# Covering Dive: Literature on One-Class vs Two-Class Jacobsthal, Covering Systems, and Interval Sieving **Date:** 2026-08-14. Deep technical literature dive (web search + primary PDFs + OEIS + MathOverflow API + erdosproblems.com). **Our object:** G₂(n) = largest gap between consecutive r mod Pₙ# with gcd(r, Pₙ#) = gcd(r+2, Pₙ#) = 1 — the sifted set for the two-class system Iₚ = {0, −2} mod p (odd p). Legend used throughout: **[PROVEN]** = published theorem with source; **[CONJ]** = published conjecture; **[ABSENT]** = we searched and found nothing published; **[INFERRED]** = our deduction from sourced facts, not itself in the literature. --- ## Q1. Iwaniec's method: the proof structure of g(q) ≪ (log q)² ### 1.1 The exact statements in the literature - Iwaniec, *On the problem of Jacobsthal*, Demonstratio Math. **11** (1978), 225–231. Universally cited form (e.g. [Erdős Problem #970](https://www.erdosproblems.com/970), verbatim: "Iwaniec [Iw78] proved h(k) ≪ (k log k)²"): **[PROVEN]** h(k) ≪ (k log k)², where h(k) = max of Jacobsthal's g(n) over n with ω(n) = k. Jacobsthal's own conjecture h(k) ≪ k² is open (Erdős #970, tagged "open"). **Say which conjecture, and mind which h: Jacobsthal made a second conjecture and it is dead, and the two conjectures are about two different functions that this repository calls by one letter.** Hajdu–Saradha, *Disproof of a conjecture of Jacobsthal*, [Math. Comp. **81** (2012), no. 280, 2461–2471](https://math.unideb.hu/sites/default/files/inline-files/jacobsrevsaradha.pdf), read at source 2026-08-18, set out the notation this line needs and the corpus lacks: **h(r) = j(p₁p₂…p_r)** (Jacobsthal at the primorial — this is OEIS A048670 and it is what the rest of this repository writes as `h(x#)`) against **H(r) = max_{ω(n)=r} j(n)** (the maximum over all n with r prime factors — this is the `h(k)` of the Iwaniec bound and of Erdős #970). Their abstract, verbatim: "In 1962 Jacobsthal conjectured that for any integer r ≥ 1, the maximal value of j(n) when n varies over N with ω(n) = r is attained when n is the product of the first r primes. **We show that this is true for r ≤ 23 and fails at r = 24**, thus disproving Jacobsthal's conjecture." So `H(r) = h(r)` is FALSE from r = 24 up (further counterexamples in Ziller, [arXiv:1903.11973](https://arxiv.org/abs/1903.11973), whose abstract says the conjecture "only applies to several small k"), while `H(r) ≪ r²` stays open. **[the extremality conjecture PROVEN FALSE; the order conjecture open]** Consequence for this repository, and it is not cosmetic: every one-class control we run — `exponent-control.md`'s 58-term fit, `PRIOR-ART.md`'s h(x#)/G₂ ratio table, Holt–Rudd's §4 table — is on **h**, A048670, and every quotation of Iwaniec's bound h(k) ≪ (k log k)² is on **H**. The two agree at r ≤ 23, which covers every level anyone here computes, so nothing measured moves; but the identification is a theorem with a known counterexample, not a definition, and should not be written as one. The implied **constant is inexplicit** — that is standard and true of Iwaniec's method, but neither #970 nor #687 says so in its own text, and the place the explicitness question is actually raised is MathOverflow [245539](https://mathoverflow.net/questions/245539). - For the primorial specialization, both Ford–Green–Konyagin–Tao ([arXiv:1408.4505](https://arxiv.org/abs/1408.4505), Annals 183 (2016)) and Ford–Green–Konyagin–Maynard–Tao ([arXiv:1412.5029](https://arxiv.org/abs/1412.5029), JAMS 31 (2018)) state verbatim, checked word for word in both PDFs: "The best upper bound known is **Y(x) ≪ x²**, which comes from Iwaniec's work [Iw78] on Jacobsthal's function," where Y(x) = the longest interval coverable choosing ONE residue class aₚ per prime p ≤ x. (`[Iw78]` expands each paper's own bibliography marker, which is `[23]` in FGKT and `[26]` in FGKMT; the sentence is otherwise identical in the two.) Since log Pₙ# ~ pₙ, this is exactly g(q) ≪ (log q)². **[PROVEN]** - Precursor, and **this is the `k^{2+ε}` result**: R. C. Vaughan, *On the order of magnitude of Jacobsthal's function*, [Proc. Edinburgh Math. Soc. 20 (1977) 329–331](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-the-order-of-magnitude-of-jacobsthals-function/BD6EBCDD8B8B3EDDF3AF69085DCD8F26). Sieve-based, slightly weaker than Iwaniec's, and Vaughan states the point in his own words on p. 329: "The purpose of this short note is to show that in (4) C can be taken arbitrarily close to 2. Iwaniec (2, Theorem 2) has shown this in the special case when n is the product of the first r primes." His theorem is `g(n) < ω(n)²(log₂ω(n))^c`. So Vaughan is the general-`n` exponent-2 statement and Iwaniec 1971 Thm 2 is the primorial case; both constants are inexplicit. **[PROVEN]** - The elementary line is **much weaker than exponent 2**, and is often miscited as though it were not. Read off Paseman's own §1, which states all three: **Kanold** (*Über eine zahlentheoretische Funktion von Jacobsthal*, Math. Ann. **170** (1967) 314–326) gives `2^k`, and `2^{√k}` for `k ≥ e^50`; **Stevens** (*On Jacobsthal's g(n)-function*, Math. Ann. 226 (1977) 95–97, Bonferroni inequalities) gives `g(n) < 2k^{2+2e·log k}`, whose `log₂` is `O((log k)²)`; **Paseman** ([arXiv:1311.5944](https://arxiv.org/abs/1311.5944)) improves Stevens to `2^{O(log k · loglog k)}`. As exponents in `k` those are `2^{√k}`, `k^{Θ(log k)}` and `k^{O(loglog k)}` — none is `k^{2+ε}`. Kanold's relevance runs the other way: Vaughan records that an exponent *below* 2 would give Linnik's theorem easily. **[PROVEN]** ### 1.2 The proof mechanism (from Granville's modern account) The clearest published modern description we found is Andrew Granville, *Sieving intervals and Siegel zeros*, **Acta Arithmetica 205 (2022), no. 1, 1–19**, DOI [10.4064/aa201002-25-6](https://doi.org/10.4064/aa201002-25-6); preprint **[arXiv:2010.01211](https://arxiv.org/abs/2010.01211)** (submitted 2 Oct 2020) ([PDF on his homepage](https://dms.umontreal.ca/~andrew/PDF/Sieve.Remark.20.09.26.pdf)). *(Both the arXiv id and the journal reference were added 2026-08-18; this line previously read "published version: Sieving intervals and Siegel zeros, 2020s" with no id, no venue and no year, which is not a checkable reference. The arXiv record itself still carries no journal-ref — the same trap Q4.2 records for Kalmynin–Konyagin, and it caught this citation too.)* Quotes below are from that paper (§1): - Setup: S(x, y, z) := #{n ∈ (x, x+y] : (n, P(z)) = 1}. This is a **linear (κ = 1) sieve** problem: "in our case each g(p) = 1 … Merten's Theorem allows us to take κ = 1, the linear sieve", with the **trivial interval error term |r(A,d)| ≤ 1** for every squarefree d | P(z), d ≤ D. - The Jurkat–Richert / Rosser–Iwaniec linear sieve gives S ≥ y·V(z)·(f(u) − o(1)) − remainder, with u = log D/log z and lower-bound function **f(u) = 2e^γ log(u−1)/u for 2 ≤ u ≤ 4, so f(u) > 0 iff u > 2**. Granville: "S(x,y,z) ≫ y/log y ⋅ f(u) fails at u = 2 since f(2) = 0. However f(u) > 0 for u > 2 and so it is of interest to understand S(x,y,z) when y = z^{2+o(1)}." - **The key theorem** (Granville, quoting Iwaniec [Demonstratio 1978]): "The key result in this range is due to Iwaniec who showed that if y ≫ z² then **S(x, y, z) ≥ (4y/(log y)²) · (log(y/z²) − O(1))**", uniformly in x. Granville's footnote 3: "A slightly weaker version of this result more-or-less follows from Iwaniec's much earlier Theorem 2 in [H. Iwaniec, *On the error term in the linear sieve*, Acta Arith. 19 (1971), 1–30]." So the engine is Iwaniec's refined linear sieve (error term analysis at the sifting limit), not merely the fundamental lemma (the fundamental lemma alone only handles u → ∞, not u near 2). - Consequence, per Granville: "Iwaniec's result above establishes that J(P(z)) ≪ z²; by the prime number theorem … J(m) ≪ (ω(m) log ω(m))², and **Iwaniec [Iw78] deduced (cleverly) that this upper bound then holds for all integers m.**" - The "clever deduction" is Iwaniec's **Lemma 1 — a shifted-sieve / transfer lemma**: per the transcription in MathOverflow [#245539](https://mathoverflow.net/questions/245539/paging-henryk-iwaniec-problems-in-lemma-1), Iwaniec posits a **multiplicative bijection l(·) between the divisors of a general squarefree Q and of a primorial-type P** with g(l(d)) ≤ f(d), and transfers the sieve estimate to the worst case where the r primes are the first r primes. (I.e., "one omitted class per arbitrary prime" reduces to "one class per smallest primes".) Notably that MO question (2016) — which claims a possible sign error inside the published proof of Lemma 1 — **has zero answers to date**; nobody has publicly walked through the 1978 proof. (Searched under Jacobsthal, the convention that owns the one-class object, whose OEIS home is A048670, and not under any house term.) There is no complete modern exposition: FGKT/FGKMT cite the result without proof, Opera de Cribro (Friedlander–Iwaniec, AMS Colloq. 57, 2010) contains the linear sieve technology but we found no worked Jacobsthal section, and Tao's large-gaps posts ([2014](https://terrytao.wordpress.com/2014/08/21/large-gaps-between-consecutive-prime-numbers/), [2014b](https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/)) discuss only the lower-bound side. ### 1.3 Where "one class per prime" enters, and what changes with two — as visible in the sources - **Entry point 1 — sifting dimension.** One omitted class per prime gives density product ∏_{p≤z}(1 − 1/p) ≍ e^{−γ}/log z: sifting dimension κ = 1, the *linear* sieve (Granville, §1, explicitly). Two omitted classes give ∏(1 − 2/p) ≍ C/(log z)²: **dimension κ = 2**, where the linear sieve's f, F functions and their u > 2 positivity threshold simply do not apply. - **Entry point 2 — the sifting limit is exactly the critical exponent only for κ = 1.** The lower-bound positivity threshold ("sifting limit" β_κ) is β₁ = 2, and this is **optimal**: Granville (§1) recounts that "Iwaniec [Acta Arith. 1971] and Selberg showed that this result is 'best possible'" via the Liouville-function examples A± = {n ≤ x : λ(n) = ∓1} — the parity obstruction. So the miracle of the one-class problem is: sifting limit (2) = twin-critical exponent (2). **[PROVEN]** - For κ = 2 the best published sifting limits are far above 2: the Diamond–Halberstam–Richert (DHR) sieve has **β₂ = 4.266** and Selberg's Λ²Λ⁻ sieve gives 4.516 at κ = 2 (table in C. S. Franze, *Sifting limits for the Λ²Λ⁻ sieve*, [arXiv:1012.3809](https://arxiv.org/abs/1012.3809), J. Number Theory 2011, Table 1; *A Higher-Dimensional Sieve Method*, Harold G. Diamond and H. Halberstam **with William F. Galway**, Cambridge Tracts in Mathematics **177** (2008), Ch. 17: β_κ ≲ 2.44κ — the book is Diamond–Halberstam–Galway; **Richert is not an author of it**, though "DHR sieve" remains the correct name for the sieve). **[PROVEN]** (the sifting-limit values; their application to G₂ is §"Where the proof breaks" below). --- ## Q2. Two-class attempts ### 2.1 Ziller–Morack (the only direct literature) - M. Ziller, J. F. Morack, *Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs*, [arXiv:1706.00317](https://arxiv.org/abs/1706.00317) (2017, unpublished preprint). Content verified from the PDF: - Defines weak divisibility k | (a,b) ⟺ k|a or k|b; the **paired Jacobsthal function** j₂(n) = smallest m such that every paired progression ⟨a,b⟩_m with 2 | (b−a) contains a pair coprime to n; h₂(n) := j₂(pₙ#) (their Def. 2.2). - **Conjecture 6 [CONJ]:** "Let n ≥ 3. Then h₂(n) < pₙ² − pₙ." Nothing about it is proven — no upper bound of any exponent is established in the paper. - **Proposition 3.2 / 3.5 [PROVEN]:** If h₂(k) < pₖ² − pₖ for all k ≥ 3, then the (tightened) Goldbach conjecture holds and, for every n and every prime p > 2n, there is a prime pair q₂ − q₁ = 2n with p < q₁ < p² — hence the twin prime conjecture and de Polignac-type prime-pairs conjecture. (Mechanism = exactly the "p² rule": an unsifted pair below pₖ² must be an actual prime pair.) - Note the pattern generality: j₂ takes the **worst case over all even offsets b−a**, so it simultaneously encodes twin-type (b = a+2) and Goldbach-type (b = a − 2N) patterns. **[INFERRED from definitions]**: our G₂(n) (fixed classes {0,−2}) is the difference-2 slice, so G₂-type quantities are ≤ the ZM h₂ up to boundary terms; their conjecture is formally stronger than the twin-relevant statement. - Companion: *A short note on the computation of the generalised Jacobsthal function for paired progressions*, [arXiv:1706.03668](https://arxiv.org/abs/1706.03668): algorithms + computed h₂(n) for the first 21 primorials (primes to 73); all values satisfy Conjecture 6. - OEIS: [A288815](https://oeis.org/A288815) (h₂-values 2, 6, 18, 30, 66, …, 2622 = a(21); comment: "If a(n) < pₙ² − pₙ holds for n ≥ 3 then Goldbach's conjecture and the twin prime conjecture hold as well") and [A072753](https://oeis.org/A072753) ("Maximum gap in two-stage prime-sieves", a(n) = (A288815(n) − 6)/6), with Giovanni Resta's ILP formulation and explicit optimal class-pairs recorded in the entry, computations by Morack/Resta/Ziller through a(21), flagged `hard,more`. - **Isolation of this literature:** Semantic Scholar lists exactly **one** citation of arXiv:1706.00317 — the authors' own computation note — and none for the note. No follow-up, no refereed version. **[ABSENT — re-checked 2026-08-18, and the sentence now needs three qualifications; see below]** 1. **The citation count is no longer verified.** `api.semanticscholar.org` returns 429 to every automated call and the HTML paper page comes back empty, so "exactly one" is a 2026-08-14 reading that has not been reproduced since. Anyone with a working Scholar or S2 session closes this in thirty seconds; until then treat the number as dated rather than current. 2. **"No follow-up" is too strong, though the author himself did not follow up.** Supporting the claim: Ziller's own two later papers — [arXiv:1903.11973](https://arxiv.org/abs/1903.11973) (2019) and [arXiv:2007.01808](https://arxiv.org/abs/2007.01808) (2020) — cite Ziller–Morack [arXiv:1611.03310](https://arxiv.org/abs/1611.03310) and each other but **not** 1706.00317 or 1706.03668. Against it: the OEIS side of the family is alive and has grown. A288815's internal record was revised **#19 Apr 12 2026** (still 21 terms, still `hard,more`, still carrying both arXiv links and the conjecture as a comment), and a neighbour this document has never seen now exists — **A384545** (`%I #22 Apr 04 2026`, Ken Clements), "Smallest prime(n)-smooth multiplier m such that both m·(prime(n)#)−1 and m·(prime(n)#)+1 are prime", with a 1000-term b-file. That is 2025/2026 activity in the primorial-twin family. 3. **A false lead, recorded so nobody re-chases it.** Two independent web-search summaries assert that 1706.00317 appeared as a chapter of *Irregularities in the Distribution of Prime Numbers: From the Era of Helmut Maier's Matrix Method and Beyond* (eds. Pintz & Rassias, Springer 2018), pp. 69–96. **It did not.** The Crossref record for that volume lists exactly ten chapters, and pp. 69–96 is Garcia–Luca, *On the Difference in Values of the Euler Totient Function Near Prime Arguments* (DOI 10.1007/978-3-319-92777-0_4). The arXiv record for 1706.00317 has one version, 1 Jun 2017, comments "10 pages, 1 figure", and no journal-ref. **"No refereed version" survives**, checked at Crossref rather than at a summariser. ### 2.2 Upper bounds for two classes — systematic negative result of the search Searches run: "Jacobsthal function two residue classes", "gaps between consecutive twin candidates", "twin Jacobsthal", "paired Jacobsthal", "sifted sets two residue classes gaps", "generalized Jacobsthal", Grimm-related literature, citation graphs of Iwaniec 1978 / Ziller–Morack / FGKMT. **We found no published upper bound for a two-classes-per-prime Jacobsthal function at any exponent — matching the audit.** The only quantitative statements are (i) ZM's Conjecture 6 [CONJ] and (ii) trivialities. **[ABSENT — confidence raised from MEDIUM-HIGH to HIGH on 2026-08-18; see the three passes below]** Every query listed above is in OUR vocabulary, which is the shape of the five-wave failure. The negative was re-run 2026-08-18 in the conventions that own the object, A144311's wording and MathOverflow 88323's bounded-number-of-residue-classes-per-prime framing, and survived. **Why the confidence moved, and what would still move it back.** Three independent passes on 2026-08-18, two of them with vocabularies this section did not originally use. 1. **The decisive query, run here and reproduced.** The one search earlier audits named as missing was the citation graph of Iwaniec 1978. OpenAlex work `W129012938` ("ON THE PROBLEM OF JACOBSTHAL", 1978) reports **82 citing works** (a same-day pass counted 83; the count drifts, the content does not). All 82 titles were read. The Jacobsthal-specific subset is: Hagedorn's *Computation of Jacobsthal's function h(n) for n<50* (2008), *A computational upper bound on Jacobsthal's function* (2012), *A short note on Jacobsthal's function* (2013), *An upper bound on Jacobsthal's function* (2014), *Jacobsthal's function and a generalisation of Euler's totient* (2012), Kalmynin–Konyagin's *A polynomial analogue of Jacobsthal function* (2024), plus the FGKT/FGKMT/Granville gap line and a long tail of finite-field, automata-theory and topology papers that use the name for something else. **Every upper bound among them is the one-class function.** If a two-class upper bound existed and were any good, it would cite Iwaniec. 2. **Expert silence, at the sharpest available point.** Kalmynin–Konyagin (2023/2024) write an entire paper generalising the Jacobsthal function to multi-class fibres and state **only** the κ = 1 upper bound, attributing it to Iwaniec. They prove no upper bound of any kind for their own j_f. Konyagin has cited Iwaniec 1978 in 2019, 2024 and 2025 and has never stated a multi-class one. 3. **A second vocabulary sweep, negative.** zbMATH Open `ti: Jacobsthal` returns 324 records; the top 100 were triaged and their number-theoretic subset is Iwaniec 1978, Kanold 1967, Vaughan 1977, Hagedorn 2009, Ziller–Morack 2016, Ziller 2019, Mercer 2018, Kalmynin–Konyagin 2024, Borosh–Hensley–Hobbs 1997, Pighizzini–Shallit 2002 — every one already in §1 above. arXiv API `abs:"Jacobsthal function"` returns 15–17 entries, all one-class; `abs:"sieved set" OR abs:"sieving system"` returns 8, none relevant. 4. **A machinery find inside the already-triaged list (2026-08-19).** Pass 1's *An upper bound on Jacobsthal's function* is Costello–Watts, Math. Comp. **84 (2015) 1389–1399**, MR3315513, and its §4 defines and bounds the ORDER-m object this corpus calls `maxsum_m`, in inverse indexing: `π_min(m,k)` = the smallest x such that every sequence of m consecutive integers contains at least x integers coprime to `P_k`, bounded for all m and k by their Theorem 4.4 — as a PARI recursion, no asymptotic in m, evaluated by the authors only at m = 1, one class per prime. The pass triaged the paper by its m = 1 conclusion and missed the machinery. The closed-form companion arXiv:1209.3464 Thm 2.3 is author-withdrawn; its page-6 error was located and refuted numerically at k = 11 (`history/staging/attack-foldL-05-maxsum-direct.md` R4). This section's negative STANDS for two classes at every order; the one-class ORDER-m object is published, and the two-class order-m absence is now calibrated with its owning convention (`SEARCH-CONVENTIONS.md` §1). **The blind spot that keeps this short of certainty is books, and it has not moved.** Diamond–Halberstam–Galway Ch. 10 ("Some applications of Theorem 9.1"), Greaves' *Sieves in Number Theory*, Halberstam–Richert Cor. 2.4.1 and Holt's 2022 *Patterns among the Primes* are not full-text searchable by any method available here. **If this claim is wrong, it is wrong inside one of those four**, most likely as an unnumbered remark. Second blind spot: every query run was in English, and the Jacobsthal problem has a German line (Jacobsthal, Kanold, Stevens) and a Russian one (Konyagin, Kalmynin). ### 2.3 FKMPT "Long gaps in sieved sets" — the lower-bound machinery and its explicit two-class disclaimer - Ford–Konyagin–Maynard–Pomerance–Tao (FKMPT), *Long gaps in sieved sets*, [arXiv:1802.07604](https://arxiv.org/abs/1802.07604), J. Eur. Math. Soc. **23** (2021) 667–700; **Corrigendum, JEMS 25 (2023), no. 6, 2483–2485** ([DOI 10.4171/JEMS/1305](https://doi.org/10.4171/JEMS/1305)). Main Theorem 1 (text-extracted verbatim from the arXiv **v4** PDF, dated 19 Sep 2022, which is the corrigendum-corrected text): for a sieving system (Iₚ)ₚ that is non-degenerate (|Iₚ| ≤ p−1), **B-bounded** (|Iₚ| ≤ B), **one-dimensional** — meaning ∏_{p≤x}(1 − |Iₚ|/p) ~ C₁/log x — and ρ-supported (the density of primes with |Iₚ| ≥ 1 equals ρ), the sifted set Sₓ contains a gap ≥ x (log x)^{C(ρ) − o(1)}, where **C(ρ) := sup{δ ∈ (0, 1/2) : 6·10^{2δ}/log(1/(2δ)) < ρ}**, and **C(ρ) > e^{−1−6/ρ}**. *(The constant is 6, not 4. arXiv v2 and v3 print (4 + δ)·10^{2δ} and e^{−1−4/ρ}. The corrigendum corrects the exponents of H in the deduction of Theorem 2 from Theorem 3, which forces M > 6; v4's own Appendix A item (1) records "the definition of C(ρ), the factor 4 + δ corrected to 6. Likewise, the corrected lower bound is C(ρ) > e^{−1−6/ρ}".* *Any 4 in this repository is the retracted value. The one number the corpus imports from this theorem is in `research/two-class-lower-bounds.md` §2a.)* **The one-dimensionality hypothesis structurally excludes Iₚ = {0, −2}** (which gives ∏(1−2/p) ≍ 1/log²x, dimension 2). **[PROVEN, incl. the exclusion]** - Their **Remark 7** is the single most explicit published statement about the two-class case (quoted verbatim from the arXiv v4 PDF; the text is byte-identical in v3, and is Remark 8 in v2. Square brackets mark one editorial expansion of the paper's own bibliography marker): > "Unfortunately our methods only seem to give good results in the one-dimensional case. Consider for instance the set {n ∈ P : n + 2 ∈ P} of (the lower) twin primes. This corresponds to a two-dimensional system in which Iₚ = {0 (mod p), 2 (mod p)} for all primes p. The 'trivial' bound coming from these methods would give a bound of ≫ log X log log X for the largest gap between lower twin primes up to X …, and one could possibly hope to improve this bound by a small power of log log X using a variant of the methods in this paper. However, a sieve upper bound (e.g., [7 = Halberstam–Richert, Cor. 2.4.1]) combined with the pigeonhole principle already gives a bound of ≫ log² X in this case." where `[7 = Halberstam–Richert, Cor. 2.4.1]` expands the source's own `[7]`: bibliography entry [7] in v2, v3 and v4 is verbatim "H. Halberstam and H.-E. Richert, *Sieve Methods*, Academic Press, London, 1974". In **v1** the marker `[7]` is a different work (the FGKT preprint), so anyone quoting the bare `[7]` must say which version. **Citation hygiene, and it costs a wrong lookup if ignored (added 2026-08-18, both PDFs pulled and diffed here).** The remark is **number 7** only in the corrected/published text — use `gaps_sievedsets_Revision2.pdf` (or the corrigendum-incorporated PDF on Ford's site, from which the block above was re-read verbatim today). In the **Dartmouth preprint** at the identical passage is **Remark 4**, its bibliography marker is **[9]**, and the trivial bound is written "≫ log X log₂X" with the closing clause shortened to "one could possibly hope to improve this bound by a small power of log₂X" — no "using a variant of the methods in this paper". Anyone checking "Remark 7" against the Dartmouth file will read a remark about polynomial coefficients instead. **The one thing that does not move between the two drafts is the attribution**: Dartmouth's `[9]` and Revision 2's `[7]` are both verbatim "H. Halberstam and H.-E. Richert, *Sieve Methods*, Academic Press, London, 1974", so the identification of Cor. 2.4.1's source is now confirmed in two independent bibliographies rather than one. **What the corrigendum actually blames, since it was challenged and the challenge failed (2026-08-18).** A same-day literature pass proposed rewriting the parenthesis above, replacing "in the deduction of Theorem 2 from Theorem 3" with "errors in the exponents of H on pages 685–686", on the strength of the publisher's abstract for the standalone corrigendum. The corrigendum-incorporated PDF was pulled and read here, and its Appendix A opens, verbatim: *"The only error which affect the results of the paper are are [sic] errors in the exponents of H **in the deduction of Theorem 2 from Theorem 3**. When corrected, these force the parameter M to be somewhat larger than claimed, namely M > 6."* The wording already on disk is the authors' own. **Change refused — and the ground given for refusing it was false, corrected 2026-08-18 by pulling the standalone corrigendum PDF.** "The page-number form is the publisher's blurb, not the paper's" is wrong: *"errors in the exponents of H on pages 685–686"* is the **body text of the published corrigendum**, JEMS 25 (2023) 2483–2485, at p. 2483, and it is the authors' own. So are v4's Appendix A words quoted above. The two artifacts describe the same defect differently — Appendix A says "in the deduction of Theorem 2 from Theorem 3" and calls it "the only error", the corrigendum locates it by page and calls it the most serious — and **both are authorial**. The wording on disk stands because it is the version-matched one for the PDF this document cites, not because the other is a blurb. A challenge refused on a false ground is still a wrong entry in the record. Its numbered items beyond (1), which this document did not carry: **(2) C(1) > 1/835** (Example 1); **(3) C(1/d) > e^{−(6d+1)}** (Corollary 1); **(4) C(1/2) > 1/325565** ((1.7) and Corollary 2 — the value `research/two-class-lower-bounds.md` §2a already uses); item (1) also gives the corrected asymptotic **C(ρ) ∼ ½·e^{−6/ρ} as ρ → 0⁺**; and (5)–(7) fix M to *6 < M ⩽ 7*, "sufficiently close to 6", with ε satisfying M < 6 + 6ε. - Tao's blog announcement ([Long gaps in sieved sets, 21 Feb 2018](https://terrytao.wordpress.com/2018/02/21/long-gaps-in-sieved-set/)) states the same limitation, **verbatim, re-fetched and confirmed at the post on 2026-08-18 by two independent passes**: *"One can consider other dimensions also, but unfortunately our methods seem to give results that are worse than a trivial bound when the dimension is less than or greater than one."* *(This line used to read "reportedly states … wording as returned by our fetch", and the sourcing caveat at the foot of this file used to warn that the Tao quotations "may be lightly paraphrased". For this quotation the hedge is retired; it is exact.)* The paper itself makes the same point in its own voice at the head of §1.1: the one-dimensional hypothesis (1.2) has counterparts in other dimensions, "but in those cases the bounds we could obtain were inferior to what could be obtained by the 'trivial' argument; see for instance Remark 7 below." - **Companion upper-bound literature for FKMPT sieved sets: none.** The JEMS paper proves only gap *lower* bounds; the natural upper-bound companion (a Jacobsthal-type theorem for general sieving systems) does not exist in print for any dimension other than the classical κ = 1 Iwaniec case. **[ABSENT — searched through FKMPT's own citation graph and the one-class Jacobsthal literature, whose OEIS home is A048670, per the query list in §2.2; tabled in `research/SEARCH-CONVENTIONS.md` §3]** - Same one-dimensionality restriction governs the follow-up literature, e.g. *Long strings of consecutive composite values of polynomials* ([arXiv:2310.20449](https://arxiv.org/abs/2310.20449)): irreducible f has one root per prime on average (Chebotarev), κ = 1. A pair pattern n, n+2 is κ = 2 and is not covered. --- ## Q3. Covering systems after Hough ### 3.1 Minimum modulus: correct attributions and state of the art (Note: the task prompt's "Hough 2015 (minimum modulus ≤ 616,000)" merges two results.) - **Hough** (Ann. of Math. 181 (2015) 361–382, [arXiv:1307.0874](https://arxiv.org/abs/1307.0874)): any covering system with distinct moduli has minimum modulus ≤ **10¹⁶**. Introduced the measure-distortion idea. **[PROVEN]** - **Balister–Bollobás–Morris–Sahasrabudhe–Tiba (BBMST)**, *On the Erdős Covering Problem: the density of the uncovered set*, Invent. Math. 228 (2022) 377–414, [arXiv:1811.03547](https://arxiv.org/abs/1811.03547). Verified from the PDF, Theorem 8.1: "Let A be a finite collection of arithmetic progressions with distinct moduli d₁,…,d_k ⩾ **616000**. Then A does not cover the integers." Also proves Schinzel's 1967 divisibility conjecture and the sharp form of Erdős–Graham uncovered-density. **[PROVEN]** - Squarefree distinct moduli: minimum modulus ≤ **118** (M. Cummings, M. Filaseta, O. Trifonov, *An upper bound for the minimum modulus in a covering system with squarefree moduli*, **Acta Math. Hungar. 175 (2025), no. 1, 1–25**, DOI [10.1007/s10474-024-01496-x](https://doi.org/10.1007/s10474-024-01496-x), published 18 Jan 2025; preprint [arXiv:2211.08548](https://arxiv.org/abs/2211.08548)). *(This line cited only the preprint until 2026-08-18. **The 118 survived to publication** — checked deliberately, because the FKMPT corrigendum in Q2.3 is this document's standing lesson that a preprint constant need not.* Their second result, also published: the k-th smallest modulus of a covering system with distinct moduli, where it is needed for the covering, is bounded by an absolute constant.*) Best lower bound: a covering system with min modulus **42** (Owens), per [erdosproblems.com covering-systems tag](https://www.erdosproblems.com/tags/covering%20systems). - Structure/enumeration: BBMST, *The structure and number of Erdős covering systems*, JEMS 2024. Number fields / function fields: [arXiv:2302.05946](https://arxiv.org/abs/2302.05946), [arXiv:2308.05378](https://arxiv.org/abs/2308.05378), [arXiv:2402.03810](https://arxiv.org/abs/2402.03810), [arXiv:2408.10460](https://arxiv.org/abs/2408.10460). ### 3.2 The distortion method (from the BBMST survey, verified full text) Survey: *Erdős covering systems*, [arXiv:2211.01417](https://arxiv.org/abs/2211.01417) (also Acta Math. Hungar. 2020; Surveys in Combinatorics 2024). Mechanism (their §3–5): work in Q = S₁×…×S_n; reveal progressions in rounds by largest fixed coordinate; maintain probability measures P_k concentrating on uncovered points, where on each fibre the covered part's measure is zeroed if its proportion α_k(x) ≤ δ and merely "capped" (multiplied by ≤ 1/(1−δ)) otherwise. Key Lemma 3.2: if (1/4δ(1−δ)) Σ_k E_{k−1}[α_k(x)²] < 1 then A does not cover Q. Second moments are bounded via parallel-class uniqueness (Lemma 4.1). Their Theorem 2.1: if lim inf |S_k|/k > 3 (sets growing a bit faster than linearly), non-parallel hyperplanes cannot cover Q unless one has all fixed coordinates ≤ C; and "there exists a sequence with |S_k| ∼ k for which the conclusion fails" — i.e., **the linear-growth regime |S_k| ≍ k (= primes!) is exactly the critical regime for the method.** **[PROVEN]** ### 3.3 Multiplicity (each modulus used ≤ k times) — explicit statement exists - **Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux**, *On the j-th smallest modulus of a covering system with distinct moduli*, **Int. J. Number Theory 20 (2024), no. 2, 471–479** (DOI [10.1142/S1793042124500234](https://doi.org/10.1142/S1793042124500234)); preprint [arXiv:2212.01299](https://arxiv.org/abs/2212.01299), [author PDF](https://dms.umontreal.ca/~koukoulo/documents/publications/covering-systems.pdf) dated 26 June 2023. *(Volume, issue and page range added 2026-08-18; this line previously gave only "Int. J. Number Theory 2024" and no author forenames.)* Verified from PDF, **Theorem 3**: "Let A be a covering system of multiplicity s [each modulus appears ≤ s times]. Then there exists an absolute constant c > 0 such that its smallest modulus is ⩽ exp(c·log²(s+1)/log log(s+2))." Proof = "suitable modification of the distortion method". This covers **s = 2 explicitly** (bounded, but no numeric constant is given for s = 2). Function-field analogs with multiplicity s: [arXiv:2308.05378](https://arxiv.org/abs/2308.05378), [arXiv:2402.03810](https://arxiv.org/abs/2402.03810). **[PROVEN]** - Translation to our setting **[INFERRED]**: two residue classes mod p for each prime p in (x, y] is a system of multiplicity 2 with prime moduli. KKL Theorem 3 says such a system **cannot cover ℤ** once x exceeds an absolute constant. It says *nothing* about covering a finite interval. ### 3.4 Finite-interval covering: what exists - **Crittenden–Vanden Eynden** (Bull. AMS 1969 / [Proc. AMS 24 (1970)](https://www.ams.org/journals/proc/1970-024-03/S0002-9939-1970-0258719-2/S0002-9939-1970-0258719-2.pdf)), proving a 1962 Erdős conjecture: **any n arithmetic progressions covering {1,…,2ⁿ} cover all of ℤ** — and 2ⁿ is essentially sharp (2^{n−1} fails via moduli 2, 4, …, 2ⁿ). They further conjectured the refinement with all moduli ≥ k (threshold k·2^{n−k+1}), proving k = 1, 2; cf. [On a conjecture of Crittenden and Vanden Eynden, J. Austral. Math. Soc.](https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/on-a-conjecture-of-crittenden-and-vanden-eynden-concerning-coverings-by-arithmetic-progressions/43D568F17A8626EC52C70627DDFFD3B1). BBMST gave a 3-page new proof: *Covering intervals with arithmetic progressions*, [Acta Math. Hungar. 161 (2020) 197–200](https://link.springer.com/article/10.1007/s10474-019-00980-z). **[PROVEN]** - **Answer to the targeted question: NO.** We found no covering-systems result that bounds the length of a finite interval coverable by ≤ 2 classes per prime from a range (x, y] at any polynomial scale. The covering-systems corpus deals with (a) covering all of ℤ (min-modulus/density/structure theorems) and (b) interval-to-ℤ transfer at the **exponential** threshold 2ⁿ, which is sharp in general and hence vacuous at Jacobsthal scale y^{O(1)}. The polynomial-length interval regime *is* the Jacobsthal/sieve regime, and there the two-class question has no literature (Q2). **[ABSENT — searched in the covering-systems convention, not in ours; tabled in `research/SEARCH-CONVENTIONS.md` §3]** --- ## Q4. The interval-covering / large-gap bridge ### 4.1 One class per prime (confirmed picture) - **Construction (lower bounds for Y(x))**: Erdős–Rankin method; current record Ford–Green–Konyagin–Maynard–Tao ([arXiv:1412.5029](https://arxiv.org/abs/1412.5029), JAMS 2018): **Y(x) ≫ x·log x·log log log x / log log x** (as stated at [Erdős #687](https://www.erdosproblems.com/687)), via random/greedy hybrid sieving plus a hypergraph covering theorem (Pippenger–Spencer generalization, Rödl nibble). So interval coverage ≈ y log y (small factors) with one class per prime ≤ y: **confirmed**. **[PROVEN]** - **Upper bound**: Y(x) ≪ x² (Iwaniec, Q1). **Gap between x·polylog and x² is open**; Erdős #687 is a **$1000 problem**: "can one prove that Y(x) = o(x²) or even Y(x) ≪ x^{1+o(1)}?" Maier–Pomerance conjectured Y(x) ≪ x (log x)^{2+o(1)} (stated in FGKT/FGKMT and at #687); FGKT: this "places a serious (albeit conjectural) upper bound on how large gaps between primes we can hope to find via lower bounds for Y(x)". **[CONJ]** - Generalized (sieved-sets) version: FKMPT JEMS 2021 (Q2.3): gaps ≥ x(log x)^{C(ρ)−o(1)} for any one-dimensional system — e.g. strings of composite values f(n) of length (log X)(log log X)^{δ}. **[PROVEN]** ### 4.2 Two-class analog — construction literature - **Published asymptotic constructions: none.** The FKMPT machinery is expressly one-dimensional (Remark 7, quoted in Q2.3); its authors state the two-dimensional variant of their method would at best gain "a small power of log log" over trivial, and note the pigeonhole/Brun bound ≫ log²X for gaps between twin primes as the current best. **A multi-class Erdős–Rankin paper DOES exist, and one of its authors is the K of FKMPT.** Alexander Kalmynin and Sergei Konyagin, *A polynomial analogue of Jacobsthal function*, arXiv:2302.00459v2 (v1 1 Feb 2023, v2 3 Dec 2023), **published as Izvestiya: Mathematics 88:2 (2024) 225-235**, DOI 10.4213/im9467e, MR4727548 (the arXiv record carries no journal-ref field, which is why a first pass here recorded it as unpublished; verified at the publisher 2026-08-18) bound j_f(P(y)) from below for f ∈ Z[x], choosing one residue x_p per prime and thereby deleting the whole fibre {i : f(i) ≡ −x_p (mod p)}. Their parameter M(f), "the average size of the maximal preimage of a point under a map f : F_p → F_p", is exactly the average number of classes deleted per prime, and **M(x²) = 2**. Their gain over Rankin is the Rankin factor raised to the M(f)-th power, so for a quadratic it is squared, landing at y·ln²y up to loglog factors — the same shape this document states as its own target in Realistic Target 4. **What survives is the narrower claim**: no Erdős–Rankin construction has been *written down* for G₂. **[the blanket sentence REFUTED 2026-08-18 by fetching the source; the narrowed absence stands, but see the two corrections immediately below]** **CORRECTION 1 (2026-08-18, later the same day): the reason first given here for the narrowing was false, and this corpus already proves it false.** This bullet used to end: *"their two classes are a **varying** fibre f^{-1}(−x_p), whereas our system I_p = {0, −2} is a **fixed pair the same for every p** and is not of the form x + f(i), so **no Erdős–Rankin construction for the fixed pair {0, −2} exists**."* The clause in bold is about the wrong object. `research/two-class-lower-bounds.md` §1 proves the opposite, **PROVEN (elementary, CRT)**, and the load-bearing line of its proof is: "the class pair is `{a_p, a_p - 2}` with `a_p = -s mod p`. As `s` runs over `Z/x#`, CRT makes `(a_p)_p` run over all of `∏ Z/pZ`." Its verdict, in that section's own summary, is that an adversary does exist and that its single restriction is a separation of exactly 2 — the residues themselves are free. So on the construction side the pair is a **free translate, one free parameter per prime**; `I_p = {0, −2}` fixed for every p is the *sifting* picture of twin primes (FKMPT Remark 7, Q2.3 above), not the *covering* picture that defines G₂. The sentence conflated the two pictures and then used the first to dismiss a construction that lives in the second. **What dies is the reason, not the conclusion.** **CORRECTION 2, which is what actually keeps the narrowing alive, and it is a different obstruction.** K–K's object is a shift of the **value**, `j_f(N) = max m : ∃x, (x + f(i), N) > 1 for all i ≤ m`; G₂'s covering formulation is a shift of the **argument**. Even at f(x) = x(x+2) these are not the same family of 2-element sets, which one line of algebra shows: `i(i+2) ≡ −x_p (mod p)` is `(i+1)² ≡ 1 − x_p`, so K–K's fibre is `{−1 ± √(1−x_p)}` — a pair with **fixed centre −1 and varying separation** — while G₂'s pair `{a_p, a_p−2}` has **varying centre and separation fixed at 2**. They coincide at `x_p = 0`, where both are `{0, −2}`, which is exactly K–K's outer-band choice. So **one may not "apply Theorem 1 at f = x(x+2)" and read off a G₂ bound**; the theorem is not about G₂. What is available is the *template*. **What the template appears to give, and the one step a reader must check. [INFERRED — not a theorem, and not written out as a proof anywhere.]** A same-day pass reports that K–K's four-band construction (§2: `p ≤ z₀ = (ln y)^A` with `x_p = 0`; `z₀ < p ≤ z₁ = exp(lll y·ln y/(A·ll y))` with a maximal fibre; `z₁ < p < y/2` with `x_p = 0` again; greedy mop-up on `(y/2, y]`) transfers to G₂ by substituting `Ω_p = {a_p, a_p−2}`, with `a_p = 0` in bands 1 and 3 and `a_p ∉ {0, 2, −2}` in band 2, and that with `ℓ_f = 2`, `h_f = 0`, `M = 2` Theorem 1's shape reads as below. **Provenance: Theorem 1 was read from the arXiv:2302.00459v2 PDF, page 3, and checked against the published Izvestiya PDF, page 226** (2026-08-18, `research/history/staging/lit-pdf-kalmynin-konyagin.md` §2); it is identical in v1, v2 and the published version. > `G₂(P(y)) ≫ y (ln y)³ (lll y)² / (ll y)⁴`, one factor of log above the INFERRED "honest analogue" at `research/two-class-lower-bounds.md` §4b and matching the exponent 3 that §4b currently carries as CONJ. **Do not quote that as proven.** Two ingredients were checked at the source here and they are the favourable half: K–K's **Corollary 1** — "Suppose that for any p ≤ z the set Ω_p ⊂ Z/pZ contains g(p) elements. Let S(X, Ω) be the number of n ≤ X such that n mod p ∉ Ω_p for all p ≤ z. Then S(X, Ω) ≪ X V(z)" — sees Ω_p **only through its cardinality**, and their proof invokes it with the words "Let g(p) = |Ω_p|"; and where K–K must settle for the *maximal* fibre size M_p(f), a distance-2 translate pair has exactly two elements at every odd p for free. **The step nobody has checked is the dichotomy, not the counting** — not here, and not in arXiv:2302.00459 itself or its published Izvestiya version, both of which were read end to end on 2026-08-18. K–K's Cases 1–3 (§2) are stated in terms of the linear and non-linear irreducible factors of f and of the condition `f(i) ≡ y_p`; substituting an arbitrary 2-element set requires re-deriving that trichotomy for `Ω_p = {a_p, a_p−2}` — in particular that Case 1's smoothness dichotomy survives with the two linear factors `i` and `i+2` when `m` has grown to `y(ln y)³`. **The substitution was carried out on 2026-08-19.** K–K's §2 trichotomy was re-derived for `Ω_p = {a_p, a_p−2}`: Case 2 is empty and `|Ω^III_p| = 2` was verified to 10⁶, the whole derivation was adversarially checked twice, and it stands, giving `G₂(P(y)) ≫ y (ln y)³ (lnlnln y)²/(lnln y)⁴` for `y ≥ 10^{134.1}`. The result is **DERIVED HERE and NOT refereed** (`research/two-class-lower-bounds.md` §4c); the referee gap that remains is Halberstam–Richert Thm 2.2 and the Selberg remainder's ξ-versus-z condition at κ = 4. Knock-on, **taken up and settled on 2026-08-18**: the transferred bound is indeed asymptotically above the measured law, and that is not a conflict. The two do not cross until `x = 10^7327` with both constants at 1, they agree on the only thing the measurement resolved (the `x`-exponent, 1 on both), and the measured law was never offered as an asymptotic. The measured law itself has since changed: §6's `c·x·ln²x` is EXCLUDED and reads `0.762 x ln²x lnln x` over `x = 11..79`. Only finishing the K–K substitution on paper at D3 decides the limit, and no finite range ever will (`history/staging/attack-lower-bound.md` §3). - **Computed values**: Ziller–Morack / OEIS A072753, A288815 are the *only* "construction data" (exact optima to n = 21 via ILP). At n = 21: h₂(21) = 2622 against the conjectural ceiling p₂₁² − p₂₁ = 5256, so the optimum two-class covering reaches 0.499 of the ceiling at the top computed term, far beyond the trivial ≈ 2pₙ origin-window covering. That ratio is a small-number coincidence and not a law: the measured growth of h₂ is **c·x·ln²x**, and x ln²x / (x²/2) → 0, so the two curves merely cross near x = 73. The direct ratio h₂/(x ln²x) over the 21 exact terms is not itself constant — it runs 1.04 at x = 11 up to **1.95 at x = 73**, so quote it with the level attached; `research/two-class-lower-bounds.md` §6 gives the coefficient in the Poisson form instead, **2.04 x ln²x** at c₂ = 0.8511, which reads 5% high against the exact h₂(73#) = 2622. Read the ≈ p²/2 as where the two curves cross, not as the shape of the optimum. **[MEASURED law; the crossing arithmetic INFERRED from sourced data]** - Baselines for G₂ itself: (i) trivial gap ≫ pₙ (integers 2…pₙ are all sieved by Iₚ = {0}) **[INFERRED, elementary]**; (ii) average gap = 1/σ₂ ≍ (log pₙ)² by Mertens **[PROVEN]**; (iii) the twin-relevant target is G₂ < p²ₙ₊₁ infinitely often. The lower side is better than trivial: twin slots are a subset of the reduced residues, so **G₂(x#) ≥ g(x#) pointwise**, and the whole Erdős–Rankin literature transfers for free — **G₂(x#) ≫ x·log x·logloglog x/loglog x** by FGKMT (JAMS 31, 2018) via Rankin 1938 and Pintz 1997, and G₂(x#) ≥ x(log x)^{2+o(1)} under Maier–Pomerance (`research/two-class-lower-bounds.md` §§1, 3). **[PROVEN, trivially, and absent from the literature and from A288815/A072753]** On the upper side there is still nothing published at any exponent (Q2.2). --- ## Q5. Expert commentary - **Erdős problems site (T. Bloom):** - [#687](https://www.erdosproblems.com/687) ($1000): estimate Y(x); "It is not clear who first formulated this problem…I offer the maximum of $1000 dollars and 1/2 my total savings" (Erdős 1980). Status: Iwaniec x² upper, FGKMT lower (JAMS 2018), Maier–Pomerance conjecture. OEIS links: A048670, A058989. - [#688](https://www.erdosproblems.com/688): almost-covering [1,n] with one class per prime (density defect ε_n; Erdős proved ε_n ≫ logloglog n/loglog n). - [#689](https://www.erdosproblems.com/689): **choose one class aₚ per prime p ≤ n so that every integer in [1,n] satisfies at least TWO of the congruences** — the double-covering cousin of the two-class problem. Status: OPEN; "a claimed solution has been posted in the comments" (30 comments); users flagging it "tractable" include **TerenceTao** and msawhney. Equals Problem 45 (with 2→10) on Ben Green's open problems list; integer-moduli version is #1205. *(Re-checked at the page 2026-08-18, every clause of that sentence still current. Sources: [Er79d], [Er80, p. 108]; banner reads "OPEN — This is open, and cannot be resolved with a finite computation"; page last edited **08 April 2026**; the status widget's **active** state is the one labelled "A claimed solution has been posted in the comments" while the banner still says open, which is the distinction to preserve when quoting this; footer counts "30 comments" and "1 claimed proof"; the tractable list has grown to TerenceTao, msawhney, Przemek, ebarschkis, MalekZ; a formalised statement exists. Note for the next agent: `erdosproblems.com` refuses an ordinary automated fetch, but every route reaches it with a browser user-agent, search included. The route is **path-encoded, not query-string**: `/search/`, `/range/-`, `/range/1-end`, `/go_to/`, `/latex/`, `/history/`, `/bibs/`. `/latex/` prints the unrendered TeX with the expanded bibliography and is the one to quote from.)* - [#970](https://www.erdosproblems.com/970): order of h(k); "That h(k) ≪ k² is a conjecture of Jacobsthal. Iwaniec proved h(k) ≪ (k log k)²." - **No Erdős problem NUMBER poses "Jacobsthal with two residue classes per prime", and that is now a swept absence rather than a sampled one.** The closest published commentary on the twin version is FKMPT Remark 7 (Q2.3). **[ABSENT, complete corpus, calibrated]** All 1217 problems were fetched in one page and swept mechanically on 2026-08-18, twice and independently: `Jacobsthal` occurs in exactly two of them, #687 and #970, each confirmed at `/latex/`, and `two congruence` in none. Calibration in the same session: `/search/Jacobsthal` returns exactly those two. So the site half of this absence no longer rests on one agent's one pass. What is *not* covered by it: whether Erdős posed the two-class extension **in prose** inside the #687 paragraph rather than as a numbered problem. That question is under primary-source verification today, in the #687 paragraph of the Erdős original rather than on the site; do not read this row as settling it. Routes and the bulk-sweep attribution gotcha are tabled in `research/SEARCH-CONVENTIONS.md` §3. - **MathOverflow** (via API search on "Jacobsthal", "Westzynthius"): - [37679 — "Erik Westzynthius's cool upper bound argument: update?"](https://mathoverflow.net/questions/37679) (Paseman, 2010) → led to Paseman's note [arXiv:1311.5944](https://arxiv.org/abs/1311.5944) (elementary Stevens/Kanold-style bounds, explicit constants). - [88323 — "Analogues of Jacobsthal's function"](https://mathoverflow.net/questions/88323): Jacobsthal-type function g_f(n) for polynomials f (f = x(x+2) would be the twin case); answers (Paseman) offer perspectives (Kanold coprimality-transfer, Cartesian-product view) but **no bounds for any deg ≥ 2 / two-class case**; asker concedes the sieve-theoretic framing (Opera de Cribro, Tao's 254B Notes 7) subsumes the question. - [67907 — "Best possible sieves for the Jacobsthal problem, linear programming, and the prime 2"](https://mathoverflow.net/questions/67907): LP/computational angle (matches the Resta ILP approach in A072753). - [245523 — "Cramér's conjecture and Jacobsthal function"](https://mathoverflow.net/questions/245523). - [245539 — "Paging Henryk Iwaniec: Problems In Lemma 1?"](https://mathoverflow.net/questions/245539) (Paseman, 2016): possible error/typo inside the published proof of Lemma 1 of Iwaniec 1978 (inequality direction in the divisor-bijection transfer). **Unanswered to this day** — evidence that the only proof of the foundational one-class bound has effectively no public expositors. - **Ford's slides**: Kevin Ford, *Large gaps between primes*, Talks 1–3, CRM Montreal workshop *Probability in Number Theory*, 2018 — [talk 1](https://ford126.web.illinois.edu/montreal_talk1_primegaps.pdf), [talk 2](https://ford126.web.illinois.edu/montreal_talk2_primegaps.pdf), [talk 3](https://ford126.web.illinois.edu/montreal_talk3_primegaps.pdf). Talk 1's "Proving large gaps: Jacobsthal's function" slide carries the trivial, FGKMT, Iwaniec, Maier–Pomerance and random-dart statements side by side, and is the source `research/maxgap-law.md` §6 quotes. Its survey content otherwise duplicates the FGKT/FGKMT intros and Granville's note. --- ## WHERE THE PROOF BREAKS (synthesis — every claim tagged) 1. **Iwaniec's (log q)² is the linear sieve run exactly at its sifting limit.** Structure: (a) linear sieve (κ = 1) with trivial per-divisor interval remainders |r_d| ≤ 1; (b) lower-bound function f(u) > 0 iff u > 2, so positivity of S(x,y,z) for y ≳ z², uniformly in x — Iwaniec's sharpened form S ≥ (4y/log²y)(log(y/z²) − O(1)); (c) a divisor-bijection transfer (Lemma 1) from general q to the primorial worst case, giving (ω log ω)². **[PROVEN — Granville; FGKT; MO 245539 transcription]** 2. **The "one class per prime" is load-bearing in exactly one place: it makes the sieve one-dimensional.** Density ∏(1−1/p) ~ e^{−γ}/log z ⟹ κ = 1 ⟹ sifting limit β₁ = 2 ⟹ critical exponent z². And 2 is *provably* the linear sieve's limit (Selberg/Iwaniec parity examples A±). The numerical coincidence [sifting limit 2] = [twin/prime-detection exponent 2] is what makes the one-class bound land exactly at the critical exponent — with nothing to spare. **[PROVEN ingredients; the "coincidence" framing is INFERRED]** 3. **With two classes the same machine still runs — but lands at exponent ≈ 4.27, not 2.** Iₚ = {0,−2} gives κ = 2; the identical trivial-remainder interval argument with the best published dimension-2 lower-bound sieve (DHR, β₂ = 4.266; Franze Table 1) yields a two-class Jacobsthal bound of shape **G₂(n) ≪ pₙ^{4.266+ε}** — i.e., g₂(q) ≪ (log q)^{4.27}. **We found no paper stating this** (consistent with the audit, and re-run 2026-08-18 in A144311's wording and in MathOverflow 88323's "bounded number of residue classes per prime" rather than in ours): it appears to be folklore-available but unwritten. **[INFERRED from PROVEN ingredients; absence sourced]** 4. **Between 4.27 and 2 stands the dimension-2 sifting-limit problem; at 2 stands parity/twin primes itself.** Improving the exponent below β₂ = 4.266 means improving 2-dimensional sifting limits — a recognized hard problem in sieve theory (Diamond–Halberstam–Galway, Cambridge Tracts 177; Franze; Selberg's Λ²Λ⁻ program). Reaching exponent 2 with the right constant would, by the p² rule (ZM Prop. 3.5 mechanism), prove the twin prime conjecture — so no "pure sieve" route can get there (parity obstruction, per Selberg's examples and the standard reading in Granville's note). The realistic frontier for upper bounds is the open interval (2, 4.266). **[Ingredients PROVEN; synthesis INFERRED]** 5. **Lower-bound (construction) machinery is also dimension-locked.** FKMPT's hypergraph-covering method "only seem[s] to give good results in the one-dimensional case" (Remark 7, verbatim); for the twin system they can't even beat the pigeonhole log²X for gaps between twin primes. The Erdős–Rankin smooth-number seeding has **no analog in print for G₂** — but it does have one for polynomial-fibre two-class systems: Kalmynin–Konyagin, arXiv:2302.00459v2, at M(f) = 2 (see Q4.2, corrected twice on 2026-08-18). *(This clause used to read "no analog in print **for the fixed pair {0, −2}**". "Fixed pair" was the wrong reason and is retired: on the covering side the pair is a free translate `{a_p, a_p−2}` per prime, PROVEN at `research/two-class-lower-bounds.md` §1. The obstruction that remains is that K–K shift the value and G₂ shifts the argument, so their theorem does not apply as stated; only their template might, and only as [INFERRED].)* **[PROVEN statements of limitation; the narrowed absence sourced]** 6. **Covering-systems technology operates at the wrong scale.** The distortion method controls covering all of ℤ (min modulus ≤ 616000 distinct; ≤ exp(c log²(s+1)/loglog(s+2)) at multiplicity s — includes s = 2), and interval-to-ℤ transfer only at the sharp exponential threshold 2ⁿ (Crittenden–Vanden Eynden; BBMST 2020). None of it constrains polynomial-length interval coverings — the regime of G₂. (Corrected 2026-08-19, `history/staging/import-distortion.md`: the old sentence here read the primes as sitting ON the critical growth. BBMST Thm 2.1 needs lim inf |S_k|/k > 3 and their |S_k| ~ k counterexample grows LINEARLY, while p_k/k ~ log k DIVERGES — the survey says so verbatim — so the primes grow strictly faster than the counterexample regime and the criterion applies; the scale-mismatch conclusion survives with the correct reason: what fails is the interval bridge, not the covering criterion.) **[PROVEN; scale-mismatch framing INFERRED]** ## REALISTIC TARGETS (for an amateur–professional collaboration) 1. **Write down the first published two-class upper bound.** Executing point 3 above (DHR κ=2 sieve + trivial interval remainders + a ZM-style transfer) would give G₂(n) ≪ pₙ^{4.27} — by our search — run in A144311's wording and MathOverflow 88323's "bounded number of residue classes per prime", not in ours, and tabled in `research/SEARCH-CONVENTIONS.md` §3 — the *first* published upper bound at any exponent for the two-class Jacobsthal function. Modest technically (a "note"), but it fills a verified gap and creates the citable baseline. A Λ²-only version (Selberg upper/lower at κ=2) with explicit constants is even more elementary. 2. **Explicit/verified Iwaniec.** MO 245539's unresolved Lemma-1 query, plus the inexplicit constant that MO 245539 is the published place to see questioned, make a modern exposition — or an explicit-constant version, or a Lean formalization — of Iwaniec 1978 a genuinely useful contribution (interest in formalization is tracked per-problem on erdosproblems.com). 3. **Computational lower-bound constructions (OEIS).** A072753/A288815 stop at n = 21 (2016-era ILP, GLPK; Morack explicitly wrote "We are looking for a GPU approach"). Extending terms, publishing b-files, and — separately — computing the **difference-2-restricted** quantity (our G₂: fixed classes {0,−2}, a pure primorial-wheel scan, no ILP needed) would be new; we found no OEIS sequence for max gap between twin candidates mod pₙ# — a candidate new sequence with clean twin-prime motivation. **[ABSENT — REFUTED 2026-08-18, and the term count was never the reason. The sequence is OEIS A144311, submitted by Andrew Carter in September 2008.]** Every search this repo ran was on `G2` itself. A144311 tabulates `G2 − 1`, which is the covering optimum and the quantity our own §1 identity says is the natural object: *"The length of the longest sequence of consecutive integers, each equal to 1 or −1 modulo at least one of the first n primes."* Their `m` is our `r + 1`, so `m ≡ 1 (mod p)` is `p | r` and `m ≡ −1 (mod p)` is `p | r+2`; `m` is covered exactly when `gcd(r(r+2), W) > 1`, that is when `r` is not a twin slot. Same fixed classes `{0, −2}`, no free translate — the same object, not an analogue. Its terms `1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709` agree with all fourteen of ours and carry **eight more**. Five waves missed it because its text contains no "Jacobsthal", "twin", "primorial" or "gap", and it does not cross-reference A072753 or A288815. Calibration held throughout: the control `2,6,18,30,66,150,192,258` returned A288815 and `2,4,6,10,14,22,26,34,40,46,58,66` returned A048670, and the `G2` searches — the fourteen-term ladder and both offset variants — still return "No results", which is why searching the right convention was the whole game. `research/oeis-G2-submission.md` is therefore a **duplicate and must not be submitted**; what survives of it is a b-file, the twin-prime motivation, and cross-references to propose on A144311. 4. **Two-class Erdős–Rankin experiments — and note the 2023 competitor before starting.** Kalmynin–Konyagin (arXiv:2302.00459v2) already do multi-class Erdős–Rankin for polynomial fibres and reach y·ln²y at M(f) = 2, so this target is no longer open ground. **Read their paper first, and read Q4.2's two corrections with it** — that target — **"re-derive K–K §2 with Ω_p = {a_p, a_p−2} and check their Case 1–3 dichotomy survives"** — was EXECUTED on 2026-08-19 and the result is `research/two-class-lower-bounds.md` §4c: Case 2 empty, `|Ω^III_p| = 2` verified to 10⁶, checked twice, DERIVED HERE and NOT refereed. What remains open is the referee gap, namely Halberstam–Richert Thm 2.2 and the Selberg remainder's ξ-versus-z condition at κ = 4. *(This item used to say "what is still untouched is the **fixed** pair {0, −2}, which is not of their form"; the fixed-pair reason is retired, see Q4.2 Correction 1.)* A referee will raise K–K, and their curve is a live comparison for our measured law. FKMPT Remark 7 states in print that beating the trivial twin-gap bound "by a small power of log log" via a two-dimensional variant of their method is plausible-but-open. A greedy/random two-class covering construction with measured asymptotics — the target form is **`c·x·ln²x·lnln x`** for `G₂`, since `c·x·ln²x` loses for it on the frame-free ranking by a factor of eight (and note the 31.8 and 9.3 that first carried this were weakened and their independence refuted on 2026-08-19, `history/staging/redteam-2026-08-18.md`), while the ZM optima for the *free* two-class object `h₂` still measure `c·x·ln²x` at `c₂ = 0.85` (`research/two-class-lower-bounds.md` §6) — is exactly the kind of computation-first problem where amateur compute + professional writeup can land. 5. **Erdős #689** (every integer in [1,n] covered at least twice by one class per prime): open, actively commented (claimed solution under review in comments), flagged "tractable" by Tao and Sawhney — adjacent to the two-class circle of ideas and with a real venue (erdosproblems forum) for incremental contributions. 6. **What NOT to attempt**: any exponent-2 two-class upper bound (equivalent-strength to twin primes via the p² rule), or improving β₂ below ~4 (the dimension-2 sifting-limit problem that has resisted DHR/Selberg-era technology). --- ## Source index **Primary papers (verified from PDF/full text):** [Granville, Sieving intervals and Siegel zeros](https://dms.umontreal.ca/~andrew/PDF/Sieve.Remark.20.09.26.pdf) · [FGKT, arXiv:1408.4505](https://arxiv.org/abs/1408.4505) · [FGKMT, arXiv:1412.5029](https://arxiv.org/abs/1412.5029) · [FKMPT, arXiv:1802.07604](https://arxiv.org/abs/1802.07604) · [BBMST survey, arXiv:2211.01417](https://arxiv.org/abs/2211.01417) · [BBMST, arXiv:1811.03547](https://arxiv.org/abs/1811.03547) · [KKL, arXiv:2212.01299](https://arxiv.org/abs/2212.01299) · [Ziller–Morack, arXiv:1706.00317](https://arxiv.org/abs/1706.00317) · [Franze, arXiv:1012.3809](https://arxiv.org/abs/1012.3809) **Secondary/metadata:** [Erdős #687](https://www.erdosproblems.com/687), [#688](https://www.erdosproblems.com/688), [#689](https://www.erdosproblems.com/689), [#970](https://www.erdosproblems.com/970) · OEIS [A288815](https://oeis.org/A288815), [A072753](https://oeis.org/A072753), [A048670](https://oeis.org/A048670), [A048669](https://oeis.org/A048669), [A058989](https://oeis.org/A058989) · MathOverflow [37679](https://mathoverflow.net/questions/37679), [67907](https://mathoverflow.net/questions/67907), [88323](https://mathoverflow.net/questions/88323), [245523](https://mathoverflow.net/questions/245523), [245539](https://mathoverflow.net/questions/245539) · [Tao blog 2018](https://terrytao.wordpress.com/2018/02/21/long-gaps-in-sieved-set/), [Tao blog 2014](https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/) · [Hough, arXiv:1307.0874](https://arxiv.org/abs/1307.0874) · [Cummings–Filaseta–Trifonov, arXiv:2211.08548](https://arxiv.org/abs/2211.08548) · [Ziller–Morack computation note, arXiv:1706.03668](https://arxiv.org/abs/1706.03668) · [Crittenden–Vanden Eynden, Proc. AMS 1970](https://www.ams.org/journals/proc/1970-024-03/S0002-9939-1970-0258719-2/S0002-9939-1970-0258719-2.pdf) · [BBMST interval note, Acta Math. Hungar. 161 (2020)](https://link.springer.com/article/10.1007/s10474-019-00980-z) · [Paseman, arXiv:1311.5944](https://arxiv.org/abs/1311.5944) · [Vaughan, PEMS 1977](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-the-order-of-magnitude-of-jacobsthals-function/BD6EBCDD8B8B3EDDF3AF69085DCD8F26) · Iwaniec, Demonstratio Math. 11 (1978) 225–231 ([De Gruyter record](https://www.degruyter.com/view/j/dema.1978.11.issue-1/dema-1978-0121/dema-1978-0121.xml); full text not freely retrievable — proof details reconstructed from Granville + MO 245539 transcription) **Caveats on sourcing:** Tao-blog quotations were extracted via automated fetch and may be lightly paraphrased — **except the Q2.3 dimension quotation, which was re-fetched at the post on 2026-08-18 by two independent passes and is exact; that one hedge is retired.** Iwaniec 1978 itself is not obtainable (De Gruyter paywall/robot-block, re-attempted 2026-08-17 and still 202/404). Its **statement** is confirmed from four independent published sources — Erdős #970 and #687, read in full; FGKMT p. 4, read from the PDF; and Vaughan 1977 p. 329 for the primorial special case via Iwaniec's earlier Acta Arith. 19 (1971) Theorem 2, which independently corroborates the Granville footnote at §1.2 above. Only the **proof** is unread, so every statement here about the paper's *internals* — Lemma 1's divisor bijection at §1.2, and the possible typo at §Q5 — rests on Granville's account and the MO 245539 transcription, and should be read at that calibration. The DHR β₂ = 4.266 figure is from Franze's Table 1 citing the DHR book (Ch. 17); we did not independently verify the book, and `research/dhr-verification.md` §1.1 supersedes the precision with Booker–Browning's rigorous 20-decimal value. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*