What “Kui yi ru Xu” actually means

The first thing any Chinese astrology chart does is divide the sky into twelve palaces. Once that division is fixed, everything follows: which palace a planet falls in, and therefore which area of life a statement is about. So where you draw the palace boundary is not a detail. It decides the whole chart.

In the publicly available material, though, that boundary is remarkably vague. The usual formulation is a four-character mnemonic:

奎一入戌 — roughly, "Kui one enters Xu."

Our first reading was the obvious one. The Xu palace starts at the beginning of the Kui lunar mansion, and 奎一 just means "the first star of Kui." Later we noticed there is at least one other reading: 奎一 means the first graduation inside Kui, so the boundary sits somewhere in the middle of the mansion rather than at its edge. On that second reading the boundary falls 1.71° of ecliptic longitude into the mansion, which is the whole difference between the two.

(One thing we are leaving on the table for now: if you take "first star of Kui" literally, meaning the determinative star itself, there is another 2.18° on top of that, from the question of where the zero point sits. That is a separate matter, and it comes back later.)

1.71° sounds negligible. But a palace is only 30° wide, and for any given chart, shifting the whole grid by δ moves roughly δ/30 of the planets into a different palace. 1.71° is about 5.7%. A planet changing palaces means a sentence that was about career suddenly has to be about money. That is not a rounding error.

So we set out to settle it.

We did not start from the classical texts. We had a third-party charting API whose conventions we wanted to reproduce, which meant finding out what its boundaries looked like first. The method was crude: read back all twelve palace cusps from a single call, because reading only the ascendant's cusp hides the widths. For four epochs — 1930, 1960, 1990, 2026 — all twelve widths came back exactly equal, minimum equal to maximum, 30.0000° each.

The phase was more interesting. The cusps taken mod 30 come out at 10.1073, 10.1071, 10.1070 and 10.1069°. Subtract the Lahiri ayanamsa for the same epoch and that phase becomes a constant, drifting only 0.0004° across forty years.

So the grid is strictly equal-division at 30°, and it is anchored to the sidereal frame — it tracks the stars, not the vernal point. That rules out the tropical frame and nothing more. Which sidereal convention it uses is a separate question, and it is the rest of this piece.

So: where does the anchor come from?

In the History of Yuan (《元史》), volume 54, there is a section titled 「黃道十二次宿度」 — the twelve ci boundaries along the ecliptic. All twelve are listed, one line each:

危，十二度六十四分九十一秒。　　　入娵訾之次，辰在亥。
奎，一度七十三分六十三秒。　　　　入降娄之次，辰在戌。
胃，三度七十四分五十六秒。　　　　入大梁之次，辰在酉。
毕，六度八十八分五秒。　　　　　　入实沈之次，辰在申。
井，八度三十四分九十四秒。　　　　入鹑首之次，辰在未。
柳，三度八十六分八十秒。　　　　　入鹑火之次，辰在午。
张，十五度二十六分六秒。　　　　　入鹑尾之次，辰在巳。
轸，十度七分九十七秒。　　　　　　入寿星之次，辰在辰。
氐，一度一十四分五十二秒。　　　　入大火之次，辰在卯。
尾，三度一分一十五秒。　　　　　　入析木之次，辰在寅。
斗，三度七十六分八十五秒。　　　　入星纪之次，辰在丑。
女，二度六分三十八秒。　　　　　　入玄枵之次，辰在子。

The answer is on the second line: 奎，一度七十三分六十三秒，入降娄之次，辰在戌 — the Xu palace (降娄) begins 1.7363 degrees into the Kui mansion. The boundary sits inside a mansion, not at its start.

Reading this requires clearing one hurdle of notation, though. In the Yuan system 1 degree = 100 fen and 1 fen = 100 miao, so 「一度七十三分六十三秒」 is 1.7363 degrees. Below fen there are also the characters 少, 半 and 太, meaning 1/4, 2/4 and 3/4 of a fen — we missed one of those, and it comes back later.

Push a little further and you reach the least visible part of the whole business. An "entry degree into a mansion" (入宿度) is a body's position relative to a mansion. Relative to which point, though? Two candidates: the determinative star itself, projected onto the ecliptic, or the intersection of the ecliptic with the hour circle through that star. The second is the traditional construction — what the ancients measured was a difference in right ascension, and a mansion boundary is essentially a plane through the celestial pole — but it is not intuitive, and we took it for granted at first.

Rather than argue from impression, we made two tables inside the same volume check each other. Volume 54 contains both a table of equatorial mansion degrees and a table of ecliptic mansion degrees. Compare the equatorial table against modern right-ascension differences and the mean discrepancy is 0.299 Yuan degrees. Compare the ecliptic table against hour-circle intersection longitudes: 0.306 Yuan degrees. Compare the ecliptic table against the stars' own ecliptic longitudes: 2.098 Yuan degrees.

The third comparison is an order of magnitude worse, so the zero point is the hour-circle intersection. At the same epoch, those two candidate points differ by 2.18° for the Kui determinative star — the debt we left on the table at the start, settled here. What reassured us is that the first two methods should agree: a determinative star defines a single hour circle, and both the equatorial and the ecliptic tables are measured from it, so their residuals ought to be the same order. They are.

(Incidentally, the two largest residuals in that comparison — Lou at −2.95, Bi at +2.76 — are exactly the two mansions whose determinative star was changed historically.)

With the twelve boundary degrees and a defined zero point, the boundaries can be added up into absolute longitudes using nothing but this one volume. The ecliptic table's four quadrant subtotals are 78.12 / 94.1075 / 83.95 / 109.08, summing to 365.2575 degrees — exactly the Shoushi calendar's tropical-year constant, so the table is internally consistent.

Then take the circular mean of the twelve boundaries mod 30. That value answers one question: what phase would an equal 30° grid need in order to sit as close as possible to these twelve boundaries? It comes out at 349.947° (taking ζ And as the Kui determinative star, the one in use in the Yuan period). The API's measured phase is 349.893°. The difference is 0.054°.

For scale: the twelve Yuan boundaries, taken on their own, deviate from equal division with a peak-to-peak spread of 4.27°. 0.054° is about one eightieth of that. So the API's equal 30° grid is highly compatible, in phase, with the twelve ci boundaries recorded in the Shoushi calendar. One step further is not in the data — and the tempting step, "so the Yuan court used equal division," is simply false.

The twelve ci are themselves unequal. Adding up the recorded degrees gives widths between 27.89° and 32.23°, the furthest 2.23° away from 30°. Equal division is a reasonable simplification, but it is not what they originally were.

The cost of that simplification can be quantified, at least. A position falling between two competing boundaries changes palaces; comparing the equal grid against the Yuan boundaries pair by pair, the absolute deviations sum to 17.217°, which over 360° gives 4.78%. That is a geometric measure, and it assumes points are spread evenly around the circle. Real planetary longitudes are not (orbits are eccentric), so it is good for order of magnitude only. It is also tempting to take one step further and say the probability that at least one of eleven bodies in a chart changes palaces is 41.7% — that calculation is wrong. Rahu and Ketu are exactly 180° apart, Mercury never exceeds 28° from the Sun, Venus never exceeds 48°, and the eleven bodies are strongly correlated, so the independence assumption fails. A real chart-level figure needs a simulation against a real ephemeris, which we have not done. What holds up is 4.78%, the geometric measure, together with its limits.

That is the conclusion. What follows is the four mistakes — each of them looked perfectly sound at the time, and two of them got written into other places as well.

The first was treating the twelve ci as whole mansions added together. In the earliest version we grouped mansions into ci and summed whole mansions: 降娄 = Kui + Lou, 实沈 = Zi + Shen, and so on. That gave widths of 11.60° to 44.30° — 实沈 only 11.6°, 大梁 44.3° — and on that basis we wrote: the twelve palaces are the twelve ci, and they are flatly incompatible with equal 30° division. The boundaries fall inside a mansion, not at its start; the twelve lines quoted above are the answer. 实沈 is half of Bi plus Zi plus Shen plus half of Jing, not just Zi + Shen. The correct widths are 27.89°–32.23°.

This one deserves a note, because at the time it looked flawless. The 28 mansions summed to 365.70 degrees against a year of 365.25 — a 0.45° discrepancy, entirely plausible — and the output was tidy. We checked whether the data was consistent; we never checked whether the model was right. Three different wrong models can all be internally consistent.

The second was reading a phase offset as a difference between two dynasties. We once wrote that the 10.1° anchor offset meant any chart whose solar longitude fell within 10.1° of a boundary would change palaces when you switched dynasty conventions — 10 of our 19 samples did. But that 10.1° is the grid's phase offset relative to the Lahiri sidereal ecliptic. It says how far this software sits from the currently standard Lahiri frame, and it has nothing to do with the difference between the Song and Yuan systems. Both dynasties anchored their twelve ci to the stars; the difference between them comes only from the precision of mansion measurements, on the order of arcminutes to about a degree. It cannot be 10° — that is a third of a palace, seven hundred years of precession. Using a millennium-scale quantity to stand in for a hundred-year difference is what the error actually was.

That kind of mistake needs no logical failure, only two numbers that look alike: both in degrees, both about conventions, sitting in the same paragraph and reading perfectly smoothly. Nor did we write it once. The same wrong number was copied into code comments, a disclosure string and three backlog items — five places.

The third was comparing J2000 star positions against a 1280 table. To get the longitude of the hour-circle intersection the formula is λ = atan2(sin α, cos α·cos ε), and the α we used the first time was the modern right ascension (J2000), while the table is from 1280. Right ascension moves with precession, about 1.28° per century, so that intersection point is not fixed in the sky. Using today's right ascension against a seven-hundred-year-old table is a mixed epoch. After the fix, the second residual above dropped from 0.618 to 0.306 Yuan degrees, and the phase agreement went from 0.44° to 0.054°. Those two numbers are not independent indicators, though: the same correction drives both, so their simultaneous improvement only shows the fix pointed in a consistent direction. It is not corroboration.

The fourth was missing a character in the base-100 notation. The last step needs the volume's ecliptic mansion table, and in the northern quadrant one entry reads 「虛九〈分空太〉」. We first recorded it as 9.00 degrees. In this system 1 fen = 100 miao and 太 = 3/4 fen, so 「分空太」 is zero fen plus 3/4 fen, or 9.0075 degrees. Missing that character makes the northern quadrant 94.10 instead of 94.1075, and the full circuit no longer matches the Shoushi calendar's 365.2575. This one is off by 0.0075° and changes no conclusion. But it is not a numerical problem, it is a misread of the notation — and notation errors are both the easiest mistake to make when transcribing classical texts and the easiest for a specialist to spot instantly.

The last two of those were caught in a cross-check. We laid out the derivation and its raw data and handed it to another party to attack; they found these two, and also offered an alternative explanation which we checked and rejected.

A few things we have not settled. The conclusion depends on one star: with ζ And as the Yuan-era Kui determinative star the offset is 0.054°, with η And it becomes 2.678°. We have two independent lines of evidence for ζ And — a reverse lookup through the volume's equatorial table, and a two-way check on the distance from the Bi mansion to Kui — but the historical sources disagree on this themselves, so we have not reached certainty.

Nor can we say the two are the same system. The statistic used above, a best-fit equal-division phase, is a constructed quantity. It can show that two things are not unrelated; it cannot show they share an origin. So the word throughout is "compatible."

The source itself carries half-a-degree-scale error per mansion — that 0.306 Yuan degree residual. The 0.054° can be smaller than it because the phase is an average over twelve boundaries, and random error shrinks as √12: 0.30/√12 ≈ 0.09°, the same order as the measured 0.054°. This is not the source being unusually precise. It is an average being insensitive to noise.

One more thing we did not check: whether 少, 半 and 太 have other uses elsewhere. We only confirmed what they mean in this particular volume.

If you know of prior work we have missed, or you find an error in how we read the source, we would like to hear about it. We will correct the article and note it here.