# Holt sweep, fifteenth manuscript: arXiv:2605.19165 *Staging note, 2026-08-19. Completes the Holt prime-gaps corpus sweep begun in `research/PRIOR-ART.md` §"Full corpus sweep, 2026-08-17", which covered fourteen of fifteen manuscripts. This file edits no live document; the edits it proposes to `PRIOR-ART.md` are listed in §6 and are not applied here.* **Provenance.** PDF pulled from `https://arxiv.org/pdf/2605.19165` (4,256,984 bytes, 14 pages, PDF 1.7); metadata from the arXiv API (`id_list=2605.19165`). Converted with `pdftotext -layout` for scanning. **Every load-bearing passage was then read as a page image** (`pdftoppm -r 150`): PDF pages 7, 9, 10, 11, 12. The prose text layer reproduces the page images word for word on pp. 10–12; the Figure 4/5 and Table 1/2 text layers do not, and nothing in this note rests on them except where a page image is named. --- ## 1. The sweep row | field | value | |---|---| | identifier | arXiv:2605.19165v1 [math.NT] | | title | *On nonconvex constellations among primes II: (J, \|s\|) = (458, 3240)* | | author | Fred B. Holt (alone; not Holt–Rudd) | | submitted | 18 May 2026, 22:40:26 UTC; paper's own dateline "19 May 2026" | | version | v1, the only version | | length | 14 pages, 7 figures (announced); printed objects are Figures 1, 2, 3, 5, 6, 7 and Tables 1, 2 | | MSC | 11N05, 11A41, 11A07 | | keywords | primes, gaps, prime constellations, Eratosthenes sieve | | code | `https://github.com/fbholt/primegaps-v2` — the repository the corpus already cloned | **Abstract**, from the arXiv API record, verbatim: > Extending our work on the $k$-tuple conjecture, we previously applied those > methods to the Engelsma counterexamples (narrow constellations) of length > $J=459$ and span $|s|=3242$. Here we extend that analysis to the $116$ > Engelsma counterexamples of length $J=458$ and $|s|=3240$. > We track the evolution of these $116$ counterexamples from inadmissible > driving terms starting in the cycle of gaps ${\mathcal G}(11^\#)$ up through > their first appearance in ${\mathcal G}(113^\#)$. We continue developing > primorial coordinates for each admissible instance through a breadth-first > exhaustive search through ${\mathcal G}(211^\#)$. > Each of the $(458,3240)$ constellations sits inside a $(459,3242)$ > constellation, which we call its {\em parent}. We show that no $(458,3240)$ > constellation occurs outside of its parent until the cycle > ${\mathcal G}(227^\#)$. The early evolution of the $(458,3240)$ constellations > is dominated by the evolution of their parents, which we have previously > studied. > For each $(458,3240)$-counterexample we calculate its asymptotic relative > population, among other constellations of length $J=458$. ### The row as PRIOR-ART.md's table wants it | paper | year | what it holds | touches our G2? | |---|---|---|---| | 2605.19165 On nonconvex constellations II: (458, 3240) | 2026 | the 116 Engelsma (458, 3240) counterexamples as heads and tails of the 58 (459, 3242) parents; the in-parent / out-of-parent population matrix; the \|s\|/2 threshold restated | no | ### The rubric, answered - **Does it touch our G₂ — the spacing between consecutive occurrences of the gap g = 2?** **No.** The string `g = 2` occurs eight times, every one of them naming the *extremal* gap of a (459, 3242) constellation, the one dropped from the front or the back to produce a (458, 3240) child ("These constellations all begin and end with a gap g = 2", p. 2). No quantity in the paper is a distance between two occurrences of g = 2. Our G₂, and its localized form M(x, x^k), are outside the paper's subject. - **Does it touch any maximum-gap upper bound?** **No.** The words *maximum*, *upper bound*, *bound*, *spacing* and *Jacobsthal* have zero occurrences in the full text layer. The only extremal statement in the paper is a *lower* search limit: "No instance of any of the 116 (458, 3240)-counterexamples occur among primes before 9.7 E73" (p. 12, page image), the same 9.7E73 the corpus already records for the (459, 3242) family from 2603.25896. - **Does "twin" appear, and in what role?** **The word does not appear at all.** A case-insensitive scan returns two hits, both of them the substring inside *entwined* / *entwined than one might expect* (p. 7). So 2605.19165 joins 2502.20470, 2308.07570 and 2309.16833 as a fourth Holt manuscript with no occurrence of the word. - **Polignac?** Zero occurrences. - **What conjecture is it about?** Hardy–Littlewood's 1923 convexity conjecture (π(x+y) ≤ π(x) + π(y)), via nonconvex admissible constellations, π(|s|) < J. --- ## 2. The theorems, with every hypothesis The paper carries exactly two numbered statements, **Lemma 1** and **Lemma 2**, plus one displayed formula, **Equation (1)**. There is no theorem, corollary, proposition or conjecture. Both lemmas were read as page images (pp. 4–6 for Lemma 1's statement and proof, p. 10 for Lemma 2 and the start of its proof, p. 11 for the end of the proof). ### Lemma 1 (p. 4), verbatim > **Lemma 1.** *Let [2ŝ2] be a (459, 3242) counterexample of index 0 ≤ j ≤ 28, > when the constellations are ordered by the prefixes of their primorial > coordinates.* > > *Then [2ŝ] is a (458, 3240) counterexample of index j, and [2ŝ] first occurs > outside of its parent in G(227#).* and, continuing on p. 5, *"And [ŝ2] is also a (458, 3240) counterexample of index j + 29, and [ŝ2] first occurs outside of its parent in G(269#)."* **Full hypothesis list.** (i) [2ŝ2] is one of Engelsma's 58 (459, 3242) counterexamples; (ii) its index j satisfies 0 ≤ j ≤ 28, i.e. it is in the half whose unique inadmissible driving term in G(11#) has γ₀ = 107, not the reversed half with γ₀ = 1271; (iii) the indexing is by the prefix of the primorial coordinates. **The proof is by computation, not by argument**: "From the computation summarized in Figure 5 we see that the unique prefixes extend into G(131#)", and "Figure 4 tabulates the numbers of admissible residues … Cells are highlighted in green when δ = 1. We observe that for all [2ŝ], this first occurs for p = 227, and for all [ŝ2] this first occurs for p = 269." So 227 and 269 are **measured on Engelsma's data**, not derived. Nothing in it is general. ### Lemma 2 (p. 10), verbatim, read as a page image > **Lemma 2.** *Let s be a (458, 3240) constellation, and denote its (459, 3242) > parent by [s2]. Let p and q be consecutive primes, and let ρ_s = q − ν_q(s) be > the number of admissible residues mod q for s.* > > *Then the population n_out of s outside of its (459, 3242) parent and its > population n_in inside its parent grow as* > > ``` > [ n_out(q#) ] [ ρ_s δ ] [ n_out(p#) ] > [ n_in (q#) ] = [ 0 ρ_[s2] ] [ n_in (p#) ] > ``` > > *Here δ = ρ_s − ρ_[s2] ∈ {0, 1}, with δ = 0 for q < 227 and δ = 1 for > q > 1621.* **Full hypothesis list, none of it optional.** (i) s is one of the 116 (458, 3240) Engelsma constellations — the lemma is stated for that family only, and its δ ∈ {0, 1} clause is justified in the proof by "direct calculations of ρ_s and ρ_[s2] across the … (458, 3240) constellations"; (ii) [s2] is the (459, 3242) parent, so the child is the parent minus one extremal g = 2; (iii) p and q are **consecutive** primes, so this is a single stage of the sieve; (iv) ρ_s = q − ν_q(s) is the count of admissible residues mod q, the ν_p(s) of 2502.20470 §3; (v) the "δ = 0 for q < 227" half is **empirical**, from the table printed on p. 7 as Table 1 (which the prose calls Figure 4); (vi) the "δ = 1 for q > 1621" half is the only part with a general argument, and its hypothesis is |s| = 3240, |[s2]| = 3242. The upper-triangular shape carries a fact worth stating in its own right: the lemma proves **n_out is fed by n_in but never the reverse** — "If the instance γ₀ is outside of its parent, then all ρ_s images of this instance in G(q#) remain outside of the parent constellation." Instances leak out of the parent, one image per stage per instance, and never leak back in. ### The |s|/2 clause, and its proof (pp. 10–11, page images) The proof's final paragraph, verbatim across the page break: > *On the other hand, under the recursion G(p#) ⟶ G(q#) [HR15], once q > 1621 all > of the fusions between adjacent gaps occur in separate images of s and [s2]. So > for q > 1621 we have ρ_s = q − 459 and ρ_[s2] = q − 460, and thereafter δ = 1.* and the marking of it on p. 11, which is the passage the 2026-08-19 officer pass already read: > *We mark the thresholds J + 1 = 459 and |s|/2 = 1620 in the graph. The first > threshold marks the point at which there are more residue classes mod p than > there are possible fusions within the constellation s. At this point bare > admissibility becomes trivial. The second threshold marks the point – half the > span of the constellation – at which all fusions of adjacent gaps must occur in > separate images of s under the 3-step recursion. Beyond this point δ = 1, and > the curves all decline by factors of (p−460)/(p−459).* **1620 = |s|/2 for the child and 1621 = |[s2]|/2 for the parent**, which is why the lemma's clause reads q > 1621 while the graph is marked at 1620: the clause has to clear both spans at once. ### Equation (1) (p. 12, page image) — where |s|/2 does load-bearing work > *From [Hol25] the asymptotic relative population of a constellation s is the > product of two factors. The first factor is the product of the number of > admissible residues up through k = J + 1, and the second factor runs from > J + 1 through half the span of s.* > > **(1)** w_{s,J}(∞) = ∏_{q ≤ J+1} (q − ν_q) · ∏_{J+1 < q ≤ |s|/2} (q − ν_q)/(q − J − 1) The second product **terminates at |s|/2** for exactly the reason the threshold exists: past |s|/2 every factor is 1, because ν_q = J + 1 there. So |s|/2 is not decorative in this paper — it is what makes an infinite product finite. ### Does anything go beyond what the officer found on p. 11? **Three things, and one of them changes an attribution.** 1. **The threshold has a stated mechanism here, not only a marking.** The p. 11 passage the officer read announces the threshold; the p. 10–11 proof of Lemma 2 *uses* it and names the recursion it comes from ([HR15] = 1510.00743, the 3-step recursion). Its consequence is quantitative: ρ_s = q − (J+1) and ρ_[s2] = q − (J+2) exactly, past the threshold. 2. **The threshold is load-bearing for the asymptotic populations**, through Equation (1)'s upper limit. That use is credited to [Hol25] = 2502.20470, so the formula is not new here. 3. **The corpus already holds this threshold, in a sharper form, under another name — and did before this paper surfaced.** `lit-pdf-holt-rudd.md` §3a records, verified against the PDF and quoted in `PRIOR-ART.md`, from arXiv:2603.25915v1 §1, PDF p. 6: > *"…we know that the minimum span between fusions is 2p_{k+1}. So provided > that |s| < 2p_{k+1}, the possible fusions in s all occur in separate images > of s."* `|s| < 2q` and `q > |s|/2` are the same inequality. The same fact appears as unnumbered prose in 1408.6002v1, PDF p. 11: *"the minimum distance between closures is 2 · p_{k+1}."* **So 2605.19165 p. 11 restates a fact the corpus already had from two other Holt manuscripts; what it adds is the numeral 1620 drawn on a graph and the ρ_s = q − 459 consequence.** The officer's candidate-1(b) reading — that this is the published ancestor of "runs die once p is large enough" — survives, but **the citation should be the 2603.25915 sentence the corpus already verified, with 2605.19165 as a secondary marking.** ### Errata noticed while reading (none load-bearing for us) - The prose calls the p. 7 residue table "Figure 4" throughout §2; it is printed as **Table 1**, and no object labelled Figure 4 is printed. - p. 11 opens "In Figure 6 we plot the proportion of occurrences…"; the plot is **Figure 7**, and its caption says it plots the ratio n_in/n_out while the plot's own title says "Proportion of occurrences … inside their parents". - p. 12 states s₂₅ has asymptotic relative population **8.657E90**; Table 2's row "25, 90" reads **2.911E90**. The stated ratio 32.73 against s₂₉'s 8.896E88 matches the table, not the prose figure, so the prose figure looks like the typo. - Lemma 2's proof says "across the 59 (458, 3240) constellations"; the family has 116, tabulated 58 at a time by reversal symmetry. --- ## 3. THE INTERACTION CHECK **Question.** Our multi-kill extinction law measures the last fold carrying an adjacent kill run L ≥ 2 in a window of length W. Holt's threshold says all fusions occur in separate images of s past |s|/2. Is our measured crossing his theorem in window coordinates? ### 3.1 The measured law, cited not recomputed From `research/history/staging/attack-foldL-06-scaling.md` §4, whose figures are the formally embedded output of `research/attack-foldL-06-scaling.js`: | W | last fold with L ≥ 2 | |---|---| | 2·10⁷ | 181 | | 2·10⁸ | 331 | | 2·10⁹ | 421 (calibration window) | | 2·10¹⁰ | 457 | Two further cited figures matter below: at W = 2·10⁹, fold 421 carries **105,790 kills** with mean spacing **m̄ = 89.1522** (§5 offset table); and at W = 2·10¹⁰ the **last fold whose θ still fits under the window's G₂ is 1021**, with the fold range stopping at **1499** in every window (§7). ### 3.2 What plays the role of s for a window of length W — the two honest readings **Reading A, the window as the constellation.** Take s to be the window's surviving word at level p. Its span is |s| ≈ W. Holt's threshold is then p > |s|/2 = W/2. **Reading B, the window in our own alphabet.** Our slots are n ≡ 5 (mod 6), so differences between slots are multiples of 6, and a fold p ≥ 5 divides a difference 6m only if p | m with m ≤ W/6. Holt's argument in our alphabet gives the one-class threshold p > W/6. The two-class branch (difference ≡ ±2 mod p, which is the twin comb's extra channel) pushes it back up to p > (W+2)/2. So reading B brackets the threshold between W/6 and W/2 and changes nothing that follows. ### 3.3 The arithmetic. It does not predict our crossing, and it is not close | W | Holt threshold W/2 | Holt threshold W/6 | measured extinction | W/2 ÷ measured | |---|---|---|---|---| | 2·10⁷ | 1.0·10⁷ | 3.33·10⁶ | 181 | 55,249 | | 2·10⁸ | 1.0·10⁸ | 3.33·10⁷ | 331 | 302,115 | | 2·10⁹ | 1.0·10⁹ | 3.33·10⁸ | 421 | 2,375,297 | | 2·10¹⁰ | 1.0·10¹⁰ | 3.33·10⁹ | 457 | 21,881,838 | **The scaling is the decisive part, not the size of the ratio.** Across W = 2·10⁷ → 2·10¹⁰ the window grows by a factor 1000 and Holt's threshold grows by the same factor 1000, because it is exactly linear in the span. Over the same three decades the measured extinction fold grows by **457/181 = 2.525**. The two laws diverge by a factor **1000/2.525 = 396 across three decades**, and the divergence is unbounded. A linear-in-W statement cannot be a grows-by-a-factor-2.5-per-thousandfold statement. **Three further checks, each independently fatal to the identification.** 1. **The threshold lies outside the computed range entirely.** Every window's fold sweep stops at 1499. At W = 2·10¹⁰ Holt's W/2 = 10¹⁰ exceeds that ceiling by a factor 6.7·10⁶. His threshold is never reached in any window we measured, so it cannot be what the measurement crossed. 2. **The kill population is in the wrong regime by five orders of magnitude.** Past p > W/2 a window carries at most one or two fold-p kills, which is why coincidences become impossible there. At our measured extinction fold p = 421, W = 2·10⁹, the window still carries **105,790 fold-p kills**. What dies at 421 is not the supply of kills; it is their *adjacency*. 3. **The relations are different relations.** Holt's is `p | (γ_j − γ_i)` — residue coincidence, a function of the difference alone, deterministic, and exactly the condition of 2502.20470 Lemma 2. Ours is *adjacency in the surviving word at level p*: every slot strictly between the two kills was deleted at some earlier fold. That is a function of the whole sieve history below p and of nothing modular. `PRIOR-ART.md` already records the general form of this: Holt's coincidence count (J+1) − ν_p(s) is blind to where in the word the coincidences sit. **Verdict: NOT the same statement. Our measured law is not his theorem.** ### 3.4 But there is a real correspondence, and it is already attributed Run Holt's threshold at the right scale and it stops being a prediction and becomes an identity. The "s" whose fusions are being tested when we ask whether two fold-p kills can be adjacent is not the window: it is the **stretch between the two candidate kills**, whose span must be at least θ_p = 2p − 2η (η = +1 for p ≡ 1 mod 6, η = −1 for p ≡ 5 mod 6). For that local s, |s|/2 = (2p ∓ 2)/2 = p ∓ 1, so Holt's threshold is p itself, at every fold, tautologically. **Holt's |s|/2 and our qualifying threshold θ_p ≈ 2p are the same fact read in two directions**: "a span below 2q admits no two fusions in one image" run forwards is "two fusions in one image are at least 2q apart" run backwards. That is precisely the 1408.6002 p. 11 sentence, *"the minimum distance between closures is 2·p_{k+1}"*. So the map between the two frames is: | Holt | ours | |---|---| | fusions in the **same image** of s | two kills at the **same fold** p | | q \| (γ_j − γ_i) | difference ≡ 0, ±2 (mod p), the two-class form | | minimum span between fusions = 2q ⟺ threshold \|s\|/2 | qualifying spacing θ_p = 2p − 2η | | — | **adjacency in the level-p word**: no counterpart | | — | **crossing of θ_p against the largest spacing W supplies**: no counterpart | `proposals-prior-art.md` §6 already splits our extinction law into exactly these two halves and attributes the first to Holt (1408.6002 p. 11) and the second to the maximal-spacing convention (Lévy → Devroye → Deheuvels). **This sweep confirms that split and adds nothing to Holt's side of it.** The extinction *fold* is set by the half that is not his. ### 3.5 Where the two statements diverge, stated for a referee 1. **Modality.** His is a deterministic impossibility past a threshold. Ours is the extinction of an event that remains possible at every fold and merely becomes rare. 2. **Scaling.** His threshold is linear in the span. Ours grows by 2.525 while the span grows by 1000. 3. **Direction of what dies.** Past his threshold, same-image coincidences are impossible. Past ours, same-fold kills at distance ≥ 2p are still abundant — what dies is their adjacency in the surviving word. 4. **What the statement is a function of.** His depends only on the difference between two positions mod q. Ours depends on the entire sieve history below p, which is what makes it a statement about the surviving word rather than about residues. --- ## 4. Bibliography check Eight references, read from the page-14 reference list: | key | reference | known to the corpus? | |---|---|---| | [Eng05] | T. Engelsma, *k-tuple permissible patterns*, 2005 | yes — `PRIOR-ART.md`, `natal-cap-04-packing-notes.md` | | [HL66] | Hardy & Littlewood, *Some problems in 'partitio numerorum' III*, Collected Papers vol. 1, pp. 561–630, Clarendon Press 1966 | yes — the corpus carries the Hardy–Littlewood k-tuple/convexity material throughout | | [Hol22] | F.B. Holt, *Patterns among the Primes*, KDP, 2022 | yes — `SEARCH-CONVENTIONS.md`, `covering-dive.md`; the self-published book the sweep flagged as read only through four proxies | | [Hol25] | F.B. Holt, *Eratosthenes sieve supports the k-tuple conjecture*, arXiv:2502.20470, 2025 | yes — row 12 of the sweep table | | [Hol26] | F.B. Holt, *On nonconvex constellations among primes I*, **arXiv:2603.25896v3**, 2026 | yes — row 13; note the **v3**, where the corpus's sweep row carries no version | | [HR15] | Holt & Rudd, *Combinatorics of the gaps between primes*, Connections in Discrete Mathematics (SFU), arXiv:1510.00743, 2015 | yes — row 7 | | [Sut13] | A. Sutherland, *Narrow admissible tuples*, math.mit.edu/~primegaps/, 2013 | yes — `natal-cap-04-packing-notes.md` | | [Sut15] | A. Sutherland, *Sieve theory and gaps between primes: narrow admissible tuples*, Oberwolfach, *Explicit Methods in Number Theory*, 2015 | yes — already cited in `PRIOR-ART.md` row 13's source | **Nothing in the bibliography is new to the corpus.** Two small deltas worth carrying: the paper cites [Hol26] specifically at **v3**, and its bibliography does **not** cite 1408.6002, 2603.25915 or 1402.1970 — for the 3-step recursion it cites [HR15] = 1510.00743 instead, which is where the corpus should point when quoting the recursion by Holt's own preferred citation. --- ## 5. What this sweep does not cover The Figure 4/5 and Table 1/2 data grids were read as page images for their shape, captions and the highlighted-cell rule, not digit by digit; no claim here rests on an individual cell. Holt's self-published *Patterns among the Primes* (2022) remains unread beyond the four proxies the corpus already holds. The `primegaps-v2` notebooks for this paper were not opened. The 2603.25896v3 revision was not diffed against the version the corpus swept. --- ## 6. Proposed edits to live documents (NOT APPLIED HERE) 1. **`research/PRIOR-ART.md`, the sweep table row for 2605.19165.** Replace the placeholder row with the completed row of §1 above, and change the section heading from "FOURTEEN OF FIFTEEN" to "FIFTEEN OF FIFTEEN", deleting the sentence "only its §3, p. 11 has been read, as page images, and the full sweep of it is OUTSTANDING." 2. **`research/PRIOR-ART.md`, THE NOVELTY BOUNDARY paragraph.** It says twin appears in the corpus in three roles and that three papers do not contain the word. Update to **four** papers: 2502.20470, 2308.07570, 2309.16833 and now 2605.19165. 3. **`research/PRIOR-ART.md`.** The boundary sentence stands unchanged for this paper: no maximum-gap upper bound, no twin-slot spacing. 4. **Citation guidance to record wherever the |s|/2 threshold is used.** Cite arXiv:2603.25915v1 §1 p. 6 (already verified in `lit-pdf-holt-rudd.md` §3a) as the primary source, with 2605.19165 p. 11 as the numeric marking. The two are the same inequality. 5. **Memory note `prime-holt-prior-art.md`.** Add: the Holt sweep is complete at fifteen of fifteen; 2605.19165 is *On nonconvex constellations II: (458,3240)*, Holt alone, 18 May 2026; it does not contain the word twin, no max-gap bound, no G₂; its |s|/2 threshold is the 2603.25915 / 1408.6002 fact restated; and **the interaction check is negative — the extinction law is not his theorem, diverging by a factor 396 across three decades of W.** 6. **`proposals-prior-art.md` §1(b) and §6**, if either is ever promoted: keep the ADJACENT verdicts, but re-point the |s|/2 citation as in edit 4 and add the interaction arithmetic of §3.3 above, which is the sharpest available demonstration that the composition is not Holt's.