A portal · five layers · the leap rule as a fraction

No Year to Round To

Every leap rule ever written is a fraction standing in for the length of the year. The trouble is that the year is not a number. Measured from the Sun's own crossings, the March equinox year, the June solstice year, the September equinox year and the December solstice year are four different lengths spanning 96 seconds, and each leap rule is aimed at one of them. Move the target and the ranking of the rules reorders: six of the ten pairs swap places, and ranking five real calendars puts the Gregorian first against the December solstice and fourth against the June one. Where this takes over from arithmetic is not a threshold but a transition, and it has a measurable width.

Two sentences this page starts after

Neither half of the idea below is new here. Both are already published in this archive, on the layers' own cards, and this page opens by quoting them so that what it adds is visible as an addition.

A calendar rule is a rational approximation to an irrational length of year, exactly as a continued fraction approximates an irrational by best fractions. Caesar's 1/4 and Gregory's 97/400 are two convergents-in-spirit to 0.2422…; the better the fraction, the longer before the seasons slip. Where the golden ratio is the number hardest to approximate, the year is just a number we are stuck approximating — and the whole history of the calendar is the search for a good enough fraction. The Year That Won't Divide, on its card to Most Irrational

The bronze solution is the same mathematics — the best whole-number approximations are the continued-fraction convergents — cut into metal. 235 months / 19 years, the Metonic cycle the machine runs on, is literally a convergent of the year-to-month ratio. Incommensurability, solved by filing teeth. The Machine Made of Months, on its card to Incommensurable

What neither says, and what this page is, is that the thing being approximated does not have a value. The same layer that published the first quotation already noticed the problem and named it rather than hiding it:

And which "year"? The mean tropical year is 365.2422 d, but the reform was really chasing the vernal-equinox year (≈ 365.2424 d), to hold Easter to its Nicene date — and against that target the Gregorian 365.2425 is better still, off by a part in ten thousand. The choice of constant changes the headline number, so the page names it rather than hiding behind one figure. The Year That Won't Divide, on the shipped page

That sentence stops at two constants, both rounded to four decimal places. This page measures how far the choice actually reaches. The answer is that it reaches past the headline number and into the ranking itself, and that beyond a certain cycle length it is the only thing left deciding.

1. The year is a curve

A tropical year is the time between two successive passages of the Sun through the same point of the ecliptic. Pick the March equinox and you get one number. Pick the June solstice and you get another. The Sun does not move at a constant rate, because the orbit is an ellipse, and the point you are measuring from is itself sliding backward under precession, so the interval depends on where in the orbit you start the stopwatch.

Nothing below is quoted from a table. Every value is the mean interval between two crossings of a fixed apparent ecliptic longitude, measured over the window 1500 to 2500 in TT, from instants found by root-finding on a VSOP87 solar position.

The four cardinal values, and the two independent implementations that produced them:

Two implementations of VSOP87 written by different people agree on these means to under a tenth of a second, and all five land within half a second of the values Meeus and Savoie published for epoch 2000 from a computation this one is not derived from. The gap between the extremes is 0.0011061 days, which is 95.6 seconds. That number is the whole argument of this page.

One honest caveat, and it cuts in a useful direction. A single equinox instant is not known nearly that well. Asked for the March equinox of 2026, the two libraries land 22.5 and 2.2 seconds from the instant interpolated out of a JPL Horizons ephemeris fetched for this page. The mean of a thousand such intervals is far better determined than any one of them, because the endpoint errors are divided by the number of years between them. The instants are uncertain by tens of seconds; the interval between them is uncertain by fractions of one.

The five year lengths also drift with epoch, which is a second axis of the same ambiguity. Measured over the thousand-year window used here the March equinox year comes out 365.2423769; over a five-hundred-year window centred on the same epoch it comes out 365.2423893, which is 1.1 seconds longer, and over two thousand years it comes out 1.5 seconds shorter still. The file research/no-year-to-round-to/data/ephemeris.json carries the whole curve from year 0 to 4000 for anyone who wants to move the window and watch the numbers move with it.

2. The bench

A leap rule is a fraction. The Gregorian rule inserts 97 leap days every 400 years, so it is 97/400. The Julian rule is 1/4. Put any fraction against any target and you get one number that matters: how many years pass before the calendar has slipped one whole day against the thing it was aimed at.

Below, every denominator from 1 to 2000 has been given its best numerator, and the staircase is the record: the cycle lengths whose best fraction beats every shorter cycle. Sitting on the staircase means nothing shorter does better. Sitting above it means somebody could have done better with a shorter cycle.

Change the target and watch the board reorder.

What the board does

Of the ten pairs of rules, six change places depending on which year you aimed at. Three different rules take first place across the four seasons. Only the Julian rule stays where it is, last, at every target, which is what a rule with a cycle of four years is entitled to.

That is not an artefact of near-ties. Across all six reversals and all four seasons, the closest any two of these rules ever come to each other is a gap of 6 per cent in their drift, and most of the comparisons are far wider than that: the widest single swing, between the 33-year cycle and the Gregorian rule, runs from the cycle leading by a factor of 2.6 at the March equinox to the Gregorian leading at the December solstice. These are reorderings of things that are genuinely apart.

The sharpest case is the calendar most of the world uses. Against the December solstice the Gregorian rule is first of the five, slipping a day in 4,205 years. Against the June solstice it is fourth of five, slipping a day in 1,152 years. Against the March equinox, the target the 1582 reform was actually protecting, it is second, at 8,121 years, behind a rule with a cycle twelve times shorter.

And against every one of the five targets, the Gregorian rule sits above the staircase. There is no year for which 97/400 is the best fraction of its size. That is not an accident, and section 4 is about why.

The famous figure, and where it comes from

Two numbers circulate for how long the Gregorian calendar takes to lose a day. This archive's own calendar layer carries the first of them:

Now the error is one day in roughly 3,200 yearsThe Year That Won't Divide, on the shipped page, against a mean tropical year of 365.2422

It is not the only figure. Four institutions give four answers, and they span a factor of three:

Every one of them is right. Three of the four follow from an assumption the source states outright, and the recomputations reproduce those to better than one per cent. The fourth needs its caveat said plainly: the NASA page prints the shrinking-year expression but never says it integrated rather than divided, so the basis in that row is this page's reconstruction of how a stated year reaches a stated figure, and it is also the loosest fit of the four at three per cent. What separates them is three choices nobody is obliged to announce: which year you meant, how many decimal places you kept it to, and whether you let the year shrink while the error accumulates. That last one is the whole distance between 3,222 and 2,500: the tropical year is getting shorter, so the calendar's error compounds, and integrating rather than dividing moves this page's own recomputation from 3,222 to 2,574, a difference of 648 years, and lands it near the figure the source printed.

And the window matters more than either. Every drift figure here comes from a year length that is a mean over a stated span, and this page ships the 1500 to 2500 column. Recompute the same rule against the same season over the other windows the ephemeris measured and the Gregorian figure against the March equinox runs from 7,842 to 9,035, a range of 1,193 years, while the 33-year cycle against the same season runs from 19,318 to 28,631, a range of 9,314. That is a larger effect than the rounding this section opened with, and it belongs beside the others rather than in a data file.

What survives all of it: across all four windows the first disagreement is still at 31, the spread is still first beaten at 29, six of ten pairs still reorder, and not one ranking changes. The numbers move and the structure does not, which is why this page leads on the structure.

The decimal places matter almost as much. Against the March equinox year written as 365.24237, the drift is 7,692 years. Against the same year as this page measures it, it is 8,121. The fifth decimal place is worth 428 years of the answer.

The rounding runs deeper than the headline. Expand a target as a continued fraction and the early terms are about the year, but the later ones are about the decimal you truncated at. The measured March equinox year has the expansion [0; 4, 7, 1, 18, …]. A year written as 365.2424 has [0; 4, 7, 1, 37, 22806269381, …], and that eleven-digit term is not astronomy, it is the rounding. Past the fourth convergent, a continued fraction of a four-decimal year is describing the truncation.

This is why the often-repeated "best" rule of 31 leap days in 128 years needs a target attached. It is a convergent of the mean tropical year, and of the rounded 365.2422, and of nothing else here: not of the March equinox year, not of June, not of September, not of December. Against the mean it slips a day in 297,660 years. Against the March equinox it slips one in 5,281, worse than the Gregorian rule it is usually offered as an improvement on.

3. Where arithmetic stops deciding

If the four years are 96 seconds apart, there is a point past which a better fraction is splitting a hair finer than the question is asked. The first draft of this section called that point a threshold and put a number on it. An adversarial pass killed the reading, and what replaced it is better, because the fork is not a line with arithmetic on one side and the seasons on the other. It is a transition, and it has a width.

• Cycle 29, where it opens. The first cycle length whose best fraction is closer to one of the four years than the four years are to each other. Below this, no fraction of any size can tell the four seasons apart: they are all the same target as far as arithmetic can see.
• Cycle 31, the first disagreement. The first cycle length at which two of the four ask for different numerators: three of them want 8 leap days and the June solstice year wants 7. It is isolated, though. From 32 to 59 all four agree again, and the slider above will show you that.
• Cycle 896, where it closes. The last cycle length at which all four seasons still ask for the same rule. Past 896 they never agree again, at any cycle length, ever.

Between those ends the share of cycle lengths at which the seasons give different instructions climbs from nothing to everything, and the climb is the finding:

There is also a line of theory to hang on it. The generic best error at cycle q falls off like 1/(2q²), and that equals the spread of the four years at q = 21.3. That value sits below the whole transition rather than inside it, which is the honest thing to say about it: it marks where a fraction first becomes capable in principle of resolving the difference, some way before any actual fraction does.

So the shape is this. Below a cycle of about thirty years the four seasons are one target. From there to about nine hundred they are increasingly two, three or four targets, depending on the cycle length you happen to have chosen. Past nine hundred they are always four.

And the two reforms sit exactly where that makes them interesting. Gregory's 400 is deep inside the transition, in the band where the seasons disagree about two cycle lengths in five. Milanković's 900 is four years past 896, the last cycle length at which the four seasons still agree about anything. Neither reformer could have known that, and neither of them was choosing on those grounds; it is simply where the arithmetic put them.

The measurement of that transition is, as far as a sweep of this archive and of the literature reachable from it could find, unpublished. The half of it that is not new is named in the sources below, and the naming matters more than the claim.

Now put the two artefacts on that scale. The Antikythera mechanism runs on 19, and its own version of this question forks much later, because the year-to-month ratio is a bigger number and its four readings are proportionally closer: the numerators there do not part company until a cycle of 167. Nineteen is nine times below its own fork. All four cardinal years agree that the best fraction at 19 is 235 months, and 235/19 is the sixth convergent of the ratio under every one of them.

19 is where the approximation gets suddenly, almost suspiciously goodThe Machine Made of Months, on the shipped page

It is also, and this is the part that layer could not have said from inside its own subject, small enough that the question it answers still has one answer. Gregory's 400 is thirteen times above the calendar's fork. The gear got a problem that arithmetic could finish. The calendar did not.

4. The family Gregory could reach

Two things have to be established before this section is allowed to say anything: that the 1582 reform really was aimed at the March equinox rather than at the mean year, and that it was not free to pick any cycle it liked. The first is in the bull itself, which names the equinox as the first of the three things the reform had to settle:

Quo igitur vernum aequinoctium, quod a patribus concilii Nicaeni ad XII kalendas aprilis fuit constitutum, ad eamdem sedem restituatur… Inter gravissimas, 24 February 1582. "So that the vernal equinox, which was fixed by the fathers of the Council of Nicaea at the twelfth day before the Kalends of April, may be restored to that same place." Roman counting is inclusive, so the twelfth day before the Kalends of April is 21 March, not 20.

Taken at its word only so far as it goes. That the Council of Nicaea fixed the equinox at 21 March is the bull's own claim about its authority; the surviving canons of Nicaea contain no such decree, and the date is an Alexandrian computistical convention. What the passage establishes, and all this page needs, is which quantity the reformers were aiming at.

The second question is the one the bench can answer. If 97/400 is beaten at every target by fractions with shorter cycles, why is it 97/400? Press centuries only above and the answer appears.

The 1582 reform did not get to choose a denominator. It kept the Julian four-year rule that was already running and phrased its correction as an exception at century years, which forces the cycle to be a whole number of centuries: skip k of every C centuries and you have (25C − k) / 100C. The reachable set is the multiples of one hundred, and nothing else.

Inside that set, against the March equinox year, 97/400 is optimal for every cycle up to 1,200 years. Gregory did not miss the staircase. He found the best point on the only shelf he could stand on.

The test of that reading is that somebody else stood on the same shelf, and the arithmetic predicts where they landed. In 1923 the Congress of Constantinople adopted Milutin Milanković's rule: century years are leap only when the year leaves 200 or 600 on division by 900, which is 218 leap days per 900 years. Same family, different aim. Against the mean tropical year, 218/900 is optimal within the family for every cycle up to 2,200 years, exactly as 97/400 is against the March equinox.

So the two reforms are not a better rule and a worse one. They are the same search run against two different targets, and each is beaten by the other at the other's target. The 1923 rule slips a day in 31,885 years against the mean tropical year where the 1582 rule takes 3,235, nearly ten times better. Against the March equinox that the 1582 reform existed to hold, the 1923 rule slips a day in 6,467 years where the 1582 rule takes 8,121: the calendar built to be more accurate is twenty per cent worse at the thing the old one was built to protect.

5. The rule with no denominator

There is a third option, and it is the one the modern Iranian calendar takes. Do not write a fraction at all. Observe. Iranian law fixes the month lengths and says the year begins at the true beginning of spring, and is silent on intercalation; there is no leap rule to legislate, because whether a year gets 366 days falls out of where the equinox lands. The operative statement of it in the literature is Borkowski's:

A Jalaali year begins on the first day of astronomically determined spring or on the day following it according to whether the exact moment of the equinox occurs before or after, respectively, 12:00 of the Teheran mean time. K. M. Borkowski, "The Persian calendar for 3000 years", Earth, Moon and Planets 74 (1996), 223

That is the rule implemented below, and "Teheran mean time" is taken as Borkowski himself takes it: he computes it by adding "its geographic longitude, 3.425 hours, to the UT1 of the equinoxes", which is 51.375 degrees east, the city's own meridian and not the 52.5 degrees Iran Standard Time is defined on. The first version of this page used the time-zone meridian and called that "exactly as written", which it was not. The adversarial pass that read the paper more carefully than the build had is the reason this sentence exists. A calendar like it has no drift against its target, by construction. What it has instead is a leap sequence that has to be computed, so it can be computed here, for the years 1600 to 2400, from the same equinox instants as everything else on this page.

Which is worth sitting with. The 33-year cycle, the one attached in the popular account to the reform of 1079 and to Omar Khayyam's name, is not an alternative to observing. It is what observing produces, twenty times out of twenty-four. The calendar with no denominator has a denominator five sixths of the time, and it is the same denominator that sits on the staircase for three of the four seasons.

Is the implementation right?

A rule stated in prose and a rule running in code are different objects, and the gap between them here is half a day wide: Julian day numbers run from noon and civil days from midnight, so getting that wrong would shift every date by one while leaving every statistic built from differences exactly as it is. The implementation is therefore scored twice, against things it did not produce.

First, dates anyone can look up. The computation puts Nowruz on 20 March in 2020, 2024, 2028 and 2029, and on 21 March in the other nine years from 2018 to 2030.

Second, and this is the stronger one. Borkowski also gives an arithmetic description of the present epoch: a Jalali year is leap when it leaves 1, 5, 9, 13, 17, 22, 26 or 30 on division by 33. He says plainly that this holds for Jalali years 1178 to 1634 and that the cycle "is sure to break sometimes". That is a published prediction with a stated expiry date, so it can be scored. Over exactly his window, this computation agrees with his arithmetic on 456 of 457 years. The single disagreement is Jalali 1634, which is the last year of the window he said it was good for.

A thirteenth-century table, and a modern ephemeris

There is a check available on that, and it is the sort this archive likes: an old record and a new computation, asked the same question separately.

Nasir al-Din Tusi's Zij-i Ilkhani lists which years of the Jalali era took a five-year interval rather than a four-year one, for the first 295 years: 31, 64, 97, 130, 163, 192, 225, 258, 291. The gaps between them are the run lengths, and they come out 33, 33, 33, 33, 29, 33, 33, 33: seven runs of 33 and one of 29, from a table compiled roughly two centuries after the reform and eight centuries before any ephemeris. (The secondary source that transmits the list prints only seven differences where nine years give eight; the eight above are this page's arithmetic on his years, not his list.)

The run lengths computed here, from VSOP87 over an entirely different span of years, are twenty of 33, three of 29 and one of 37. Both are modal at 33, at 87.5 per cent and 83.3 per cent. A medieval astronomer watching the sky and a modern planetary theory produce the same shape of sequence, which is what you would expect if both are tracking the same equinox and neither is applying a rule.

Attribution, held loosely on purpose, because the history here is genuinely contested and the arithmetic does not depend on it. There is no direct documentary evidence that the 1079 reform adopted a fixed 33-year cycle; Khayyam left no surviving account, and specialists disagree, with Heydari-Malayeri defending the 33-year reading and Akrami and others favouring a 2820-year scheme. What the medieval sources do attest, in Khazini and in Tusi, is intercalation every fourth year with an occasional fifth-year interval decided by observation, which is a different kind of thing from a cycle applied mechanically. The familiar arithmetic version, leap when the year leaves 1, 5, 9, 13, 17, 22, 26 or 30 on division by 33, is Borkowski's description of the current epoch and he states plainly that it is valid for roughly 1799 to 2256 and that the 33-year cycle "is sure to break sometimes". The year length of 365.24219858156 often attributed to Khayyam appears in no historical source and is the value of 683/2820. This page claims none of that history. It computes the arithmetic of the fraction 8/33, and the run lengths the observational rule produces, and both stand whoever wrote what in 1079.

Over the same 800 years the observational rule inserts 193 leap days. The March equinox year predicts 193.9, and a count over a finite window is entitled to be off by one at the ends, so that is agreement, not a discrepancy.

The price

The arithmetic rule drifts, but you can print its calendar for the year 3000 today. The observational rule does not drift, and you cannot. Its answer depends on where a particular meridian is pointing at a particular instant, which depends on how fast the Earth is turning, and the Earth's rotation is measured, not derived.

That is not a hedge. It is measurable, and here it is measured three ways. Every one of these is a place where the rule's own wording, or the world's own unpredictability, moves a date.

The last row is the one to read carefully. Two independent implementations agree on all 801 dates, and that agreement is not evidence that the future is known. They agree because they read the same published extrapolation of the Earth's rotation. Agreement between two readers of one table is a fact about the table.

So the honest experiment is to shift that extrapolation and see what moves. Below, a fixed offset is applied to the Earth's rotation for every year after 2026 only, since the record up to now is measured, and the count is of Nowruz dates that land on a different day.

The accumulated difference between the Earth's rotation and uniform time runs from about 120 seconds in 1600 to about 1,057 seconds in 2400 under the model used here, and the part after today is an extrapolation of a quantity driven by the coupling between the planet's core and its mantle. An hour of error in it moves 16 of the 374 future dates. This archive has a whole layer on why that quantity cannot simply be predicted, built from 23,560 daily measurements of how long the Earth actually took to turn.

A calendar can be predictable or it can stay put. It cannot be both, and the reason is not astronomy. It is that the Earth's rotation is not a clock.

What this page is claiming

Not that leap rules are rational approximations: that is on a member's card, quoted at the top. Not that the Antikythera runs on a convergent: that is on a member's page, quoted in section 3. Not that the choice of constant changes the headline figure: the calendar layer said so itself and this page opens by quoting it.

What is new is the size of that choice and where it takes over. The four seasonal years are 96 seconds apart, which is more than most of the candidate rules are apart from each other, so six of the ten pairs reorder and three different rules take first place. The crossover is at a cycle of about thirty years, locatable three ways. Below it, arithmetic decides, and the Metonic 19 sits comfortably in that regime along with Caesar's 4. Above it, the season decides, and every one of the five rules here that was meant to improve on Caesar sits above it: 33, 128, 400 and 900. That is the shape of the thing. The moment a calendar tries to do better than one day in four, it has walked past the point where the arithmetic can settle the question on its own.

Show the check

The equinox and solstice instants come from root-finding on a VSOP87 solar position, twice, with two independently written libraries pinned to exact versions in an isolated dependency island so that nothing in the site's own build depends on an ephemeris. Their raw output is committed at research/no-year-to-round-to/data/ephemeris.json. The JPL Horizons response used as a third-party anchor is committed verbatim beside the script that fetched it, with its sha256, so the comparison can be rerun offline.

The arithmetic is separate from the astronomy by design: decide.mjs knows nothing about where its target came from, and is the same file the instrument above runs on. The member quotations are not retyped, they are extracted from those layers' own committed bytes by build-data.mjs, and the verifier requires the blockquotes on this page to match what it extracts. If a member edits a sentence this page quotes, this layer goes red rather than going stale.

Every number the prose marks as a number is recomputed by the verifier with arithmetic written separately from the engine, and then matched back against the bytes this page serves: the verifier pulls every marked figure out of the shipped HTML and requires each one to be in the register of things the lab computed. A figure that is on the page and nowhere in the computation turns the layer red.

That check exists because it was missing. The first version of this section made the same claim while the verifier held about forty hand-written string comparisons, and an adversarial pass falsified seventy-three numbers on the page one at a time and found sixty-nine of them went straight through. The claim was true of the arithmetic and false of the coupling, which is the failure this archive names most often, and it happened here anyway.

What was already known, and by whom

Section 1 of this page is a re-derivation. C. Quigley, Mean tropical year length at arbitrary ecliptic longitude, arXiv:2605.02239, submitted 4 May 2026, computes the same quantity: the mean interval between successive returns of the apparent geocentric solar longitude to a fixed value, averaged over a multi-millennium window, giving "eight mean years against which calendar leap rules can be tuned", four cardinal and four cross-quarter. That paper reports the curve as near-sinusoidal and spanning about 98 seconds peak to peak; this page measures 95.6 for the cardinal four, from a different solar theory. It makes the same argument for working in TT rather than UT, and it covers the epoch dependence over seven millennia.

This page did not know about it while building, and found it because an adversarial pass went looking for prior art the build had not. What that paper does not do, on a full-text search, is the rational-approximation half: no ranking of leap rules, no staircase, no crossover. So section 1 here is a second measurement of a published quantity, which is worth something and is not new, and sections 2 to 5 are the part that is. Its own closing result, that the shrinking of the tropical year gives any fixed rule a quadratic cumulative error reaching a day in about 5,700 years, is the same effect this page meets in the NASA row of section 2, approached from the other side.

The qualitative half of the ranking argument also has a home outside this archive. Irv Bromberg's The Lengths of the Seasons and Calendar Leap Cycles distinguish the mean northward equinoctial year from the mean tropical year and hold that a fixed arithmetic cycle tuned to one of them cannot hold another. This page could not reach that site while building, so it is named here as a lead rather than quoted, and anyone continuing this work should close it properly.

Sources

• J. Meeus and D. Savoie, "The history of the tropical year", Journal of the British Astronomical Association 102, 1 (1992), 40 to 42. The four seasonal year lengths for epoch 2000 quoted in section 1 are that paper's table on p. 42, deduced by Meeus from the VSOP87 theory of Bretagnon and Francou; those four are the ones this page compares against, and they match. The mean tropical year 365.242190 quoted alongside them is a different quantity from a different fit, and the value used for the comparison here is Laskar's, so read that row as an adjacency rather than as a fifth entry in their table.
• Ephemeris: astronomy-engine 2.1.19 and astronomia 4.1.1, both VSOP87-based, pinned in research/no-year-to-round-to/ephemeris/package.json. Third-party anchor: JPL Horizons, target Sun (10) from DE441, geocentric, apparent ecliptic longitude, response committed and hashed.
• Inter gravissimas, 24 February 1582. Latin text as transmitted by Clavius; consulted at thelatinlibrary.com and Wikisource. The often-cited ISO trilingual PDF no longer resolves.
• K. M. Borkowski, "The Persian calendar for 3000 years", Earth, Moon and Planets 74 (1996), 223 to 230, for the operative form of the Iranian rule and for the mod-33 arithmetic description and its stated validity window.
• M. Akrami, arXiv:1111.4926, for Tusi's list of pentaennial years and for Khazini's count, and for the statement that no document directly attests the Jalali intercalation. M. Heydari-Malayeri, arXiv:astro-ph/0409620, for the opposing reading and for the provenance of the year length attributed to Khayyam.
• T. Viik, "On the calendar reform by Johann Heinrich Madler", for the 31-in-128 rule, published anonymously in Das Inland at Tartu in 1858 and reported to the Russian Ministry of Education in 1864.
• N. Gajic, "The curious case of the Milankovitch calendar", History of Geo- and Space Sciences 10 (2019), 235 to 243, for the 1923 Congress of Constantinople, the 900-year rule, and the mean tropical year it was aimed at.
• Drift figures in section 2: NASA GSFC eclipse site (Doggett, reprinted from the Explanatory Supplement), the US Naval Observatory leap-year page, and the English Wikipedia article on the Gregorian calendar with its stated bases. All four quotations were read at source. Three of the four figures are recomputed from a basis the source states; the NASA row's basis is this page's reconstruction, as section 2 says.
• C. Quigley, Mean tropical year length at arbitrary ecliptic longitude, arXiv:2605.02239 (2026), for the prior computation of section 1's quantity. Found by an adversarial pass after this page was built, and discussed above rather than buried here.
• T. Freeth et al., Nature 444 (2006), 587 to 591, and Nature 454 (2008), 614 to 617, for the Antikythera mechanism's Metonic dial: a five-turn spiral of 47 cells per turn, and the back-door inscription naming 19 years and 235 subdivisions. Note that the final gear of the Metonic train is reconstructed rather than recovered, so the dial geometry and the inscription, not the gearing alone, are what carry the 19 here.

Offline: 256 checks. Run node research/no-year-to-round-to/verify.mjs.

The full working record, including the three things this page could not settle, is research/no-year-to-round-to/facts.md.