# applied-Q.md — wave 4, partition Q: the OEIS drafts get an instrument Files owned and edited: `research/oeis-G2-submission.md`, `research/oeis-seam-submission.md`, `research/audit-numbers.js`. Nothing outside that list was touched. **Gate.** Before: refs 0, quotes 0, crosslinks 0, scripts 0, transfers 0, calibration 0; `qc/selftest.js` exit 0. After: identical, all six at zero, selftest exit 0. `audit-numbers.js` went from 37/39 to 76/78: the two pre-existing failures are unchanged and are reported in §6, and all 39 new checks pass. **Lead with the finding.** Three defects, all in `oeis-G2-submission.md`, all in COMMENTS: a conjecture cited in the wrong direction, a claim of openness the repository's own theorem contradicts, and the exponent of a *different sequence* quoted as this one's. Details in §3, and the third arrived by routing from partition P and is written up in §8. The seam draft, by contrast, came through the audit with no correctness defect at all, which is a result and is recorded as one. --- ## 1. Method: nothing was read, everything was recomputed Every number below was produced in this session from an independent implementation, and every A-number was fetched from oeis.org rather than taken from the repository's description of it. `WebFetch` returns HTTP 403 on oeis.org; `curl -A "Mozilla/5.0" "https://oeis.org/search?q=id:AXXXXXX&fmt=text"` works and returns the full internal record. **The negative searches were controlled.** Three OEIS searches returned "No results", and a null search result proves nothing until the endpoint is shown to return a positive. Two positive controls were run through the identical URL form: `seq:2,6,18,30,66,150,192,258` returns A288815, and `seq:1,1,3,15,135,1485,22275` returns A059861. Both fired. Only then were the three nulls trusted. Scratch scripts used, all under the session scratchpad, none added to the repo: `seam.py` (two independent primality tests), `g2num.py` (the G2 COMMENTS numbers and the Hardy-Littlewood sums), `extra.py` (the retired index fit and the a(21)..a(30) extension). --- ## 2. Every A-number, checked at the source Fetched and read in full. The middle column is what oeis.org actually says, not what the drafts or the corpus say it says. | A-number | what oeis.org has | draft's use | verdict | |---|---|---|---| | A002110 | "Primorial numbers: product of first n primes", **offset 0**, a(0)=1 | "P = A002110(n) is the n-th primorial" | correct; offset 0 still makes A002110(n) the product of the first n primes | | A059861 | `a(n) = Product_{i=2..n} (prime(i) - 2)`, offset 1 | twin candidates per period | correct, and confirmed by Tener's own formula on the page: `a(n) = \|{r : 0 <= r < primorial(n), gcd(r,P)=gcd(r+2,P)=1}\|`. This is a fourth independent proof of the `(n)` indexing, at the source | | A048669 | "The Jacobsthal function g(n): maximal gap in a list of all the integers relatively prime to n" | the one-class analogue | correct | | A048670 | "Jacobsthal function A048669 applied to the product of the first n primes", 58 terms in DATA | pointwise lower bound; the 58-term control | correct, and the DATA field really does hold **58** terms, counted | | A288815 | "Paired Jacobsthal function applied to the product of the first n primes", Ziller, ref Ziller & Morack arXiv:1706.00317 | dominating comparison | correct, including the Ziller-Morack attribution | | A072753 | "Maximum gap in two-stage prime-sieves", **offset 3** | grouped with A288815 as the paired function | correct, and stronger than the draft claimed: the page carries `a(n) = (A288815(n) - 6)/6`, so the two are one object in two normalisations. Verified arithmetically over n = 3..12 | | A192870 | max M with no prime **n-tuplet** between M^2 and (M+1)^2; a(2) = 122 is the twin case | labelled "twins between squares" | loose, not wrong. Sharpened, and A091592 added, which is the twin case directly | | A091592 | "Numbers n such that there are no twin primes between n^2 and (n+1)^2" | not previously cited | added to CROSSREFS; the script already computed it in part `a091592` | | A087732 | "Smaller of twin primes of the form j*P(i)#-1 and j*P(i)#+1 with **0 < j < P(i+1)**" | open-range row lengths | correct, open range confirmed verbatim | | A087651 | primorials P# such that j*P# has twin prime neighbours for some 0 < j < next prime, **offset 0** | "the primorials that produce them" | correct | | A057706 | "Smaller of twin primes whose average is a primorial number", terms 5, 29, 2309 | the endpoint indicator | correct; 5, 29, 2309 are the primorials 6, 30, 2310, so n+1 = 2, 3, 5 exactly as the draft says | | A014545 | k with 1 + (product of first k primes) prime | primorial + 1 | correct | | A057704 | m with m-th primorial minus 1 prime | primorial - 1 | correct | | A060256 | "Smallest multiple a(n) of n-th primorial q(n) such that a(n)*q(n)-1 and a(n)*q(n)+1 are twin primes" | the companion, deliberately not submitted | correct | **No A-number in either draft is wrong.** That is a null result and it is the answer to the highest-consequence question asked. ### The two absence claims, checked - The seam count sequence is absent from OEIS in both conventions. `seq:2,4,4,3,4,6,2,1,7,1,1,2,0,1,1,1,3,1,0,4` (closed range) and `seq:1,3,4,2,4,6,2,1,7,1,1` (open range) both return "No results", against the two positive controls above. - The G2 sequence is likewise absent: `seq:2,6,12,30,42,66,108,150,204,258` returns "No results". The draft does not claim this, but it is worth having on the record before submission. --- ## 3. The two defects found, both in oeis-G2-submission.md ### 3.1 The Maier-Pomerance conjecture was cited in the wrong direction The draft read: > a(n) >= p * log(p)^(2+o(1)) under the Maier-Pomerance conjecture for > Jacobsthal's function. The conjecture is an **upper** bound in the source the corpus itself pinned. `research/maxgap-law.md`:415, checked against FGKMT (arXiv:1412.5029) on 2026-08-17, quotes it verbatim: *"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 cannot yield the `>=` the draft deduced from it. The equality form `J(T) = T(log T)^{2+o(1)}`, which does yield it, is Ford's Montreal slides, and `maxgap-law.md`:426 says so in terms and locates them. So the corpus already held both forms, correctly separated, and the draft collapsed them into the one that does not support the sentence. Fixed by naming both forms and attributing each. Ford's slides added to LINKS. Note the shape of this: the draft was internally consistent, consistent with the loose reading of A048670's OEIS comment, and wrong against the primary source. Only the corpus's own source-pinned section caught it. ### 3.2 "No upper bound is published, at any exponent" is contradicted by the repository The draft read: > 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 a table row for the twin Jacobsthal G2(p#) with sieve dimension 2, sieve limit **4.2665**, and a proven Jacobsthal exponent of **4.2665**, pointing at `paper/beta2-note.md`. That note's status line is "THEOREM, sieve input FULLY VERIFIED against the primary source, no outstanding items", and its §1 states `G2(n) <= C(eps) * p_n^(beta_2 + eps)` for all n. Its input, `beta_2 ≈ 4.266`, is quoted as **in print** at Diamond and Halberstam, Cambridge Tracts 177, p. 79. The sentence is defensible on the narrow word "published", since beta2-note is unpublished and calls itself "the first published upper bound of any exponent" if it ever appears. But "at any exponent" reads to an OEIS user as "nothing is known", when the submitter holds a theorem giving 4.2665 from an in-print sieve limit by a standard argument. That is the campaign's dominant defect exactly: a scoped result restated without its scope, in the one place with an external consequence. Rewritten to say what is true: no analogue is in print, the obstruction is sieve dimension rather than missing input, the dimension-2 limit gives `a(n) << p^(beta_2 + eps)`, and the open band on the exponent is (2, 4.2665]. This also matches `THE-DIALS.md`:288 word for word on the band. --- ## 4. Everything recomputed, term by term ### 4.1 G2 draft, all twelve DATA terms Rebuilt from scratch by `audit-numbers.js` parts `ladder` and `g2big`, which construct the twin-slot cycle directly to 29#, lift one prime to 31# and two primes to 37#, and settle copy-boundary gaps separately. Every term matches: 2, 6, 12, 30, 42, 66, 108, 150, 204, 258, 348, 528 a(1) and a(2) were also done by hand: mod 2 the only slot is r = 1, mod 6 the only slot is r = 5, so each cycle has a single slot and the cyclic gap is the modulus, giving 2 and 6. Both agree with A059861(1) = A059861(2) = 1. The EXAMPLE was checked independently: mod 30 the slots are 11, 17, 29, cyclic gaps 6, 12, 12, so a(3) = 12. The provenance table's primorials were checked against A002110 term by term, including 200560490130 at n = 11 and 7420738134810 at n = 12, and the slot counts against A059861, including 6226553025 and 217929355875. All correct. A002110(13) = 304250263527210, so "3.0e14 positions" and "about 40x" are right (the ratio is 41.0). ### 4.2 G2 draft, the COMMENTS numbers | claim | recomputed | verdict | |---|---|---| | a(n) < 0.32 * prime(n+1)^2 | max over twelve terms is 0.3141 | holds | | ratio 0.222 at n = 1 to 0.314 at n = 12 | 0.2222 and 0.3141 | correct (not monotone in between, but the draft says "runs from", not "increases") | | a(n)/A048670(n) = 2.00 ... 8.00 at n = 3..12 | matches, already checked in part `onecls` | correct | | c averages 0.48 over n = 5..12, cv 10% | mean 0.4814, sample sd 0.0490, **cv 10.2%**, range 0.4463..0.5939 | correct | | a(n)/(p log^2 p) between 0.66 and 1.13, mean 0.89 | **over n = 3..12**: 0.6640, 1.1318, mean 0.8929 | correct but unscoped; over n = 5..12 the same three read 0.66, 1.09, 0.86 and the sentence would be wrong. Scope added | | 58-term one-class control | A048670's DATA field holds exactly 58 terms, counted | correct | | exponent near 1.57, bracket 1.3 to 1.9, floor 1 | **wrong number for this sequence**, see §8 | 1.57 is A288815's; G2's is 1.54 | | Iwaniec O(log(P)^2) | corpus states it as `g(x#) << x^2` with an inexplicit constant; log P = theta(x) ~ x, so the forms agree | correct | | a(n) <= A288815(n), the listed pairs | 12<=18, 30<=30, 42<=66, 528<=708 and all twelve | holds | ### 4.3 The reduction to the twin prime conjecture: checked, and it survived a wrong objection The draft states that if `a(n) < prime(n+1)^2 - prime(n)` for infinitely many n then TPC follows. My first derivation made this look off by two: the argument needs a twin candidate r with both r and r+2 below prime(n+1)^2, so r must be at most `prime(n+1)^2 - 3`, and I read the draft's threshold as one too generous. That was my error, not the draft's. A twin candidate r has no prime factor at or below prime(n), so r <= prime(n) forces r = 1, and r = 1 is a candidate only when 3 does not divide P, that is only at n = 1. So for n >= 2 the lower end of the window costs nothing and the correct threshold is `a(n) < prime(n+1)^2 - 2`, which is exactly what `research/two-class-lower-bounds.md`:619 and `THE-DIALS.md`:274 already carry. Since prime(n) >= 2, the draft's `prime(n+1)^2 - prime(n)` is at least as strong a hypothesis as `prime(n+1)^2 - 2`, so the draft's implication is valid and conservative. **No change made.** Recorded because the objection is the natural one and the next reader will raise it. The supporting clause "any composite in (prime(n), prime(n+1)^2) has a prime factor <= prime(n)" is correct: a composite m has least prime factor at most sqrt(m) < prime(n+1). ### 4.4 G2 draft, the PROG entry Hand-traced, since no PARI is installed on this machine. The state machine is correct: `first` latches the first slot without contributing a gap, `prev` carries the previous slot, and the final `max(g, first+P-prev)` closes the cycle. Traced at n = 1 (slots {1} mod 2, returns max(0, 1+2-1) = 2) and n = 3 (slots 11, 17, 29 mod 30, internal max 12, wrap 11+30-29 = 12, returns 12). Both correct. `primes(n)`, `vecprod`, `gcd` and `max` are all valid PARI. The trailing comment said a(10) came from a segmented sieve and stopped there, while DATA runs to a(12). Corrected to name all three. ### 4.5 Seam draft, all twenty DATA terms and every witness Recomputed from scratch in Python with **two independent primality tests**: a Miller-Rabin with 40 random bases, and a Baillie-PSW (Miller-Rabin base 2 plus a strong Lucas probable-prime test, with its own Jacobi symbol implementation). The two were asserted to agree on **every** call, not only on the accepted ones. Independently reimplemented a third time in BigInt inside `audit-numbers.js`. All twenty terms reproduce: `2, 4, 4, 3, 4, 6, 2, 1, 7, 1, 1, 2, 0, 1, 1, 1, 3, 1, 0, 4`, summing to 48. Every witness list in the provenance table reproduces exactly, including the two empty rows at n = 13 and n = 19. **Four independent confirmations from OEIS itself**, none of which the draft claims and all of which hold: 1. A087732's DATA, decoded, gives the open-range witnesses row by row. 3 is j=2 of 2#; 5, 11, 17 are 6#; 29, 59, 149, 179 are 30#; 419, 1049 are 210#; 2309, 9239, 11549, 25409 are 2310#; 180179 through 420419 are the six terms of 30030# at k = 6, 9, 10, 11, 13, 14, matching a(6) = 6 witness for witness; 4084079 and 8678669 are 510510# at k = 8, 17; 106696589 is 9699690# at k = 11; and the seven terms from 892371479 to 6023507489 are 223092870# at k = 4, 11, 16, 19, 20, 22, 27, matching a(9) = 7 exactly. 2. A087651's run lengths are 1, 3, 4, 2, 4, 6, 2, 1, 7, 1, 1 for n = 1..11, which are precisely a(n) minus the endpoint indicator. The identity the draft states holds at every one of the eleven. 3. A057706 = {5, 29, 2309} gives endpoint hits at n+1 = 2, 3, 5, so the indicator fires at n = 1, 2, 4, and 2 = 1+1, 4 = 3+1, 3 = 2+1 are exactly a(1), a(2), a(4). Correct. 4. A060256(n) equals the least witness for every n = 1..30 with a(n) > 0, and the n with a(n) = 0 (13, 19, and 21, 29 in the extension) are exactly those where A060256(n) exceeds prime(n+1). Thirty-for-thirty. ### 4.6 Seam draft, the ceiling and the two exceptions `a(n) <= prime(n+1) - 2 for n >= 3` verified at all computed n. The stated proof is correct: k*P sweeps every residue mod q = prime(n+1) once, so exactly two k give a member divisible by q, and for n >= 3 that member exceeds q and is therefore composite. The n = 1, 2 exceptions are right and were checked individually: at n = 1 the pair (3,5) at k = 2 counts because the divisible member is 3 = prime(2) itself, at n = 2 the pair (5,7) at k = 1 because it is 5 = prime(3). Both give exactly one extra over the bound, and a(1) = 2 = (q-2)+1, a(2) = 4 = (q-2)+1. The EXAMPLE at n = 4 was checked: among k = 1..11 the twin pairs are k = 2 (419,421), k = 5 (1049,1051) and k = 11 (2309,2311), so a(4) = 3. The PROG entry is correct as written, including at n = 1 where k = 1 gives `isprime(1)` = 0. ### 4.7 Seam draft, the Hardy-Littlewood heuristic, rederived from scratch The wave-3 correction is right, and it was rederived here rather than accepted. Write `W = prod_{q<=x} q` and `D = prod_{3<=q<=x} (q-2)`. Then W/D = 2 * prod_{3<=q<=x} q/(q-2) and since `C2 = prod_{q>=3} (1 - 1/(q-1)^2)` with `1 - 1/(q-1)^2 = q(q-2)/(q-1)^2`, substituting gives prod_{3<=q<=x} q/(q-2) = (1/C2) * prod_{3<=q<=x} (q/(q-1))^2 ~ (1/C2) * (e^gamma ln x / 2)^2 by Mertens with the q = 2 factor removed. Hence W/D ~ 2 * e^{2 gamma} ln^2 x / (4 C2) = e^{2 gamma} ln^2 x / (2 C2) which is `2.4026`, not `1.2013`. The draft's stated form is correct and the twin constant does cancel exactly, leaving no surviving factor of 2. Numerically, the per-seam probability summed over k = 1..prime(n+1) and over n = 1..20 gives **52.51** by exact per-k evaluation and **57.32** by the asymptotic form, against **48** observed. The retired form gives 26.25, which does not bracket 48, and 26.25 doubling to 52.51 is what identifies the defect as exactly a factor of two. The asymptotic total's n = 1 term is 9.52 on its own, because there P = prime(1) = 2 and `(log prime(n) / log P)^2` degenerates to 1; this is now said in the draft so the number is not read as a clean prediction. The decay claim was checked too: `log P = theta(p_n) ~ p_n`, so the expectation decays like `log^2(p_n)/p_n`, whose sum over n behaves like `sum log n / n` and diverges. The draft's conclusion, infinitely many positive terms but density zero, follows heuristically as stated. ### 4.8 The extension a(21)..a(30), computed and verified Computed with both primality implementations and cross-checked against A060256: n: 21 22 23 24 25 26 27 28 29 30 a: 0 2 1 1 2 1 1 1 0 1 witnesses none, {20,75}, {83}, {91}, {35,72}, {80}, {48}, {47}, none, {2}. The sum over n = 1..30 is 58. Recorded in the draft's provenance section rather than appended to DATA, because DATA is the submitter's field and the moratorium means nothing is going anywhere today. Appending is a one-line change. --- ## 5. The instrument, extended `research/audit-numbers.js` gains a part `oeis`, 39 checks, running in 0.1 s. It was added, nothing was removed, the banner title is intact, and the existing retired-value checks are untouched. Design constraint kept: the part reads no repo script and no draft. A059861 is recomputed by a **direct residue sieve** over Z/P for n = 1..7 and only then compared to its closed form, so the closed form is tested rather than assumed. The seam terms come from a self-contained BigInt Miller-Rabin written for this part. The G2 terms are cross-referenced to parts `ladder` and `g2big`, which build the cycle from nothing. Retired values now inside the audit's reach, each of which will fail the run if reintroduced: - `c` fitted with **A059861(n-1)**: mean 0.0406 at cv 69.4%, against 0.4814 at cv 10.2% for the correct A059861(n). The retired indexing collapses the fit by more than a factor of ten, so this is a loud check. - The seam heuristic with the retired `/2`: 26.3 against the live 52.5. - `e^{2gamma}/(4 C2) = 1.2013` against `e^{2gamma}/(2 C2) = 2.4026`. This pair already existed at line 63 in part `const`, which is the point: it was the right regression test in the wrong scope, and it now also stands inside the part that covers the drafts, where the same slip actually happened. Also newly checked: the G2 DATA string, A002110(12) and A059861(12), both margin columns, the p ln^2 p row with its n = 3..12 scope written into the label, the A048670 and A288815 pointwise inequalities, the `A288815 = 6*A072753 + 6` identity over n = 3..12, all thirty seam terms, the seam ceiling, the A060256 agreement, and the zero-term characterisation. One bug of my own was caught by the run and fixed before it landed: the first version of the A072753 check appended two terms to the claimed side that A072753's offset-3 indexing does not reach. The audit failed, which is the instrument working on its author. --- ## 6. Outside my files, reported not fixed `audit-numbers.js` failed two checks **before** any of my edits, and still fails exactly those two. Both are live disagreements in documents I do not own, and both are already labelled as such in the script: - `G2(41#) x=37-as-outlier`: the script computes **488**; `two-class-lower-bounds.md` and `maxgap-law.md` say **487**. - `G2(41#) band low`: the script computes **476**; `U-FRAME.md` says **475**, while the other two files say 476. Each is a rounding-level discrepancy of 1, and each is a real disagreement between the script and a document. They are outside partition Q's scope and are handed on untouched. --- ## 7. What was NOT checked, and why - **The PARI PROG entries were not executed.** No PARI or gp on this machine. Both were hand-traced against DATA instead, at two values of n each, which catches state-machine errors but would not catch a PARI syntax error. Running them under a real gp before submission is cheap and is the one verification gap I could not close. - **a(13) of the G2 sequence was not computed.** It needs about 40x a 54-minute run and no term depends on it. - **The a(10), a(11), a(12) G2 runs were not re-executed at full scale.** The audit rebuilds all twelve terms by an independent route (cycle construction plus one- and two-prime lifts with separate boundary handling), which is a stronger check than repeating the original segmented sieve would be. - **The Iwaniec 1978 and Rankin 1938 bibliographic details were not fetched from a primary source.** They are consistent with the corpus, which has them at `THE-DIALS.md`:256 and `ZONE-POSTULATE.md`:270, but internal consistency is precisely what this campaign has twice found insufficient. The FGKMT details (JAMS 31 (2018), 65-105) *were* verified, at `maxgap-law.md`:412, which records a source check against arXiv:1412.5029 dated 2026-08-17. - **The novelty claims were not re-checked against the literature.** "Absent from OEIS" was checked and holds. "No analogue in print" now rests on the audit recorded in `paper/beta2-note.md` and was not independently re-searched; wave 3 already flagged the repository's remaining novelty claims as needing an expert literature read before anything leaves. --- ## 8. Addendum, routed from partition P: the exponent was another sequence's P reported that the draft might quote 1.57 where G2's figure is 1.54, and warned against copying 1.54 in on its say-so because the home's own arithmetic had been questioned elsewhere. Both ends were checked here and the routing is **correct**. **The draft did carry it.** `oeis-G2-submission.md` read "the growth exponent in p is near 1.57". The draft is about G2. 1.57 is not G2's number. **The home is explicit.** `research/exponent-control.md`:198 says in terms: *"Quote 1.57 for h2 and 1.54 for G2"*. Its §5 table at :169 gives "control-corrected, G2 | **1.54 ± 0.09**" against "control-corrected, h2 | **1.57 ± 0.06**" at :168. `research/U-FRAME.md`:156 corroborates independently with "1.54 ± 0.09 for our own G₂ on 10". So the instruction is not a stray line. **Neither figure was copied.** Both were rebuilt from the control. The estimator is the raw log-log slope in the x-frame minus the one-class control's own bias at the *matching window width*, and every input reproduces: - The raw G2 slope over the ten exact terms x = 5..37 is **1.801**, which the audit's part `fits` already computed independently. - The control A048670, 58 terms, fitted over every sliding window, reproduces `exponent-control.md` §1 **digit for digit**: width 10 gives 49 windows, mean 1.262, sd 0.095, min 1.126, max 1.549; width 12, 47 windows, mean 1.267; width 19, 40 windows, mean 1.280; width 21, 38 windows, mean 1.282; width 30, 29 windows, mean 1.283. Every one of those matches the printed table. - G2: 1.801 − 0.262 = **1.539**, which is the 1.54 the home instructs. - h2: 1.847 − 0.280 = **1.567**, which is 1.57. So the §5 table reproduces exactly and the two figures are genuinely different numbers about genuinely different sequences: G2 on ten terms, A288815 on nineteen. P's caution about the home was worth raising and does not bite here. The arithmetic P had seen questioned, "1.924 − 0.282 = 1.642 against 1.567 printed", belongs to §6's margin discussion and pairs a different raw slope with the width-21 bias; the §5 exponents pair 1.801 with width 10 and 1.847 with width 19, and both land on the printed values. **One loose end, outside my files and not chased.** `exponent-control.md` §1's *nested* prefix readings, "1.191 on ten terms, 1.238 on nineteen, 1.282 on fifty-six", do not reproduce from a prefix starting at x = 2: I get 1.176, 1.198 and 1.262. The sliding-window table, which is what the corrected exponents actually use, reproduces perfectly. The difference is almost certainly a starting-x convention in the nested row. Reported, not fixed, and it does not touch 1.54 or 1.57. **Fixed in the draft**, with the scope and the distinction written in rather than a bare number swapped: the raw slope, the control, the bias, 1.54 ± 0.09, the bracket and floor kept, an explicit sentence that A288815 reads 1.57 ± 0.06 over nineteen terms and that the two should not be quoted at a common exponent, and the home's instruction to note the long certificate ladder at about 1.2 with a third of the gap priced and the residual unexplained. **Audit coverage added**, eight checks: the control's 58 terms, its bias at widths 10, 19 and 21, the raw G2 slope, both corrected exponents, and an explicit check that the two differ by more than rounding. The comment block names 1.57-for-G2 as the retired value, so reintroducing it fails the run. **This is the third distinct error class in the drafts**, after the A059861 index off-by-one and the seam factor of two, and all three were in COMMENTS fields of documents no instrument reached. It is the single strongest input to the readiness judgement.