Summary (See also tight precis of entire case at inside cover of DIO 16 [2009].)
At the end of the 20th century, only three precise ancient lunar period-relations yet remained unsolved.
[“Period-relations” are integral ratios of celestial speeds, the central ancient one being the very accurate equation of 251 synodic months with 269 anomalistic months. The better known Meton→Easter relation, equating 19 years with 235 months, is far less accurate.
(Neither of these are among the triple-quarry sought below.)]
Testimony on these 3 relations had come down to us from sources spread across nearly half a millennium: from the 3rd century BC to the 2nd century AD.

Yet in 2002-2003, all three  period-relations were found to be precisely consistent with common-prime-defactored integral eclipse-cycles and thereby explainable by the theory that the period-relations' classical-era discoverers had (using the only method attested in antiquity for finding such period-relations — that of Almajest 4.2&6.9) simply compared their own eclipse observations to records of eclipse observations within plus-or-minus 1/2 century of about 1240BC.
Thus, one single fruitful theory — rooted in one single hitherto utterly unconsidered century — simultaneously reveals the origins of all three of the only hitherto-unsolved highly accurate ancient lunar period-relations.
[When this case is looked back upon someday, it will be apparent that a prime reason that this approach was not considered was that scholars (including DR) had uniformly discounted the very idea of such early eclipse records surviving for a millennium.]
Our time-bases' vastness — over 1000y in each case — can also account for the previously-inexplicable astonishing accuracy of all three of these period-relations.
[The Almajest (e.g., 3.1, 4.6, 9.2) explicitly recommends the obvious: long time-bases, for those seeking accuracy and thus long-term tabular reliability.]
Each's accuracy is ordmag one part in a million or better — ten million in the draconitic case. (So since there are three equations & three super-accuracies, we are actually solving not 3 but 6 mysteries at one Occamite swoop.)
[Note, also, that other anciently attested lunar period-relations (also founded on ordmag 1000y time-bases) are equally accurate, e.g., 251 syn.mo. = 269 anom.mo., 1979 syn.mo. = 160 sid.yr. (Both discussed at DIO 6 [1996].)]

DR's theory is indeed shocking: classical-era access to accurate eclipse-times over half a millennium older than the previously-accepted limit of eclipse-data availability (more reliable than the Ammizaduga data) — so shocking, that some historians reject the idea purely on the basis of the enormity of the remoteness.
[Ptolemy refers (Almajest 4.11) to a series of eclipses brought over from Babylon, but does not give the duration of it. Some, e.g., Toomer 1984 p.166 n.59, point to the Almajest 3.7 statement that only from the 8th century BC on, are records on the whole preserved down to Ptolemy's time. But Ptolemy seems only to be saying that pretty complete then-surviving records start around then; he never says that no astronomical data at all survived from eariler. (We know some data did: the Ammizaduga record.) Also: note that Hipparchos' time isn't Ptolemy's time: one is 1100y later than the 13th century BC; the other, 30% longer — later by 3 centuries of potential time-ravages. Caesar's burning of part of the Alexandria Library occurred during that time.]

However, doubters of the eclipse solutions ought to recognize that not one but several possibilities inherently reside in the remoteness-enormity of DR's proposed Babylonian eclipse-records:

1. The very idea is crackpot.
   [Easy to undo that complaint, since older data are known, and the method proposed here to explain the three integral equations is common-sense, highly accurate, and very naturally emits integral relations — self-evidently apt for solving the sources of equations which are entirely expressed in integral form. Our method is also attested and standard for the era, since the best ancient scientists understood its advantages.]

2. The proposal is possible but inherently unlikely.
   Gauging probability-inherence is discussed below — and this vague complaint is contrasted with several soft & hard probability-tests that have already been applied: DIO 11.1 [2002] ‡3 §E [pp.24-25]; and DIO 13.1 [2003] ‡2 §C [p.13]. Results: odds against chance are statistically significant, if not lock-certain. Below, we have chain-linked these various surprisingly confirmatory indicia, four such in all:
   just keep on clicking upon each's glowing link, starting with this link, to go right on to the next (ultimately returning here).

3. The very enormity reflected in historian-shock tells us that if the proposal is true, it represents a clarifying advance in our perception of the devoted preservation (and possibly precision) of early Assyrian-Babylonian astronomical records — long before the Persian and Greek conquests of Babylonia.

4. If the theory is ultimately accepted, it also may be of some ancillary use — as a non-extrapolated snapshot, for one remote century — to expert extrapolative researches on ΔT behavior, possibly assisting in suggesting an upper bound on ΔT for that century.

Keep in mind: if the hypothesized 13th century data had survived, there would be no inductive achievement here. I.e., there is no sense in attacking an unexpected induction of lost science — on the grounds that it's lost! Inductions of lost data obviously must be judged on merits other than the self-evident fact of lostness.
[DIO 11.1 [2002] ‡2 n.7 [p.12]: “If we cannot accept any finding in ancient science without direct attestation, then: should we all park our brains at the entrance to the ancient science field? Is it forbidden to induce beyond the texts?”
Note that there are several pieces of ancient astronomy which are not mentioned in extant classical-era materials, yet which we know existed: e.g., a valid Alexandria winter solstice solar altitude observation (DR Isis 1982); Hipparchos' two obliquities 23°11/12 & 23°2/3 (PASP 1982), the latter triply verified; his early and late solar orbits (the latter recognized since 2001 in the Encyclopedia of Astronomy & Astrophysics); not to mention the Ammizaduga Venus data (which still exist).]

Integral Relation→Integral Relation
— From Visible Empirical ECLIPSE-CYCLE to Tabular PERIOD-RELATION:
We know that all pre-Ptolemy lunar speeds were based upon period relations found from observed (visible) eclipse-cycles of ordmag 1000y, where any prime factor common to both sides of each empirical eclipse-cycle was naturally extracted before the period-relation (usually invisible outdoors) became established as the convenient computational basis of astronomers' tables, and thereby came down to us in the pleasingly compact integral format which we are about to start sampling.

Ptolemy explains the centrality of just this process at Almajest 4.2, when he discusses the most important (DIO 11.1 [2002]) ‡1) of empirical ancient periodicities, the visible 345y-long eclipse cycle

which was repeatedly verified empirically as of a duration (126007d01h) virtually independent of position (proving its anomalistic cyclicity). Thus, division by common-prime 17 provided the canonical and extremely accurate period-relation:

Inverting the Foregoing Process — to Untransform
   Three Long-Mysterious PERIOD-RELATIONS→ECLIPSE-CYCLES:
Three famous but hitherto-unsolved integral lunisolar period-relations are found on texts of the classical period. (No selectivity here: as of the start of 2002, these were the only outstanding ancient lunisolar period-relations remaining to be solved.)

[1] 6247 synodic months = 6695 anomalistic months (System A, 3rd century BC)

[2] 5458 synodic months = 5923  draconitic  months (Hipparchos, 2nd century BC)

[3] 3277 synodic months = 3512 anomalistic months (PlanHyp, 2nd century AD)
[Note that in what follows we will often use abbreviations for lunar periods: u for synodic, v for anomalistic, w for draconitic.
Thus: [1] 6247u = 6695v, [2] 5458u = 5923w, [3] 3277u = 3512v.]

Conservative Math Automatically Yields WayUnconservative TripleDiscovery:
What could possibly be more obvious and conservative than to use ancients'

Therefore, DR recently (2002/3/18-2003/1/26) just probed via the simplest, most astronomically-sound, and only-attested technique: Ptolemy's common-sense method — which we just invert for each equation. For all three variously disparate cases, the unambiguous recovery-math was mere multiplication by an integer (or half-integer) — hey, how hard was that?!
[The three DR inductions' directness — leading (unambiguously in all three cases) to data in the range 1292BC-1190BC is what others must verify before they either accept or reject the theory. After DR's exhaustive 2001-2003 investigations, he has consistently contended that no post-1500 BC data, other than the c.13th century BC eclipse records cited, will [a] directly (by attested ancient procedure) and [b] exactly produce all six integers of the three quarry period-relations, connecting to eclipses of the very later eras when extant testimony for each of the 3 relations first appears.
(NB: no opposing theory can satisfy either [a]or[b]. Indeed, again: no other contending theory automatically produces integers at all — especially ones of appropriate dimensions — and in exact agreement with every one of the six attested ancient lunar-speed integers.)
In the years since 2002-2003 publication, not one alternate eclipse has been found by any of the several committed cultist knee-jerk disbelievers in the DR theory. This, despite raging desire among them to find some way — any way — to kill off DR's eclipse theory. Note that those who reject the theory because of the great antiquity of the proposed eclipses are engaging in literally preposterous logic: one of the great imports of the discovery IS the antiquity which is rigorously indicated by the uniqueness of the three solutions (all pointing towards classical-era use of 13th century BC data). It is the reverse of reason to attack an evidential argument-towards-a-conclusion by rejecting the conclusion a-priori merely in order to devalue the argument by then going backwards to it. Such mental behavior is typical of math-challenged cultists who (selectively) refuse to go beyond what is directly attested.]
In each case, the DR solution remarkably LED RIGHT TO the close vicinity of the 13th century BC. The underlying eclipse-relations were (length in years, respectively, about 1010y, 1103y, 1325y):

[1] 12494 synodic months = 13390 anomalistic months (System A, 3rd century BC)

[2] 13645 synodic months = 14807 1/2 drac months (Hipparchos, 2nd century BC)

[3] 16385 synodic months = 17560 anom months (PlanetaryHyp, 2nd century AD)

Verifications:

1. To check the connexions, divide each of these 3 hitherto-unknown eclipse-cycles by its corresponding period-relation to see how simple the ratios are:
   [1] two;
   [2] five halves;
   [3] five;
   again: how hard was that?

2. To verify the 13th century BC clustering, simply subtract the relations' durations from their (parenthetical) dates for century-rounded estimates:
   c.300 BC − 1010y ≈ 1300 BC;
   c.100 BC − 1103y ≈ 1200 BC;
   c.100 AD − 1325y ≈ 1200 BC;
   again, this isn't higher math. For most of us.

That, in skeletal outline, is what has been uncovered. And probably no set of discoveries in ancient astronomy has been more ordinary (in method or math-execution) or more unexpected (in result) by all — certainly including DR. Since history of astronomy archons are reluctant learners, there will be disbelief in officialdom at how easily such treasure was captured.

It is true that the the implications here may subtract from the rep of decaying post-Alexander Babylon. But we should have been realistic all along. There is, e.g., no sign in late Babylon of the sophistication we find in contemporary Hellenistic math — i.e., there's no known Babylonian Euklid.
[DR's bluntness about Babylonian astrology's mathematical inferiority to Hellenstic astronomy (after 300BC) has not won him any archonal friends. Babylon: no trig, no transit circle, no vertical observations, no solstice or equinox observations (Neugebauer HAMA p.366), no awareness of lunar parallax, or precession, observations of needlessly rough accuracy (see F.R.Stephenson's consistent findings or DIO 1.3 [1991] n.223 [p.152]), no knowledge of (or even evident interest in) Babylon's geographical latitude, planets taken in good-to-bad astrological order instead of the Greeks' physical order. (Summation: DIO 1.2 [1991] §E3 [p.112].)]
But DR's present theory that Greeks used Babylonian 13th century BC eclipse records adds to the glory of the height of early Babylonian civilization — a thousand years before Greece became serious about astronomy — a revelation that ought to appeal to cliques who have long affected a love of Babylonian astronomy. (A passion which may chill when Babylon is exalted from external quarters.)

[There is little doubt that 13th century BC Babylonia was more than advanced enough to care to record simple eclipse-data. E.g., the Hilprecht Collection in Jena possesses a strikingly precise clay tablet map of Nippur, Babylonia, from c.1300 BC\@.]

To assist archonal realization of what has been found, the following less condensed discussions may be of enlightening value.

Contextual Considerations:

1. It is not at all controversial that Greek astronomy had access to Babylonian records of eclipses and planetary data. And Babylonian celestial data much older than the 13th century BC have survived far longer (c.1500BC→present for the Ammizaduga Venus tablets: 3500y) than the data-survivals proposed here (1010y, 1103y, 1325y).
   [And over 99% classical-era science mss have been lost. (Less than 10% of even Hipparchos' writings survive. Under 1% [perhaps 0% of primary writings] for Aristarchos.) Which makes it hard to be confident — as confident as certain moderns, anyway — that we would know of classical-era references to 13th century data if astronomers of that time had used them. After all, there are no such references to the earlier Ammizaduga data — which we know existed.]

2. All 3 ancient month-estimates are accurate to ordmag 1 in (at least) a million, not accomplishable (in a naked-eye era) without vast time-bases.
   

3. All pre-Ptolemy lunar speeds are attested by Ptolemy as having been based upon period-relations obtained from long eclipse cycles. (Almajest 4.2 & 6.9; Toomer 1984 Almajest pp.176 & 309.)
   [Note that Ptolemy's top modern defenders reject his direct testimony here on method. See, e.g., O.Neugebauer HAMA pp.310-311 & 391, or Toomer 1984 Almajest p.176 n.10. E.g., 251 syn mos = 269 anom mos: its accuracy — 1 part in millions — was based upon the 345y eclipse-cycle (which is 17 times the 251-month relation), just as Ptolemy says at Almajest 4.2. This is obvious to any astronomer familiar with how astronomical periods are determined accurately: record a huge time base T for a known number N of cycle-repeats, then the mean-motion's period P is found by the equation P = T/N. The idea is that the observational errors at each end of the span are automatically reduced to relative insignificance after division by the large number N.
   E.g., when errors of 1h (3600s) at each end of a 345y cycle are divided by 4267, the period (month M) is determined to ordmag 1 time-second.]

4. All 3 are anciently reported in integral-ratio form, exactly the form which empirical eclipse cycles would emit, simply & naturally — merely through dividing by an integer (or half-integer).

5. When we attempt to reconstruct the ancients' three hitherto-unsolved ratios (from System A, Hipparchos, & Ptolemy) by undoing such division (i.e., multiplying by an integer), we find: all three reconstructions lead into the same narrow 10-decade temporal window, the 13th century BC, and all three do so unambiguously — despite the fact that they come from eras centuries apart (spread over nearly 1/2 a millennium). [I.e., the three results all zero-in on an early theoretical span (c.100y: 1292 BC to 1190 BC) which is about four times narrower than the later empirical base: 3 different observers, spanning over 400y: mid-3rd century BC to late 2nd century AD.]
   This provocative over-arching statistical point is one which the history-of-astronomy community has yet to take in (a point independent of whether it will be persuasive, even if&when finally understood). And said general point is merely prelude to presentation of some specific matches yet to come (below: as the earlier-cited chain-links continue), which multiplicatively shrink even further the statistical likelihood of chance-explanation here.

6. Starting in 1980, historians refused for nearly 1/4 century to accept that all Greek planet speeds were also based upon cycles, even though DR had shown that three planets' mean motions (Mercury, Venus, & Saturn) positively were — and that the numbers generating their speeds were right in the Almajest 9.3 preface to their tables. Then (DIO 11.2 [2003]) Alex Jones showed that the mean motions of Mars & Jupiter were also based upon cycles cited in the same preface — ending that controversy.

7. DR has traced (idem) all five planets' Almajest mean motions to integral sidereal cycles of ordmag 1000y. (Note the Jupiter case's glaring indication that the Almajest's tropical or Metonic planet relations must have come from larger sidereal relations: DIO 11.2 [2003] ‡4 §H4 [p.45].) And we note that Ptolemy's Planetary Hypotheses explicitly provides such huge sidereal cycles for the planets (Neugebauer HAMA p.906), sidereal-integral being the sort of speeds which ancient observers would find from centuries of raw stationary-point planet data (ibid p.390) — just as synodic-integral are the sorts of lunar speeds one would find from eclipse-cycles, and are precisely the format of the lunar speeds we find in pre-Ptolemy records.
   So DR has proposed his General Theory of Ancients' Cyclicities (DIO 11.1 [2002] ‡2 §H [p.19]): all ancient celestial mean motions were ultimately based upon integral cycles — the Moon, the 5 planets, even the Sun's accurate sidereal motion (DIO 6 [1996] ‡1 eqs.23&31 [pp.22&24]), and Aristachos' durable inaccurate precession (arising out of his 4868y-cycle Great Year: DIO 11.1 [2002] ‡1 eq.11 [p.8]; DIO 9.1 [1999] ‡3 eq.16 [p.37]). Since there is no longer any question that ancients' planet motions were based upon cycles, isn't the analogy obvious regarding ancient scientists' likely approach to determining lunar motions?

What Led DR to His Eclipse-Cycle Triple?
We repeat the surprisingly-ultra-conservative answer: DR simply took ancient scientists' STANDARD METHOD — that is, their use of very long eclipse cycles — to search out the sources of three previously unsolved problems.

In no case was there serious ambiguity: ALL 3 arithmetically-indicated early eclipses directly plopped down into the 13th century BC or barely after. We examine these shockers one-by-one, in order of date and (equivalently, as it happens) of DR discovery.
[According to the 1992 canons of Meeus-Mücke and Liu-Fiala (both works based upon F.R.Stephenson's pioneering researches), each of the ten eclipses used here were above the appropriate horizon for at least part of the event's duration.
(The relation of visibility and ΔT questions are touched on in a 2008 note appended to the online pdf of DIO 13.1 ‡2 §G [p.17].)
In any case, the σ in ΔT is roughly 1000 time-sec at that epoch: Stephenson, conversation, 2007/11/29. Stephenson warns at JHA 39:229-250 (2008) p.230 that, before 1000BC, extrapolation using his induced acceleration should treat it as “a first approximation”. The closest eclipse to being eliminated by such extrapolation is that of 1245BC/11/13 (which is the front half of the most convincing of the pairings). Note that, though we cannot on such a basis prove whether or not this eclipse was seen, that very situation reflects the present paper's potential modern scientific utility.]

Solution [1]: System A:
The characteristic integral equation defining System A (found only on Babylonian tablets) is a period-relation of length c.505y (Neugebauer HAMA pp.478 & 501):

[See, e.g., J.Britton Arch Hist Exact Sci 61:83-145 [2007] p.124. (The relation is mis-typed there.) There is no citation of DR's proposal that the 6247u relation was obtained, not by Britton's wildly speculative theory of massive averaging of elaborately-elicited Lunar-Four-based anomalistic estimates (there's not-a-jot of ancient testimony that anyone computed lunar speeds so), but simply through dividing an eclipse-cycle by a small integer: below; or DR DIO 11.1 [2002] ‡2 §A4 [p.12]. (Non-citation of this simple, conventional proposal occurs even though the same DIO issue is cited for carping purposes at Britton n.66.)]

Well, given [a] Ptolemy's ascription of all pre-Ptolemy lunar speeds to integral eclipse-cycle bases; [b] his account (based upon division by an integer) of the ancestry (above) of the 251-month period-relation; [c] the fact that eclipses are the most obvious, naturally integral, and accurate way to establish such ratios — it is reasonable to explore whether the above System A period-relation is (like the 251-month period-relation) also an eclipse-cycle simply divided by an integer. But, since the equation is over 500y long, our options are (quite) limited — i.e., the integer 2 is all there is. (If it fails, then: game-over.)
But when (2002/3/18) DR tried multiplication by 2, he was delighted to find that (against the a priori odds) the result was a valid eclipse-cycle (c.1010y long, our above cycle [1]) — which is recalled here:

The 22° remainder is so near the limit that allows eclipse-pairs to occur at all, that such pairs are rare — fortunate in helping delimit possibilities for those which underlay period-relation [1].
[Only three post-400BC pairs could have been used. These are listed at DIO 11.1 [2002] ‡2 §B4 [p.13], (Due to a DR book-keeping error, a half-invisible 1418BC-408BC pair was accidentally listed there. However, the pair was not used in any analysis originally or latterly. The online version of DIO 11.1 [2002] has the deleted the pair, as well as righting an equally harmless but equally disgraceful typo at ‡3 n.11 [p.22].)]
The search for pairs that are suspect of thus triggering the invention of System A find (between 336BC and 67AD) merely two pair:

(DIO 11.1 [2002] ‡2 §B4 [p.13].) Presumably the to-the-hour agreement between the two pairs' time-gaps suggested an anomalistic return. The possibility that 13th century BC eclipse reports specified the time (not just date) of the eclipse could be a hint of early precision-astronomy; however, the fact that most (all but the 1st) of the five older eclipses (underlying the present triple-analysis) were near the horizon suggests the possibility that near-horizon eclipse reports were preferentially selected in order to narrow the event's time-uncertainty (a technique still used today by experts in ΔT research) — which further implies that enough records survived (contra DR's initial 2002 impression) to permit later scientists' selectivity. Finally, when we learn (ibid [2002] §E6 [p.18]) that the earliest cuneiform tablet firmly connected to System A starts its computations (of lunar positions) from only a few months after the 263BC eclipse, the coincidence is most gratifying.

Solution [2]: Hipparchos' Draconitic Month:
At Almajest 6.9 (Toomer 1984 Almajest p.309), we find Ptolemy reporting in-effect the Hipparchos ratio or — period-relation — that the draconitic month equals 716/777 of a synodic month, for expressing the draconitic (eclipse) month discovered by Hipparchos.

(For simplicity in this equation, we use the remainders implied by his parameters, which are close to reality.)
[The “draconitic month” or “eclipse month” is the time the mean Moon takes to return to a node. A visible (umbral) eclipse can only occur within about a dozen degrees of a node.] Though this is not an eclipse cycle as its stands, we could (if this is all we had testimony for) recover the eclipse cycle empirically underlying the equation merely by finding when a multiple of the equation produces regularly-repeating eclipses. Here, multiplying by 10 does it, producing the following eclipse cycle:

(Accurate to about 1 part in 2 million.) Here, Ptolemy provides us the reverse of our upcoming 3  recoveries: he (loc cit) reports the latter not the former of the above two equations. But the 1st of these equations (716 mos) is of course an obvious consequent of the 2nd, via simple division by common primes — 2 and 5 in this case (thus the 1st equation is 1/10th the 2nd equation).

Now comes the disjunct-impediment (which DIO 11.1 [2002] ‡3 finally [2002/4/3-4] penetrated past): Ptolemy says (loc cit) that the foregoing 7160 mo eclipse cycle produces Hipparchos' famous and amazingly accurate draconitic period-relation (already given earlier at Almajest 4.2: Toomer 1984 p.176), which was accurate to 1 part in ordmag 10 million:

But in fact Ptolemy is (partly) wrong: this equation positively does not follow from the above 7160 month eclipse cycle.
[The disagreement is about 1 part in 2 million. Small, yes. But the two ratios are definitely incompatible, so 5458u = 5923w cannot have arisen from the 7160u = 7770w cycle.]
DR found that the above 5458 month cycle follows instead from cycle-pairing an earlier eclipse with the very same −140/1/27 eclipse which Hipparchos had initially used to find the 7160 month cycle.
[Hipparchos perhaps made a similar decision (much less successfully) after recording his 135BC Summer Solstice, when he used it to gauge the length of the tropical year by comparing to an earlier solstice. He evidently switched from using Meton's 432BC solstice as his older datum (DIO 1.1 [1991] ‡6), to using Aristarchos' 280BC solstice instead (Almajest 3.1). In both cases, use of the older datum gave better results (despite large errors in both antique data), simply because of the longer time-base T.]

The search for a parent eclipse-cycle that is proportional to this period-relation is again narrowly restricted, since the latter is already long: 441y. But, again, a little testing elicits the solution (DIO 11.1 [2002]) ‡3 §§B2-B3 [p.21]); multiplication by 5/2 finds an eclipse-cycle ([2]) about 1103y long:

The half-integral anomalistic term requires: only apogee-perigee pairs need apply.
[Any pair (of the type specified by Ptolemy) with identical anomaly would have sufficed; but Hipparchos apparently preferred to try eliminating (as completely as possible) differences between true and mean longitude, by using events near the lunar orbit's major axis.]
It happens that the nearest-to-perigee eclipse during Hipparchos' lifetime that works with the 13645-month cycle is that of 141BC/1/27 (anomaly −1°), which we know he recorded at Rhodos — and is the very eclipse which Almajest 6.9 says he once paired with the apogee eclipse of 720BC/3/8 (to produce the above 7160-month eclipse-cycle). But if we instead apply the 13645-month eclipse-cycle (behind Hipparchos' actual 5458-month period) to the very same 141BC eclipse, we find his choice of earlier apogee-eclipse and see that the chosen eclipse-pair was:

— and we find that modern theory indicates that the end of the 1245BC eclipse was probably just above the horizon in Babylon and was near-apogee as required (anomaly = 171°).
[DIO 11.1 [2002] ‡3 §C2 [p.21] suggests the 1239BC/7/12-136BC/9/24 pair, alternatively.]

The fact that the astronomy of the above 13645-month eclipse-cycle requires a perigee-apogee pair is striking because Hipparchos is the only astronomer in history who is known to have tried such a ploy (Almajest 6.9). That is, the 13645-month cycle's half-integral anomaly shouts Hipparchos. And, though Ptolemy is not always reliable, it is worth noting that he (Almajest 4.2) explicitly credits Hipparchus with the wonderfully accurate 5458-month relation.
[Firmly establishing the 4267-month eclipse-cycle or cycle [1] or cycle [3] requires multiple eclipse-pairs. But once the synodic and anomalistic motions are nailed down (preferably by the 251-month period-relation, or cycle [3]), one needs but a single well-chosen equal-magnitude eclipse-pair to establish draconitic-return cycle [2]. This is why DIO's establishment of all three cycles has required five eclipse pairs: 2+1+2 = 5.]

Solution [3]: Ptolemy's Anomalistic Equation :
At c.160AD, Ptolemy in the Planetary Hypotheses 1.1.6 provides the following period-relation

Here there are two possible integers (which will by multiplication produce an eclipse cycle not too outlandishly long): 3 and 5. But the former produces no eclipse pairs after 36BC until the mid-3rd century AD, while the latter connects to two eclipses reported in the Almajest, compiled by the very author (Ptolemy) whose Planetary Hypotheses left us relation [3]. Which suggests that the associated pairs helped found the above period-relation. So, multiplying it by 5, we have — at c.1325y! — the longest cycle ever seriously proposed for determining accurate ancient parameters (of course, it would figure that the longest cycle would originate from later rather than earlier antiquity):

Again against chance expectation (continuing the train of indicia suggesting we are on the right track here throughout), Ptolemy reports eclipses in 125/4/5 and 136/3/6 that work with the above eclipse cycle. (The odds are 20-to-1 against his selection occurring by chance: computed at DIO 13.1 [2003] ‡2 §C1 [p.13].) Thus, applying the 13645-month eclipse-cycle to those, we find two pairs that could have handed him that cycle's durations (which produce exactly period-relation [[3]):

Note that it makes sense that Ptolemy would not attempt to find a period-relation better than the famous one of 251 months unless he were using a huge time-base — which he appears also to do elsewhere (DIO 11.2 [2003] §L [pp.49-50]) in the same PlanHyp.

Temporal Coherence:
The earlier eclipses just derived are only about a century later than the earliest we adduced (above) for System A — emphasizing the tightness of the temporal window here, where all five of our remote eclipses fall in a one-century space, though arising from period-relations established throughout high antiquity over a span of over four centuries: 281BC to 136AD. We list those used-eclipses (recovered here) in the order of chronology and duration.
System A 1010y 281BC&263BC↔1292BC&1274BC
Hipp 1103y 141BC↔1245BC
Ptol 1325y 125AD&136AD↔1201BC&1190BC

Ancients may, if several relations seemed comparably useful, have chosen the one where both sides of the equation contained a common prime, which rendered the final relation less cumbersome.
[Hipparchos' alternate draconitic eclipse-cycle for 7160 syn months was divisible on both sides by 10, leading to a 716 mo period-relation. If there were no common prime factor, then the most convenient period-relation was identical to the eclipse-cycle and thus directly visible. (Also, some period-relations [e.g., 800 sidereal years = 9895 syn mos] were not anomalistically right for conversion to period-relation usability, even though visible.)]

Responses to Objections:
As with so many of DR's discoveries, at first only the most enlightened are taking this seriously. Centrists imagine (hope?) that there is some impediment that all-by-itself makes the hypothesis flat-impossible. After a century of Babylonian-worship, the findings here are obviously an unwelcome shock.
[Five years after publication, not one professional historian has yet reported checking even the easy arithmetic, much less the more laborious task (which may be beyond the abilities of some among the set) of confirming the delicate behavior of the eclipse-cycles we discuss. I.e., historians-of-astronomy are again learning nothing from their own history.
(But, then, there's the old saying that the one thing we learn from history is that most folks learn nothing else from it.)]
So let us look critically at the obvious objections:
[a] Data so remote couldn't have survived.
Yet the Ammizaduga Venus data have come through to us, and they are quite pre-13th century. Recall too that the availability of pre-747BC lunar records is indicated by Geminos' knowledge of the 800y sidereal lunar cycle: DIO 11.1 [2002] ‡2 §G3 [p.19]. And it is hardly outré to suggest that Babylonians were observing in the 13th century BC: see, the 1350 BC estimate at Isis 83:474 (1992). I.e. (despite cultist tendencies to Ptolemaically assume Right-Think before the bother of looking at mere data), there is nothing here that is a-priori inherently-impossible — which thereby dispenses with cemental minds' prime (unconscious) impediment to acceptance of our proposed theory.
[b] Thousand-year cycles aren't in the Almajest .
No, but they're in Ptolemy's Planetary Hypotheses (see, e.g., O.Neugebauer, HAMA p.906) for at least (DIO 11.2 [2003] ‡2 §L6 [p.50]) three planets: Mercury (993y), Venus (964y), & Mars (1010y). Note: these cycles are sidereal — which is just what raw empirical observations yield. (Stationary-point returns: method well explained at O.Neugebauer HAMA 1975 p.390.)
[c] These dates could not have been ascertained reliably enough before the Babylonian calendar became regular.
Yet Ptolemy's eclipse trio (721BC-720BC) is correctly dated, for events well before said calendar became regular. Why assume that early Babylonian astronomers were not up to Ptolemy's smart habit of dating all by the unambiguously rigid Egyptian calendar?
[d] There are cuneiform records using the 5458-month cycle c.200BC, well before Hipparchos.
But this objection falls, upon detection of a flaw in tablet-dating firmness (discussed elsewhere: DIO 11.1 [2002] ‡3 sect;§D1-D2 [pp.22-23]) — a point which shows that our findings have important implications (for Bablylonian datings) — well beyond the issue of the eclipses analysed here.
[It turns out that the only explicitly dated Babylonian tablet that uses 5458u = 5923w was written a quarter century after Hipparchos.]

Creativity vs Creationism:
For now, this paper may prove a vain exercise.
In dealing with those (in the present case, a seething clique, which has spent years straining to draw high-precision astronomy out of Babylonian not Greek records) who already know the wrongness of alternate theories even before looking at the evidence, DR feels a kinship with those scientists who (like himself) have debated creationists. The cementality is the same: no interest in what is the most reasonable explanation of the admittedly fragmentary data (though the present theory's mechanism is [unlike Darwin's] anciently attested), but an obsession instead with rejecting anything but a pre-arrived-at sacred conclusion (in this case: Babylonian astrologers were the ultimate pre-Ptolemy astronomical genii) for which there is no evidence whatever on method — fragmentary or otherwise.

Mental Archaeology and the Future:
Throughout his career, DR has enjoyed numerous vindications by emergent new evidence. The above theory will probably not be one of them, since 13th century BC cuneiform eclipse records are unlikely to turn up at this late date. But the case as it stands is sufficiently coherent, direct, and simple that (even if the theory is never 100% established) its merits will be gingerly appreciated by balanced scholars in the not-too-remote future.

A theorist specializes in envisioning beyond available data, so the creator's vision of that future will for-now suffice as vindication in this matter — which is in any case of lesser importance, compared to DR's 2002-2003 combination of delight and initial-disbelief when experiencing the three discoveries themselves.

Appendix on Method: Standard Greek vs Double-Secret Babylonian

DR does not at all object to any scientist who wishes to question the foregoing on logical or empirical grounds. But it is disappointing to observe the unscientifically passionate resentment of the theory, by a clique that has long been committed (professionally & emotionally) to ascribing the discovery of high-accuracy ancient lunar speeds to Babylonia, though:
[1] The only proven relevant inter-cultural transmission of a mathematical parameter based upon known dated observations was in the direction Greece→Babylonia: the sole Babylonia-attested yearlength, found upon the key System B cuneiform tablet BM55555 (a DR discovery, now honored by the tablet's permanent display at the British Museum), based upon Summer Solstices by Meton (432BC) and Hipparchos (135BC).
[B.L. van der Waerden's evaluation began the realization of this finding's import, and the discovery is now so universally accepted that it has long since entered the Encyclopedia of Astronomy and Astrophysics (2000, Hipparchos entry).]
[2] Even DR-critics' most adventurous unattested (purely hypothetical) Lunar-4or6 schemes would not automatically produce the integral-ratio format of all the high-accuracy extant pre-Ptolemy ancient lunar speeds. The cultists' pretzel-processes have never been able convincingly to account for the extant lunar speeds' precise integral ratios (which automatically issue from eclipse-cycles by attested methodology) and amazing accuracy. Because based on data that are less steady and less directly related to anomaly than eclipses' neat periodic returns, Babylonianist schemes require heavy use of averaging, thus negating the use of merely two bounding events separated by a huge time-base — which all astronomers know is the preferred method for accurately determining steady temporal periods. (Not just for eclipses, but also for, e.g., rotations of planets or pulsars. Note that some cultists propose explaining ancient period-relations by combining multiples of shorter relations — needlessly jettisoning the accuracy-advantage of using long ones. Astronomers understand this point without being told.) The Greek-standard eclipse method (used by DR) fills all the above requirements perfectly naturally — indeed, effortlessly.
DR has long contended that the “Babylonian” month's 1-in-millions accuracy was based simply upon the 4267 syn.mo. eclipse cycle's empirical invariance within under 1h (DIO 6 [1996] ‡1 fnn.18&56 [pp.6&13; DIO 11.1 [2002] ‡1 §A3 [p.6]; DIO 13.1 [2003] ‡2 §§E6-E7&F7 [pp.15&17]), since 1h/4267u ≈ 1/3,000,000. J.Britton, after for years questioning this basis for the Bab.mo., now agrees to it, and sets forth the same narrow sub-1h range of variation in the 4267u cycle: Arch. Hist. Exact Sci. 61:83-145 [2007] pp.123-124. Yet:
[a] JB does not cite DR's confirmation here, though his calculations find and use the very same range which DR was (2002 ‡A3 [p.6]) perhaps first to quantify (in hours and in equations), tabulate, and use in this connexion (see above DIO citations).
[Nor does JB cite DR elsewhere except at his n.66, where he cavils about periphery, not noting online-DR 2002 p.7's admiration of his alternate rounding-approach (that is, DR will cite a Britton priority, though the reverse isn't happening here [JB admits only DR's BM55555 solution]), nor P.Tannery's well-known undoing [Heath 1913 p.315] of JB's attestation-complaint — a quibble which avoids facing the reality that there has long been no serious doubt that Aristarchos used a Great Year of 2434y or 4868y, which DR showed (DIO 11.1 [2002] ‡1 §A [pp.6-7]) leads mathematically to the precise “Babylonian” month — decades before any extant record of it in Babylon.]
[b] How does one explain (except by the 345y eclipse cycle) how ancients pinned-down and settled-upon the wonderfully accurate period-relation 251u = 269v, by contrast to the very-VERY-nearby period-relations 237u = 254v or 265u = 284v, whose anomalistic speeds differ from the 251u relation's by only 1 part in ordmag 100,000?
[There's no record that any ancient realized that 355/113 is a better approximation to π (good to 1 part in better than 10 million) than Almajest 1.11's 377/120. (The two values are relatively further apart than the cited lunar ratios surrounding 269/251.).]
JB (still p.124) ascribes this scalpel-like precision to sinuous but (to quote Animal House's Dean Wormer) double-secret Babylonian analyses of Lunar Four data — a process so shaky (connectively & empirically) that laborious ultra-massive data-processing must be hypothesized. (Britton loc cit; note p.125: “considerable averaging must have played a role”. Classic Emperor-of-China folly.) JB even contends that the relation's accuracy is evidence for use of Lunar Four analyses. (As if such an approach is scalpelly superior to using the 4267u = 4573v eclipse cycle?) All this, even though JB (now) simultaneously agrees (still p.124) that the Bab.mo. came from that very cycle which includes discovery of 251u = 269v (from division by 17).
[c] Thus, taking his grouplet's dreamed-up non-eclipse methods as an Expert conclusion (though no fundamental astronomer's support is cited), JB continues privately to replace demonstration with scorn, since he can find no mathematical fault with DR's 2002-2003 eclipse-cycle solutions to the parallel problem of the other period-relations considered here. Where is the detailed math of Britton's alternate explanation for their remarkable accuracy? (DR has here provided his simple math in detail.) And where do Babylonian records testify to use of his group's proposed bizarre methods for finding lunar orbital elements? How is it that neither Ptolemy nor Geminos nor anyone else in antiquity left any record of use of these schemes? As repeatedly noted here, Ptolemy provides a method for elementary use of eclipse-cycle bases (i.e., the very theory we are proposing to explain ancients' adopted lunar speeds — both anomalistic and draconitic), so: where is the parallel Babylonian testimony to Lunar-Four inductions of lunar period-relations — accurate to 1 part in ordmag 1,000,000? (I.e., silence on Babylonianists' fantasized methods is not just Greek.)
And (Muffiosi, are you sitting down?): HOW COULD LUNAR-FOUR ANALYSES FIND A DRACONITIC PERIOD-RELATION AT ALL? — much less to 1 part in ordmag 10,000,000?
[Lunar-four data of course have nothing whatever to do with eclipses. This point parallels R.Newton's comment on laborious Muffia attempts to explain-away Ptolemy's solar fakes by assuming (against the testimony of Almajest 1.12) that the equinoxes were made on a krikos: a krikos is for equinox but not solstice observations, yet Ptolemy's SSolstice “observation” was just as fraudulently accordant with Hipparchos' solar data as his equinox “observation”. In each case (Britton's speculations & defenses of Ptolemy's solar data — Britton, ironically, being the only Muffioso commenting on the latter without making a fool of himself): the Muffia theory is not only baselessly speculative, it doesn't even cover all the outstanding problems — while the hated, proscribed competing theory (cycles for luni-solar period-relations, and fraud for solar data) does coherently solve ALL said problems.]

The rock-base mental blocks preventing Muffia&co acceptance of DR's solution are pretty obvious.
[a] We note first that the three-eclipse-cycles solution (to the mystery of super-accurate ancient lunar speeds) discussed here was not thought-of earlier by Muffiosi and rejected for cause. No, these sadly limited volk (while proposing wildly unsupported speculations of their own) never even thought of an eclipse-cycle explanation for system A, or the Hipparchos draconitic cycle, or Ptolemy's final equation — and never checked to find out whether it would work. (Since DR's 2002-3 publication of such solutions, no one has found any eclipse cycles other than those he cited, which would explain the lunar relations they perfectly solve. So: silence. Such generosity of spirit.)
[b] And, beyond cultism: once a problem is solved, it's no longer available as a money-cow. So there is a simple economic motive for not admitting that anything in the field has been solved. (Even one of DR's associates has explicitly said that nothing is ever proved.) But as DR has repeatedly asked (DIO 4.3 [1994] ‡14 [p.119] and DIO 7.1 [1997] ‡5 n.40 [p.33]): “if even the most logically & evidentially one-sided controversies are to be decreed (e.g, by N.Swerdlow DIO 2.3 [1992] ‡8 §§C20&C25 [pp.109&111]) as indefinitely irresolvable, then — why investigate anything?”