# CHANGELOG entries from wave-2 partition C (`paper/`) *(Staged for the parent to merge into `research/history/CHANGELOG.md`. House form: dated, newest first, grouped by the document changed.)* --- ## 2026-08-17, wave 2: the paper suite corrected against its own research layer ### paper/moire-primes.md **Door 2 of §7, both load-bearing numbers.** RETRACTED and replaced. The door claimed the certified window-discrepancy budget "grows like 2ⁿ" and missed certifying the p = 11 zone "by only 18%", citing `research/attack-04`. Both numbers came from that script's `certified` column, which its own correction banner of 2026-08-14 disclaims: it indexed local factors at k mod p and omitted the CRT twist y_p = (W/p)^{−1} mod p, moving pointwise |S| by up to N/2, so the column was not a certificate. The corrected artifact `research/natal-cap-02-fourier-budget.js` reverses the conclusion: at x = 11 a perfect per-window oracle already fails (cap 117.3 against a census of 90), from x = 13 the gross strike total alone exceeds the census (1,135 against 990), and the real near miss is 15% at x = 7 and is moot. The door now states the mechanism, the per-prime toll it does certify (deviation below N/q for every q ≤ 263 at x = 17; gross(19) ≤ 1,685 against a true 1,563), and where it stops, with the retraction left visible in place. Forced by `research/natal-cap-02-fourier-budget.js` readings 1 to 5, carried as a standing note in `research/NATAL-CAP-CAMPAIGN.md` since 2026-08-14 and never inherited by the paper. This was the campaign's only false claim as opposed to an internal inconsistency. **Door 5 of §7, the covering theorems.** CORRECTED. Hough (2015) and BBMST (2022) require **distinct** moduli and our classes {0, −2 mod q} use each modulus twice, so neither theorem applies to our object. The door now cites Klein–Koukoulopoulos–Lemieux (2024), Theorem 3, which covers multiplicity s = 2 explicitly and gives **no numeric constant at s = 2**, and marks our translation inferred. The two-class shortfall is now p/ln²p from Ziller and Morack's measured law 1.90·x·ln²x rather than the one-class p/ln p, and the "covering phenomenon of a fundamentally new kind" claim is reduced to what an ABSENT marker supports. A visible note asks for an expert read. Forced by `research/covering-dive.md` §§3.1, 3.3, 4.2. **§7 preamble, "five independent routes, each rigorous until its last step".** SUPERSEDED. Doors 1 and 4 are one classical object at two truncation depths (Door 4's union bound is the first line of Brun 1919), and Door 2 fails structurally at x = 11 rather than at a last step. All five doors are now stated in three labelled parts, mechanism / toll / where it stops, each with its own calibration, so a door can no longer lend the authority of its mechanism to its toll. Same fix to the abstract's "five independent doors". **§7A Face 4, "missing exactly one ingredient, which we name Lemma V".** SUPERSEDED. At the working point the ladder measures, s/u ≈ 1.2 is outside the range s ≤ u that Lemma V is stated in, so the operative assumption is an unproven Gaussian maximal law for the sawtooth, which is the whole price of the route. The phrase "maximal law" had appeared in no paper file. Face 4 also now carries the ladder's ceiling (exact full-period suprema 1.9524, 1.9477, 2.0018, 2.0476 at z = 19, 23, 29, 31) and drops "nothing in print blocks it" for the κ = 2 sifting limit that does. Forced by `research/sift-limit-attack.md` §§3, 4.5 and `research/theta-ladder.md`. **§8's "an unexplained structural bias in the helpful direction".** RETIRED. The anchored calm's mechanism is proven and the calm is uncorrelated with survival, which §7A Face 2 already said forty lines earlier. Forced by `research/natal-cap-31-calm-vs-kill.md` reading 1 and `research/natal-cap-19-calm-lemma.md`. **The abstract's "the matching lower bound".** RETIRED at all three corpus sites (this one and `paper/PAPERS.md`:30; `README.md`:28 belongs to another partition). The bounds are exponent 4.2665 against 1 + o(1). The note itself says "The point of this note is not sharpness". **The abstract's one novelty theorem.** The hypothesis p ≥ 17 had been dropped, and an o(1) inequality was quantified over each p. Both statements, abstract and §6, now give the finite form (p² − p)/(K − 1) with K = π(p²) − π(p) and derive the (2 + o(1))·ln p reading from it. **Six further dropped hypotheses and scopes**, all invisible to anything but a word diff: the 4,190× certificate at x = 11 needs moments through order 6 and not degree 4 (degree 4 gives 80×); Face 2's z = −0.30 and −2.71 are the diagonal ensemble, not Face 1's window ensemble; corr(X, S) = 0.99 and the 97% hold at x = 17, the ladder being 0.48, 0.91, 0.99; "the anchor is in deficit" printed a 45% surplus as its first value; K* is a function of the moduli pool as well as of x and its home withdraws the word "law"; the certificate ladder costs 5μ per **two** degrees, only even degrees existing. **Attribution.** FKMPT's Remark 7, not FGKMT's; arXiv:1402.1970 is Holt **and Rudd**; Klein–Koukoulopoulos–Lemieux and Brun added to the references, and the FKMPT corrigendum (JEMS 25 (2023), 2483–2485) cited beside the paper; H. J. S. Smith 1857 now carries PRIOR-ART's "per Dickson's *History*" hedge; our "fusion" is distinguished from Holt's; Maier (1985) and Brun (1920) cited at their points of use in §7A, which had cited nothing outside this repository. **Honesty items the notes carried and the flagship did not.** The certified floor at x = 29 now prints the truth beside it (31,327 against 12,307,838, being 0.25%); Face 1 gains the parity floor's measured gap (2 against the needed 1.28 at x = 17) and the z-ladder's nine-level scope; the §7A preamble now states that the faces overlap the doors, Face 2 being Door 4's overlap credit and Door 3's certificates seen again. ### paper/anchored-note.md **§10's "the anchored-calm lemma, two legs from closed" and its five-level ladder.** SUPERSEDED. The certified aggregate 30-skeleton bound now reads six levels, @11 through @29, at 0.2132, 0.1113, 0.1011, 0.1259, 0.0945, 0.1176, the @29 row reproduced during this campaign, so the leg is a theorem for x ≤ 29 and no longer for x ≤ 23. The reading "the deepest level also the lowest" is **false** at six levels, the minimum being 0.0945 at @23, and the worry it softened is back. The paragraph no longer names a lemma, since its parts carry four calibrations, and it now states that the uniform-in-q form of the anticorrelation is **refuted** (595 of 599 scour primes anticorrelate; four cataloged exceptions) and that only the aggregate form is claimed. Forced by `research/natal-cap-36-skeleton-door.js --at29` and `research/natal-cap-23-covadj-proof.md`:175-179. **The X-limitation theorem, "from @13 on".** SCOPE CORRECTED at both paper sites. `research/natal-cap-31-calm-vs-kill.js` reading 3 marks the proof "per level, from L2 + enumerated max VR", and the enumerated maxima exist at @11, @13 and @17 only. The theorem is stated as proven at @13 and @17, with the extension upward resting on two measured trends (max VR bounded at 2.78, 2.35, 2.14; the driver S̄/√(K·V̄) growing 3.1, 4.9, 9.8). **§7's "the entire Twin Prime Conjecture lives inside that single ratio".** QUALIFIED: positivity suffices, no converse is claimed, so Assumption A is at least as strong as the conjecture and may be strictly stronger. **§11's "the proven layer around it (… Proposition 1 …)".** CORRECTED to Proposition 1's part (i), part (ii) being an extrapolation of two measured trends, which the same section says four lines later. ### paper/staircase-note.md **§9's quotation of Fan and Pomerance.** CORRECTED. The source reads Φ(x,y) < .6x/log y when y ≤ √x; the note printed 6x/log y for y ≤ x, a dropped decimal point and a dropped square root, in a bound the note leans on. Forced by the arXiv:2306.03339 text quoted in `research/history/staging/qc-refs.md` W5. **Honesty item 1, "four finite computations".** CORRECTED to six; the note's own Theorem 8 table has had six rows since 2026-08-15. **The title's name.** The status block now states that "the Staircase Theorem" names Theorem 3 and nothing else, the certified floors being Theorem 8 at six levels and the tail bound Theorem 6, the note's one non-elementary ingredient. ### paper/beta2-note.md **Two "FGKMT Remark 7" attributions.** CORRECTED to FKMPT (Ford, Konyagin, Maynard, **Pomerance**, Tao, *Long gaps in sieved sets*), one of them at :152 which the reference audit did not list. The corrigendum is now cited beside it. ### paper/PAPERS.md **Paper II's title.** The note governs: "An upper bound for the twin Jacobsthal function", replacing "Two-sided bounds for …", with the reason recorded in place. **"Iwaniec's contested transfer lemma".** The word DROPPED. One unanswered 2016 MathOverflow post is not a controversy. Replaced by the note that the route avoids the lemma entirely and that an explicit-constant or formalised exposition of it would be useful. **"the matching lower bound".** Retired, as in the flagship. **Zone Equivalence listed among "the spine theorems".** CORRECTED to a framing device presented as a proposition, which is what `research/PRIOR-ART.md` asks for and what `paper/moire-primes.md` §6 already does. **"OEIS: G₂ sequence (drafted, 11 terms)".** CORRECTED to twelve, through a(12) = G₂(37#) = 528. **The Lemma V map's positioning.** Now carries the qualification that travels with it: the operative assumption at the measured working point is the Gaussian maximal law, not Lemma V.