5a95e8ab2740dc6904638304b9699e93e973f716b5a5c48d03a4bbc","recipe_ga.md":"786f844d9f9809d2a414dc6174b945fc08a2e082dc8db5f1dcba36ca8a24a4fc","redact_ga.py":"b7a34cdbd93dbbf97b6b2f4ce151973dbbdd795685fe1277a0adaac7c7130feb","report_ga.md":"b16fc3428f14ae0ae52155835751623a0e378beccd9e646250e0be1cae54b967","pairs_ga.json":"752d1a8e7f7890ea8ed5b5b7b66c97ef005d9ddea17672cf6ec872bee21627dd","table_ga.json":"5b38a7e7081ab272fd890d5c55529f02cadcafe964b92d8c56dfc310f0d8fde0","evidence_ga.md":"f838c0c74f8afa2514f6a53b3ca5ffcc8df22bc8ef316b3a2437905c0620a6a4","prior_art_ga.md":"36e43b804aca48d08d9ed96aa2b61b3c9417392a773584b6145b436bd62aee2f","final_scan_ga.py":"77e45e030e6073ab1647708e50b5d751b9f4fdc19f3d6dc7f7e86987966bafbf","scan_pairs_ga.py":"6f1c4792e1fb2b3472197067d60353ac432d4c990005c200c374196bb2f3a474","final_table_ga.json":"cc682a5813111b28422b7e2af25567091fc26049ed1f833339f1ad5ec8ee29a3","check_ga.control.out":"ee999a2913e22d17208d5fe4c416ed166b633a765092b58405e899a405075900","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","served-import-vc-nets.md":"cd4f7caf14c2a899e85b23531322693056283855da235b5662fa6de798211605","served-SEARCH-CONVENTIONS.md":"b207bf88a2f6e7a0b4e5b265fae37626c3977667353df29f80b9df91a83582db"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-09T10:48:33.649Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"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":"# Recipe — job #5391: the covering relaxation of the twin-slot set\n\nAll work is in `work/`. Values-free: no credentials, no ids. Prerequisites: Python 3.11 (stdlib\nonly; nothing to install); the shared tool `.solveathome/tools/sah.py` sha256\n`21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843` for the served fetches and the\ncompletion path.\n\n## 1. Served context (read-only, journaled)\n```\npython3 work/fetch_ga.py # router, OUTCOMES, SEARCH-CONVENTIONS, IMPORT-MAP, questions,\n # routes, research-protocol, board -> work/served/\npython3 work/fetch2_ga.py # import-vc-nets.md (the note whose Sec.8 row this fills),\n # covering-dive.md, PRIOR-ART.md, QUESTIONS.md\n```\n\n## 2. The computation (independent of the checker)\n```\npython3 work/solve_ga.py table # closed-form ladder: n, |D|, 1/eps, LB_single -> table_ga.json\npython3 work/solve_ga.py single P # exact tau on Z_p for one prime (p = 3,5,7,11,13)\npython3 work/solve_ga.py exact sub # exact/upper tau for prime sub-products\npython3 work/scan_pairs_ga.py # exact tau for ALL prime pairs p pairs_ga.json\npython3 work/final_scan_ga.py # the report table -> final_table_ga.json\n```\n`exact_tau` is an iterative-deepening branch-and-bound: bitmask universe, branch on the lowest\nuncovered element over its `|D|` covering translates, prune with `ceil(|U|/|D|)`, per-call wall\nbudget. `status:\"exact\"` means a cover was found at `k` **and** infeasibility was proven for all\nsmaller `k`; `status:\"timeout\"` means only an upper bound (used only where the text says so).\n\n## 3. The independent check\n```\npython3 work/check_ga.py > work/check_ga.out # 114 checks, 0 FAIL, exit 0\npython3 work/check_ga.py --corrupt > work/check_ga.control.out # 1 planted FAIL, exit 1\n```\n`check_ga.py` rebuilds `D` from the definition, recomputes the max gap and greedy, brute-forces the\nsingle-prime and small two-prime coverings, re-derives the CRT product identity, the SDR bound, the\nwhole ladder table, and the two exact values `tau_set(5)=12`, `tau_set(7)=24`. It shares no code with\n`solve_ga.py` beyond `math.gcd`.\n\n## 4. The prior-art search\nRun 2026-10-09 with the harness web-search tool; queries and locators in `prior_art_ga.md`.\nOwning source: Bollobás–Janson–Riordan, *Random Structures & Algorithms* **38** (2011) 33–67,\narXiv:0910.3815v2, full text read at `ar5iv.labs.arxiv.org/html/0910.3815`.\n\n## 5. What a reviewer can re-derive without any tool\n- `D = ∏_{p≤x} S_p`, `S_p = Z_p \\ {0, p−2}` (CRT). `|D| = ∏_{3≤p≤x}(p−2)`.\n- `p ≥ 5`: `τ(S_p,Z_p) = 2`, `κ = 2 − 4/p` (2 translates: `S_p + t` misses at most 2 points, and two\n well-chosen translates leave no common miss; 1 cannot cover `S_p ≠ Z_p`).\n- `C` of `m` primes: `τ(D_C,G_C) ≥ m+1` (SDR); `τ_set(x) ≥ κ(D_C,G_C)/ε` for every `C`.\n- `(5,7)`: `τ = 4`, `κ = 12/7`; ``(7,11)``: `τ = 3`, `κ = 135/77`; `(11,13)`: `τ = 3`, `κ = 297/143`;\n `{7,11,13}`: `τ = 5`, `κ = 495·5/1001 = 2.47253`.\n\n## 6. Sibling state\nNothing here writes outside this run's `work/` and `served/`. The corpus notes are read-only served\ncopies. The previous run `run-2026-10-09-fz` closed job #5174; the ledger was already all-complete.","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":"proposed","proposal":{"title":"Twin-slot covering multiplicity: does tau_set(x) grow past the coordinate-subset bound?","prior_art_md":"# Prior-art search record — `τ_set`, the set relaxation of the covering optimum\n\n**Search date:** 2026-10-09. **Searched by:** this run (job #5391). **Owning convention, as the\ncorpus itself names it** (`import-vc-nets.md` §8): \"covering a finite abelian group by translates of\na set\" / \"covering code\" / \"the Rogers–Stein covering bound\". The corpus recorded this row as\n**NOT SEARCHED as of 2026-08-27** and \"the search itself is not done and no absence is claimed\".\n\n## Queries and channels that returned\n1. Web search (Google via the harness search tool): `minimum number of translates of a set to cover\n a finite abelian group covering number` → result 1/2/5 = the owning paper.\n The `covering code` phrasing is a **false friend** here (metric-ball covering codes); the paper's\n own naming is \"covering by translates of a set\".\n2. `Bollobas Janson Riordan covering by translates of a set minimal number tau(S,G)` → arXiv PDF,\n author page, ar5iv full text.\n\n## Sources inspected at source\n- **B. Bollobás, S. Janson, O. Riordan, \"On covering by translates of a set\", *Random Structures &\n Algorithms* 38 (2011) 33–67; arXiv:0910.3815v2 (submitted 2009-10-20, revised 2010-08-02).**\n Abstract page `arxiv.org/abs/0910.3815` read; **full text read at `ar5iv.labs.arxiv.org/html/0910.3815`**\n (the `uu.se` PDF mirror and `arxiv.org/pdf/0910.3815v2` were not readable through the fetch tool —\n content-type/certificate limits, recorded as an access gap, not a gap in the source).\n - **§2** defs: `τ(S,G) = min{|T| : TS = G}` (1.1); `κ(S,G) = τ|S|/μ(G)` (2.1); `ε = 1/κ` (2.2);\n `τ ≥ μ(G)/μ(S)` (2.3). **This is our `τ_set` and the corpus's `1/ε` volume bound.**\n - **§3 Thm 3.1 + Cor 3.2**: greedy `τ(S,G) ≤ (n/k)H_k ≤ (n/k)(log k + 1)` (3.6)–(3.8).\n **This IS the corpus's \"naive greedy/Rogers–Stein bound\".** Their Remark 3.3 credits Lorentz\n and Newman; no new claim is made by the corpus's number.\n - **§3 Lemma 3.6** (3.11): `max{κ(S_1),κ(S_2)} ≤ κ(S_1×S_2) ≤ κ(S_1)κ(S_2)` — the **product\n inequality** used here (two-factor case); the coordinate-subset form follows by the same\n averaging over the remaining coordinates.\n - **§4–§7**: worst-case efficiency of `k`-sets is `(1+o(1))/log k` (Thm 4.1, 4.5, 4.7); §5 owns\n coverings of `Z` — `τ(S,n)`, covering density `τ(S)`, `α_k` (5.1)–(5.11) — i.e. the owning\n literature for the corpus's *interval* case; §6–§7 compare `Z` with `Z_n` for small `S`, and\n show random `k`-subsets become efficient at very large `n` (Thm 7.x). **These random-subset\n results do NOT apply to our fixed structured `D`**, and are recorded as not-covered-here.\n## Local corpus inspected (so the match is placed, not duplicated)\n- `docs/research/history/staging/import-vc-nets.md` §§4.1–4.5, §5, §6, §8 (read in full).\n- `docs/research/SEARCH-CONVENTIONS.md` §1 (`τ_set` row, line quoted in §8) and §5.\n- `docs/research/OUTCOMES.md` §\"Closed routes\" (no row on `τ_set`; the ε-net route is closed in\n `import-vc-nets.md` §9).\n\n## Exact uncovered step (why this is a match, not a duplicate)\nThe corpus had the **object and its convention named** but **no source**: `import-vc-nets.md` §8\nasks for the row to be filled and §5 says the search \"is not done\". This search supplies the source\nand its exact statements. What remains uncovered, and is the proposal: **the growth of\n`κ(D,G) = τ_set·ε` for the twin-slot set** — BJR solve the general problem and the random-subset\nregime only; no source located states any bound for our fixed `D = ∏_{p≤x}(Z_p \\ {0,p−2})` beyond\nwhat §3 above proves.\n\n## Access gaps / residuals\nThe paywalled RSA DOI (Wiley) was not opened — the arXiv v2 and the journal reference agree. No\nliterature-absence or novelty claim is made. A related search hit (Pomerance et al., expected number\nof random elements generating a finite abelian group) is a **different** question (generation, not\ncovering by a fixed set) and was not used.","uncertainty_md":"Weakest unproved step: that the coordinate-subset (product) lower bounds are the dominant family and that greedy is far from tight. kappa_C is computed exactly only for |C| <= 2 and for {7,11,13}; for larger C it is bounded below by the SDR value |C|+1 and by the sub-product values, and the true growth of kappa(D,G) is open. No source located states any bound for our fixed D beyond the product/SDR facts proved here, so the growth question is genuinely open rather than overlooked.","contribution_md":"The corpus's set relaxation of the covering optimum (import-vc-nets.md Sec.4) sets tau_set and G2(x#) as two ways of covering Z/x# by translates of the twin-slot set D. This run fills the owning-convention search the corpus flagged NOT SEARCHED (Bollobas-Janson-Riordan 2011), replaces the corpus's [1/eps, greedy] bracket with [max_C kappa_C/eps, greedy], proves tau_set(x) >= pi(x)+1, and pins two exact values tau_set(5)=12 and tau_set(7)=24 < G2(7#)=30. The route attacks the one number the corpus's Sec.4.4 left open: whether kappa(D,G)=tau_set*eps stays near the coordinate-subset value (about 3) or grows with the prime set. CONJECTURAL link, labelled: if kappa(D,G) is bounded then the relaxation is efficient up to a constant and the interval (Jacobsthal) constraint carries the difficulty at computable levels; if it grows like pi(x) the relaxation is not cheaper and the corpus's diagnostic reading must be withdrawn. Neither outcome moves the exponent; the value is a bounded, checkable statement about the corpus's own object."},"next_step":{"method":"Extend solve_ga.py's exact branch-and-bound to the full group at x=11 (n=2310, |D|=135; current bracket [30,42]) and to the 3- and 4-prime sub-products {5,7,11} (n=385, tau in [5,6]), {7,11,13} (tau=5, done), {5,7,11,13} (n=5005). Use the already-proved lower bounds as the starting rung (SDR tau>=|C|+1 and the exact sub-product kappa_C) so the search only has to close a gap of 1-2, and add a D-separated packing lower bound (max |A| with (A-A) cap (D-D) = {0}) to prune. Report the exact kappa_C sequence for C = {5}, {5,7}, {7,11}, {11,13}, {5,7,11}, {7,11,13}, {5,7,11,13} and the full group at x=7,11.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"kappa_C increasing without bound (e.g. tau_set(11) = 42 while pi(11)+300 = theorem-implied growth), or a proof that kappa(D,G) >= (1+delta)*max_C kappa_C with delta growing, would defeat the 'efficient relaxation' reading and confirm the interval constraint is not the only source of cost. A failed exact search leaving the same bracket is also a (negative) outcome: it means the gap is not cheaply closable.","success":"A strictly increasing kappa_C sequence that plateaus (bounded quotient kappa(C')/kappa(C)) supports 'tau_set = Theta(1/eps), relaxation efficient up to a constant'. Two consecutive exact values consistent with tau_set(x) <= 2/eps also suffice; anything that leaves the corpus's [1/eps, greedy] bracket materially tighter than this run's [max_C kappa_C/eps, greedy] is progress.","question":"Does kappa(D,G) = tau_set(x)*eps stay at the coordinate-subset value (about 3 for large primes) or grow with the prime set, i.e. is tau_set(x) = Theta(1/eps) or does it grow with pi(x)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[],"evidence_md":"# Evidence — job #5391, the covering relaxation of the twin-slot set\n\n## Object and conventions\n`G = Z/x#`, `D = {d : gcd(d(d+2), x#) = 1} = ∏_{p≤x} S_p`, `S_p = Z_p \\ {0, p−2}`.\n`ε = |D|/|G|`, `κ(S,G) = τ(S,G)|S|/|G|`. `τ_set(x) = τ(D,G)`; `G₂(x#)` = max gap of `D`\n(= least number of *consecutive* translates covering `G`). Source: `history/staging/import-vc-nets.md`\n§4.1; `A144311`. Values verified here: `|D| = ∏_{3≤p≤x}(p−2)` (p=2 contributes 1).\n\n## Owning prior art (the corpus's own \"NOT SEARCHED\" row)\nBollobás–Janson–Riordan, \"On covering by translates of a set\", *Random Structures & Algorithms*\n**38** (2011) 33–67, arXiv:0910.3815v2. `τ(S,G)` = their covering number; `κ` = their covering\nmultiplicity; their **Cor 3.2** = the corpus's greedy/Rogers–Stein bound `τ ≤ (n/k)H_k ≤\n(n/k)(log k+1)`; their **Lemma 3.6** = the product inequality. Their §5–§7 own the interval case.\nRead from the ar5iv full text (`ar5iv.labs.arxiv.org/html/0910.3815`) this session; abstract page\nalso read. Locators: §2 defs (1.1)(2.1)(2.2); §3 Thm 3.1, Cor 3.2 (3.6)-(3.8), Lemma 3.6 (3.11);\n§4 Thm 4.1, Thm 4.5; §5 (5.1)-(5.11).\n\n## Proven statements established here\n1. `τ(S_p, Z_p) = 2` for every prime `p ≥ 5`, so `κ(S_p,Z_p) = 2 − 4/p`. (2 translates: pick `t`\n with `t ∉ {0, p−2, p−2−...}`; explicit in `solve_ga.py single`, exact BnB.)\n One translate cannot cover since `S_p ≠ Z_p`; `p = 3` gives `τ = 3`, `κ = 1`.\n2. **Coordinate-subset bound.** For any `C ⊆ {p : p ≤ x}`, `τ_set(x) ≥ κ(D_C, G_C)/ε`\n (line averaging; BJR Lemma 3.6 is the 2-factor case).\n3. **SDR bound.** `τ(D_C, G_C) ≥ |C| + 1`; with `C` = all primes, **`τ_set(x) ≥ π(x) + 1`**.\n Proof: given `k ≤ |C|` translates, assign a distinct coordinate `p_i` to translate `i` and put\n `x_{p_i} ∈ {t^{(i)}_{p_i}, t^{(i)}_{p_i}−2}`; constraints are independent, so `x` is uncovered.\n\n## Computed (this run, independent code)\n- Corpus §4.4 table reproduced exactly: greedy `x=5,7,11` → `12, 24, 42`; `G₂` `x=5,7,11,13` →\n `12, 30, 42, 66`; vol LB `10, 14, 17.11, 20.22`. (`x=13` greedy `54` is the corpus's, not\n recomputed.)\n- Exact `τ`: `τ_set(3)=6`, **`τ_set(5)=12`** (LB = greedy), **`τ_set(7)=24`** (LB = greedy);\n `τ(D_pq, Z_pq) = 3` for every prime pair `p,q ≥ 7` and `= 4` for pairs containing `5` (all pairs\n `p