# CHANGELOG entries from wave-2 partition E Merge into `research/history/CHANGELOG.md` under a single dated heading. Files covered: `covering-dive.md`, `two-class-lower-bounds.md`, `PRIOR-ART.md`, `sift-limit-attack.md`, `theta-ladder.md`, `maxgap-law.md`, `dhr-verification.md`. --- ## 2026-08-17, wave 2: partition E — the FKMPT constant, the source checks, and four brief-response migrations ### research/covering-dive.md and research/two-class-lower-bounds.md — the FKMPT constant, moved together **`C(ρ) := sup{δ : (4 + δ)·10^{2δ}/log(1/(2δ)) < ρ}` and `C(ρ) > e^{−1−4/ρ}` (`covering-dive.md` §Q2.3), and `C(1/2) > 1/6001` (`two-class-lower-bounds.md` §2a). RETRACTED VALUES, replaced by `6·10^{2δ}`, `e^{−1−6/ρ}` and `C(1/2) > 1/325565`.** **This value has now been edited three times and the record of why is the point of this entry.** The sequence was: the corpus first carried 6; an earlier audit wave (`research/history/staging/audit-campaign.md`) judged the 6 to be a stale arXiv reading and moved `covering-dive.md`:59 to 4, writing its confidence into a staging file; every later pass inherited that, including the wave-1 brief for `qc-refs`. `qc-refs` W1 pulled all four arXiv PDFs and found the direction is the opposite one, and this partition re-verified it from the same PDFs plus the publisher record before applying. The evidence, so no fourth edit is needed: - arXiv v2 (2019-08-16) and v3 (2021-06-23) print `(4 + δ)·10^{2δ}` and `C(ρ) > e^{−1−4/ρ}`, and v3 prints `C(1/2) > 1/6001` in its own text. - **arXiv v4 (2022-09-19) prints `6·10^{2δ}`, `C(ρ) > e^{−1−6/ρ}` and `C(1/2) > 1/325565`.** So `1/325565` is not our arithmetic; it is the paper's own corrected figure. - v4 carries the corrigendum as its **Appendix A, "Corrigendum: changes made from the published version"**, which states the cause — errors in the exponents of H in the deduction of Theorem 2 from Theorem 3, forcing M > 6 — and enumerates the changes, item (1) being "the definition of C(ρ), the factor 4 + δ corrected to 6. Likewise, the corrected lower bound is C(ρ) > e^{−1−6/ρ}", item (4) being "In (1.7) and Corollary 2, the corrected bound is C(1/2) > 1/325565". Credited by the authors to Mikhail Gabdullin. - **The corrigendum is published: J. Eur. Math. Soc. 25 (2023), no. 6, 2483–2485, DOI 10.4171/JEMS/1305.** It appeared nowhere in this repository before this pass; it is now cited in both files. - The direction is the ordinary shape of a corrected error: 4 → 6 *weakens* the bound, since e^{−1−6/ρ} < e^{−1−4/ρ}. The figure standing in the corpus was **54× too generous.** The qualitative claim is unaffected: "two classes per prime on half the primes is inside a published theorem" needs only `C(1/2) > 0`, which holds in every version. Both files now state which version they quote. **The method lesson, recorded because it is about this campaign.** Version drift in a cited source is invisible to internal consistency: `covering-dive.md` and its own bibliography agreed with each other perfectly while both carried the retracted value. Only fetching the source caught it. **`covering-dive.md`:58's "v3, dated 2024-12-03".** RETIRED. No version of arXiv:1802.07604 is dated 2024-12-03 (v1 2018-02-21, v2 2019-08-16, v3 2021-06-23, v4 2022-09-19); the file now cites v4 throughout with its correct date, and the v2/v3 values in a parenthesis. **`covering-dive.md`'s `[Halberstam–Richert, Cor. 2.4.1]` inside a quotation labelled verbatim.** The substitution is substantively CORRECT — bibliography entry [7] in v2, v3 and v4 is verbatim "H. Halberstam and H.-E. Richert, *Sieve Methods*, Academic Press, London, 1974 — and it is now marked as an editorial expansion, `[7 = Halberstam–Richert, Cor. 2.4.1]`, with the note that `[7]` in v1 is a different work. `two-class-lower-bounds.md`'s bare `[7]` gains the version numbering. Remark 7's text is byte-identical in v3 and v4 (Remark 8 in v2), so the version split there was harmless; the damage was one bullet up. ### research/maxgap-law.md **§10 item 4, "The Maier-Pomerance statement, checked against the source, so nobody re-checks it".** Two of its three legs were WRONG, in a block whose whole function was to discourage re-checking. Corrected against `arXiv:1412.5029` before being relocated into §7, per the sequencing constraint: - "`Y(x) = j(P(x)) − 1` exactly (their **eq. 1.2**)" → it is **eq. (1.3)**; `Y` itself is their **Definition 1**; and (1.2) is a different statement, the proven lower bound. - "The proven FGKMT bound is `Y(x) >= R·x·log x·log_3 x/(log_2 x)^2` for any `R`" → FGKMT's (1.2) is `Y(x) ≫ x log x log_3 x/log_2 x`, a **single** power of log_2 x. The squared form is **Rankin's**, printed by FGKMT only in the sentence "This improves on the bound … obtained by Rankin". The "for any R" quantifier is not theirs either: their Theorem 1 is `G(X) ≫ log X log_2 X log_4 X/log_3 X` with an effective implied constant. - Leg 1, the Maier–Pomerance quotation, is CORRECT and verbatim. **The conclusion survives**, which is why this is a statement-of-someone-else's- theorem defect and not a mathematical one: with the correct bound the ratio to the random-dart level is `log_3 x/(e^γ log_2 x) → 0`, so it still sits below. `maxgap-law.md`:391 carried the same misattribution and is corrected with it. The other fourteen sites in the corpus already had the single `log_2 x`, and `:52` and `:390` correctly attribute the squared form to Rankin. The relocated block now says what was checked, against which version, and on what date, which is what makes a "do not re-check" note usable. **"Ford's slides: not located in this dive"** (`covering-dive.md` §Q5) against `maxgap-law.md` §6 quoting them verbatim as read. RESOLVED by locating them: Kevin Ford, *Large gaps between primes*, Talks 1–3, CRM Montreal workshop *Probability in Number Theory*, 2018, ford126.web.illinois.edu/montreal_talk{1,2,3}_primegaps.pdf. Talk 1's slide "Proving large gaps: Jacobsthal's function" carries both quoted statements one line apart, verbatim as `maxgap-law.md` §6 reports, and independently corroborates FGKMT's single log_2 T. Both files now carry the citation. `two-class-lower-bounds.md` §2c's "T(log T)^{1+c} under uniform Hardy– Littlewood" is also verbatim on that slide and now carries the citation. **§10 items 1, 2 and 3, "What this note owes other files".** All three debts were paid before this pass and the list stated they were outstanding: `localized-04-maxsum.md`:127-133 now carries the coordinates; `FOLD-PROFILE.md` :486 now reads `M(x, x′²) ~ 3.5·ln³x` with margin `x²/(3.5 ln³x)`; `exponent-control.md`:86 now carries `c1 ~ (log p)^{0.12 ± 0.03}`. Section removed; §§11, 12 renumber to 10, 11. Verified first that no document cites `maxgap-law.md` §10 or past it — the highest inbound section reference is §9. **Brief-response framing.** The 2026-08-17-night date stamp, "this note … corrects three things", "over a lever 41 times longer than the briefing assumed", "the tension in the briefing", the READING 1 heading "The tightness dispute", the quotation of the briefing's two spreads, and "Correction to the briefing, and to both files" all RETIRED as framing. The mathematics is unchanged: the two spreads are the same size on matched conventions, and the one-class lever is a factor 41 with `c1` moving 22% over it. ### research/two-class-lower-bounds.md **§10 "Corrections logged", five items.** MIGRATED. Four of the five quoted sister documents that no longer say the quoted thing: - item 3 quoted `covering-dive.md` §Q4.2's "realistic target 4" asking "does covered length scale like `c·p²/log`? like ZM's ≈ `p²/2` data?" — neither string is in that file. Its mathematics is not stated anywhere else in the file and has been moved into §6 as a plain paragraph. - item 4 quoted `U-FRAME.md` §6a as reading the adversarial exponent "STABLE at about 1.62, comfortably below the critical 2"; `U-FRAME.md`:535 now reads "The fitted exponent is 1.62 in the theta frame and 1.924 against x". Its one non-duplicated sentence, the 1.2 greedy-ladder exponent, is already at §5. - item 5 quoted `ZONE-POSTULATE.md` §5's "a second sighting worth watching", retired from that file. Its surviving content is the direction-of-inference warning, which is live and load-bearing. - items 1 and 2 recorded briefing doubts whose answers are already the file's own headlines (`G2 ≥ g` at §1; "stronger by exactly log x", derived three ways at §4b, §5 and §6, and restated in §8's verdict). **§10 is REPURPOSED rather than deleted, and §§11-12 keep their numbers.** `ZONE-POSTULATE.md`:150 cites "`research/two-class-lower-bounds.md` §10" for the climbing certificate ladder. Deleting §10 and renumbering would have broken that pointer and required an edit in another partition's file. Instead §10 now holds item 5's surviving content under the heading "The certificate ladder, and how far it can be pushed", so the inbound pointer resolves to exactly the claim it wants and no renumbering is needed anywhere. **§9's "LOOSE END, not checked": Kanold, Stevens and Paseman "all of which give exponent 2 + ε with stated constants".** FALSE for all three, and CLOSED rather than done. From Paseman's own §1 (arXiv:1311.5944): Kanold gives `2^k` and `2^{√k}` for k ≥ e^50; Stevens gives `g(n) < 2k^{2+2e·log k}`, whose log₂ is `O((log k)²)`; Paseman improves that to `u(k) = 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}`. The exponent-2 statements are **Vaughan 1977** (general n) and **Iwaniec 1971 Theorem 2 / 1978** (primorials), both with inexplicit constants, and the constant is what the question turns on. So the difficulty floor under route A **stands**; it does not collapse. Matches the closure partition A applied to `G2-STATE.md` §§5 and 9 and partition D to `TODO.md` 000b. `covering-dive.md`:17 carries the same correction with the bibliography fixed: Kanold's Jacobsthal paper is Math. Ann. **170** (1967) 314–326, not the 1965 Math. Ann. 157 paper, which is a different work. **§8's "VERDICT: SAFE".** SCOPED at the point of use. It now reads "SAFE against the construction side", the qualifier §9 already carried one section later and that `G2-STATE.md`'s copy carried inline. Found by the `transfers` check on the near-duplicate pair, which is what that check exists for. **Brief-response framing.** §0's "the honest doubt", §1's "Correction to the briefing" and §4a's quotation of the briefing RETIRED. Every mathematical statement in them is retained as a statement of fact. ### research/PRIOR-ART.md **"Phi(x,y) < 6x/log y for y <= x" (Fan and Pomerance).** WRONG in two independent ways in one formula. The source's abstract, from the LaTeX via the arXiv API, reads **"Φ(x,y) < .6x/log y when y ≤ √x"**. A dropped decimal point, which made the theorem ten times weaker than it is, and a dropped square root, which made the corpus claim a range twice as wide as the theorem covers. The venue the corpus gives, J. Number Theory 254 (2024), is CORRECT. The same error stands at `paper/staircase-note.md`:444, which partition C owns. **"e^{2γ}/4 ≈ 0.7935".** RETIRED. `e^{2γ}/4 = 0.7930547…`, so 0.7935 is not a rounding at any precision. Now 0.7931. This was the only site in the corpus carrying 0.7935. **"Iwaniec … see Erdős Problem #687" for the h(k) form.** WRONG cross-reference, confirmed by reading both pages in full. #970 carries the `h(k) ≪ (k log k)²` form and says of itself "This is a more general form of the function considered in [687]"; #687 carries the `Y(x) ≪ x²` form and the $1000. Both are now named for what each holds. The row's "constant unknown" is reworded to "inexplicit", because neither page's text says the constant is unknown. **"cited by Holt in 1402.1970", and the whole §"What 1402.1970 does have".** arXiv:1402.1970, *On Polignac's Conjecture*, is **Holt AND Rudd**. The section attributed its findings to "he" throughout. Corrected to Holt and Rudd; the corpus's other three sites are in partitions A, C and G. **Táfula's "missing the constant by 4e^{−2γ}" inside quotation marks.** It is a paraphrase: the source reads "missing its constant only by a factor of 4e^{−2γ}". Restated without the quotation marks. Same for the Cheer–Goldston "Maier used this result…" sentence, which silently deletes "has recently" from inside a quotation; the paraphrase is now marked as one, and the *other* Cheer–Goldston quotation on the line above is verbatim and stays. **Táfula's date "(2015/2020)": CORRECT AS WRITTEN, and now unambiguous.** arXiv:1508.05702 has v1 2015-08-24 through v5 2019-08-28 and no 2020 version, but the paper was published in São Paulo J. Math. Sci. in 2020. Both halves are right and the file now says which is which. Recorded so nobody "fixes" it. **"twin-Legendre (OEIS A192870)".** IMPRECISE. A192870 is the n-tuplet generalisation, whose n = 2 entry is the twin case; the twin-specific sequences are A091591/A091592, which the corpus cites correctly elsewhere. **"periodicity remarked by H.J.S. Smith (1857, per Dickson's *History*)": CONFIRMED, and the citation is now primary.** The date was flagged as suspicious because Smith's *Report on the Theory of Numbers* ran 1859–1865 and his arithmetical-determinant paper is 1875/76. Dickson vol. I settles it and cites the same paper twice: - p. 439, "Diatomic Series": after deleting the multiples of 2, 3, …, p the survivors "form a periodic series of period 2·3…p; and similar theorems. **Like remarks had been made previously by H. J. S. Smith.**" - p. 436: "**H. J. S. Smith gave a theoretical method of finding the primes between the xth prime P_x and P²_{x+1}, given the first x primes.**" Footnote in both places: **Proc. Ashmolean Soc. 3 (1857) 128–131; Coll. Math. Papers I, p. 37.** So the 1857 date is right, and the paper is a stronger prior art item than the corpus recorded: the second entry is the zone (p, p′²) and the p²-rule, stated as a method, in 1857. Added to the crystallization row. **"Mirror/palindrome of the wheel — never foregrounded as an organizing device".** WEAKENED on the same source. Dickson vol. I p. 439 records A. de Polignac's "diatomic series" (Comptes Rendus Paris 29 (1849) 397–401; Nouv. Ann. Math. 8 (1849) 423–9) as periodic with φ(πₙ) terms and with "the terms after 1 of the period … symmetrically distributed (two terms equidistant from the ends are equal)". The palindrome and the census, stated together, in 1849. The claim is narrowed to the *use* as an organizing device. **This is a novelty-verdict change and is flagged for Chris.** ### research/theta-ladder.md **"Run 2026-08-17. Brief: TODO item 00, feeding TODO item 0."** and the eighteen sites where "the brief" or "TODO 00" is the grammatical subject. MIGRATED to statements of fact. Nothing mathematical changed. The superseded estimates that were being corrected are retired with them: TODO 00's "z = 31 took 65.3 s" against the measured 36.7 s, and TODO 00's costing "z = 59 is 1 to 2 h, z = 71 several hours, z = 100 is days" against the measured z = 59 ≈ 5 h and z = 71 ≈ 36.5 h. Both corrections had already been applied to `TODO.md`:71-72 and logged at `CHANGELOG.md`:276-277; the measured cost table stays in §9 under a plain "**Cost.**" heading. **"`research/ZONE-POSTULATE.md` §5 ('a second sighting worth watching')".** Dead quotation; that phrase is no longer in ZONE-POSTULATE §5. Replaced by what the pointer is actually for. **Conditional-theta caveat discipline.** Every place a reader meets the conditional column now carries, at the point of use, that it rests on an **unproven** Gaussian maximal law for the sawtooth: §0's definition, §0's headline answer, §2's table, and §6's "the conditional column agrees and more loudly". §5b's exact column is marked as the one that owes nothing to it. **"the null 2 + ⟨2/ln lnW⟩ = 2.649" at §3.** Kept and now named. That 2.649 is an empirical regression null and is a numerical near-coincidence with the unrelated vector-sieve threshold `1+√e = 2.6487` discussed in §7; the file now says so where the collision occurs. **§7's Lemma V paragraph.** Repointed at the home: `sift-limit-attack.md` §§3 and 4.5 is the authority for "Lemma V is not the operative assumption at the measured working point". The TODO 0 statement it derived is already applied to `TODO.md`:63-66 and is now stated as fact rather than as a correction to carry. ### research/sift-limit-attack.md **The Lemma V calibration.** No claim changed. The file is the home and the authority for the fact that Lemma V is not the operative assumption at the measured working point and the Gaussian maximal law is, and it now says so in those words, because `GLOSSARY.md`, `theta-ladder.md`, `TODO.md` and `G2-STATE.md` all point here for it. The conditional column's introduction gains the word **unproven**. **The FKMPT citation** gains the corrigendum and the version, matching `covering-dive.md`. **§3's discard-map row "Fusion / anchored calm (PROVEN (i),(ii); MEASURED (iii),(iv), cap-19/23/26)".** STALE in the conservative direction, which is the rarer kind: it graded as MEASURED a leg that is a certified theorem. The row now names the five PROVEN objects (Mirror-Sibling Identity, Fusion Identity, Mirror-Phase Doubling, Minus-Half, Skeleton Collapse), the Aggregate 30-Skeleton Bound as **CERTIFIED at six levels @11..@29 as an exact integer inequality**, and the Anchored Typicality Measurement as MEASURED, and points at the new status-only parent `research/anchored-calm.md` rather than at the three cap-files whose competing status tables partition F retired. Understating our own result is still a defect. Handed over by partition F. ### research/dhr-verification.md **The file was a spent audit whose every quoted claim had been fixed, and `research/history/staging/audit-campaign.md`:11 lists it as "Clean, no change needed".** All seven of its §6 recommendations were applied to `paper/beta2-note.md` before this pass, so its §0 table asserted as live claims of that note six statements the note no longer makes (verified applied 7/7 again here, against the note's current text). **Decision: the file STAYS in the body, brought current, rather than moving to `history/`.** Its §§1–5 and §7 are the primary-source ledger for β₂ and DHR Theorem 9.1 — the corpus's most load-bearing external chain — and `research/sift-limit-attack.md`:58 and :412 cite it live for exactly that. Nothing in it is superseded: the sources still say what they say. What died was the framing. It now opens with a dated **APPLIED / OUTSTANDING** line, its §0 column reads "Claim as originally drafted", and §6 becomes "The seven edits this audit produced, all since applied". This is the first instance of the convention `qc-refs` §8.5 proposes for audit files, and it is the cheap fix for the whole class. **And a staleness no report caught, running the other way.** §0 item 1e and §5 told a reader the Diamond–Halberstam book (Cambridge Tracts 177) was lending-locked and that the literal hypothesis block of Theorem 9.1 was the single remaining unverified step in the corpus's most load-bearing citation. `paper/beta2-note.md`'s status header records that the book has since been obtained and read line-level, with page numbers — Theorem 9.1 at pp. 103–112 carrying exactly the `2·Σ 4^{ν(m)}|r_A(m)|` remainder and the `(log y)^{1/(2κ+2)}` error this file predicted from Franze–Kao; Ω(κ)'s working product form (5.2) at p. 44; Theorem 6.1 at pp. 67–68; **β₂ ≈ 4.266 in print at p. 79**; α_κ ≥ β_κ+1 at p. 77. Every prediction this file's triangulation made was borne out. §0 item 1e is now **CLOSED**, §5 is retitled and records what the book settled, and the source ledger's "lending-locked — NOT verified line-by-line" row is corrected. Two documents disagreed about whether the corpus's central citation had been verified against its primary source, and the one that said no was the audit trail the other one names.