# The Hawkins literature, read at source *Staging-layer record of a primary-source read of the whole Hawkins random-sieve tradition, run because `proposals-prior-art.md` §3 identified the owning convention through Rivoal's survey alone and left the tradition itself unopened. It edits no live document. Its findings for `paper/proposals/prop-thinning-null.md` and for `research/SEARCH-CONVENTIONS.md` §1 are stated as proposals in §8 and §9.* ## 1. What the tradition is, and how big it is **The Hawkins literature is small and closed.** The forward-citation walk converges on eleven items and stops. OpenAlex `cites:W2138929421` (Neudecker 1975) returns 6 citing works and `cites:W919582814` (Wunderlich 1974) returns 14, of which 12 are inside the tradition and 2 are unrelated back-matter records; the same eleven items recur in the reference lists of Neudecker–Williams (1974), Heyde (1976), Heyde (1978), Bui–Keating (2006), Lorch (2007) and Rivoal (2008), each read here at source. Nothing after 2009 cites any of it except Rivoal. That closure is what makes the negatives below usable. It is a bounded corpus, not a keyword sweep. ## 2. Bibliography, corrected, with read status | item | bibliographic data, verified | status | |---|---|---| | Hawkins, *The random sieve* | Math. Mag. **31** (1957/58) 1–3; DOI 10.2307/3029322 and 10.1080/19300980.1957.12467521. Neudecker–Williams and Heyde both date it 1957/58; Crossref says 1957 | **NOT REACHED** (JSTOR/T&F paywall). Its construction is quoted verbatim in four items read here | | Hawkins, *Random sieves, II* | J. Number Theory **6** (1974) 192–200; DOI 10.1016/0022-314X(74)90013-4 | **NOT REACHED** (ScienceDirect blocked the fetch). Its content is stated by Rivoal p. 800: a prime number theorem in `L¹` and in probability | | Wunderlich, *A probabilistic setting for prime number theory* | Acta Arith. **26** (1974) 59–81; DOI 10.4064/aa-26-1-59-81; received 22.4.1973 | **READ IN FULL**, own OCR of the Bronze-OA scan at impan.pl | | Wunderlich, *A general class of sieve generated sequences* | Acta Arith. **16** (1969) 41–56; DOI 10.4064/aa-16-1-41-56 | **READ IN FULL**, own OCR of the impan scan. Found in Wunderlich 1974's own reference list | | Wunderlich, *The prime number theorem for random sequences* | J. Number Theory **8** (1976) 369–371; DOI 10.1016/0022-314X(76)90084-6 | **NOT REACHED** | | Neudecker and Williams, *The "Riemann hypothesis" for the Hawkins random sieve* | Compositio Math. **29** fasc. 2 (**1974**) 197–200; MR 399029; Zbl 0312.10033 | **READ IN FULL**, numdam PDF | | Neudecker, *On twin "primes" and gaps between successive "primes" for the Hawkins random sieve* | Math. Proc. Cambridge Philos. Soc. **77** (1975) 365–367; DOI 10.1017/S0305004100051185 | **ABSTRACT AND EXTRACT ONLY** at Cambridge Core; closed access, no OA location in OpenAlex. Its two results are stated by Rivoal at p. 808 | | Heyde, *On asymptotic behavior for the Hawkins random sieve* | Proc. Amer. Math. Soc. **56** (1976) 277–280; DOI 10.1090/S0002-9939-1976-0404177-X | **READ IN FULL**, free AMS PDF | | Heyde, *A log log improvement to the Riemann hypothesis for the Hawkins random sieve* | Ann. Probab. **6** (1978) no. 5, 870–875; DOI 10.1214/aop/1176995433 | **READ IN FULL**, own OCR of the Project Euclid scan | | Bui and Keating, *On twin primes associated with the Hawkins random sieve* | J. Number Theory **119** (2006) 284–296; arXiv:math/0607196v3, 10 pp. | **READ IN FULL** from the arXiv PDF | | Lorch, *A generalized Hawkins sieve and prime k-tuplets* | Rocky Mountain J. Math. **37** (2007) no. 2, 533–550; DOI 10.1216/rmjm/1181068765 | **READ IN FULL**, free Project Euclid PDF | | Lorch and Ökten, *Primes and probability: the Hawkins random sieve* | Math. Mag. **80** (2007) 112–119 (Lorch's own reference list gives 116–123) | **NOT REACHED**. A survey; both other Lorch-tradition items were read | | Deheuvels, *Asymptotic results for the pseudo-prime sequence generated by Hawkins' random sieve: twin primes and Riemann's hypothesis* | Proc. Seventh Conf. on Probability Theory (Braşov 1982), VNU Sci. Press 1985, 109–116; DOI 10.1515/9783112314036-013 | **NOT REACHED**. De Gruyter refused the fetch. Found in Lorch's reference list, absent from every other list in the tradition | | Williams, *A study of a diffusion process motivated by the sieve of Eratosthenes* | Bull. London Math. Soc. **6** (1974) 155–164; DOI 10.1112/blms/6.2.155 | **NOT REACHED**. The continuous ancestor Neudecker–Williams say they are mirroring | | Rivoal, *On the distribution of Hawkins' random "primes"* | J. Théor. Nombres Bordeaux **20** (2008) 799–809; DOI 10.5802/jtnb.651 | **READ IN FULL**, numdam PDF (the officer pass had read pp. 800–801 only) | **One correction to the record: the formalisation paper is 1974, not 1979.** `prop-thinning-null.md` §2, `proposals-prior-art.md` §3a and §8, and the `SEARCH-CONVENTIONS.md` §1 thinning row all date Neudecker–Williams to 1979. The paper is Compositio Math. 29 fasc. 2, 1974, pp. 197–200, submitted 30.V.1974, and that date is confirmed independently in the reference lists of Heyde 1978, Bui–Keating, Lorch and Rivoal. No 1979 item exists in this tradition. ## 3. The null law, as the tradition actually states it Neudecker–Williams p. 199 make the process `{(X_n, Y_n)}` Markovian and say of the gap law only that *"Elementary properties of geometric distributions now make things particularly neat."* Heyde 1976 p. 278 writes the law out, following Williams: > *"P(X_{n+1} − X_n = j | X_n, Y_n) = Y_n^{−1}(1 − Y_n^{−1})^{j−1}, j ≥ 1"* with `Y_n = ∏_{k≤n}(1 − X_k^{−1})^{−1}`. Rivoal p. 800 restates it as `P(p_{n+1} − p_n = j | F_n) = (1/m_n)(1 − 1/m_n)^{j−1}` with the same Mertens product `m_n`. **The parameter of the geometric is the reciprocal of the Mertens product, which is the mean gap**, and that identification is explicit in print. It is the one-class instance of our `α = ∏ q/(q−2) = m̄/6`. The officer pass's verdict on the null law survives the primary read intact and gets sharper: the geometric parameter is not merely "some rate", it is named as the Mertens product in 1974. Heyde 1976 also derives, exactly and with no error term, the first two conditional moments of the gap: > *"E(X_{n+1}|X_n,Y_n) = X_n + Y_n"* and > *"E[(X_{n+1} − X_n − Y_n)² | X_n, Y_n] = Y_n² − Y_n"* These are stage-to-stage identities of the null, not of the arithmetic sieve. ## 4. Question (a): a two-class or k-residue variant — NOT FOUND **Nothing in the Hawkins tradition deletes residue classes at all.** Every member of it deletes each surviving element independently, with a probability attached to the sieving number. The two generalisations that exist go in different directions: - **Lorch 2007 generalises the RATE.** He replaces Hawkins' `1/n` by an arbitrary `p : N≥2 → [0,1]`, calls the survivors *Hawkins' p-primes*, and proves the PNT and Mertens analogues under decay conditions. Theorem 2.1 needs only `Σ p²(k) < ∞` and `Σ p(k) = ∞`, and then `P(S_n) ∼ (Σ_{k≤n} p(k))^{−1}`. Our two-class deletion rate `2/p` satisfies both hypotheses, so **a random sieve running at exactly our rate is inside a published family**. What it produces there is density `1/(2 log n)`, not the twin density: the rate alone does not make a two-class object, because Lorch's sieving numbers are the survivors of the same sieve, whereas our moduli are the primes and our comb is separate. Lorch's own twin model is `p(n) = log n / n`, not `2/n`; his §6 shows `p(n) = log^k n / n` gives `P(S_n) ∼ (k+1)/log^{k+1} n`, "which is (more or less) the conjectured asymptotic density of prime (k+1)-tuplets". - **Bui–Keating 2006 generalise the PATTERN counted**, from `{m, m+1}` to `{m, m+k}` and then to arbitrary `l`-tuples. The sieve is unchanged. - **Wunderlich 1969 generalises the ELIMINATION SEQUENCE** of a deterministic sieve: the process is fixed by sequences `{s_n(k)}` with `s_n(k) ∼ k a_n`, so again a rate, one class, no modulus. Calibrated negative, run over the six full texts read here: the strings `residue`, `congruen`, `modul` and `two classes` return **zero** hits in Neudecker–Williams, Heyde 1976, Heyde 1978, Lorch 2007, and the whole of Bui–Keating except its two references to Green–Tao on arithmetic progressions. The single hit in Wunderlich 1974 is about elementary sets in his σ-algebra. **The owning convention searched is "Hawkins' random sieve", the row already in `research/SEARCH-CONVENTIONS.md` §1, plus Lorch's own term "Hawkins' p-primes"; no two-class or k-residue variant is published under either.** Rivoal p. 802 says why in one sentence, and it is the strongest possible confirmation of the negative: *"it is not surprising that Hawkins' sieve cannot detect arithmetical facts such as coprimality or Dickson's conditions."* A residue-class variant is not a gap in this tradition; it is outside its design. ## 5. Question (b): a stage-to-stage moment identity — FOUND IN THE NULL, NOT IN THE FOLD This is the finding that changes the picture, and it cuts both ways. **An exact moment recursion between consecutive stages is in print, twice.** Wunderlich 1974 eq. (5), for `a_n` the survival product `∏_{j *"E[a_{n+1}^k] − E[a_n^k] = ((1 − 1/n)^k − 1) E[a_n^{k+1}]"* holding for all integral `k` with boundary `E[a_2^k] = 1`. Lorch 2007 Lemma 3.3 is the same identity at an arbitrary rate: > *"E[y_{n+1}^k] − E[y_n^k] = ((1 − p(n))^k − 1) E[y_n^{k+1}]"* **And a joint moment recursion mixing the survival density with a pair count is in print once.** Bui–Keating eq. (4), with `Π̂` their auxiliary twin count: > *"∆E(y_{m+1}^i Π̂(m)) = (1 − 1/(m+1))^{i+1} E(y_{m+1}^{i+1}) − [1 − (1 − 1/(m+1))^i] E(y_{m+1}^{i+1} Π̂(m))"* with the `l`-tuple version at their eq. (7). These are exact, error-term-free, stage-indexed recursions on mixed moments, closed by taking `i` downward. **The technique our Fold Moment Identity uses is the standard technique of this tradition.** Any draft that presents the shape as new will be told so. **What is still not in print is the identity itself.** Three differences, and each is load-bearing: 1. Wunderlich's and Lorch's moments are power moments of the survival *density*. Ours is an exponential moment of the *gap word*. No exponential-moment generating variable appears anywhere in the tradition. 2. Their recursions close in the moment index `k` at a single stage. Ours mixes four distinct objects, `Φ`, `Ψ`, `Ω` and `Δ`, across two stages. 3. **Every one of them is an identity for the random sieve.** Wunderlich's, Lorch's and Bui–Keating's recursions all describe the null. Ours describes the arithmetic fold. That is the whole point of `Ψ − Φ²`, and the tradition has no counterpart because it has no arithmetic sieve to compare against. **The pair-correlation half.** Bui–Keating do carry a joint two-point probability expanded about the independent product. On their p. 2: > *"P(m ∈ α, m + 2 ∈ α) = (1 − 1/m)[y_m(α)² − (1/(m+1))(1 − 1/m) y_m(α)³]"* and in Lemma 2 the general `l`-tuple version, `P_m = y^{l+2} − (k − l − 1) y^{l+3} + O(y^{l+3}/m) + O(y^{l+4})`. The leading term is exactly the product of marginals and the next term is the first-order departure from it. **So an expansion of the shape "joint minus the product of marginals, first correction named" exists in print for the random sieve.** It is not `Ψ − Φ²`: their correction measures the cost of forcing the intermediate integers out, inside the null, and neither `Φ` nor `Ψ` is an exponential moment. The word `correlation` does not occur in any text read here. ## 6. Question (c): the analogue question — RAISED REPEATEDLY, ANSWERED ONLY AS CONSTANTS The tradition asks our question out loud and leaves it at the level of a missing constant every time. - **Bui–Keating p. 2**, on the twin count: *"The absence of the twin prime constant factor here is due to the drawback of the probabilistic setting of the random sieve that it contains little arithmetical information about the primes."* The deviation is named. It is the singular series. No expansion computes it. - **Neudecker–Williams p. 199**, on Mertens: *"(One could not expect the e^γ factor which is the rather tantalising feature of the real Mertens Theorem.)"* Lorch p. 534 repeats the observation and adds *"We are reminded that the Hawkins' model is not entirely ideal."* - **Rivoal Theorem 2** is the sharpest instance and the only quantitative one: `#{k ≤ n : p_k ∈ aN + b} = n/a + O(n/log n)` almost surely, against `n/φ(a)` for the primes, with the remark quoted in §4 above. That is a published, exact comparison of a random-sieve density to the true density, at one observable. - **Hawkins' own defence of the model**, as Lorch p. 534 reports it: *"Like the real primes, the events of Hawkins primality are interdependent."* **So the question our identity answers is a question this literature poses and does not answer.** Three separate authors record the discrepancy as `e^γ`, as the twin constant, and as `a` against `φ(a)`. Nobody writes an identity for it. Under the owning convention "Hawkins' random sieve" and Lorch's "Hawkins' p-primes", **no expansion of the deviation between the random sieve and the arithmetic sieve was found in print.** ## 7. The twin analogue, and the gap analogue, exactly as proven This was the direct question in the brief and it has a full answer. **The random-sieve twin conjecture is a theorem, and has been since 1974.** Wunderlich 1974 Theorem 4: with `t_n(α)` the count of consecutive integers in the sequence below `n`, for almost all `α`, `t_n(α) ∼ n/log²n`. Wunderlich's own framing, in his introduction: *"This result is analogous to Hardy and Littlewood's conjecture regarding the density of twin primes."* His "twin" is a pair `k−1, k` of consecutive elements, since the random sieve has no notion of parity. Then, in order: - **Neudecker 1975** gives a short proof of Wunderlich's theorem by the Markov route, per its own abstract: *"give a short proof of Wunderlich's result on the distribution of twin 'primes' for the Hawkins random sieve"*. - **Deheuvels 1985** revisits twin primes and the Riemann hypothesis for the sieve. NOT REACHED; known only from Lorch's reference list. - **Bui–Keating 2006** prove `Π_{X,X+k}(x) ∼ x/(log x)²` almost surely for every fixed `k`, and `Π_{X,X+k₁,…,X+k_{l−1}}(x) ∼ x/(log x)^l` for every `l`-tuple. Corollary 1 gives arbitrarily long arithmetic progressions of Hawkins primes. - **Rivoal 2008 Theorem 1** adds the error term Bui–Keating did not give: `#{1 ≤ k ≤ n : p_{k+α} = p_k + α} = n/log^α n + O(n log log n / log^{α+1} n)` almost surely. **The gap analogue is also a theorem, and it is the one most directly material to this corpus.** Rivoal p. 808 states Neudecker's second result: > *"Neudecker [11] showed that lim sup_{n→+∞} (p_{n+1} − p_n)/log(p_n)² = 1 > a.s., which is an analogue of Cramér's conjecture."* The constant is 1, the limsup is attained, and it is almost sure. **Under independent thinning the largest gap is exactly `log²p`, proven, since 1975.** Anything this corpus says about a maximal gap under the thinning null has a published answer to be measured against, and any claim that the null's extreme behaviour is being derived here for the first time is false. ## 8. What this does to `prop-thinning-null.md` — proposed regrade: PARTIALLY RE-UPGRADE, and re-scope Not applied here. The proposal file is outside this brief's write scope. **Direction: hold the grade at WEAKENED but move two of its three headline claims from "unchecked" to "checked and standing", and add one concession.** Standing, and now confirmed against the primaries rather than a survey: - **The Fold Moment Identity is not in the Hawkins literature.** All six full texts read. The identity is about the arithmetic fold; the tradition has only the null's own recursions. - **The `Ψ − Φ²` reading answers a question this literature asks and leaves open.** That is a stronger position than "not found": §6 gives three published statements of the discrepancy as an unexplained constant. - **No two-class or k-residue variant exists**, and Rivoal states the structural reason it does not. The check `prop-thinning-null.md` §2 says "decides what a draft could still claim" has now been run and it came back clean. New concessions the proposal must absorb: - **The technique is theirs.** Exact stage-to-stage mixed-moment recursions with no error term are how Wunderlich, Lorch and Bui–Keating all work. §1.2's derivation style is a re-derivation of a standard method and must be presented as one. - **A joint-minus-product expansion already exists for the null** (Bui–Keating p. 2 and Lemma 2). The novelty in `Ψ − Φ²` is which two objects are being differenced, not the move of differencing them. - **The two-class RATE is inside a published family.** Lorch's `p`-primes admit `p(n) = 2/n`. What is ours is the two-class structure on a comb of twin slots sieved by the actual primes, not the rate. - **The maximal-gap side is owned.** Neudecker 1975 has `limsup (p_{n+1} − p_n)/log²p_n = 1` a.s. The extinction half of the proposal, and anything in `attack-foldL-06-scaling.md` about how large a gap the null supplies, must cite it and must be checked against it. **Downgrade trigger to add:** retire the extinction half if its measured null behaviour disagrees with Neudecker's `log²p` limsup at constant 1. **No change proposed to the §4 Daley–Vere-Jones exposure**, for the reason in §10. ## 9. Proposed rows and edits elsewhere — not applied **`research/SEARCH-CONVENTIONS.md` §1, correction to the existing thinning row.** Change `Neudecker–Williams 1979` to `Neudecker–Williams 1974`, and extend the "where it lives" cell to the closed list: Hawkins, Math. Mag. 31 (1957/58) 1–3 and J. Number Theory 6 (1974) 192–200; Wunderlich, Acta Arith. 26 (1974) 59–81; Neudecker–Williams, Compositio Math. 29 (1974) 197–200; Heyde, Proc. AMS 56 (1976) 277–280 and Ann. Probab. 6 (1978) 870–875; Bui–Keating, J. Number Theory 119 (2006) 284–296; Lorch, Rocky Mountain J. Math. 37 (2007) 533–550; Rivoal, JTNB 20 (2008) 799–809. **`research/SEARCH-CONVENTIONS.md` §1, three new rows.** | object | our name | canonical | OWNING convention — search THIS | where it lives | |---|---|---|---|---| | the same ladder run at a rate other than one class per prime | the two-class thinning rate `2/p` | — | **"Hawkins' p-primes"**, a random sieve whose sieving number `n` deletes with a general probability `p(n)`, admissible when `Σ p²` converges and `Σ p` diverges | Lorch, *Rocky Mountain J. Math.* **37** (2007) 533–550, Thm 2.1 and §6 | | the maximum gap the thinning null supplies | the extinction fold, `ln(kills)` crossing | Cramér analogue for a random sieve | **`lim sup (p_{n+1} − p_n)/log²(p_n) = 1` almost surely for Hawkins' primes** | Neudecker, *Math. Proc. Cambridge Philos. Soc.* **77** (1975) 365–367, via Rivoal JTNB 20 (2008) p. 808 | | twins and `k`-tuples under the null | the null's pair count | random analogue of Hardy–Littlewood | **"k-difference twin primes associated with the Hawkins random sieve"**, `Π_{X,X+k}(x) ∼ x/log²x` a.s. with no singular series | Wunderlich Acta Arith. 26 (1974) Thm 4; Bui–Keating arXiv:math/0607196 Thms 1–3; Rivoal JTNB 20 (2008) Thm 1 | **`research/PRIOR-ART.md` rows.** Six proposed, each ADJACENT rather than overlapping, since none of them is about a two-class sieve: | result | ours | theirs | relation | |---|---|---|---| | gaps exactly geometric at every stage, parameter the Mertens product | the null law, `import-thinning.md` §1.2 | Neudecker–Williams 1974; Heyde 1976 p. 278 | **OWNED BY THEM.** One class per prime; ours is the two-class instance | | exact stage-to-stage moment recursion, no error term | Fold Moment Identity technique | Wunderlich 1974 eq. (5); Lorch 2007 Lem. 3.3 | **TECHNIQUE OWNED BY THEM**, applied to the null; ours applies it to the arithmetic fold | | joint survival minus the product of marginals, first correction named | `Ψ − Φ²` | Bui–Keating 2006 p. 2 and Lem. 2 | **ADJACENT.** Same move, different pair of objects, and theirs is null-internal | | the deviation of the random sieve from the true sieve | `Ψ − Φ²` as the first-order deviation | Bui–Keating p. 2; Neudecker–Williams p. 199; Rivoal Thm 2 | **QUESTION OWNED BY THEM, ANSWER NOT.** Recorded as `e^γ`, the twin constant, and `a` against `φ(a)`; no identity | | a random sieve at a general deletion rate | two-class rate `2/p` | Lorch 2007 Thm 2.1 | **ADJACENT.** `p(n) = 2/n` satisfies his hypotheses | | maximal gap under the null | extinction fold | Neudecker 1975 | **OWNED BY THEM**, with constant 1 | ## 10. NOT REACHED, and what that costs - **Hawkins 1957 and Hawkins 1974** at source. Both paywalled here. Their construction is quoted verbatim in four items read at source and their results are stated in three, so the cost is low, but neither has been opened. - **Neudecker 1975** beyond its abstract and Cambridge Core extract. This is the one that matters, because the `log²p` limsup is being carried second-hand from Rivoal p. 808. It is closed access with no OA location. - **Deheuvels 1985** and **Lorch–Ökten 2007**. Both twin-adjacent, neither reached. Deheuvels appears in no reference list in the tradition except Lorch's, which is itself a reason to open it. - **Williams, Bull. LMS 6 (1974) 155–164**, the diffusion ancestor. - **Wunderlich 1976**, J. Number Theory 8, 369–371. - **Daley and Vere-Jones remains UNVERIFIED**, and the attempt to fix it here failed. Springer's authentication wall blocked both the volume front matter and the chapter pages. What was established: volume II (2nd ed., 2008, 978-0-387-49835-5) has seven chapters and **none of them is titled for thinning**, so `import-thinning.md` §1.1's citation still names no section a reader could turn to. The substitute route the officer pass already found, Gnedenko–Kovalenko 1968 via Gorenflo–Mainardi arXiv:1808.06563, was re-checked here and Gorenflo–Mainardi turns out not to cite Daley–Vere-Jones at all, so it cannot corroborate that attribution either. Treat §1.1's Daley–Vere-Jones citation as unusable until a section number is obtained. - **MathSciNet and zbMATH review text** for any of the above. - **Google Scholar** was unreachable; the forward walks here were run on OpenAlex and Crossref only. *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [../CHANGELOG.md](../CHANGELOG.md), indexed by document.*