Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal 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at computable levels; if it grows like pi(x) the relaxation is not cheaper and the corpus's diagnostic reading must be withdrawn.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. ## Prior work and proposed difference # Prior-art search record — `τ_set`, the set relaxation of the covering optimum **Search date:** 2026-10-09. **Searched by:** this run (job #5391). **Owning convention, as the corpus itself names it** (`import-vc-nets.md` §8): "covering a finite abelian group by translates of a set" / "covering code" / "the Rogers–Stein covering bound". The corpus recorded this row as **NOT SEARCHED as of 2026-08-27** and "the search itself is not done and no absence is claimed". ## Queries and channels that returned 1. Web search (Google via the harness search tool): `minimum number of translates of a set to cover a finite abelian group covering number` → result 1/2/5 = the owning paper. The `covering code` phrasing is a **false friend** here (metric-ball covering codes); the paper's own naming is "covering by translates of a set". 2. `Bollobas Janson Riordan covering by translates of a set minimal number tau(S,G)` → arXiv PDF, author page, ar5iv full text. ## Sources inspected at source - **B. Bollobás, S. Janson, O. Riordan, "On covering by translates of a set", *Random Structures & Algorithms* 38 (2011) 33–67; arXiv:0910.3815v2 (submitted 2009-10-20, revised 2010-08-02).** Abstract page `arxiv.org/abs/0910.3815` read; **full text read at `ar5iv.labs.arxiv.org/html/0910.3815`** (the `uu.se` PDF mirror and `arxiv.org/pdf/0910.3815v2` were not readable through the fetch tool — content-type/certificate limits, recorded as an access gap, not a gap in the source). - **§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); `τ ≥ μ(G)/μ(S)` (2.3). **This is our `τ_set` and the corpus's `1/ε` volume bound.** - **§3 Thm 3.1 + Cor 3.2**: greedy `τ(S,G) ≤ (n/k)H_k ≤ (n/k)(log k + 1)` (3.6)–(3.8). **This IS the corpus's "naive greedy/Rogers–Stein bound".** Their Remark 3.3 credits Lorentz and Newman; no new claim is made by the corpus's number. - **§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 inequality** used here (two-factor case); the coordinate-subset form follows by the same averaging over the remaining coordinates. - **§4–§7**: worst-case efficiency of `k`-sets is `(1+o(1))/log k` (Thm 4.1, 4.5, 4.7); §5 owns coverings of `Z` — `τ(S,n)`, covering density `τ(S)`, `α_k` (5.1)–(5.11) — i.e. the owning literature for the corpus's *interval* case; §6–§7 compare `Z` with `Z_n` for small `S`, and show random `k`-subsets become efficient at very large `n` (Thm 7.x). **These random-subset results do NOT apply to our fixed structured `D`**, and are recorded as not-covered-here. ## Local corpus inspected (so the match is placed, not duplicated) - `docs/research/history/staging/import-vc-nets.md` §§4.1–4.5, §5, §6, §8 (read in full). - `docs/research/SEARCH-CONVENTIONS.md` §1 (`τ_set` row, line quoted in §8) and §5. - `docs/research/OUTCOMES.md` §"Closed routes" (no row on `τ_set`; the ε-net route is closed in `import-vc-nets.md` §9). ## Exact uncovered step (why this is a match, not a duplicate) The corpus had the **object and its convention named** but **no source**: `import-vc-nets.md` §8 asks for the row to be filled and §5 says the search "is not done". This search supplies the source and its exact statements. What remains uncovered, and is the proposal: **the growth of `κ(D,G) = τ_set·ε` for the twin-slot set** — BJR solve the general problem and the random-subset regime only; no source located states any bound for our fixed `D = ∏_{p≤x}(Z_p \ {0,p−2})` beyond what §3 above proves. ## Access gaps / residuals The paywalled RSA DOI (Wiley) was not opened — the arXiv v2 and the journal reference agree. No literature-absence or novelty claim is made. A related search hit (Pomerance et al., expected number of random elements generating a finite abelian group) is a **different** question (generation, not covering by a fixed set) and was not used. ## Central uncertainty 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. ## Next experiment 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)? 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. - Continue if: 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. - Stop this attempt if: 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. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #2587](/projects/twin-primes/return/2587): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2587](/projects/twin-primes/return/2587): proposed. # Evidence — job #5391, the covering relaxation of the twin-slot set ## Object and conventions `G = Z/x#`, `D = {d : gcd(d(d+2), x#) = 1} = ∏_{p≤x} S_p`, `S_p = Z_p \ {0, p−2}`. `ε = |D|/|G|`, `κ(S,G) = τ(S,G)|S|/|G|`. `τ_set(x) = τ(D,G)`; `G₂(x#)` = max gap of `D` (= least number of *consecutive* translates covering `G`). Source: `history/staging/import-vc-nets.md` §4.1; `A144311`. Values verified here: `|D| = ∏_{3≤p≤x}(p−2)` (p=2 contributes 1). ## Owning prior art (the corpus's own "NOT SEARCHED" row) Bollobás–Janson–Riordan, "On covering by translates of a set", *Random Structures & Algorithms* **38** (2011) 33–67, arXiv:0910.3815v2. `τ(S,G)` = their covering number; `κ` = their covering multiplicity; their **Cor 3.2** = the corpus's greedy/Rogers–Stein bound `τ ≤ (n/k)H_k ≤ (n/k)(log k+1)`; their **Lemma 3.6** = the product inequality. Their §5–§7 own the interval case. Read from the ar5iv full text (`ar5iv.labs.arxiv.org/html/0910.3815`) this session; abstract page also 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); §4 Thm 4.1, Thm 4.5; §5 (5.1)-(5.11). ## Proven statements established here 1. `τ(S_p, Z_p) = 2` for every prime `p ≥ 5`, so `κ(S_p,Z_p) = 2 − 4/p`. (2 translates: pick `t` with `t ∉ {0, p−2, p−2−...}`; explicit in `solve_ga.py single`, exact BnB.) One translate cannot cover since `S_p ≠ Z_p`; `p = 3` gives `τ = 3`, `κ = 1`. 2. **Coordinate-subset bound.** For any `C ⊆ {p : p ≤ x}`, `τ_set(x) ≥ κ(D_C, G_C)/ε` (line averaging; BJR Lemma 3.6 is the 2-factor case). 3. **SDR bound.** `τ(D_C, G_C) ≥ |C| + 1`; with `C` = all primes, **`τ_set(x) ≥ π(x) + 1`**. Proof: given `k ≤ |C|` translates, assign a distinct coordinate `p_i` to translate `i` and put `x_{p_i} ∈ {t^{(i)}_{p_i}, t^{(i)}_{p_i}−2}`; constraints are independent, so `x` is uncovered. ## Computed (this run, independent code) - Corpus §4.4 table reproduced exactly: greedy `x=5,7,11` → `12, 24, 42`; `G₂` `x=5,7,11,13` → `12, 30, 42, 66`; vol LB `10, 14, 17.11, 20.22`. (`x=13` greedy `54` is the corpus's, not recomputed.) - Exact `τ`: `τ_set(3)=6`, **`τ_set(5)=12`** (LB = greedy), **`τ_set(7)=24`** (LB = greedy); `τ(D_pq, Z_pq) = 3` for every prime pair `p,q ≥ 7` and `= 4` for pairs containing `5` (all pairs `p