# lit-pdf-fgkt-maier — arXiv:1408.4505 and the Maier literature, checked against PDFs Read-only literature verification, 2026-08-18. **No repo file was edited except this one.** No commit, no push. Every verdict below comes from a PDF I downloaded and read in this session, not from an abstract page, an ar5iv/arXiv HTML rendering, a search snippet, or a restatement inside a third paper. ## PDFs obtained (all of them; nothing in this report rests on HTML) | short | identity confirmed from the PDF itself | pages | how obtained | |---|---|---|---| | **FGKT** | arXiv:1408.4505**v2** [math.NT] 9 Nov 2015, *Large gaps between consecutive prime numbers*, **Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao** (four authors, no Maynard) | 32 | `https://arxiv.org/pdf/1408.4505`, HTTP 200, 346,253 B, `PDF document, version 1.4` | | **Maier 85** | *Primes in short intervals*, Helmut Maier, **Michigan Math. J. 32 (1985), 221–225**; footer "Received May 11, 1984" | 6 | Project Euclid `journalArticle/Download?urlId=10.1307%2Fmmj%2F1029003189`, HTTP 200, 328,501 B. **Image-only scan, no text layer** — read visually, page by page | | **MP 90** | *Unusually large gaps between consecutive primes*, Helmut Maier and Carl Pomerance, **Trans. Amer. Math. Soc. 322, Number 1, November 1990, 201–237** | 37 | `ams.org/.../S0002-9947-1990-0972703-X.pdf`, HTTP 200, 2,793,314 B | | **GS (preprint)** | arXiv:math/0406018**v1**, *An uncertainty principle for arithmetic sequences*, Granville and Soundararajan — this is the file behind `scratchpad/holt/gs.txt` | 21 | already on disk from an earlier session; I re-extracted from the PDF myself | | **GS (published)** | **Annals of Mathematics 165 (2007), 593–635**, same title and authors | 43 | `annals.math.princeton.edu/wp-content/uploads/annals-v165-n2-p06.pdf`, HTTP 200, 890,630 B | | **FGKMT** | arXiv:1412.5029**v3** 14 Jul 2016, *Long gaps between primes*, Ford, Green, Konyagin, **Maynard**, Tao | — | `https://arxiv.org/pdf/1412.5029`, HTTP 200 — pulled because item 4 turned up a provenance problem that spans both Ford papers | **PDF NOT OBTAINED, and therefore not verified here:** Cheer and Goldston, *A differential delay equation arising from the sieve of Eratosthenes*, Math. Comp. 55 (1990). The `"introduced by Buchstab in connection with an asymptotic formula for the number of uncanceled terms in the sieve of Eratosthenes"` quotation at `research/PRIOR-ART.md`:399 is **UNVERIFIED in this pass** — I did not read that paper. What I *can* say from Maier 85's own reference list is that the phrase closely tracks the title of Maier's reference [1], de Bruijn, *On the number of uncancelled elements in the sieve of Eratosthenes*, Nederl. Akad. Wetensch. Proc. 53 (1950), 803–812 — note **"uncancelled … elements"** there against our **"uncanceled … terms"**. That is a flag for whoever re-checks, not a verdict. **Note on method.** `pdftotext` flattens superscripts and subscripts, which is precisely the distinction that decides whether FGKT wrote `log² x` or `log₂ x`. Every such glyph in item 1 was resolved by `pdftotext -bbox-layout` and comparing the y-coordinate of the digit against the baseline of the adjacent `log`. Those readings are reported explicitly below. --- ## Item 1 — `research/maxgap-law.md`:363, the `a_p` passage ### (a) Verdict: **VERBATIM**, within its two marked elisions. The common value is confirmed to be **zero**. ### (c) Location FGKT §1 Introduction, the paragraph beginning "We turn now to a discussion of the proof of Theorem 2." It starts at the **foot of printed page 3** and runs onto **printed page 4** (printed page n = PDF page n in this file). Our header at :360 says "arXiv:1408.4505 §1, verbatim" — §1 is correct. ### (b) Source beside ours Source, transcribed from the PDF: > Prior authors divided the sieving into different steps, **a key to all of them being to take a common > value of a_p for "large" p, say a_p = 0 for z < p < δx,** where δ > 0 is a small constant and > z = x^{c log₃ x / log₂ x} for some constant c > 0. The numbers in [y] surviving this first sieving > either have all of their prime factors ⩽ z (i.e., they are "z-smooth") or are of the form pm with p > prime and m ⩽ y/δx. One then appeals to bounds for smooth numbers, e.g. [3], to see that there are very > few numbers of the first kind, say O(x/log² x). By the prime number theorem there are ∼ y log₂ x / log x > unsieved numbers of the second kind. By contrast, if one were to take a random choice for a_p for > z < p < δx, then with high probability, the number of unsifted integers in [y] would be considerably > larger, about y log z / log x. A word-level diff against our block at :363–372 returns exactly these differences and no others: | # | source | ours | reading | |---|---|---|---| | 1 | `Prior authors divided the sieving into different steps,` | *(absent, no leading ellipsis)* | our quote opens mid-sentence on a lowercase word, so it reads as a fragment; **unmarked front truncation**, cosmetic | | 2 | `, where δ > 0 is a small constant and z = x^{c log₃ x/log₂ x} for some constant c > 0.` | `...` | **properly marked elision** | | 3 | `, e.g. [3],` | `...` | **properly marked elision** | | 4 | `δ`, `⩽`, `∼` | `delta`, `<=`, `~` | house ASCII transliteration, used consistently corpus-wide | | 5 | *(no emphasis)* | last sentence set in `**bold**` | **emphasis added, not marked as added** | ### The sub/superscript question, resolved from glyph coordinates (page 4, y increases downward) | source text | `log` baseline yMin | digit yMin | digit sits | therefore | our rendering | |---|---|---|---|---|---| | `O(x/ log 2 x)` | 156.70 | **154.46** | **higher** | **superscript** → `log² x` | `O(x/log^2 x)` ✅ | | `y log 2 x/ log x` | 156.70 | **161.54** | **lower** | **subscript** → `log₂ x` | `y log_2 x/log x` ✅ | | `log x/(log 2 x) O(1)` | 515.50 | `2` at **520.34** / `O(1)` at **513.74** | lower / higher | `(log₂ x)^{O(1)}` | `(log_2 x)^{O(1)}` ✅ | | `z = x c log 3 x/ log 2 x` (p. 3) | exponent block at 682.10 vs baseline 683.86; `3`,`2` at 686.09 | — | whole block raised, digits lowered inside it | `z = x^{c log₃ x/log₂ x}` | `z = x^{c log_3 x/log_2 x}` at :393 ✅ | **Our document gets all four right.** This is the exact class of error an HTML rendering produces, and the corpus did not make it here. ### (d) Is the surrounding claim supported? **Yes, and the quantifier the brief flagged is the strong reading.** The brief asked what the common value is set to, because the covering-identity work turns on `a_p` being a free translate rather than a fixed pair. The source is unambiguous: **`a_p = 0`**, a single fixed value, held common across the whole range `z < p < δx`. Not a free translate, not a per-prime choice — zero. FGKT's own contrast sentence makes the point structural rather than incidental: a *random* `a_p` on that same range leaves `≈ y log z / log x` survivors instead of `∼ y log₂ x / log x`, and since `log z = c x log₃ x / log₂ x` while `log x` is the ambient scale, the aligned choice is the entire gain. Our §7 reading — that `T1 = log z / log₂ x` "is the gain of an ALIGNED `a_p` over a random one" (:390–391) — is exactly what the source says, in the source's own two numbers. The arithmetic our document builds on top also closes: `T1 = log z/log₂ x = (c log₃ x/log₂ x)·log x/log₂ x = c·log x log₃ x/(log₂ x)²`, which is what :393 states. **Second quote block, :375–378** (`Assuming that V is a "random" subset…`): source is on **printed page 4**, same paragraph run. Diff returns two differences only — source `V ∩ (a (mod p))` against our `V n (a mod p)` (dropped inner parenthesis, cosmetic), and source **`≈ |V|/p`** against our **`~ |V|/p`**. The second is worth one line: our block renders source `∼` and source `≈` with the same `~`, so the two FGKT blocks use one glyph for two different relations. Immaterial to the claim; noted for exactness. --- ## Item 2 — `research/maier-matrix.md`:550, "(1.7) … α ≥ 60 log log log x / log log x" ### (a) Verdict: **VERBATIM.** Constant 60 confirmed, iterated-log structure confirmed, η-cap confirmed. ### Which paper: **Granville and Soundararajan, not Maier.** Our §8 heading names it ("Granville-Soundararajan Corollary 1.4") and cites `scratchpad/holt/gs.txt` lines 313–325. I traced that file: it is `pdftotext` output of **arXiv:math/0406018v1**, *An uncertainty principle for arithmetic sequences*, Granville and Soundararajan. I verified against **that PDF** and, independently, against the **published Annals of Mathematics 165 (2007), 593–635** version. **The numbering is identical in both** — (1.7) is (1.7), Corollary 1.4 is Corollary 1.4 — so the corpus is not exposed to a preprint/published renumbering here. ### (c) Location **Published: Annals 165 (2007), printed page 599 = PDF page 7** of the Annals reprint, §1b "General results", immediately after Example 5. Preprint: gs.txt lines 313–325 — our citation is **exact**, Corollary 1.4 opens at 313 and alternative (ii) ends at 325. ### (b) Source beside ours Published Annals, p. 599, transcribed from the PDF: > **Corollary 1.4.** Let A, S, f_q and γ_q be as above. Suppose that 0 ≤ h(n) ≤ 1 for all n. **Suppose that > (1.7) holds for some α ≥ 60 log log log x/ log log x and set η = min(α/3, 1/100). Then for each > 5/η² ≤ u ≤ η(log x)^{η/2} at least one of the following two assertions holds:** Ours (:550–552): > *"Suppose that (1.7) holds for some α ≥ 60 log log log x/ log log x and set η = min(α/3, 1/100). Then for > each 5/η² ≤ u ≤ η(log x)^{η/2} at least one of the following two assertions holds …"* **Character-for-character identical.** The only difference is the trailing `:` replaced by `…`. The constant is **60**. The structure is **60·log log log x / log log x** — triple log over double log, exactly as printed. ### What (1.7) is in the source Stated in **Corollary 1.3**, printed page 598 (preprint gs.txt:262): > (1.7) Σ_{p ≤ log x} [(1 − h(p))/p] · log p ≥ α log log x, where `h` is the non-negative multiplicative function of the setup (1.5), `A_d(x) ≈ (h(d)/d)A(x)`, and `S` is "a finite set of 'bad' primes". GS's own gloss (p. 598): "The condition (1.7) ensures that h(p) is not always close to 1; this is essential in order to eliminate the very well behaved Example 1." ### (d) Is the surrounding claim supported? **Yes — the vacuity derivation is correct, and I re-derived it.** - η = min(α/3, 1/100) ⇒ **η ≤ 1/100 for every sequence, for every α** ✅ (:554) - ⇒ 5/η² ≥ 5·10⁴ = **50,000 always** ✅ (:555) - upper bound u ≤ η(log x)^{η/2} ≤ (1/100)(log x)^{1/200}, monotone in η in the right direction ✅ - non-empty range ⇒ (log x)^{1/200} ≥ 5·10⁶ ⇒ **log x ≥ (5·10⁶)²⁰⁰** ✅ (:556) - (5·10⁶)²⁰⁰: log₁₀ = 200·6.69897 = 1339.79, so **≈ 10^{1340}** ✅ (:557, "roughly x > exp(10^{1340})") The `α = 1 + o(1)` claim at :565–568 also checks. With `h(p) = 0` for `p ≤ x`, (1.7)'s summand is `(1−0)·log p/p`, and `Σ_{p ≤ log X} log p/p ~ log log X` by Mertens, giving α = 1. GS themselves write that Corollary 1.3 "applies to the sequences of primes (with α = 1 + o(1))" — same value, same reason. ### Two things to record against §8, neither of them a quotation defect 1. **The cited source file no longer exists.** `scratchpad/holt/gs.txt` lives in another session's ephemeral scratchpad (`/private/tmp/claude-501/.../2a01d02b-…/scratchpad/holt/gs.txt`), not in the repo. §8 also never names the paper's title, journal, year or arXiv id — only "Granville-Soundararajan Corollary 1.4". The full citation *is* carried at `research/PRIOR-ART.md`:414, but §8 does not point there. A reader of `maier-matrix.md` alone cannot reach the source. 2. **GS's own framing of Corollary 1.4 strengthens our §8 conclusion and is not quoted.** Immediately after the corollary (p. 600): *"Corollary 1.4 is our general version of Maier's result; it is a weak form of the more technical Theorem 2.5."* So the vacuity we compute is the vacuity of the *weak* form, and GS's Theorem 2.5 is the sharp one. §8's verdict "Corollary 1.4 is not an obstruction to anything in this repo" is correct as written and about the right object, but the sentence "it is a weak form of the more technical Theorem 2.5" means the corollary's vacuity does not by itself clear Theorem 2.5. Worth a line in the body, since §8 currently reads as though the whole GS obstruction had been disposed of. --- ## Item 3 — `research/maier-matrix.md`:154, `ρ_y(S) := [D_y(S)/S] / [D_y/W_y]` ### (a) Verdict: **OURS-NOT-THEIRS.** This is a house definition, not a quotation from Maier. It appears in **no** Maier or GS source I read. Maier 85 has no ρ. GS use `γ_q`, `f_q(a)`, `h(d)`, `Φ`. The `:=` is a definitional colon-equals: our own object, being introduced. ### Is the blockquote reading as attribution? Partly — and the risk is real but structural, not deceptive. In this corpus `>` carries **two** jobs, and `maier-matrix.md` uses both within a few hundred lines: - **house result box** — `> **Transfer identity (PROVEN).**` (:107), `> **Origin Excess Lemma…**` (:199), `> **Proposition.**` (:422), `> **e^{2γ}/4 = 0.79305.**` (:394) - **external quotation** — `> *"Suppose that (1.7) holds…"*` (:550), always **italic + double quotes**, and always preceded by a naming line ("Checked against … Verbatim:") Line 154 is `> **bold**` with `:=` and **no italics, no quote marks, no naming line**. By the file's own convention it is unambiguously a box, and the same convention is honoured at :550 for the one real quotation in the file. So this is **not** a house definition dressed as an attributed quotation. **But the surrounding prose does most of the damage the brief was worried about.** The definition sits under the heading `### 3a. What the identity says in Maier's language` (:150), and the sentence immediately after it reads `This is Maier's mechanism exactly, with all error terms zero.` (:157). A reader skimming heading → box → next sentence can carry away that `ρ_y(S)` is Maier's notation. It is not. **Recommended fix (not applied): leave the box, and change "in Maier's language" to something that does not imply the notation is his** — the mechanism analogy is fair, the vocabulary attribution is not. ### (d) And there is a substantive over-claim two lines later, which matters more than the formatting `:157–160` reads: *"ρ_y(S) is the two-dimensional survival function … and its oscillation away from 1 is the Buchstab oscillation Cheer and Goldston computed and Maier used."* Against the PDF of record, **Maier's ω and our ρ are different functions.** Maier 85, p. 222: > **LEMMA 3 (Buchstab).** Let λ > 1. Then lim_{z→∞} z^{−λ} W(z)^{−1} Φ(z^λ, z) = e^γ ω(λ), > where ω(u) is defined by (2.1) ω(u) = u^{−1}, 1 ≤ u ≤ 2; d/du(u ω(u)) = ω(u−1), u ≥ 2 with `Φ(x,y) = |{n ≤ x : (n, P(y)) = 1}|` and `W(z) = ∏_{p limsup [pi(x + Phi(x)) - pi(x)] / [Phi(x)/log x] > 1, and liminf < 1 Maier 85, **printed page 221**, the paper's only numbered THEOREM: > **THEOREM.** Let Φ(x) = (log x)^{λ₀}, λ₀ > 1. Then > lim sup_{x→∞} [π(x+Φ(x)) − π(x)] / [Φ(x)/log x] > 1 and lim inf_{x→∞} [π(x+Φ(x)) − π(x)] / [Φ(x)/log x] < 1. > For the range 1 < λ₀ < e^γ we have even lim sup_{x→∞} [π(x+Φ(x)) − π(x)] / [Φ(x)/log x] ≥ e^γ/λ₀, > where γ denotes Euler's constant. Our statement is **exactly Maier's**, with `λ` for his `λ₀`. `paper/wall-note.md`:210–212 and `paper/moire-primes.md`:645–648 state the same thing in prose and are equally correct. **One caveat worth recording.** We repeatedly say our window is "the case λ = 2" (`PRIOR-ART.md`:384, `moire-primes.md`:647). λ = 2 is inside Maier's **qualitative** range (λ₀ > 1) ✅ but **outside his quantitative range 1 < λ₀ < e^γ = 1.7811**. So the qualitative irregularity is ours to cite at λ = 2; the explicit constant `e^γ/λ₀` is **not available at λ = 2** from this theorem. Nothing in the corpus currently claims the constant, so no live claim breaks — but anyone reaching for `e^γ/2` at λ = 2 would be reaching past what p. 221 licenses. ### 4.2 The Maier matrix method — **correct, and better sourced than our own citations suggest** `PRIOR-ART.md`:362–365 says the method "constructs its intervals by *first selecting a primorial and then using the distribution of integers coprime to that primorial*", and credits the explicit matrix layout to GS. **Both halves verify, and the matrix is in Maier's own paper, printed page 224, not only in GS:** > We consider the matrix 𝔐 = (a_{rs}), where a_{rs} = s + rP(z), 1 ≤ s ≤ U, P(z)^{D−1} < r ≤ 2P(z)^{D−1}. > … The **rows** of 𝔐 are **intervals** of U consecutive integers, whereas the **columns** of 𝔐 are > **arithmetic progressions** with common difference P(z). Only those columns for which (s, P(z)) = 1 > contain primes. We call such columns **admissible**. with `P(z) = ∏_{p