Abstract
A year is not $365$ days. This paper asks what a calendar is doing——the answer is approximating with an integer, correcting the residue with another integer, and correcting that. No new mathematical theorem and no new law is claimed.
First, a year is $\mathbf{365.24219}$ days (Section 2).
Second, this is the core of the paper. The Julian slips a day in $\mathbf{128}$ years, the Gregorian in $\mathbf{3226}$——better by $\mathbf{25.19}$ (Section 3).
Third, the rule is three nested corrections. Divisible by $4$, by $100$, by $400$ (Section 3).
Fourth, $\mathbf{97}$ leap days per $400$ years give $\mathbf{365.242500}$ (Section 3).
Fifth, the length of a year is itself moving. What is being corrected does not hold still (Section 4).
Sixth, the separator is where to stop, and that is a promise (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the length of the tropical year, the Julian and Gregorian leap rules, continued-fraction approximation and leap seconds are all standard. We do not build astronomy——all we use is one subtraction and one division. We do not write calendar history——the politics of reform and the dates of adoption are not entered. We do not forecast the tropical year——tidal slowing of the rotation is quoted as fact, and no future value is computed. We do not discuss leap-second policy——the debate about ending them is not entered. We do not compare other calendars——lunar, lunisolar and Islamic calendars are not treated. Relation to earlier papers: Paper 306 showed that how many base units there are is a promise——the leap rule is a promise too, and where it is cut has been decided. Paper 295 showed the molar mass of carbon-12 is no longer exactly $12$——the same shape, of the fixed definition and the moving reality, appears in the calendar. Paper 314 showed there is only one kind of constant one may ask “has it changed” about——the tropical year is dimensionful, so its change cannot be separated from the choice of unit. This paper writes everything as the dimensionless ratio of day to year. Paper 82 read time as an accumulation of events——a calendar is exactly a vessel for counting events (sunrises). Paper 123 showed that what differs between the twins is a worldline, not a lagging clock——all time here is terrestrial coordinate time and no relativistic effect is treated. What is added is computing the years to a day’s slip from each calendar’s mean year, counting the leap days per $400$ years to confirm the mean, listing the continued-fraction convergents and showing that $97/400$ is not among them, and putting the separator at where the approximation is cut.
1. Notation and the Question
Setting: The time for the Earth to go once round the Sun and the seasons to return is the tropical year. One rotation of the Earth is a day.
Observation 1.1 The two do not divide. Revolution and rotation are separate motions with no reason to stand in an integer ratio——every calendar problem comes from here.
Question: What, then, is a calendar doing?
2. A Year Is $365.24219$ Days
Check 2.1:
| Quantity | Value |
|---|---|
| Tropical year | $\mathbf{365.24219}$ days |
| Excess over $365$ | $0.24219$ d $=5$ h $48$ m $45$ s |
Observation 2.2 About six hours are left over each year. Left alone that is about a day every four years——a month in $120$ years, half the seasons in $720$.
Observation 2.3 $0.24219$ is nearly $1/4$. So the first approximation is “add a day every four years”——the Julian calendar.
3. The Core: $128$ Years and $3226$
Check 3.1: Each calendar’s mean year against the tropical year.
| Calendar | Mean year | Error | A day slips in |
|---|---|---|---|
| Julian | $365.250000$ d | $+0.007810$ | $\mathbf{128.0}$ yr |
| Gregorian | $365.242500$ d | $+0.000310$ | $\mathbf{3225.8}$ yr |
Observation 3.2 This is the core of the paper. $128$ years became $3226$——better by $\mathbf{25.19}$.
Check 3.3: The Gregorian rule has three stages.
| Stage | Rule | Per $400$ years |
|---|---|---|
| First | divisible by $4$ is a leap year | $+100$ |
| Second | divisible by $100$ is not | $-4$ |
| Third | divisible by $400$ is | $+1$ |
| Total | — | $\mathbf{97}$ |
Observation 3.4 $\mathbf{97}$ per $400$ years. The mean year is $(400\times365+97)/400=\mathbf{365.242500}$ days——only $0.00031$ from the tropical year.
Observation 3.5 Every stage only adds or subtracts. A hundred were put in, four taken out, one put back——each stage picks up the error of the one before.
Check 3.6: The continued-fraction convergents of $0.24219$.
| Convergent | Decimal | Error | Meaning |
|---|---|---|---|
| $1/4$ | $0.25000000$ | $+0.00781000$ | Julian |
| $7/29$ | $0.24137931$ | $-0.00081069$ | — |
| $8/33$ | $0.24242424$ | $+0.00023424$ | the Persian $33$-year cycle |
| $31/128$ | $0.24218750$ | $\mathbf{-0.00000250}$ | — |
| $97/400$ | $0.24250000$ | $+0.00031000$ | Gregorian (not on the list) |
Observation 3.7 $97/400$ is not on that list. The convergents are $1/4$, $7/29$, $8/33$, $31/128$, and $97/400$ appears nowhere among them——the Gregorian rule stands off the ladder of best approximations.
Observation 3.8 And two rungs back, $8/33$ is closer. The errors are $0.00023424$ against $0.00031000$——the Persian $33$-year cycle is more accurate than the Gregorian calendar.
Observation 3.9 $31/128$ is in another class again. An error of $\mathbf{0.00000250}$, so a day slips in $\mathbf{400{,}000}$ years——$31$ leap days per $128$ years, that is “every fourth year, except drop one every $128$”.
Observation 3.10 And $400$ was chosen anyway. $400$ is a multiple of $100$ and rides straight on decimal counting——$33$ and $128$ both need a separate decision about where to start counting. What was chosen was not accuracy but ease of use.
4. What Is Being Corrected Does Not Hold Still
Observation 4.1 The tropical year is itself changing. Tidal friction slows the rotation and lengthens the day——so the answer to “how many days in a year” drifts.
Observation 4.2 So the figure of $3226$ years is not good for $3226$ years. The denominator moves, so it is an extrapolation from today’s value——this paper does not assert that extrapolation.
Observation 4.3 A leap second is a correction at another layer.
| Correction | What it reconciles | Timing |
|---|---|---|
| Leap year | year with day | fixed by rule |
| Leap second | day with second | no rule; decided by observation |
Observation 4.4 The second row has no rule. When and by how much the Earth’s rotation will lag cannot be predicted, so it is measured and then decided——a leap year is computed, a leap second is observed.
Observation 4.5 Paper 295 wrote that the molar mass of carbon-12 is no longer exactly $12$. The same shape——beside the side fixed by definition, the real thing moves.
5. Separator: Where to Stop
Observation 5.1 Separator——where the approximation is cut. And nature does not decide it.
Check 5.2: More stages make it as good as one likes.
| Rule | Mean year | A day slips in |
|---|---|---|
| Every fourth year only | $365.250000$ | $128$ yr |
| $+$ drop at $100$ | $365.240000$ | $456.6$ yr |
| $+$ restore at $400$ | $365.242500$ | $\mathbf{3226}$ yr |
| $+$ drop at $3200$ (hypothetical) | $365.242188$ | $\mathbf{400{,}000}$ yr |
Observation 5.3 The fourth row does not exist. It can be written down and nobody has adopted it——because $3226$ years was judged enough.
Observation 5.4 Paper 306 wrote that how many base units there are is a promise. The number of leap stages is a promise too——three is not correct; three is where it was decided to stop.
Observation 5.5 Applying Paper 300’s criteria to the two corrections.
| Criterion | Leap year against leap second |
|---|---|
| Do they reconcile the same quantities? | no. Year with day, day with second |
| Are they decided the same way? | no. Computed, observed |
| Can they be predicted? | no. One can, one cannot |
| Do both correct calendar drift? | yes |
Observation 5.6 Three noes and one yes——two distinct roots under the one phrase “calendar correction”.
Observation 5.7 One last thing——calendars are not hard because astronomy is hard. Both the tropical year and the day are measured to ample precision——what is hard is having to approximate a non-integer with a rule made of integers. There is no error on nature’s side; the error is on the side of the promise.
6. Roll Call and Closing
- $365.24219$ days (Check 2.1): $5$ h $48$ m $45$ s left over.
- Julian: a day in $128$ years (Check 3.1): Error $+0.007810$.
- Gregorian: a day in $3226$ (Check 3.1): Better by $25.19$.
- $97$ per $400$ years (Check 3.3): $100-4+1$.
- Mean year $365.242500$ (Observation 3.4): Off by $0.00031$.
- $97/400$ is not a convergent (Observation 3.7): $31/128$ gives $400{,}000$ years.
- Ease of use was chosen (Observation 3.10): Not accuracy.
- The day is lengthening (Observation 4.1): $3226$ is an extrapolation.
- Leap seconds are observed (Observation 4.4): Leap years are computed.
- Three noes, one yes (Check 5.5): Corrections have distinct roots.
Closing: Revolution and rotation are separate motions, so there is no reason for a year to be a whole number of days. The tropical year is $\mathbf{365.24219}$ days and leaves $5$ h $48$ m $45$ s over every year. The Julian calendar slips a day in $\mathbf{128}$ years, the Gregorian in $\mathbf{3226}$——better by $\mathbf{25.19}$. The rule is three nested corrections——add at $4$ ($+100$), drop at $100$ ($-4$), restore at $400$ ($+1$), giving $\mathbf{97}$ leap days per $400$ years and a mean year of $\mathbf{365.242500}$, each stage picking up the error of the one before. And yet $97/400$ is not a continued-fraction convergent at all——the convergents are $1/4$, $7/29$, $8/33$, $31/128$. $8/33$, two rungs back, is already closer ($0.00023424$ against $0.00031000$), and $31/128$ would slip a day only in $\mathbf{400{,}000}$ years. $400$ was chosen because it rides on decimal counting: what was chosen was ease of use, not accuracy. And what is being corrected does not hold still——tidal friction lengthens the day, so the figure of $3226$ is itself an extrapolation from today. A leap second is a correction at another layer——a leap year reconciles year with day by rule; a leap second reconciles day with second with no rule, by observation. One is computed, the other observed, and by Paper 300’s criteria they are distinct roots. One thing separates them——where the approximation is cut. And nature does not decide it. One last thing——calendars are not hard because astronomy is hard. Both quantities are measured to ample precision; what is hard is approximating a non-integer with a rule made of integers, and the error lies not on nature’s side but on the side of the promise.
References
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- Yamagishi, Y. (2026). Time as a Renko Chart [Paper 82]. Zenodo. 10.5281/zenodo.21735260
- Yamagishi, Y. (2026). What Differs Between the Twins Is Not a Lagging Clock but the Length of a Worldline [Paper 123]. Zenodo. 10.5281/zenodo.22055231
- Yamagishi, Y. (2026). The Molar Mass of Carbon-12 Is No Longer Exactly 12 [Paper 295]. Zenodo. 10.5281/zenodo.22171659
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). How Many “Base” Units There Are Is a Promise [Paper 306]. Zenodo. 10.5281/zenodo.22172149
- Yamagishi, Y. (2026). There Is Only One Kind of Constant You Can Ask “Has It Changed?” About [Paper 314]. Zenodo. 10.5281/zenodo.22175192
Reference
Yamagishi, Y. (2026). A Calendar Is Three Approximations Stacked [Paper 319]. YOI: https://www.yamagishiyuuki.com/yoi/319
DOI: 10.5281/zenodo.22208933
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 319. This paper is a preprint and has not been peer reviewed.