# changelog-add-Q.md — the history migrated out of partition Q's files Wave 4, partition Q. Every block below was removed from or rewritten in a working document and is reproduced verbatim with the file it came from. Nothing here is a claim about the mathematics; it is the record of how the mathematics got to where it is. To be folded into `research/history/CHANGELOG.md` by the shepherd, indexed by document. --- ## research/oeis-G2-submission.md **COMMENTS, the Maier-Pomerance clause (RETIRED, conjecture cited in the wrong direction).** Verbatim: > and > a(n) >= p * log(p)^(2+o(1)) under the Maier-Pomerance conjecture for > Jacobsthal's function. Maier and Pomerance state an **upper** bound. `research/maxgap-law.md`:415, which records a source check against FGKMT (arXiv:1412.5029) dated 2026-08-17, quotes them verbatim as *"It is conjectured by Maier and Pomerance that in fact `Y(x) << x(log x)^{2+o(1)}`"*, with `Y(x) = j(P(x)) - 1` their eq. (1.3). A `<<` statement does not give the `>=` this sentence drew from it. The equality form `J(T) = T(log T)^{2+o(1)}`, which does, is Ford's 2018 Montreal slides, and `maxgap-law.md`:426 separates the two forms explicitly and locates each. The corpus therefore already held the distinction and the draft collapsed it. Replaced by a version that names both forms and attributes each, and Ford's slides were added to LINKS. **COMMENTS, the upper-bound paragraph (RETIRED, an openness claim the repository contradicts).** Verbatim: > No upper bound is published. Iwaniec's bound O(log(P)^2) for Jacobsthal's > function has no published analogue for two forbidden classes per prime, at any > exponent. `research/THE-DIALS.md`:257 carries the twin Jacobsthal G2(p#) at sieve dimension 2 with a proven Jacobsthal exponent of **4.2665**, citing `paper/beta2-note.md`, whose status line is "THEOREM, sieve input FULLY VERIFIED against the primary source, no outstanding items" and whose §1 gives `G2(n) <= C(eps) * p_n^(beta_2 + eps)` for all n. The sieve limit `beta_2 ≈ 4.266` is quoted there as in print at Diamond and Halberstam, Cambridge Tracts 177, p. 79. The retired sentence is defensible on the narrow word "published", since beta2-note is itself unpublished. It is not defensible on "at any exponent", which reads as "nothing is known" while the submitter holds the bound. Replaced by a statement of what is true: no analogue is in print, the obstruction is sieve dimension rather than missing input, and the open band on the exponent is (2, 4.2665], matching `THE-DIALS.md`:288. **NAME, the residue range (RETIRED, half-open interval written as closed).** Verbatim: > Largest gap between consecutive integers r in [0, P] with gcd(r, P) = > gcd(r+2, P) = 1 The period is `0 <= r < P`. Writing `[0, P]` names P itself, which is not a residue of the cycle. Harmless arithmetically, since gcd(P,P) = P excludes it anyway, but an OEIS NAME field should not need the reader to notice that. A059861's own OEIS formula uses `0 <= r < primorial(n)`. **COMMENTS, the p log^2 p ratio (RESCOPED, not retired).** The sentence > The direct ratio a(n)/(p * log(p)^2) wobbles between 0.66 and 1.13 with > no trend, mean 0.89 carried no range, and followed a sentence scoped to n = 5..12. The numbers are those of **n = 3..12**: 0.6640, 1.1318, mean 0.8929. Over n = 5..12 the same three read 0.66, 1.09 and 0.86, so the inherited scope would have made the sentence wrong. The range is now written into the sentence. **PROG, the feasibility comment (SUPERSEDED, stopped short of the DATA).** Verbatim: > \\ feasible through n=9; a(10) computed by segmented sieve (see link) DATA runs to a(12). Corrected to name a(10) and a(11) as segmented-sieve terms and a(12) as the mod-30 lattice walk, which is what the provenance table has always said. **COMMENTS, the growth exponent (RETIRED, another sequence's number).** Verbatim: > Calibrated against the 58-term one-class control, the growth > exponent in p is near 1.57, with a practical bracket of 1.3 to 1.9 and a hard > floor of 1. 1.57 is the control-corrected exponent of **h2**, Ziller and Morack's paired Jacobsthal function A288815, measured over nineteen terms. This sequence's figure is **1.54 ± 0.09** over its ten exact terms. `research/exponent-control.md`:198 gives the instruction in terms, *"Quote 1.57 for h2 and 1.54 for G2"*, its §5 table at :168-169 carries both with their error bars, and `research/U-FRAME.md`:156 carries 1.54 ± 0.09 for G2 independently. Both figures were rebuilt from the control rather than copied. The raw x-frame slope over x = 5..37 is 1.801; the control A048670 fitted over all sliding windows reproduces `exponent-control.md` §1 digit for digit, giving bias +0.262 at width 10 and +0.280 at width 19; and 1.801 − 0.262 = 1.539 while 1.847 − 0.280 = 1.567. The replacement states the raw slope, the control, the bias and the corrected value with its scope, and says explicitly that A288815 reads 1.57 over nineteen terms so the two are not one exponent. **CROSSREFS, the A192870 label (SUPERSEDED, loose).** Verbatim: > A192870 (twins between squares) A192870 is the maximum M with no prime **n-tuplet** between M^2 and (M+1)^2, an irregular sequence over k-tuplet types whose twin case is the single term a(2) = 122. A091592 is "numbers n such that there are no twin primes between n^2 and (n+1)^2" and is the sequence this draft's danger-line discussion actually concerns. A091592 added, A192870 relabelled, and the A288815 / A072753 grouping strengthened with the identity `A288815(n) = 6*A072753(n) + 6` that A072753's own page carries. --- ## research/oeis-seam-submission.md No claim in this file was retired. The audit recomputed all twenty DATA terms, every witness, the ceiling, the two low-n exceptions, the EXAMPLE, the PROG entry and the Hardy-Littlewood heuristic from scratch, and found no correctness defect. The wave-3 factor-of-two correction was rederived independently from Mertens rather than accepted, and it stands. Two additions, neither superseding anything: - The heuristic now states its numbers: 52.5 by exact per-k sum and 57.3 by the asymptotic form against 48 observed, with a note that the n = 1 term carries about 8 of the asymptotic total because P = prime(1) there and the form degenerates. Written down so that a future reintroduction of the retired factor of two shows up as 26.3 against 48 in prose as well as in the audit. - a(21)..a(30) were computed and verified and are recorded in the provenance section: `0, 2, 1, 1, 2, 1, 1, 1, 0, 1`. They are deliberately **not** appended to DATA, which is the submitter's field. --- ## research/audit-numbers.js Exempt from history migration by house rule: a correction banner in a script header is that artifact's current status. Recorded here only as a scope change, since it is the enforcement behind the two blocks above. The script gained a part `oeis`, 39 checks. Nothing was removed and no retired value was weakened. The pair at line 63, `e^{2gamma}/(4 C2) = 1.2013` against `e^{2gamma}/(2 C2) = 2.4026`, labelled in code "the retired factor-2-slipped value", was the correct regression test for the seam draft's error and could not reach it, because the script's scope was the research corpus. That same pair now also stands inside the part that covers the drafts, alongside two further retired values: the `c` fit under the retired `A059861(n-1)` indexing (mean 0.0406 at cv 69.4%, against 0.4814 at cv 10.2%), and the seam heuristic under the retired `/2` (26.3 against 52.5). A third retired value joined them at the end of the wave: the exponent 1.57 quoted for G2, where 1.57 is A288815's. The part now rebuilds both corrected exponents from the control and checks that they differ by more than rounding, so collapsing the two sequences to one exponent fails the run.