# Audit staging: campaign archive (2026-08-17) Files: `research/natal-cap-*.md`, `research/NATAL-CAP-CAMPAIGN.md`, `research/ATTACKS.md`, `research/ATTACKS2.md`, `research/ATTACKS3.md`, `research/anchored-windows.md`, `research/covering-dive.md`, `research/bv-import-survey.md`, `research/dhr-verification.md`, `research/sift-limit-attack.md`, `research/two-moire-argument.md`, `research/d2-d4-bijection.md`, `research/oeis-G2-submission.md`, `research/oeis-seam-submission.md`, `research/README.md`. Clean, no change needed: `bv-import-survey.md`, `dhr-verification.md`, `two-moire-argument.md`, `d2-d4-bijection.md`, `oeis-seam-submission.md`, and the natal-cap notes 04, 12, 19, 21, 23, 26, 31, 32. --- ### research/covering-dive.md **"C(ρ) > e^{−1−6/ρ}" in the FKMPT main theorem.** WRONG NUMBER, now settled at the source. Text-extracted from the arXiv:1802.07604 v3 PDF, Theorem 1 reads `C(ρ) := sup{δ ∈ (0,1/2) : (4 + δ)·10^{2δ}/log(1/(2δ)) < ρ}` and `C(ρ) > e^{−1−4/ρ}`. The constant is **4**, and it appears twice, once in the sup and once in the bound. The 6 came from an earlier arXiv version, which ar5iv still renders; the PDF text is the authority. Both the definition and the bound are now quoted in full, with the non-degeneracy and ρ-supportedness hypotheses spelled out. This closes the item the brief flagged UNRESOLVED. **"Ford–Green–Konyagin–Maynard–Tao (FGKMT)" as the author list of *Long gaps in sieved sets*.** ATTRIBUTION ERROR, eight occurrences. arXiv:1802.07604 / JEMS 23 (2021) is Ford, Konyagin, Maynard, **Pomerance**, Tao: FKMPT, no Green. The file's own source index already had it right, so the body contradicted its own bibliography. Remark 7, the JEMS 2021 citations and the "machinery is expressly one-dimensional" line are now FKMPT; the JAMS 2018 large-gaps references stay FGKMT. Same fix applied in `natal-cap-10-sieve-cap.md` (4 places) and `sift-limit-attack.md` (2 places). **"the optimum two-class covering already reaches ≈ p²/2".** SMALL-NUMBER ARTIFACT. h₂(21) = 2622 against p₂₁² − p₂₁ = 5256 is 0.499 at the top computed term and nowhere near a law. The measured growth over all 21 exact terms is **c·x·ln²x with c ≈ 1.90** (1.90 × 73 × ln²73 = 2552 against the exact 2622), and x ln²x/(x²/2) → 0. Stated as a crossing near x = 73, not a shape. The same figure was driving the target form in REALISTIC TARGETS §4 ("does covered length scale like c·pₙ²/log? like ZM's ≈ p²/2 data?"); that now names c·x·ln²x. **"known bounds bracket G₂ between ~pₙ (trivial) and — nothing".** SUPERSEDED on the lower side. `two-class-lower-bounds.md` §1 proves G₂(x#) ≥ g(x#) pointwise in one line, which imports the whole Erdős–Rankin literature: G₂(x#) ≫ x·log x·logloglog x/loglog x by FGKMT (JAMS 31, 2018) via Rankin 1938 and Pintz 1997, and x(log x)^{2+o(1)} under Maier–Pomerance. The upper side is still empty at any exponent. ### research/sift-limit-attack.md **"reports one live route" / "the one live route" / "the single most promising entry".** REPRICED, not deleted. The vector sieve is still the only mechanism in the literature that consumes any structure the set has, but it is an exponent-improvement programme and not a road to TPC. Added the theta ladder's verdict in the summary, in §4.5 and in §6: the exact full-period suprema of the all-positions exponent are **1.9524, 1.9477, 2.0018, 2.0476 at z = 19, 23, 29, 31**, with a 4·10⁹-position prefix supremum holding it above 2.05 through z = 71. The Gap Reformulation needs theta < 2; the requirement fits the zone budget only at the two smallest points available and misses by at least half again from z = 43 on. Conditional column 2.159 to 2.482 across z = 19 to 47, still rising, margin below the full-decoupling target 2.649 now 0.17 and closing at 0.57 per unit ln z. Forced by `research/theta-ladder.md` §§2, 7, 8. **Lemma V presented as the operative assumption of the theta column.** FRAME ERROR. With D₁ = D₂ = z^s, H = z^u and θ_total = 2s/u, the ladder's family (s = 3.0, u = theta ≈ 2.16 to 2.48) sits at θ_total ≈ 2.4 to 2.8, i.e. ABOVE full decoupling, with **s/u ≈ 1.2 to 1.25 > 1** — outside Lemma V's stated range s ≥ (0.63+δ)u up to s = u − ε. So the column is not "Lemma V suffices"; it assumes a Gaussian maximal law for the sawtooth, which §3 of the same file already names as the parity wall in concentration clothing. The two open items, the drift and the maximal law, are not independent halves: the maximal law is the whole price. Recorded in the summary, in §4.5 and in the closing calibration, and the "first lemma to try" list now points at the exact-supremum direction instead of more conditional rows. **"within a factor of about two of the measured 0.8·pₙ·ln²pₙ" (§3).** RETIRED as an asymptotic law. The direct ratio G₂/(x ln²x) wobbles between 0.66 and 1.13 with no trend, mean 0.89; the law is c(x, lnD)·m̄·lnD with c a surface, and the control-corrected growth exponent is **1.57** central, bracket 1.3 to 1.9, floor 1. From `exponent-control.md` §5 and `maxgap-law.md` §1. **"below the Ziller-Morack p² line" (pilot table reading).** NARROWED to "below the Ziller-Morack conjectural p² ceiling", with the exact-supremum ladder attached: the true all-positions requirement at z = 19 and 23 is 1.9524 and 1.9477, inside the zone budget, and it crosses 2 by z = 29 and never returns. ### research/oeis-G2-submission.md **"Computed terms satisfy a(n) < 0.31 * prime(n+1)^2 (the ratio stays near 1/4)".** ARITHMETIC ERROR. a(12)/41² = 528/1681 = **0.3141**, which exceeds 0.31. The twelve ratios run 0.222, 0.240, 0.245, 0.248, 0.249, 0.228, 0.299, 0.284, 0.243, 0.268, 0.254, 0.314. Restated as < 0.32 with the endpoints given. **"Empirically a(n) is close to 0.8 * p * log(p)^2".** RETIRED. Replaced with the extreme-value form c·m·(log P − log m) where m = P/A059861(n−1), c averaging 0.48 over n = 5..12 at cv 10%; the direct-ratio wobble 0.66 to 1.13, mean 0.89, flagged as a description of the computed range and not an asymptotic law; and the control-corrected exponent 1.57, bracket 1.3 to 1.9, floor 1. This is the constant G2-STATE §6.2 named as needing a decision, decided in favour of the Poisson form. **No lower bound stated.** ADDED, as the brief required. a(n) ≥ A048670(n) pointwise, offsets verified to align (A048670 offset 1, terms 2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66 against our 2, 6, 12, 30, 42, 66, 108, 150, 204, 258, 348, 528; ratios 2.00, 3.00, 3.00, 3.00, 4.15, 4.41, 5.10, 5.61, 6.00, 8.00 at n = 3..12), with the one-line subset argument, and the imported FGKMT/Rankin/Pintz and Maier–Pomerance consequences. FGKMT added to LINKS, A048670 relabelled in CROSSREFS as a term-by-term lower bound. Still under moratorium; not submitted. **Verified unchanged:** the DATA line 2, 6, 12, 30, 42, 66, 108, 150, 204, 258, 348, 528 matches the repo ladder exactly, and the A288815 comparison (12 ≤ 18, 30 ≤ 30, 42 ≤ 66, …, 258 ≤ 450) checks against the eight terms `h2-scoping.md` reproduces from Ziller and Morack's table 1. ### research/README.md **"G₂ ≈ 0.8·p·ln²p, only ~25% of the zone, through 29#" (measurements table).** RETIRED. Now G₂(x#)/p'² runs 0.222–0.314 over the twelve exact terms, never near 1, with the growth exponent ~1.57, bracket 1.3–1.9, and the explicit note that it is not a constant times p·ln²p. **"the early '≈0.2 constant' reading was refuted by deeper computation".** CONVENTION. The claim was already current; the sentence carried its own history. Rewritten to state Var/E's drift and the ln(Var/E) ≈ −(0.24u² + 0.13u) law without reference to what it used to say. Same for "its sub-Poisson ≈0.2 constant" in the prior-art paragraph. **"then divergent enrichment" (anchored origin).** REFUTED, and refuted by our own `anchored-windows.md` §3 on the same archive. Anchored enrichment is bounded by e^{2γ} = 3.172 and mortal, crashing to 0 as p_n → x; the trough has asymptotic depth e^{2γ}/4 = 0.79305 at the zone edge; and at S = x'² the origin carries **21% LESS** than mean density. **Prior art paragraph listed three surviving novelties with no mention of Holt.** NARROWED. Added: Holt and Rudd own the cycle of gaps, the fold recursion, fusions, the closure theorem, the transfer operator, the population models, the interval of survival and the one-class discrepancy; Maier owns the tile-as-matrix; Buchstab owns ω(u). Also added that the lower side of the Gap Reformulation is not open, since G₂ ≥ g imports Erdős–Rankin for free. ### research/ATTACKS.md **Attack 10, "divergent enrichment".** REFUTED, same fix as README: trough of asymptotic depth e^{2γ}/4 = 0.79305, enrichment bounded by e^{2γ} = 3.172, then a crash to 0; and the origin's 21% deficit at S = x'² noted. **Attack 2, "universal dip ~0.90 near x~10³, suspiciously ≈ e^γ/2".** RESOLVED in the archive itself and never carried back. The 0.89–0.90 is a finite-size blend; the asymptotic dip constant is e^{2γ}/4 = 0.79305 and the resemblance to e^γ/2 = 0.8905 is a coincidence (`anchored-windows.md` §4). ### research/ATTACKS2.md **Attack 1, "Seams are real twin hotspots — as POINTS".** NARROWED. The measured E(P) = 2·∏p/(p−2) enrichment is against random integers and is entirely the singular-series factor, i.e. slot-level. Conditional on being a twin slot a seam is not enriched: survival 1.016, 0.980, 1.007, 0.986 × the tile mean at half-widths 300 to 3e5 against 400 random controls each, and 1 genuine twin pair from 20 seam slots against 2.25 expected. A seam position is 25× more likely than average to be a SLOT and no more likely than average to be a twin PRIME. Forced by `fold-profile-12-anatomy-survival.js`, 2026-08-17. ### research/ATTACKS3.md **The wave was written as a live plan and never carried its outcomes.** Ten "Landed" verdicts added, sourced from `U-FRAME.md` §§10-15, `a3-01-misalignment-ledger.js` and `THE-DIALS.md`. The five that matter: - **A5, "the attack that matters most", bounding L.** Theorem B is PROVEN and is the first unconditional bound on L, and it is **structurally capped at the linear branch** — maxsum_m ≥ G₂ always, so it can never prove L below G₂/(3p) ≈ 0.18x, and Theorem C inherits the ceiling. Bounding L is no longer the target and the naive (L+1)·G₂ recursion is closed (A4's win condition was also not met). The alternation condition is worth exactly 3/2 and no more. - **A9's transfer operator.** PRIOR ART, and the same object rather than a near miss: Holt and Rudd, arXiv:1408.6002 §5 (2014), transfer matrix M_J on the cycle of gaps, eigenstructure and binomial eigenvectors included. Marked "nothing here may be presented as new structure"; what is ours is the exact head engine and its numbers. - **A7's pane.** CLOSED, twice. The pane sits at u → 1 and needs u > 4.26645 against the zone's 2: "4.267 proven, 1 needed" (`THE-DIALS.md` §§1-2). - **A10's exactness.** ANSWERED NO. 3 of 9 folds, 0.927 over 329 cells, and the ladder is pinned in the worst 2p/m̄ band, reaching the safe band only near x ≈ 130 (width 10⁵⁷). - **A6's origin recursion.** CLOSED, via `origin-excess.md`: the (ln x/ln y)² factor is capped by an absolute constant ~2.2, measured max 1.372; the real advantage is G₂/F ≈ 7 because window/F is capped at x' and saturated. **"House rules apply: … refutations stay visible."** CONVENTION. The house rule is now the doc convention: bodies state current understanding, refutations live in the changelog. Header updated. **"corrected 2026-08-16 by A7; first written 4.95, too big by 3/2" (A7).** CONVENTION. Inline correction removed; the corridor width 5·C2/ln²n = 3.301/ln²n now stands on its own. ### research/natal-cap-10-sieve-cap.md **Verified, no change to the mathematics.** The parity floor is stated as **2** throughout, at §1.5, §2 Regime 2, §4's ordering and §5's caveat list; 4 is correctly identified as the 2/θ rule at the BV level θ = 1/2, a technology wall breached by Chen since 1978 and by Lichtman's 3.29956 in 2025. The campaign verdict also stands: C*(x) = 1/Σ_{x