The Derivations of the Heavenly Stems and Earthly Branches and the Transformations of the Escaping Jia—A Unified Examination of the Information Capacity of Bazi and Qimen Dunjia from Mathematical and Metaphysical Perspectives

Author: Xuanji Editorial Department

General Preface

The greatest virtue of Heaven and Earth is to give birth; the greatest treasure of the Sage is position. Since Fuxi in antiquity observed the patterns above and the laws below, drawing the trigrams and establishing the images, the ancient people of Huaxia have connected Heaven and humanity through the methodology of "Images and Numbers" (Xiang Shu). Their Dao is vast and profound, and their techniques are numerous and diverse. As for the two arts later referred to as "Bazi" (Eight Characters) and "Qimen Dunjia" (Mysterious Gates and Escaping Jia), although their forms and operational methods differ, their foundations all stem from the grand system of the Heavenly Stems, Earthly Branches, Yin, and Yang, Five Phases. Their origins can all be traced back to the great achievements of the ancient Sage-Kings in establishing the calendar and charting the celestial phenomena during the pre-Qin period and earlier.

Among contemporary practitioners of the arts of calculation and divination, there is often a dispute: between Bazi and Qimen Dunjia, which one carries a greater volume of information$1 This question appears to be a mere technical distinction, but it actually touches upon the deep structure of mathematical principles and the fundamental doctrines of metaphysics. The concept of "information capacity" is not new to modern times. The Book of Changes, Great Treatise (Xici Zhuan) states: "The Yi as a writing is vast and fully equipped: there is the Way of Heaven, there is the Way of Man, there is the Way of Earth. Combining the Three Powers and multiplying them by two, thus we have Six." This "vast and fully equipped" (guangda xibèi) refers to the ultimate expression of information capacity. The Sages established the trigrams and observed the images with the aim of "encompassing the Dao of Heaven and Earth" (mílún tiāndì zhī dào); how immense, then, is the information capacity of the hexagrams! The arts of Stems and Branches, Dunjia, and others all derive from this fundamental principle of "encompassing" and are adapted therefrom. The size of their information capacity ultimately depends on their respective mathematical structures and layers of symbolic meaning.

The purpose of this article is to trace the origins, starting from pre-Qin classics and ancient transmissions, using the mathematical structure as the warp and metaphysical doctrine as the weft, to conduct a systematic comparative study of the information capacity of Bazi and Qimen Dunjia, ultimately attempting to arrive at a unified conclusion based on both mathematics and metaphysics. All references cited herein are based on classics dating from the pre-Qin period up to the end of the Han Dynasty, striving for evidential rigor and doctrinal grounding.

What is "information capacity"$2 This is not a concept invented by modern people. The Xici Zhuan says: "Writing does not exhaust words; words do not exhaust meaning" (shū bù jìn yán, yán bù jìn yì). And also: "The Sage established images to exhaust meaning, set up hexagrams to exhaust emotion and falsehood, and attached statements to exhaust their words." There is an "inexhaustible" gap between words and meaning, which requires supplementation by "images" (xiàng) and "hexagrams" (guà)—this is the classical expression of information encoding and capacity. The capacity of a system to bear "meaning" (yì), to express "emotion and falsehood" (qíngwěi), and to encompass the "juncture" (jǐ) of transformations, lies precisely in its information capacity.

We must ask:

How is it that the Stems and Branches can record the movements of Heaven and Earth$3 And how is it that Dunjia can unfold the transformations of military strategy$4 From where do the "abilities" (néng) of these two arts derive$5 And how is the extent of this "ability" to be measured$6

This is the core proposition of this paper.

Part One: Tracing the Origins

Chapter 1: The Initial Judgment of Heaven and Earth and the Beginning of Mathematical Principles—The Creation of Images and Numbers by the Ancient Sage-Kings

Section 1: Taiji Gives Birth to the Two Modes: Yin and Yang as the Origin of Information

The foundation of all arts of calculation lies in Yin and Yang.

The Book of Changes, Great Treatise (Part I) states:

"The Yi has the Supreme Ultimate (Taiji); this gives birth to the Two Modes (Liangyi); the Two Modes give birth to the Four Images (Sixiang); the Four Images give birth to the Eight Trigrams (Bagua)." (Yì yǒu tàijí, shì shēng liǎngyí, liǎngyí shēng sìxiàng, sìxiàng shēng bāguà.)

These few sentences are the grand outline of Chinese mathematical thinking. Taiji is the undifferentiated "One" of chaos; the Liangyi are the "Two" resulting from the differentiation of Yin and Yang. From "One" to "Two" is the beginning of information. Why is this so$7 Taiji is chaos, undifferentiable, thus there is no information to speak of. When Yin and Yang are distinguished, there is light and dark, hard and soft, movement and stillness; only then can things be "distinguished"—and "distinction" is the essence of information.

Chapter 42 of the Laozi states:

"The Dao produces One; One produces Two; Two produces Three; Three produces all things. The myriad things carry Yin on their back and embrace Yang in their arms; harmony is achieved through the blending of vital energy (chong qì)." (Dào shēng yī, yī shēng èr, èr shēng sān, sān shēng wànwù. Wànwù fù yīn ér bào yáng, chōng qì yǐ wéi hé.)

Laozi's "One produces Two" corresponds to the Xici Zhuan's "Taiji gives birth to the Two Modes." Once Yin and Yang are differentiated, the myriad things can be categorized, named, and inferred—this is the prerequisite for the emergence of information.

Consider deeply: Why can the "Two Modes" give birth to information$8 For Yin and Yang do not merely refer to light/dark or cold/heat; they are the general terms for all complementary opposites. With this opposition comes relationship; with relationship comes distinction; with distinction comes recognition; with recognition comes judgment—the flow of information begins here.

The Guanzi, Inner Cultivation Chapter (Neiye) states:

"The essence (jīng) of all things, this becomes life. Below it gives birth to the five grains; above it becomes the array of stars. Flowing between Heaven and Earth, it is called Ghost and Spirit (guǐshén); hidden within the chest, it is called the Sage." (Fán wù zhī jīng, cǐ zé wéi shēng. Xià shēng wǔ gǔ, shàng wéi liè xīng. Liú yú tiāndì zhī jiāijiān, wèi zhī guǐshén; cáng yú xiōngzhōng, wèi zhī shèngrén.)

This "essence" (jīng) is the subtle matter (jīngwēi) transformed by the interaction of Yin and Yang. Within this subtlety is contained the information of all things under Heaven, thus it can "below give birth to the five grains, above become the array of stars." Without the content of this information, how could the five grains differ$9 How could the stars be distinct$10

When we discuss the information capacity of Bazi and Qimen Dunjia, we must first realize: both take Yin and Yang as their fundamental informational units. The "position" (wèi) of one Yin and one Yang constitutes the smallest unit of information—like the modern "bit" (bǐtè)—a single position can carry two values: Yin or Yang. The complexity of all arcane arts arises from the superposition, combination, and transformation of these Yin and Yang units.

Section 2: The Two Modes Give Birth to the Four Images: The First Expansion of Information

The superposition of the Two Modes gives birth to the Four Images (Sixiang).

The original meaning of "Four Images" in the Xici Zhuan refers to Greater Yang, Lesser Yin, Lesser Yang, and Greater Yin—the dual-position combinations of Yin and Yang. If we denote Yang as "—" and Yin as "--", then:

• Greater Yang: Yang above, Yang below (— —)
• Lesser Yin: Yin above, Yang below (-- —)
• Lesser Yang: Yang above, Yin below (— --)
• Greater Yin: Yin above, Yin below (-- --)

The Four Images represent all possible arrangements of two Yin/Yang positions. Mathematically, $2^2 = 4$, which is the mathematical basis of the Four Images.

Why are the Four Images important$11 Because the Four Images correspond to the Four Seasons—Spring, Summer, Autumn, Winter—which is the fundamental framework for the operation of the Heavenly Way. The Book of Documents, Canon of Yao (Shangshu: Yao Dian) records:

"He then commanded Xi and He, to respectfully follow the vast Heaven, calculate the ephemerides of the sun, moon, and stars, and diligently present the proper times to the people. Xi Zhong was separately commanded to reside in Yí (Eastern lands), called the Valley of Yang. At sunrise in the Yin position, they regulated the work in the East. At midday, the star Niao (Heart), marking the middle of Spring... Xi Shu was separately commanded to reside in the Southern Border. They regulated the flourishing in the South, showing reverence. When the sun is at its zenith, the star Huǒ (Sharp), marking the middle of Summer... He Zhong was separately commanded to reside in the West, called the Valley of Darkness. At sunset in the Yin position, they regulated the harvest in the West. When the night is at its midpoint, the star Mǎo (Pleiades), marking the middle of Autumn... He Shu was separately commanded to reside in the Northern Quarter, called Yōu Dū. They regulated the transition in the North. When the day is short, the star Mǎo (Pleiades), marking the middle of Winter..."

This records Emperor Yao commanding the four sons of Xi and He to be in charge of the Four Seasons. The Four Seasons constitute the first layer of the framework for the information of the Heavenly Way. The four pillars (Year, Month, Day, Hour) of Bazi are based on the succession of the Four Seasons; the layout of Qimen Dunjia also takes the Four Seasons as the basis for initiating the chart. Thus, the principle of the Four Images is the common mathematical foundation for both arts.

Section 3: The Four Images Give Birth to the Eight Trigrams: The Second Expansion of Information

The superposition of the Four Images, by adding one more line, gives birth to the Eight Trigrams (Bāguà).

The Xici Zhuan states: "The Four Images give birth to the Eight Trigrams." The Eight Trigrams represent all possible arrangements of three Yin/Yang lines. $2^3 = 8$, which is the mathematical basis of the Eight Trigrams.

The names and images of the Eight Trigrams:

Qian is three connected lines (☰), Kun is six broken lines (☷), Zhen is an upturned bowl (☳), Gen is an inverted bowl (☶), Li is hollow in the center (☲), Kan is full in the center (☵), Dui has a gap on top (☱), Xun has a break below (☴).

The Shuogua Zhuan details the images of the Eight Trigrams:

"Qian is Heaven, Kun is Earth, Zhen is Thunder, Xun is Wind, Kan is Water, Li is Fire, Gen is Mountain, Dui is Lake."

"Qian, vigorous. Kun, yielding. Zhen, moving. Xun, entering. Kan, sinking. Li, adhering. Gen, stopping. Dui, pleasing."

"Qian is the horse, Kun is the ox, Zhen is the dragon, Xun is the cock, Kan is the swine, Li is the pheasant, Gen is the dog, Dui is the sheep."

"Qian is the head, Kun is the abdomen, Zhen is the foot, Xun is the thigh, Kan is the ear, Li is the eye, Gen is the hand, Dui is the mouth."

The information capacity of the Eight Trigrams lies not only in their number (eight symbols) but also in their "Images" (Xiàng). Each trigram can be associated with infinite images—this is the core feature of symbolic information encoding in arcane arts: to bear infinite symbolic meaning through a finite set of symbols.

Here we must ask a critical question:

Why is the information capacity of the Eight Trigrams far greater than their mathematical combination number (eight)$12

The answer is: because the Eight Trigrams are not merely numerical symbols but symbols of "Images." An "Image" (Xiàng) is a category (lèi). The Xici Zhuan states: "The Sage perceived the intricacy (zé) of the world below, and modeled it on its forms, symbolizing its suitability for things; therefore, it is called Image." To "model it on its forms, symbolizing its suitability for things" means using the trigram images to "simulate" the forms and laws of all things in Heaven and Earth. The image of one trigram can encompass a category of things—and the capacity of a "category" is far greater than that of an "individual." Therefore, the information capacity of the Eight Trigrams cannot be simply counted as "8," but should be counted as "8 × N," where N is the number of image categories encompassed by each trigram, theoretically tending toward infinity.

This principle is crucial for comparing the information capacities of Bazi and Qimen Dunjia—one must not only count the number of mathematical combinations but also measure the breadth of the symbolic semantic space mapped by each combination.

Section 4: The Eight Trigrams Doubled to Form Sixty-Four Hexagrams: The Third Expansion of Information

The doubling of the Eight Trigrams results in the Sixty-Four Hexagrams (Liùshísì Guà).

The Xici Zhuan states:

"The Eight Trigrams, when made small, are extended and stretched, categorized and enlarged; the matters of the world below are thereby completed." (Bāguà ér xiǎo chéng, yǐn ér shēn zhī, chùlèi ér zhǎng zhī, tiānxià zhī néngshì bìyǐ.)

"And then doubled, the lines (yáo) are contained within. Hardness and softness push and change each other, transformation is contained within. Statements are attached to name them, movement is contained within." (Yīn'ér chóng zhī, yáozài qízhōng yǐ. Gāngróu xiāngtuī, biànzài qízhōng yǐ. Xì cí yān ér mìng zhī, dòngzài qízhōng yǐ.)

The doubling of the Eight Trigrams is the superimposition of the upper and lower primary trigrams. $8 \times 8 = 64$, which is the mathematical basis for the Sixty-Four Hexagrams. And each hexagram has six lines, each line having two values (Yin or Yang), meaning the total number of information positions in the Sixty-Four Hexagrams is $64 \times 6 = 384$ lines.

The Xici Zhuan also states:

"The divination sticks for Qian number two hundred and sixteen, and for Kun one hundred and forty-four; in total three hundred and sixty, corresponding to the days of the year. The sticks of the two sections total eleven thousand five hundred and twenty, corresponding to the number of the myriad things." (Qián zhī cè èrbǎi yīshíyù, kūn zhī cè yīshíwùyù, fán sānbǎi yǒu liùshí, dāng qī zhī rì. Èr piān zhī cè wàn yī qiān wǔbǎi èrshí, dāng wànwù zhī shù yě.)

This refers to the number of sticks used in the divination process. The Qian hexagram has six lines, each line assigned thirty-six sticks, so $6 \times 36 = 216$. The Kun hexagram has six lines, each assigned twenty-four sticks, so $6 \times 24 = 144$. $216 + 144 = 360$, corresponding to the days of the year. This is the unification of mathematical principles and the Heavenly Way. And "the sticks of the two sections total eleven thousand five hundred and twenty," meaning the total information capacity expressed mathematically for the sixty-four hexagrams is 11,520—this "corresponds to the number of the myriad things."

Why does 11,520 "correspond to the number of the myriad things"$13 This requires deep reflection. The 360 days of the year are the cycle of the Heavenly Way; the 11,520 sticks are the mathematical expression of the total information capacity of the sixty-four hexagrams. Looking at $11,520 / 360 = 32$, this happens to be half of the sixty-four hexagrams—which implicitly agrees with the principle that Yin and Yang are balanced. Such a mathematical structure is not accidental; it was established by the Sages after deeply discerning the mathematical laws of the Heavenly Way.

The significance of the Sixty-Four Hexagrams lies in this: they constitute a complete "Cosmic Information Encoding System." What the Xici Zhuan refers to as "vast and fully equipped" (guǎngdà xī bèi) means precisely this. All things under Heaven can be encoded, categorized, and inferred using the framework of the Sixty-Four Hexagrams.

Both Bazi and Qimen Dunjia can be seen as "adaptations" (huàcái) of the Sixty-Four Hexagram System—transforming the cosmic information carried by the sixty-four hexagrams into operational arcane forms through different methods.

Section 5: The Five Numbers of Heaven and Five Numbers of Earth: The Mathematical Foundation of the River Chart and Luo Writing

The Xici Zhuan states:

"Heaven is One, Earth is Two; Heaven is Three, Earth is Four; Heaven is Five, Earth is Six; Heaven is Seven, Earth is Eight; Heaven is Nine, Earth is Ten. The numbers of Heaven are five, the numbers of Earth are five; the five combine in pairs, each having a total. The sum of the numbers of Heaven is twenty-five, the sum of the numbers of Earth is thirty; in total, Heaven and Earth together number fifty-five. This is how change is accomplished and spirits and ghosts are moved." (Tiān yī dì èr, tiān sān dì sì, tiān wǔ dì liù, tiān qī dì bā, tiān jiǔ dì shí. Tiān shù wǔ, dì shù wǔ, wǔ xiāng dé ér gè yǒu hé. Tiān shù èrshíyǒu wǔ, dì shù sānshí, fán tiāndì zhī shù wǔshíyǒu wǔ. Cǐ shì yǐ chéng biànhuà ér xíng guǐshén yě.)

This is the "Numbers of Heaven and Earth." The Heavenly Numbers (odd): 1, 3, 5, 7, 9, summing to 25; the Earthly Numbers (even): 2, 4, 6, 8, 10, summing to 30. The total sum of Heaven and Earth numbers is 55.

These 55 numbers are closely related to the River Chart (Hétú) and Luo Writing (Luòshū).

The numbers of the River Chart:

Heaven generates Water by One, Earth completes it by Six (North). Earth generates Fire by Two, Heaven completes it by Seven (South). Heaven generates Wood by Three, Earth completes it by Eight (East). Earth generates Metal by Four, Heaven completes it by Nine (West). Heaven generates Earth by Five, Earth completes it by Ten (Center).

The Generating Numbers (1, 2, 3, 4, 5) sum to 15; the Completing Numbers (6, 7, 8, 9, 10) sum to 40; the total sum is 55—which is the sum of the Heavenly and Earthly Numbers.

The numbers of the Luo Writing:

Nine on top, One below; Three on the left, Seven on the right; Two and Four form the shoulders, Six and Eight form the feet; Five resides in the center.

The numbers in the Luo Writing sum to fifteen in all rows, columns, and diagonals—this is the mathematical structure of a third-order magic square.

The distinction between the River Chart and Luo Writing corresponds precisely to different facets of the mathematical foundations of Bazi and Qimen Dunjia.

The River Chart governs "Generation" (Shēngchéng)—Heaven One generates Water, Earth Two generates Fire, etc. This represents the sequential generation and completion of the Five Phases in the time series, matching the characteristic of Bazi, which takes the flow of time (Year, Month, Day, Hour) as its main axis. The essence of Bazi is to grasp the pattern of "generation" (shēngchéng) of Yin/Yang and Five Phases at the moment of a person's birth—thus its mathematical foundation leans toward the River Chart.

The Luo Writing governs "Arrangement" (Bùliè)—the fixed positions of the Nine Palaces, with their intersecting numerical relationships expressed spatially—matching the characteristic of Qimen Dunjia, which uses the Nine Palaces as a board and arranges various symbols in spatial orientations. The essence of Qimen Dunjia is to grasp the "arrangement" (bùliè) pattern of the Heaven, Man, and Earth plates in a specific time-space—thus its mathematical foundation leans toward the Luo Writing.

However, the River Chart and Luo Writing are not entirely separate entities. The Xici Zhuan states:

"The River produced a chart, Luo produced writing; the Sages took them as models." (Hé chū tú, Luò chū shū, shèngrén zé zhī.)

The Sages "took them as models," meaning they established arcane arts based on the River Chart and Luo Writing. The numbers of the River Chart and the numbers of the Luo Writing correspond, transform, and interweave. Although Bazi leans toward time (River Chart), it also contains spatial information; although Qimen Dunjia leans toward space (Luo Writing), it also uses time to initiate the chart. The comparison of their information capacities first manifests as a comparison of the mathematical capacities of the River Chart system versus the Luo Writing system.

Let us ask: Is the mathematical capacity of the River Chart greater, or is the capacity of the Luo Writing greater$14

Core data of the River Chart: 5 pairs of generating numbers, totaling 10 numbers, summing to 55. Core data of the Luo Writing: 9 palace positions, summing to 45, with only 1 arrangement structure for the third-order magic square (8 including rotations and reflections).

From a purely combinatorial mathematics perspective, the "generative relationship" of the River Chart contains the mutual generation and overcoming relationships of the Five Phases; its information capacity lies in "relationship" rather than "arrangement." The "nine-palace arrangement" of the Luo Writing contains the intersecting relationships of spatial directions; its information capacity lies in "structure" rather than "generation."

Here we see a deeper pattern: A temporal information system (e.g., River Chart $\rightarrow$ Bazi) determines its information capacity mainly by "sequential relationships"; a spatial information system (e.g., Luo Writing $\rightarrow$ Qimen) determines its information capacity mainly by "structural relationships." Sequential relationships are one-dimensional, while structural relationships are multi-dimensional—this might suggest that the information capacity of Qimen Dunjia could, in some respects, be greater than that of Bazi.

However, this inference requires more detailed argumentation, which will be unfolded in the subsequent sections.

Section 6: The Great Extrapolation Number and the Information Theory of Divination

The Xici Zhuan further states:

"The Great Extrapolation Number is Fifty, and forty-nine are used. Divide them into two to symbolize the Two Modes; suspend one to symbolize Three; strip them four by four to symbolize the Four Seasons; return the remainder to the pile to symbolize the intercalary month. After five years, there are two intercalary months, so after two remainders are returned, one is suspended." (Dàyǎn zhī shù wǔshí, qí yòng sìshíyǒu jiǔ. Fēn ér wéi èr yǐ xiàng liǎng, guà yī yǐ xiàng sān, shiè zhī yǐ sì yǐ xiàng sìshí, guī qí yú lèi yǐ xiàng rùn. Wǔ nián zài rùn, gù zài lèi ér hòu guà.)

This describes the method of divination using yarrow stalks (shéshī). The Great Extrapolation Number is Fifty, but Forty-nine are used. This process of dividing, suspending, stripping, and returning the 49 stalks ultimately yields four results for a single line: Elder Yang, Lesser Yin, Lesser Yang, Elder Yin.

Why is the Great Extrapolation Number Fifty$15 The Xici Zhuan does not explain, and there are many debates. From a mathematical perspective: $55$ (Heaven and Earth Numbers) $- 5$ (Heaven's Five) $= 50$. Heaven's Five resides in the center and is unused, taking the remaining 50, and then discarding one unused stalk (49). The meaning of this "omission of One" is profound—Laozi says "Dao produces One"; this "One" is the number of Taiji, omitted because it is untouchable. Thus, the usable number is 49.

$49 = 7 \times 7 = 7^2$. Seven is the number of Lesser Yang (Heaven Seven). Multiplying the number of Lesser Yang by itself yields the base number for divination, implying the meaning that Lesser Yang governs generation and activity.

In each round of divination (three changes), the 49 stalks are distributed according to specific numerical values, and the resulting probability distribution is:

• Elder Yang (Value 36): Probability 3/16
• Lesser Yin (Value 32): Probability 5/16
• Lesser Yang (Value 28): Probability 7/16
• Elder Yin (Value 24): Probability 1/16

This probability distribution is not uniform—Lesser Yang (7/16) is most frequent, Elder Yin (1/16) is least frequent, implicitly agreeing with the principle that "Yang governs emergence, Yin governs concealment."

From an information theory perspective, a non-uniform distribution has lower information entropy than a uniform distribution. If all four outcomes were equally probable, the entropy per line would be $\log_2(4) = 2$ bits. Under the non-uniform probabilities of yarrow divination, the entropy ($H$) is:

$H = - (3/16 \cdot \log_2(3/16) + 5/16 \cdot \log_2(5/16) + 7/16 \cdot \log_2(7/16) + 1/16 \cdot \log_2(1/16))$

This value is approximately $1.749$ bits, which is less than 2 bits.

The total information entropy for a complete hexagram (six lines) is approximately $6 \times 1.749 \approx 10.49$ bits.

This is the amount of information obtained from one complete divination—about $10.5$ bits.

Using this as a reference, we can similarly calculate the information capacities of Bazi and Qimen Dunjia.

We must ask: Why did the Sages establish such a non-uniform probability distribution$16 Wouldn't a uniform distribution be more "fair"$17

The answer is: The Heavenly Way is inherently not "uniform." The lengths of Spring, Summer, Autumn, and Winter are different; the waxing and waning of day and night are unequal; the rise and fall of the Five Phases change constantly—this is the actual state of the "non-uniformity" of the Heavenly Way. The Sages' divination method simulates precisely this "non-uniformity" of the Heavenly Way. The information encoding of arcane arts does not seek "uniformity" but rather "conformity to the Dao" (hézō*). This differs from the pursuits of pure mathematics later on. The information structures of Bazi and Qimen Dunjia also possess their own forms of "non-uniformity"; this "non-uniformity" is precisely the characteristic of their respective information.

Chapter 2: The Origin of the Stems and Branches: An Encoding System for the Circulation of Heaven's Way

Section 1: The Origin and Meaning of the Ten Heavenly Stems

The Ten Heavenly Stems (Tiāngān) are: Jia (甲), Yi (乙), Bing (丙), Ding (丁), Wu (戊), Ji (己), Geng (庚), Xin (辛), Ren (壬), Gui (癸).

The names of the Stems already existed in antiquity. The Book of Documents, Oath of Gan (Shangshu: Gan Shi) records the words of Xia Qi:

"The great battle at Gan, then summoned the Six Ministers. The King said: 'Alas! Men of the Six Offices, I now declare to you...'"

Although the Stems are not explicitly mentioned here, in the oracle bone inscriptions of the Shang Dynasty, the Stems were already widely used for dating days and for ancestral temple names, such as "Da Jia," "Pan Geng," "Wu Ding," etc., all using Heavenly Stems as names. The fact that the Yin people named their ancestral kings using Stems shows the sacred status of the Stems in the Shang period.

The meaning of the Stems, explained in the Erya, Distinctions of Heaven (Shi Tian):

"Jia, the beginning of the year. Yi, rubbing/grinding (yǎ). Bing, bright. Ding, mature/reaching. Wu, flourishing (mào). Ji, regulating (jì). Geng, changing (gēng). Xin, new (xīn). Ren, responsible/bearing (rèn). Gui, measuring (kuí)."

Although this method of naming may have been established by Han Confucian scholars, the meanings revealed can be traced back earlier. The sequence of the Ten Stems actually reflects a complete cycle of growth and storage:

• Jia (Beginning to grow) $\rightarrow$ Yi (Emerging crookedly) $\rightarrow$ Bing (Clearly visible) $\rightarrow$ Ding (Flourishing) $\rightarrow$ Wu (Peak flourishing) $\rightarrow$ Ji (Regulating and concluding) $\rightarrow$ Geng (Changing and becoming austere) $\rightarrow$ Xin (Hardship and decline) $\rightarrow$ Ren (Concealing and nurturing) $\rightarrow$ Gui (Measuring for the next generation).

These ten phases reflect the entire process of growth and storage of all things throughout the year.

From a mathematical perspective, the Ten Stems constitute a "decimal cycle." Why is Ten the number$18

The Guanzi, Five Phases Chapter states:

"Heaven takes time (shí) as its standard; Earth takes material (cái) as its standard. When the material standard and the time standard align, life occurs; if they do not align, life does not occur." (Tiān yǐ shí wéi zhèng, dì yǐ cái wéi zhèng. Cái shí shùn, zé shēng; bù shùn, zé bù shēng.)

Heaven's "Time" is based on the Five Phases, and each Phase is divided into Yin and Yang, thus $5 \times 2 = 10$, which is the mathematical basis for the Heavenly Stems.

Five Phases: Wood, Fire, Earth, Metal, Water. Yin/Yang: Yang Stems, Yin Stems.

Correspondence:

• Jia (Yang Wood), Yi (Yin Wood)
• Bing (Yang Fire), Ding (Yin Fire)
• Wu (Yang Earth), Ji (Yin Earth)
• Geng (Yang Metal), Xin (Yin Metal)
• Ren (Yang Water), Gui (Yin Water)

Information encoding dimensions of the Ten Heavenly Stems:

1. Five Phase attributes (5 types)
2. Yin/Yang attributes (2 types)
3. Sequential position (10 places)
4. Directional correspondence (East, South, Center, West, North—5 directions)
5. Seasonal correspondence (Spring, Summer, Autumn, Winter, including Long Summer)
6. Five Tones correspondence (Jue, Zhi, Gong, Shang, Yu)
7. Five Colors correspondence (Blue-green, Red, Yellow, White, Black)

Each Heavenly Stem simultaneously carries multiple pieces of information—this is the core feature of the mapping of "Images" (Xiàng): multi-dimensional mapping.

Section 2: The Twelve Earthly Branches: Origin and Meaning

The Twelve Earthly Branches (Dìzhī) are: Zi (子), Chou (丑), Yin (寅), Mao (卯), Chen (辰), Si (巳), Wu (午), Wei (未), Shen (申), You (酉), Xu (戌), Hai (亥).

The use of the Branches is also evident in Shang oracle bones, used for the earthly branch component of dating days, and for dating months.

The number Twelve originates from observations of the Heavenly Way. The Zuo Zhuan, Duke Zhao Year 7 quotes an ancient saying:

"Heaven has ten Suns; Man has ten ranks." (Tiān yǒu shí rì, rén yǒu shí děng.)

These "Ten Suns" refer to the Ten Heavenly Stems. The "Twelve" of the Earthly Branches originates from observations of Jupiter (the Wood Star) taking about twelve years for one revolution, as well as the division of the year into twelve months and the day into twelve shí chén (double-hours).

The Erya, Distinctions of Heaven explains the names of the Earthly Branches:

"When the Great Year (Tàisuì) is in Yin, it is called Shè tí gé; in Mao, Dān é; in Chen, Zhí xú; in Si, Dà huāng luò; in Wu, Dūn zāng; in Wei, Xié xiá; in Shen, Tún tān; in You, Zuò è; in Xu, Yān mào; in Hai, Dà yuān xiàn; in Zi, Kùn dūn; in Chou, Chì fèn ruò."

These are the twelve names for dating the year by Jupiter, corresponding one-to-one with the Twelve Earthly Branches. The method of dating by Jupiter existed in antiquity and was mature by the Spring and Autumn period and the Warring States period.

Information encoding dimensions of the Twelve Earthly Branches:

1. Five Phase attributes: Yin, Mao (Wood); Si, Wu (Fire); Chen, Xu, Chou, Wei (Earth); Shen, You (Metal); Hai, Zi (Water).
2. Yin/Yang attributes: Zi, Yin, Chen, Wu, Shen, Xu (Yang); Chou, Mao, Si, Wei, You, Hai (Yin).
3. Directional correspondence: Twelve directions (30° per direction).
4. Monthly Correspondence: The first month is Yin, the second Mao... the twelfth Chou.
5. Time Correspondence: Zi hour (11 PM – 1 AM) ... Hai hour (9 PM – 11 PM).
6. Zodiac correspondence: Zi Rat, Chou Ox... Hai Pig (this system is also ancient).
7. Hidden Stems (Zàng Gān) correspondence: Each Earthly Branch "hides" one to three Heavenly Stems within it.

The concept of "Hidden Stems" is very important here. Take Zi, for example: it hides Gui Water. Take Chou: it hides Ji Earth, Xin Metal, and Gui Water. The Heavenly Stems are concealed within the Earthly Branches—this signifies "Heaven hidden within Earth," meaning the information capacity of the Earthly Branches is greatly increased, far exceeding their superficial twelve symbols.

Section 3: The Sixty Jiazi: The Great Cycle of Stems and Branches

By pairing the Ten Heavenly Stems with the Twelve Earthly Branches, pairing Stems and Branches of the same Yin/Yang quality, we obtain sixty combinations of Stems and Branches, called the "Sixty Jiazi" or "Sixty Hua Jia."

Jia Zi, Yi Chou, Bing Yin, Ding Mao, Wu Chen, Ji Si, Geng Wu, Xin Wei, Ren Shen, Gui You, Jia Xu, Yi Hai... cycling until Gui Hai, totaling sixty.

Why sixty$19

Mathematically, the Least Common Multiple of 10 and 12 is 60. Thus, only after the Heavenly Stems cycle 6 times and the Earthly Branches cycle 5 times does it return to the starting Jia Zi—this is the mathematical necessity of the Sixty Jiazi.

In terms of the Heavenly Way, sixty years constitute one "Great Cycle." The Zuo Zhuan, Duke Xiang Year 9 records:

"The eleventh month, Jia Zi. The Odes say: 'The foundation was martial like those of Zhou; in every generation there were wise kings.' Jia Zi is the Great Number of Heaven." (Jiǎzǐ, tiān zhī dà shù yě.)

Although this does not directly mention the Sixty Jiazi cycle, the status of "Jia Zi" as the "Great Number of Heaven" shows its importance in the minds of pre-Qin people.

Information structure analysis of the Sixty Jiazi:

Sixty combinations of Stems and Branches, each combination carrying the following layers of information:

1. Heavenly Stem Information: Five Phases, Yin/Yang, Ten Gods relationship (relative to a reference Stem).
2. Earthly Branch Information: Five Phases, Yin/Yang, Direction, Monthly Command, Time, Hidden Stems.
3. Stem-Branch Relationship: The Heavenly Stem sits upon the Earthly Branch, and the two have a "generation-overcoming" relationship—Stem generates Branch, Stem overcomes Branch, Branch generates Stem, Branch overcomes Stem, Stem and Branch share the same Qi.
4. Nayin Five Phases: Each pair of Stems and Branches has a unique Nayin (Sound-Intonation)—Jia Zi and Yi Chou are "Metal in the Sea," Bing Yin and Ding Mao are "Fire in the Furnace," etc. This Nayin system adds another dimension of information to the Sixty Jiazi system.

Information capacity calculation of the Sixty Jiazi (preliminary):

If we only count the number of combinations, the information capacity of 60 symbols is $\log_2(60) \approx 5.91$ bits.

However, the Sixty Jiazi are not isolated symbols—they have sequential relationships (Yi Chou must follow Jia Zi), hierarchical relationships (Stem, Branch, Hidden Stems, Nayin), and relationship dynamics among them (Harmonies, Clashes, Overcomings, Harmonies, Destructions). Therefore, their actual information capacity far exceeds $5.91$ bits.

Here, a key point is raised: The calculation of information capacity in arcane arts cannot rely solely on simple combinatorial logarithms; it must also consider the information embedded in the network of relationships among the symbols.

Section 4: The Secret of Nayin: A Deeper Information Dimension of the Sixty Jiazi

The Nayin Five Phases are the deep information encoding of the Sixty Jiazi. Every two adjacent Stems and Branches share one Nayin, so the Sixty Jiazi have thirty Nayin, divided into six subcategories for each of the Five Phases:

• Metal: Metal in the Sea, Metal of Sword Edge, Metal of White Wax, Metal in Sand, Metal Foil, Metal Hairpin.
• Wood: Wood of the Great Forest, Wood of Willow, Wood of Pine and Cypress, Wood on Flat Ground, Wood of Mulberry and Ziziphus, Wood of Pomegranate.
• Water: Water of the Torrent Bed, Water of the Great Stream, Flowing Water of the Vast River, Water of the Heavenly River, Water of the Great Sea, Water of Well and Spring.
• Fire: Fire of Thunderclap, Fire in the Furnace, Fire of Overturned Lamp, Fire in the Heavens, Fire under the Mountain, Fire on the Mountaintop.
• Earth: Earth by the Roadside, Earth of City Walls, Earth on Rooftops, Earth of Walls, Earth of the Great Relay Station, Earth in Sand.

The method for deriving Nayin is rumored to originate from the pre-Qin method of "Nà Jiǎ" (incorporating Jia). The Zuo Zhuan, Duke Xi Year 15 quotes Han Jian:

"The tortoise is an image; divination is number. Things are born and then have images; after images, there is growth; after growth, there is number." (Guī, xiàng yě; shì, shù yě. Wù shēng ér hòu yǒu xiàng, xiàng ér hòu yǒu zī, zī ér hòu yǒu shù.)

"Image" precedes "Number"—this is a basic concept in pre-Qin Yi learning. The Nayin method is to merge "Number" with "Image"—transforming the numerical relationships of the Sixty Jiazi into concrete material images of the Five Phases, thereby greatly enriching the information capacity of the Stems and Branches system.

"Metal in the Sea" and "Metal of Sword Edge" both belong to Metal, yet one is in the sea, the other is on a blade—the nature, strength, and use of the "Metal" are entirely different. This method of "material objectification" encoding allows the information capacity of the Sixty Jiazi to extend beyond the framework of "Five Phases $\times$ Yin/Yang" into the specific domain of countless things and phenomena.

From an information capacity perspective, Nayin adds 30 unique "material image symbols" to the Sixty Jiazi, each symbol capable of mapping to a class of concrete things. This additional information capacity is approximately $\log_2(30) \approx 4.91$ bits.

Thus, the total information capacity of the Sixty Jiazi is at least:

Basic Combination Information ($\approx 5.91$ bits) + Nayin Information ($\approx 4.91$ bits) + Relationship Information (Clash, Harmony, etc., to be calculated separately).

This total volume is already considerable—it is at least 10.82 bits plus relationship information.

Section 5: The Complete System of Stem-Branch Timekeeping

The dating of Years, Months, Days, and Hours by the Sixty Jiazi forms the data foundation of Bazi.

A complete point in time is expressed by the Four Pillars of Bazi:

• Year Pillar: One of the Sixty Jiazi (Year dating)
• Month Pillar: One of the Sixty Jiazi (Month dating, delimited by solar terms)
• Day Pillar: One of the Sixty Jiazi (Day dating)
• Hour Pillar: One of the Sixty Jiazi (Hour dating, Twelve shí chén $\times$ Heavenly Stems)

The Four Pillars comprise eight characters (four Heavenly Stems, four Earthly Branches), which is the origin of the name "Eight Characters."

Calculating the theoretical combination number of Bazi from a mathematical perspective:

• Year Pillar: 60 possibilities
• Month Pillar: Constrained by the Year Pillar's Stem (Method of deriving months from the year), theoretically 60 possibilities, but practically about 60.
• Day Pillar: 60 possibilities
• Hour Pillar: Constrained by the Day Pillar's Stem (Method of deriving hours from the day), 12 hours per day, pairing with Stems yields 60 possibilities, but only 12 are used per day.

Considering actual astronomical constraints—the Heavenly Stem of the Year determines the start of the Heavenly Stem of the Month, the Heavenly Stem of the Day determines the start of the Heavenly Stem of the Hour—the actual combination number of Bazi requires a more precise calculation:

Year Pillar (60) $\times$ Month Pillar (12 possibilities per year, constrained by the Year Stem) $\times$ Day Pillar (60) $\times$ Hour Pillar (12 per day)

That is: $60 \times 12 \times 60 \times 12 = 518,400$ combinations.

However, not all 518,400 combinations actually occur in the astronomical calendar due to factors like long/short months and leap months; the actual number of occurring Bazi combinations is slightly lower—but the order of magnitude remains around half a million.

In terms of information capacity: $\log_2(518,400) \approx 18.98$ bits.

This is the "static combination information capacity" of Bazi—about 19 bits.

But this is only the starting point. The information capacity of Bazi is far greater than this.

Here we introduce a critical point: The calculation of information capacity in arcane arts cannot rely merely on simple combinatorial logarithms, but must consider the information contained in the network of relationships among the symbols.

Chapter 3: The Origin of Qimen Dunjia: An Encoding System for Military Strategy and the Heavenly Way

Section 1: The Origin of the Nine Palaces and the Number of the Luo Writing

The spatial foundation of Qimen Dunjia is the Nine Palaces (Jiǔ Gōng). The Nine Palaces originate from the Luo Writing.

The Xici Zhuan states:

"The River produced a chart, Luo produced writing; the Sages took them as models." (Hé chū tú, Luò chū shū, shèngrén zé zhī.)

The arrangement of the Nine Numbers of the Luo Writing:

4  9  2
3  5  7
8  1  6

This is a third-order magic square, where the sum of numbers in every row, column, and main diagonal is fifteen.

The names of the Nine Palaces, according to pre-Qin documents like the Lüshi Chunqiu, Records of Origin (You Shi Lan), relate to the concept of eight directions plus the center. The Nine Palaces correspond to the Eight Trigrams plus the Central Palace:

• Kan (One Palace, North)
• Kun (Two Palace, Southwest)
• Zhen (Three Palace, East)
• Xun (Four Palace, Southeast)
• Zhong (Five Palace, Center)
• Qian (Six Palace, Northwest)
• Dui (Seven Palace, West)
• Gen (Eight Palace, Northeast)
• Li (Nine Palace, South)

This assignment of the Eight Trigrams to the Nine Palaces perfectly matches the numbers of the Luo Writing—Kan trigram in the northern first palace, Kun in the southwestern second palace... Li in the southern ninth palace.

Mathematical characteristics of the Nine Palaces:

1. Magic Square Property: Rows, columns, and diagonals all sum to fifteen—this implies that the numerical relationship among the three palaces on any line is "balanced," and the information distribution is uniform.
2. Symmetry: Opposite palace numbers sum to ten (1+9, 2+8, 3+7, 4+6)—this implicitly agrees with Yin/Yang symmetry.
3. Centrality: Five resides in the center, being half the sum of all diagonals—this is the numerical expression of "Center."
4. Rotational Invariance: The Luo Writing has four rotational transformations (0°, 90°, 180°, 270°), forming the cyclic group $C_4$.

The information capacity of the Nine Palaces is based on nine positions $\times$ the content that can be placed in each position = capacity determined by the richness of the content placed.

Section 2: The Origin of the Three Curiosities and Six Instruments

The core symbolic system of Qimen Dunjia includes the "Three Curiosities" (Sān Qí) and the "Six Instruments" (Liù Yí):

Three Curiosities: Yi (乙), Bing (丙), Ding (丁) Six Instruments: Wu (戊), Ji (己), Geng (庚), Xin (辛), Ren (壬), Gui (癸)

Totaling nine Heavenly Stems (Jia 甲 is not used because Jia is the commander, hidden beneath the Six Instruments, hence "Dunjia" or "Escaping Jia").

The method of Jia Hiding:

• Jia Zi hides under Wu.
• Jia Xu hides under Ji.
• Jia Shen hides under Geng.
• Jia Wu hides under Xin.
• Jia Chen hides under Ren.
• Jia Yin hides under Gui.

These six Jia (Jia Zi, Jia Xu, Jia Shen, Jia Wu, Jia Chen, Jia Yin), which are the six leaders of the Sixty Jiazi cycle, are collectively called the "Six Jia." Each of the Six Jia hides under one Instrument, meaning the Six Instruments are the "incarnations" of the Six Jia.

Why must Jia "hide"$20 The Zuo Zhuan, Duke Xuan Year 3 records:

"The position of the Son of Heaven cannot be vacant." (Tiānzǐ zhī wèi, bùkě kuàng yě.)

Jia, the head of the Ten Stems, is like the Son of Heaven. The Son of Heaven does not personally lead the troops but entrusts the generals to carry out his orders—this is the symbolic basis for "Dunjia." Jia hides and does not appear, using the Three Curiosities and Six Instruments as vanguards, like the Son of Heaven hidden in the deep palace dispatching his generals.

From an information science perspective, "hiding" (dùn) means "hiding information"—the information of Jia is not directly presented but expressed indirectly through the Instrument under which it hides. This adds a layer to the system's information hierarchy: the surface information (the Stems displayed by the Six Instruments) and the deep information (the identity of the Jia hidden by the Six Instruments) form a dual-layer encoding, causing the same symbol to carry a double meaning.

The character Qí in the Three Curiosities means "unusual" or "surprise troops." The Guanzi, Minor Officials Chapter (Yòu Guān) states:

"Govern the state with the Orthodox (zhèng); employ troops with the Unusual (qí)." (Yǐ zhèng zhì guó, yǐ qí yòng bīng.)

Laozi, Chapter 67 (in common editions Chapter 57) also states:

"Governing the state with the Orthodox, employing troops with the Unusual, conquering the world without action." (Yǐ zhèng zhì guó, yǐ qí yòng bīng, yǐ wú shì qǔ tiānxià.)

The Three Curiosities are the surprise troops of military strategy. Yi, Bing, and Ding have a special auspicious status in Qimen Dunjia—any palace occupied by the Three Curiosities generally indicates advantage.

Information dimensions of the Three Curiosities and Six Instruments:

1. Five Phase and Yin/Yang attributes of the Stems.
2. Special identity (Curiosity or Instrument).
3. The identity of the Jia hidden by the Instrument (which of the Six Jia).
4. The relationship among the Three Curiosities and Six Instruments.

Section 3: The Origin of the Eight Gates

Qimen Dunjia has eight "Gates" (Mén):

Open Gate (Kāimén), Rest Gate (Xiūmén), Life Gate (Shēngmén), Hurt Gate (Shāngmén), Block Gate (Dùmén), Scene Gate (Jǐngmén), Death Gate (Sǐmén), Shock Gate (Jīngmén).

The Eight Gates are assigned to the Eight Palaces (the Central Palace has no gate, temporarily assigned to the Kun Two Palace or Gen Eight Palace).

Meanings of the Eight Gates:

• Open Gate (Qian Six Palace): Initiation, opening.
• Rest Gate (Kan One Palace): Rest, ease.
• Life Gate (Gen Eight Palace): Growth, generation.
• Hurt Gate (Zhen Three Palace): Harm, damage.
• Block Gate (Xun Four Palace): Blockage, sealing off.
• Scene Gate (Li Nine Palace): Scenery, brightness.
• Death Gate (Kun Two Palace): Extinction, ending.
• Shock Gate (Dui Seven Palace): Fear, shock.

Among the Eight Gates, the Open, Rest, and Life Gates are auspicious; the Hurt, Block, and Scene Gates are neutral (some classify Scene Gate as inauspicious; views vary). The Death and Shock Gates are inauspicious.

Information dimensions of the Eight Gates:

1. Auspicious/Inauspicious attribute.
2. Five Phase attribute (determined by the Palace it occupies).
3. Directional attribute (one of the eight directions).
4. Temporal attribute (changes with the rotation of the chart).
5. Combinational relationship with other symbols (Stars, Spirits, Curiosities/Instruments).

Why establish "Gates"$21 Gates are the pivots of entry and exit. Chapter 1 of the Laozi states:

"Nameless is the beginning of Heaven and Earth; named is the Mother of the myriad things. Thus, constantly without desire, one contemplates their mystery; constantly with desire, one contemplates their manifestations. These two spring from the same source but differ in name; united they are called darkness. Darkness within darkness, the gateway to all mystery." (Wú míng tiāndì zhī shǐ, yǒumíng wànwù zhī mǔ. Gù cháng wú yù yǐ guān qí miào, cháng yǒu yù yǐ guān qí jiǎo. Cǐ liǎng zhě tóng chū ér yì míng, tóng wèi zhī xuán, xuán zhī yòu xuán, zhòng miào zhī mén.)

"The gateway to all mystery" (zhòng miào zhī mén)—the Gate refers to the crucial point for accessing profound principles. Qimen Dunjia is named after "Gate" precisely because it captures the meaning of "pivot." On the human level, the Eight Gates represent the direction and manner of human action—open when needed, rest when needed, generate when needed, block when needed—this is the information of decision-making.

Section 4: The Origin of the Nine Stars

Qimen Dunjia has nine "Stars" (Xīng):

Heavenly Canopy Star (Tiānpéng), Heavenly Root Star (Tiānrúi), Heavenly Rush Star (Tiānchōng), Heavenly Assistant Star (Tiānfǔ), Heavenly Fowl Star (Tiānqín), Heavenly Heart Star (Tiānxīn), Heavenly Pillar Star (Tiānzhù), Heavenly任 Star (Tiānrèn), Heavenly Hero Star (Tiānyīng).

Each of the Nine Stars resides in one Palace:

• Tianpeng (Kan One Palace)
• Tianrui (Kun Two Palace)
• Tianchong (Zhen Three Palace)
• Tianfu (Xun Four Palace)
• Tianqin (Zhong Five Palace)
• Tianxin (Qian Six Palace)
• Tianzhu (Dui Seven Palace)
• Tianren (Gen Eight Palace)
• Tianying (Li Nine Palace)

The names of the Nine Stars are related to the Nine Stars of the Big Dipper. The Seven Stars of the Big Dipper plus the two hidden stars, Left Assistant and Right Auxiliary, combine to make Nine Stars. The Big Dipper held a supreme position in pre-Qin astronomical beliefs.

The Heguanzi, Circular Flow Chapter states:

"When the handle of the Dipper points East, the world enters Spring. When the handle points South, the world enters Summer. When the handle points West, the world enters Autumn. When the handle points North, the world enters Winter." (Dǒu bǐng dōng zhǐ, tiānxià jiē chūn. Dǒu bǐng nán zhǐ, tiānxià jiē xià. Dǒu bǐng xī zhǐ, tiānxià jiē qiū. Dǒu bǐng běi zhǐ, tiānxià jiē dōng.)

The Big Dipper is the pivot of Heaven. Taking the Nine Stars of the Big Dipper as elements of the "Heaven Plate" (Tiānpán) of Qimen Dunjia captures the meaning that the Big Dipper governs the movement of Heaven.

Information dimensions of the Nine Stars:

1. Auspicious/Inauspicious attribute.
2. Five Phase attribute.
3. Original Palace position.
4. Position after rotation.
5. Combinational relationship with Gates, Curiosities/Instruments, and Spirits.

Auspiciousness of the Nine Stars: Tianxin, Tianren, Tianqin, Tianfu are auspicious stars; Tianchong is neutral (or slightly auspicious); Tianpeng, Tianrui, Tianzhu, Tianying are inauspicious stars.

Section 5: The Origin of the Eight Spirits

Qimen Dunjia has eight "Spirits" (Shén):

Value Symbol (Zhífù), Coiled Serpent (Téngshé), Great Yin (Tàiyīn), Six Harmony (Liùhé), White Tiger (also called Gou Chen 勾陈), Black Tortoise (also called Zhu Que 朱雀), Nine Earths (Jiǔdì), Nine Heavens (Jiǔtiān).

The origin of the Eight Spirits is closely related to ancient astronomical beliefs and the system of "Auspicious and Inauspicious Markers" (Shénshā) derived from Yin and Yang and the Five Phases.

The Zuo Zhuan, Duke Xi Year 5 records the words of the diviner Yan:

"A child's rhyme says: 'On the day of Bing, the Dragon's Tail sets; the uniform vestments are grand; they seize the banner of Guo. The Partridge is blazing, the Heavenly Commander is powerful; fire forms an army in the middle; the Duke of Guo will flee.'"

Terms like "Dragon's Tail" and "Heavenly Commander" reflect the pre-Qin tradition of using astronomical constellations to predict human affairs. The Eight Spirits of Qimen Dunjia inherit this tradition, transforming the astronomical "Spirits" into the arcane "Spirits."

Information dimensions of the Eight Spirits:

1. Auspicious/Inauspicious attribute.
2. Yin/Yang attribute.
3. Intrinsic characteristics (Zhifu governs nobility, Tengshe governs startling changes, Taiyin governs concealment...).
4. Palace occupied.
5. Combinational relationship with Gates, Stars, Curiosities/Instruments.

Section 6: The Complete Chart Structure of Qimen Dunjia

A single chart in Qimen Dunjia is formed by the superposition of the following layers:

1. Earth Plate (Dìpán, Fixed) Nine Palaces $\times$ Fixed Curiosities/Instruments (The Earth Plate Curiosities/Instruments are determined by the Bureau number, but once set, they do not move).

2. Heaven Plate (Tiānpán, Rotating) Nine Stars + Heaven Plate Curiosities/Instruments (Rotate with the Star carried by the Value Symbol).

3. Man Plate (Rénpán, Rotating) Eight Gates (Rotate with the Gate carried by the Value Embodiment).

4. Spirit Plate (Shénpán, Rotating) Eight Spirits (Rotate with the Value Symbol).

Thus, in a single Qimen Dunjia chart, each palace simultaneously carries four layers of information:

• Earth Plate Curiosities/Instruments (Bottom layer)
• Heaven Plate Star + Heaven Plate Curiosities/Instruments (Second layer)
• Man Plate Gate (Third layer)
• Spirit Plate Spirit (Fourth layer)

Adding the inherent information of the palace itself (Trigram attribute, Five Phase attribute, etc.), each palace is effectively an "information pillar"—vertically superimposed with multiple encodings.

Nine Palaces $\times$ Four Layers of Information = The total information volume of one chart.

This preliminary structure already reveals the information complexity of Qimen Dunjia far exceeds that of ordinary arcane arts. A detailed mathematical comparison will be made in the following sections.

Section 7: Setting the Bureau Number: Eighteen Bureaus for Yin and Yang Escapes

The number of Bureaus in Qimen Dunjia is $18 + 18 = 36$ Bureaus (some argue for repetition; the number of independent bureaus is fewer).

Yang Bureau (Yáng Dùn): Nine Bureaus (from Winter Solstice onwards, Bureaus One to Nine). Yin Bureau (Yīn Dùn): Nine Bureaus (from Summer Solstice onwards, Bureaus Nine down to One).

The establishment of these 18 Bureaus is closely related to the 24 Solar Terms and the 72 Five-Day Periods (Hòu).

Each Solar Term (15 days) is divided into three Yuan periods (Upper, Middle, Lower Yuan), each Yuan lasting 5 days. $24$ Solar Terms $\times 3$ Yuans $= 72$ divisions, corresponding to the 72 Hòu.

Yang Bureau Nine $\times$ Upper/Middle/Lower Yuan $= 27$ Yuan divisions (though in practice, only a portion of these $3 \times 9 = 27$ five-day segments are used). Yin Bureau Nine $\times$ Upper/Middle/Lower Yuan $= 27$ Yuan divisions.

Totaling 54 Yuan divisions$22 The mathematical relationships here are complex and require careful tracing.

In practice, the method for setting the Qimen chart is: Starting from the Yang Bureau from the Winter Solstice, progressing sequentially according to the Yuan period and Solar Term. Starting from the Yin Bureau (Nine) from the Summer Solstice, progressing sequentially.

A new Yuan begins every five days, and the same Bureau is used within that Yuan—but the chart for every shí chén (double-hour) is different (because the Value Symbol rotates with the shí chén). Thus, the information variation in Qimen Dunjia is reflected not only in the "Bureau" level but also in the "Time" level.

Within one Bureau, the chart for each of the 12 shí chén forming a day is different. In fact, one five-day Yuan (60 shí chén) produces 60 different charts.

The mathematics here is crucial:

There are approximately 72 Yuan divisions in a year (5 days each), with 60 shí chén per Yuan. $72 \times 60 = 4,320$ different shí chén charts.

However, because there are 18 Bureaus (Yang Nine + Yin Nine), and each Bureau has variations depending on the shí chén, the actual number of unique charts needs a more precise calculation.

In the mathematical comparison section later, we will analyze this in detail.

Chapter 4: Traces of Stems and Branches and Dunjia Thought in Pre-Qin Classics

Section 1: Stems and Branches in Oracle Bone Inscriptions

Shang Dynasty oracle bone inscriptions (Jiǎgǔwén) are the earliest material evidence of Stems and Branches dating. Oracle bones contain numerous records of Stems and Branches dating days, such as:

"Gui Mao divination, Ke asks, is there misfortune during the ten-day period$23" (Guǐmǎo bǔ, Ké zhēn, xún wáng huò$24) "Ding You divination, Zheng asks, will it rain tomorrow, Wu Xu$25" (Dīngyǒu bǔ, Zhēng zhēn, yì Wùxū yǔ$26)

Such divinations clearly date days using the Sixty Jiazi system, proving that the Sixty Jiazi system was fully mature by the Shang Dynasty at the latest.

The forms of the Heavenly Stems and Earthly Branches in oracle bones:

Jia (Shield shape), Yi (Bent shape), Bing (Platform shape), Ding (Nail shape), Wu (Spear shape), Ji (Bow shape), Geng (Musical instrument shape), Xin (Torture device shape), Ren (工 shape), Gui (Variant of the spear shape).

Zi (Infant shape), Chou (Finger shape), Yin (Arrow shape), Mao (Door shape), Chen (Conch shape), Si (Snake shape), Wu (Pestle shape), Wei (Branch shape), Shen (Lightning shape), You (Wine vessel shape), Xu (Spear shape), Hai (Kernel shape).

The original meanings of these characters reveal that the Stems and Branches may have held concrete material significance in antiquity—rather than being purely abstract symbols as later understood. This implies that the original information content of the Stems and Branches was richer than later understood.

The concept of "Xun" (Ten-day period) also appears in oracle bones—ten days per xun, corresponding to the Ten Heavenly Stems. Divination inscriptions often asked "Is there misfortune during this xun$27"—this is prediction based on the cycle of the Heavenly Stems—a rudimentary form of Bazi thinking.

Section 2: Time and Space Information Encoding in the Book of Changes

The textual content and commentaries of the Book of Changes (Zhou Yi) contain the joint origin of Bazi and Qimen Dunjia thought.

Firstly, the hexagram sequence in the Book of Changes implies temporal information. The Xugua Zhuan states:

"Only after Heaven and Earth came into existence did the myriad things begin to grow. What fills the space between Heaven and Earth is only the myriad things, thus it is followed by Tun. Tun means proliferation; it is the beginning of the growth of things. Things that begin to grow must be nurtured, thus it is followed by Meng. Meng means covering; it is the immaturity of things. Immature things must be nourished, thus it is followed by Xu. Xu is the way of eating and drinking. Eating and drinking must lead to contention, thus it is followed by Song. Contention must lead to the gathering of a crowd, thus it is followed by Shi. Shi means a multitude. A multitude must have something to align with, thus it is followed by Bi. Bi means alignment. Alignment must involve accumulation, thus it is followed by Xiao Chu. Only after things are accumulated can there be rites, thus it is followed by Lü. ..."

The sequence of these sixty-four hexagrams is essentially a temporal narrative from the "Initial differentiation of Heaven and Earth" to the "Completion of all things"—this is similar to the approach of Bazi, which centers on temporal flow.

Secondly, the Shuogua Zhuan's directional assignments of the Eight Trigrams imply spatial information:

"The Emperor emerges from Zhen (East), flourishes in Xun (Southeast), meets in Li (South), commands service in Kun (Southwest), expresses pleasure in Dui (West), battles in Qian (Northwest), labors in Kan (North), and completes the discourse in Gen (Northeast)."

This is the "Later Heaven Bagua Orientation" (Wen Wang Eight Trigrams sequence)—Zhen East, Xun Southeast, Li South, Kun Southwest, Dui West, Qian Northwest, Kan North, Gen Northeast—which differs from the spatial assignments in the Former Heaven Trigrams.

The existence of two spatial reference systems, Former Heaven and Later Heaven, implies two "reference frames" for spatial encoding—this dual reference system provides the intellectual basis for the "rotating disk" mechanism of Qimen Dunjia. The Earth Plate uses the Later Heaven positions (fixed), while the Heaven Plate and Man Plate rotate upon this fixed framework—analogous to the interaction between Former and Later Heaven.

The Xici Zhuan states:

"The Way of Yi is not fixed. Its Way shifts repeatedly; movement does not cease; it circulates through the six positions; its top and bottom are never constant; hardness and softness interchange; it cannot be made into a fixed canon, only adapting to change." (Yì zhī wéi shū yě, bùkě yuǎn. Wéi dào yě lǚ qiān, biàndòng bù jū, zhōuliú liùxū, shàngxià wú cháng, gāngróu xiāngyì, bùkě wéi diǎnyào, wéi biàn suǒ shì.)

This passage vividly reveals the essential characteristic of the Book of Changes—"Change" (Biàn). Unfixed, unconstant, interchanging, adapting only to change—this is the dynamic nature of information. The "Great Cycles" (Dàyùn) and "Flowing Years" (Liúnián) in Bazi reflect "change," just as the "rotating plates" and "flying layouts" in Qimen Dunjia reflect "change." The richness of this "change" directly determines the size of the information capacity.

Section 3: The Book of Documents, Grand Plan and the Five Phases in Mathematical Logic

The Shangshu: Hongfan (Grand Plan) is a classic document on Five Phase theory. The "Nine Categories" (Jiǔ Chóu) presented by Jizi to King Wu have profound parallels with the structure of the Nine Palaces in Qimen Dunjia.

The Hongfan states:

"First, the Five Phases; second, respectfully using the Five Matters; third, using the Eight Political Tasks for agriculture; fourth, harmonizing with the Five Temporal Records; fifth, establishing the Grand Ultimate (Huángjí); sixth, governing with the Three Virtues; seventh, clarifying with Divination to Resolve Doubt; eighth, considering all responses; ninth, aspiring to the Five Blessings and using the Six Extremes."

The Nine Categories use the "Grand Ultimate" (the Fifth Category) in the center, just as the number Five resides in the center of the Luo Writing. This structure of the "Nine Categories" shares an uncanny similarity with the Nine Palaces of Qimen Dunjia—both use Nine as a framework, with the center as the pivot.

It is particularly noteworthy that the Seventh Category, "Clarifying with Divination to Resolve Doubt" (Qímì Jié Yí):

"Select and establish diviners, then order the tortoise and yarrow for divination. They say: Rain, Clear weather, Obscurity, Delay, Success, Affirmation, Regret—seven in all. Five are from tortoise divination, two are used from yarrow, for extension and reversal. Establish people to perform divination; three persons interpret, then follow the words of two."

This details the methods of divination—tortoise divination has five signs (Rain, Clear weather, Obscurity, Delay, Success), and yarrow divination yields two results (Affirmation, Regret), plus "extension and reversal" (extension of variation). This is a description of the information system of pre-Qin divination.

"Three persons interpret, then follow the words of two"—this is the principle of "majority rule," a classical expression of information redundancy coding and error correction mechanisms. Why require three diviners$1 Because the judgment of a single person might contain error ("noise"); following the majority of three reduces the probability of misjudgment—this is completely consistent with modern principles of error-correcting codes in information theory.

Section 4: Examples of Divination in the Zuo Zhuan and Guoyu

Pre-Qin classics preserve numerous examples of divination, which provide valuable material for understanding the information processing methods of the pre-Qin era.

Example 1: Divination of Marquis Li of Chen in the 22nd Year of Duke Zhuang (Zuo Zhuan)

"The Marquis of Chen sent someone to divine. They obtained Guan (Observing) transforming into Pi (Obstruction). The diviner said: 'This is called "Observing the light of a state, which is beneficial for being a guest to the King." Will he replace Chen in ruling$2 No, it will be in another state. If not in his own person, it will be in his descendants. The light is far away and shines from elsewhere. Kun is Earth. Xun is Wind. Qian is Heaven. Wind acts as Heaven over Earth, which is Mountain. If one has the material of a mountain illuminated by heavenly light, he then resides on the earth, thus it is said, "Observing the light of a state, beneficial for being a guest to the King." The courtyard is filled with a hundred sacrifices, presented with jade and silk, the beauty of Heaven and Earth is complete, thus it is said, "Beneficial for being a guest to the King." There is still observation, thus it is said, perhaps in the future. Wind moves and leaves its mark on the earth, thus it is said, perhaps in another state. If in another state, it must be of the Jiang surname. Jiang is descended from the Great Yue. Mountains and peaks are matched with Heaven; nothing can be truly great twice. When Chen declines, will this be its prosperity$3'"

This record is extremely precious. The divination obtained Hexagram Guan transforming into Pi—the original hexagram is Guan (Wind over Earth), the transforming hexagram is Pi (Heaven over Earth), meaning the fourth line changed from Yin to Yang.

The diviner's interpretation process:

1. Based on the line text, "Observing the light of a state, beneficial for being a guest to the King."
2. Then analyzing the hexagram images: Kun for Earth, Xun for Wind, Qian for Heaven.
3. Further inferring: Wind acting as Heaven over Earth suggests the image of a Mountain $\rightarrow$ the image of a State.
4. Then relating to "the light is far away and shines from elsewhere" $\rightarrow$ in another state.
5. Then deducing the Jiang surname based on the Five Phases and directions.

In this divination process, the extraction of information is multi-layered and multi-dimensional—it involves textual information from the line texts, symbolic information from the hexagram images, and inferential information from the Five Phases and directions. This is the pre-Qin mode of information processing in divination—far more complex than later imagined.

Example 2: The Battle of Qin and Jin at Han (Zuo Zhuan, Duke Xi Year 15)

"Initially, Duke Xian of Jin divined about marrying Bo Ji to Qin; he obtained Gui Mei (Slight Mismatch) transforming into Kui (Strife). Shi Su interpreted it: 'Inauspicious. The changing line says: 'The official cuts the sheep, yet there is no remnant. The woman carries the basket, yet there is no abundance. The western neighbor blames and speaks, which cannot be repaid.' Gui Mei transforming into Kui, still lacking mutual support.' ..." "...When Duke Hui was in Qin, he said: 'If the former lord had followed Shi Su's divination, I would not have suffered this fate.'"

In this record, Shi Su's divination was not followed, and later the negative outcome was realized. This shows that the divination information in pre-Qin times possessed predictive power—but this predictability depended on the diviner's correct interpretation of the information.

Example 3: The Discussion of Musical Pitch by Ling Zhoujiu (Guoyu, Zhou Yu)

"Ling Zhoujiu said: 'Governance is like music; music follows harmony; harmony follows equilibrium. Sounds are used to create music, pitch (lǜ) is used to ensure equilibrium. Metal and stone are used to move it; silk and bamboo are used to carry it. Poetry is used to convey it; singing is used to chant it; the gourd is used to express it; ceramics are used to supplement it; leather and wood are used to regulate it. When things attain their norm, it is called the ultimate of music (yuè jí); where the ultimate gathers is called sound (shēng); sound responding to each other in protection is called harmony (hé); small and large not exceeding bounds is called equilibrium (píng).'"

Although this passage does not directly discuss arcane arts, it reveals the pre-Qin concept of "Pitch and Calendar Unity"—musical pitch is connected to the calendar. This idea is crucial for the mathematical foundation of Qimen Dunjia.

The Twelve Musical Pitches (Lǜ) correspond to the Twelve Earthly Branches:

• Huang Zhong $\leftrightarrow$ Zi
• Da Lü $\leftrightarrow$ Chou
• Tai Cu $\leftrightarrow$ Yin
• Jia Zhong $\leftrightarrow$ Mao
• Gu Xi $\leftrightarrow$ Chen
• Zhong Lü $\leftrightarrow$ Si
• Rui Bin $\leftrightarrow$ Wu
• Lin Zhong $\leftrightarrow$ Wei
• Yi Ze $\leftrightarrow$ Shen
• Nan Lü $\leftrightarrow$ You
• Wu She $\leftrightarrow$ Xu
• Ying Zhong $\leftrightarrow$ Hai

This correspondence means that the Earthly Branches not only carry temporal and spatial information but also auditory (frequency) information—this greatly expands the information dimensions of the Stems and Branches system.

Section 5: The Five Phases and Chronology in Guanzi

While the compilation date of the Guanzi is debated, many chapters reflect pre-Qin Five Phase thought and arcane concepts.

The Guanzi, Five Phases Chapter states:

"At the Summer Solstice, one observes Jia Zi, and the Wood Phase governs... Seventy-two days complete this. Observe Bing Zi, and the Fire Phase governs... Seventy-two days complete this. Observe Wu Zi, and the Earth Phase governs... Seventy-two days complete this. Observe Geng Zi, and the Metal Phase governs... Seventy-two days complete this. Observe Ren Zi, and the Water Phase governs... Seventy-two days complete this."

This passage divides the 360 days of the year into five segments (72 days each), governed by one Phase in each, marked by the starting Stems and Branches—this is a method combining timekeeping by Stems and Branches with the assignment of the Five Phases to time—a precursor to the concept of Five Phases governing the Month Command in Bazi.

Furthermore, the Guanzi, Four Seasons Chapter states:

"Therefore, Yin and Yang are the great principles of Heaven and Earth. The Four Seasons are the great framework of Yin and Yang. Punishment and Virtue (Xíng Dé) are the conjunction of the Four Seasons. When Punishment and Virtue align with the seasons, blessings arise; when they deviate, misfortune arises." (Shì gù yīnyáng zhě, tiāndì zhī dàlǐ yě. Sìshí zhě, yīnyáng zhī dàjīng yě. Xíngdé zhě, sìshí zhī hé yě. Xíngdé hé yú shí zé shēng fú, guǐ zé shēng huò.)

The chain of inference "Yin/Yang $\rightarrow$ Four Seasons $\rightarrow$ Punishment and Virtue" clearly shows the path of pre-Qin arcane arts: starting from the most fundamental Yin/Yang duality, moving through the framework of the Four Seasons, to arrive at judgments of human fortune and misfortune. This logic is entirely inherited by Bazi and Qimen Dunjia.

The Guanzi, Minor Officials Chapter offers a more detailed description of seasonal arcane calculations:

"Spring acts with the politics of Winter—austere; acts with the politics of Autumn—thunder; acts with the politics of Summer—castration i.e., severe restriction. After twelve days, the Earth Qi erupts, cautioning Spring affairs. After twelve days, the minor Mao early spring, plowing begins. After twelve days, Heavenly Qi descends, giving rewards. After twelve days, Righteous Qi arrives, repairing gates and doors. After twelve days, Purity and Brightness, prohibitions are lifted. After twelve days, the main Mao late spring, mating begins."

This subdivides the affairs of Spring into periods of "twelve days," with a change every twelve days—similar to the Qimen Dunjia concept of a "Yuan" (five days per Yuan), involving fine temporal segmentation.

Section 6: The Lunar Months in Lüshi Chunqiu and the System of Mathematical Principles

The Twelve Records (Shí'èr Jì) in the Lüshi Chunqiu represent the culmination of pre-Qin lunar month theory.

The Record of Early Spring states:

"In the month of Early Spring, the Sun is in the Camp House (Yíngshì), at dusk it is in the Heart (Shēn), at dawn it is in the Tail (Wěi). Its Stems are Jia and Yi; its Emperor is Taihao; its Spirit is Gou Mang; its creatures are scaled; its tone is Jue; its pitch is Tai Cu; its number is Eight; its taste is sour; its odor is rank; its sacrifice is the Door; sacrifice to the Spleen first. The East Wind thaws the frost, hibernating insects begin to stir, fish rise under the ice, the otter offers fish to the heron, the migrating geese fly north."

This passage contains extremely rich information: the attributes of just one month include:

1. Celestial observation (Sun in Yingshi, dusk in Shen, dawn in Wei).
2. Heavenly Stems (Jia and Yi—Wood).
3. Heavenly Emperor (Taihao—Emperor of the East).
4. Heavenly Spirit (Gou Mang—Spirit of Wood).
5. Animal category (Scaled creatures).
6. Musical Tone (Jue tone).
7. Pitch Name (Tai Cu).
8. Number (Eight).
9. Taste (Sour).
10. Odor (Rank).
11. Sacrifice (Sacrifice to the Door, sacrifice to the Spleen first).
12. Phenology (East wind thawing, etc., five observations).

One month has twelve or more dimensions of information! The Twelve Records detail the attributes for all twelve months, forming an extremely vast "Annual Information Matrix."

This "Lunar Month" system is precisely the shared knowledge basis for the Month Pillar information in Bazi and the Solar Term initiation of Qimen Dunjia charts.

Especially noteworthy is "Its Number" (qí shù)—the number for Early Spring is Eight, for Mid-Spring is Eight, for Late Spring is Eight (Number of Wood); for Early Summer is Seven, Mid-Summer is Seven, Late Summer is Seven (Number of Fire)... This numerical system corresponds to both the generating numbers of the River Chart and the positional numbers of the Luo Writing.

Section 7: "Number" and "Juncture" in the Zhuangzi

Although the Zhuangzi is famous for Daoist philosophy, it contains profound reflections on "Number" (shù) and "Juncture" (jī).

The Zhuangzi, Transcending Things Chapter (Tianxia) states:

"Hui Shi had many methods, his writings filled five carts. His Dao was discordant, his words did not hit the mark. His intent in charting things was: 'The greatest without outside is called Great One (Dàyī). The smallest without inside is called Small One (Xiǎoyī). Without thickness, it cannot be accumulated, yet it can span a thousand li. Heaven and Earth are low, mountains and marshes are level. The sun at its zenith is already declining; things in the process of being born are already in the process of dying. Great Equality differs from Small Equality; this is called Small Difference. All things are equally the same and equally different; this is called Great Difference.' ..."

Hui Shi's "Ten Points of Charting Things" involve concepts of limits, infinity, and relativity—numerical philosophy. If information capacity can tend toward "the greatest without outside," then the information capacity of any arcane system has the potential for infinite expansion—the problem is how much can be utilized in actual practice.

The Zhuangzi, Discussion on Making Things Equal (Qíwù Lùn) states:

"Heaven and Earth were born together with me, and the myriad things are one with me. If they are already one, can words still exist$4 If one has already spoken of 'One,' can one refrain from speaking$5 One and words become two; two and one become three. From here onward, the skilled calculator cannot arrive, let alone the ordinary person! Thus, from nothing adapting to something, one reaches three, how much more so from something adapting to something$6 There is no adaptation, thus one rests here." (Tiāndì yǔ wǒ bìngshēng, ér wànwù yǔ wǒ wéi yī. Jì yǐ wéi yī yǐ, qiě dé yǒu yán hū$7 Jì yǐ wéi yī yǐ, qiě dé wú yán hū$8 Yī yǔ yán wéi èr, èr yǔ yī wéi sān. Zì cǐ yǐ wǎng, qiǎolì bùnéng dé, ér kuàng qí fán hū$9 Gù zì wú shì yǒu, yǐ zhì yú sān, ér kuàng zì yǒu shì yǒu hū$10 Wú shì, yīn shì yǐ.)

"One and words become two; two and one become three"—this is the classical expression of the self-referential paradox. "From here onward, the skilled calculator (qiǎolì) cannot arrive"—Qiǎolì, meaning those skilled in chronology and mathematics, cannot exhaust this recursive count, let alone the ordinary person. This reveals a deep information theory proposition: The self-referential nature of information leads to an infinite increase in information capacity.

When we discuss the information capacity of Bazi or Qimen Dunjia, are we not facing a similar dilemma$11 The interpretation of a Bazi chart can generate new information, and that new information can be further interpreted... This recursive process is theoretically endless. Furthermore, the multi-layered superposition of Qimen Dunjia, where each layer interacts with others to generate new information—is its depth of recursion greater than that of Bazi$12 This is a question requiring deeper exploration later.

The Zhuangzi, Heaven and Earth Chapter also states:

"In the primal beginning there was nothingness, and nothingness was without name. The stirring of the One brought forth form, but it was not yet shaped. Things obtained their generation, called Virtue (Dé). That which was unshaped had divisions, yet still without separation; this is called Fate (Mìng). When stillness moves and things are born, things achieve their inherent principle, called Form (Xíng). Form preserves Spirit, and each has its standard (yí zé); this is called Nature (Xìng)."

The appearance of the word "Fate" (Mìng) is crucial—"That which was unshaped had divisions, yet still without separation, this is called Fate." Even before things take shape, they already possess inherent differentiation—this is the original meaning of "Fate." What Bazi seeks to divine is precisely this "Fate"—the innate pattern predetermined at birth.

And "each has its standard (yí zé)"—the "standard" or "rule" (yí zé)—this is the "principle" (lǐ) upon which arcane arts rely. The multi-layered structure of Qimen Dunjia is intended to capture the multi-dimensional expression of these "standards."

Section 8: Pre-Qin Military Treatises and Dunjia Thought

The tradition of Qimen Dunjia has always been closely linked to military strategy. Pre-Qin military texts contain rich elements of Dunjia thought.

The Art of War, Laying Plans Chapter (Shǐjì) states:

"Sunzi said: Warfare is a matter of supreme importance to the State; the way of life or death; the road to survival or ruin; it is mandatory that it be studied. Hence the five fundamental factors must be compared when making plans: First, the Way (Dào); second, Heaven (Tiān); third, Earth (Dì); fourth, the Commander (Jiāng); and fifth, Method (Fă)." (Sūnzǐ yuē: Bīng zhě, guó zhī dàshì, shēngsǐ zhī dì, cúnwáng zhī dào, bùkě bù chá yě. Gù jīng zhī yǐ wǔ shì, jiào zhī yǐ jì ér suǒ qí qíng: yī yuē dào, èr yuē tiān, sān yuē dì, sì yuē jiāng, wǔ yuē fǎ.)

This framework of the "Five Factors"—Dao, Heaven, Earth, Commander, Method—corresponds to the multi-layered structure of Qimen Dunjia:

• "Heaven" corresponds to the Heaven Plate (Nine Stars + Heaven Plate Curiosities/Instruments).
• "Earth" corresponds to the Earth Plate (Nine Palaces + Earth Plate Curiosities/Instruments).
• "Commander" corresponds to the Man Plate (Eight Gates).
• "Dao" and "Method" correspond to the overall rules of application.

Sunzi also says:

"Heaven refers to Yin and Yang, cold and heat, and the constraints of the seasons. Earth refers to distance and proximity, hazard and ease, breadth and narrowness, life and death." (Tiān zhě, yīnyáng, hánshǔ, shí zhì yě. Dì zhě, yuǎnjìn, xiǎnyì, guǎngxiǎo, shēngsǐ yě.)

The information of "Heaven" includes Yin/Yang, cold/heat, and temporal constraints—this is the information in the temporal dimension; the information of "Earth" includes distance/proximity, hazard/ease, breadth/narrowness, life/death—this is the information in the spatial dimension. Qimen Dunjia takes the Heaven Plate as primarily temporal and the Earth Plate as primarily spatial, which coincides exactly with Sunzi's classification of "Heaven" and "Earth."

And the words "life and death" (shēngsǐ) are crucial—Qimen Dunjia has the "Life Gate" and the "Death Gate," and the terminology used directly stems from the military tradition.

The Art of War, Maneuvering for Position Chapter states:

"Attack where the enemy is unprepared to defend; advance where they do not expect. To march a thousand li without exhaustion, march where there are no men. To attack and be certain of success, attack where they do not defend. To defend and be certain of security, defend where they do not attack. Thus, the skillful attacker conceals his point of defense from the enemy; the skillful defender conceals his point of attack from the enemy. Subtlety, subtlety, to the point of invisibility (wú xíng); mystery, mystery, to the point of inaudibility (wú shēng), thus one can become the disposer of the enemy's fate."

"Subtlety, subtlety, to the point of invisibility (wú xíng)"—this is the meaning of "Dùn" (Escape/Concealment). Jia hides beneath the Six Instruments, not revealing its form, making it impossible for the enemy to grasp—this is perfectly consistent with the concept of "concealment of form" in military strategy. And "mystery, mystery, to the point of inaudibility (wú shēng)"—this is the concealment and encryption of information. "Dùn" in Qimen Dunjia, from an information science perspective, is an information encryption mechanism—the superficially presented information (the Heavenly Stems shown by the Six Instruments) differs from the deeply hidden information (the identity of the Six Jia)—decoding the deep information requires "decryption" (knowing which Jia hides under which Instrument).

This information encryption mechanism makes the information hierarchy of Qimen Dunjia richer—it has not only "explicit information" (what is superficially visible) but also "implicit information" (what is hidden beneath the surface). Bazi is relatively "transparent"—the Stems and Branches are directly presented, although there is the concept of Hidden Stems, it is not as clearly layered as the "hiding" of Dunjia.

The Nine Grounds Chapter of Sunzi states:

"Dispersive ground, light ground, contention ground, intersecting ground, open ground, heavy ground, obstructing ground, encircled ground, death ground."

These nine types of terrain can also be mapped analogously to the Nine Palaces of Qimen Dunjia—different palaces present different "terrain" characteristics in different Bureaus, with pros, cons, life, and death.

The Six Strategies (attributed to Lü Wang, though its dating is debated, its military thought inherits pre-Qin tradition) also contains extensive application of celestial timing and terrestrial advantage:

"King Wu asked Grand Tutor: 'Leading troops deep into the territory of the feudal lords, facing the enemy army head-on. The two formations face each other, the strength of the multitudes is equal. I do not dare move first. I wish to make the enemy general fearful, and the soldiers confused, so that if they wish to fight they dare not, if they wish to defend they cannot, their fronts and rears separated, their left and right lost—is this possible$13' Grand Tutor replied: 'It is possible.'"

Although this passage does not explicitly mention Dunjia, the military effect described—"making the enemy general fearful and the soldiers confused"—is the ideal application scenario for Qimen Dunjia in military affairs.

Section 9: The Legend of the Yellow Emperor vs. Chiyou and the Origin of Dunjia

The most famous legend concerning the origin of Qimen Dunjia is the story of the Yellow Emperor fighting Chiyou.

The Records of the Five Emperors in the Records of the Grand Historian (Shiji) records:

"During the time of Xuanyuan (Yellow Emperor), the era of Shennong declined. Feudal lords attacked each other, oppressing the common people, and Shennong could not subdue them. Thus, Xuanyuan practiced the use of weapons and mobilized troops to conquer those who did not submit, and all the feudal lords came to assist him. Chiyou was the most violent, and no one could defeat him. The Flame Emperor wished to invade the feudal lords, and all the lords defected to Xuanyuan. Xuanyuan then cultivated virtue and mobilized troops, managed the Five Qi, cultivated the Five Grains, comforted the people, measured the Four Directions, taught the Bear, Grizzly, Panther, Leopard, Tiger, to fight the Flame Emperor at the field of Banquan. After three battles he achieved his aim. Chiyou rebelled and ignored the Emperor's commands. Thus, the Yellow Emperor mustered the feudal lords and fought Chiyou at the field of Zhuolu, finally capturing and killing Chiyou."

While Dunjia is not mentioned here, "cultivated virtue and mobilized troops, managed the Five Qi (wǔ qì), cultivated the Five Grains, comforted the people, measured the Four Directions (dù sì fāng)"—"managing the Qi of the Five Phases" and "measuring the Four Directions" (spatial orientation)—already contain the basic elements of Dunjia.

Legend has it that the Yellow Emperor was trapped by Chiyou's great fog, and the Heavenly Emperor sent the Nine Heavens Mysterious Lady to teach him the Dragon Jia Divine Chapter (also called the Dunjia Heavenly Book). Based on this, the Yellow Emperor invented the South-Pointing Chariot to break the fog and ultimately defeated Chiyou. Although this legend is mythological, it reflects an important concept: the art of Dunjia is closely related to "breaking confusion" (making correct decisions under conditions of incomplete information)—this is the classical expression of "signal processing."

Chiyou's "great fog" is, from an information theory perspective, "information interference" or "obscured vision"—i.e., a noisy environment. The function of the Dunjia art is to extract useful information in a noisy environment and make correct judgments—this is the classical expression of "signal processing."

The Yellow Emperor's legend also includes the achievement of "regulating the calendar and clarifying time":

"Obtained the treasure tripod, pushed the stalks according to the sun's arrival." (Huò bǎodǐng, yíngrì tuī cè.) (Shiji: Wǔdì Běnjì)

"Pushing the stalks according to the sun's arrival" means establishing the calendar. The calendar is the encoding system for temporal information. The dual roles of the Yellow Emperor in "regulating the calendar" and "employing troops" correspond precisely to the division between Bazi (application of the calendar) and Qimen Dunjia (application of military strategy)—both arts originate from the creations of the ancient Sage-Kings.

Part Two: Mathematical Analysis

Chapter 5: Detailed Examination of the Mathematical Structure of Bazi

Section 1: Combinatorial Mathematics of the Four Pillars of Bazi

In the previous section, we roughly calculated the theoretical combination number of Bazi to be about 518,400. We will now analyze this more precisely.

Year Pillar:

Since the Sixty Jiazi cycle repeats, the Year Pillar has 60 possible combinations, each appearing once in a 60-year cycle.

Month Pillar:

The Heavenly Stem of the Month Pillar is determined by the Heavenly Stem of the Year Pillar (Method of the Five Tigers Hiding the Month):

• Jia or Ji Year starts with Bing Yin month.
• Yi or Geng Year starts with Wu Yin month.
• Bing or Xin Year starts with Geng Yin month.
• Ding or Ren Year starts with Ren Yin month.
• Wu or Gui Year starts with Jia Yin month.

There are twelve months per year; the Earthly Branch of the Month is fixed sequentially (Yin, Mao, Chen, etc.). The Heavenly Stem of the Month is determined by the starting Stem of the Year Stem and follows sequentially.

Thus, for any given Year Pillar, there are 12 possibilities for the Month Pillar (corresponding to the twelve months). Because different Year Stems correspond to different starting Month Stems, five groups of Year Stems (Jia-Ji, Yi-Geng, Bing-Xin, Ding-Ren, Wu-Gui) $\times$ 12 months = 60 types of Month Pillars—coincidentally also sixty combinations of the Jiazi cycle.

Year Pillar $\times$ Month Pillar: $60 \times 12 = 720$ Year-Month combinations.

However, a structural constraint exists: once the Heavenly Stem of the Year is given, the Heavenly Stem of the Month is determined. Therefore, the independent information of the Month Pillar only lies in the choice of the Month Branch (12 possibilities), i.e., $\log_2(12) \approx 3.58$ bits of independent information.

Day Pillar:

The Sixty Jiazi cycle of the Day Pillar is independent of the Year and Month—on any day of any month, the Day Pillar can be any of the Sixty Jiazi. Thus, the Day Pillar has 60 possibilities, completely independent of the Year/Month Pillars.

Information capacity of the Day Pillar: $\log_2(60) \approx 5.91$ bits.

Hour Pillar:

The Heavenly Stem of the Hour Pillar is determined by the Heavenly Stem of the Day Pillar (Method of the Five Rats Hiding the Hour), similar to the Month Pillar:

• Jia or Ji Day starts with Jia Zi hour.
• Yi or Geng Day starts with Bing Zi hour.
• Bing or Xin Day starts with Wu Zi hour.
• Ding or Ren Day starts with Geng Zi hour.
• Wu or Gui Day starts with Ren Zi hour.

There are twelve shí chén (double-hours) per day; the Earthly Branch of the Hour is fixed sequentially (Zi, Chou, Yin...). The Heavenly Stem of the Hour is determined by the starting Stem of the Day Stem and follows sequentially.

Thus, for every Day Pillar, there are 12 possibilities for the Hour Pillar.

Day Pillar $\times$ Hour Pillar: $60 \times 12 = 720$ Day-Hour combinations.

Similarly, the independent information of the Hour Pillar only lies in the Hour Branch (12 possibilities), i.e., $\log_2(12) \approx 3.58$ bits.

First Layer of Total Independent Information of Bazi:

= Year Pillar Information + Independent Month Pillar Information + Day Pillar Information + Independent Hour Pillar Information = $\log_2(60) + \log_2(12) + \log_2(60) + \log_2(12)$ = $5.91 + 3.58 + 5.91 + 3.58$ $\approx 18.98$ bits

This is the "first layer" static combination information capacity of Bazi—about 19 bits.

Section 2: Relational Information of Bazi

The information in Bazi extends far beyond static combinations. A complex network of relationships exists among the Four Pillars:

1. Relationships among Heavenly Stems:

The four Heavenly Stems can have the following relationships with each other:

• Five Harmonies (Wǔ Hé): Jia-Ji, Yi-Geng, Bing-Xin, Ding-Ren, Wu-Gui.
• Overcoming (Kè): Jia/Yi Wood overcomes Wu/Ji Earth, Bing/Ding Fire overcomes Geng/Xin Metal...
• Generation (Shēng): Water generates Wood, Wood generates Fire, Fire generates Earth, Earth generates Metal, Metal generates Water.

There are $C(4, 2) = 6$ pairs of relationships among the four Stems. Each pair can be: Harmony, Generation, Overcoming, Being Generated, Being Overcome, or Comparison/Similarity (Bi He)—a total of 6 basic types.

6 pairs $\times$ approximately 6 types per pair = 36 possible relational combinations.

In reality, Stem relationships are more complex—harmony leading to transformation, harmony without transformation, overcoming within generation, generation within overcoming—these subtleties further increase the relational information.

2. Relationships among Earthly Branches:

The relationships among the four Earthly Branches are particularly complex:

• Six Harmonies (Liù Hé): Zi-Chou, Yin-Hai, Mao-Xu, Chen-You, Si-Shen, Wu-Wei.
• Three Combinations (Sān Hé): Shen-Zi-Chen combine to form Water Bureau; Hai-Mao-Wei combine to form Wood Bureau; Yin-Wu-Xu combine to form Fire Bureau; Si-You-Chou combine to form Metal Bureau.
• Three Meetings (Sān Huì): Yin-Mao-Chen meet as Wood; Si-Wu-Wei meet as Fire; Shen-You-Xu meet as Metal; Hai-Zi-Chou meet as Water.
• Six Clashes (Liù Chōng): Zi-Wu, Chou-Wei, Yin-Shen, Mao-You, Chen-Xu, Si-Hai.
• Three Punishments (Sān Xíng): Yin punishes Si, Si punishes Shen, Shen punishes Yin (Unkind Punishment); Chou punishes Xu, Xu punishes Wei, Wei punishes Chou (Presumptuous Punishment); Zi punishes Mao, Mao punishes Zi (Shameless Punishment); Chen-Chen, Wu-Wu, You-You, Hai-Hai (Self-Punishment).
• Six Harms (Liù Hài): Zi-Wei harm, Chou-Wu harm, Yin-Si harm, Mao-Chen harm, Shen-Hai harm, You-Xu harm.
• Breach (Pò): Zi-You breach, Chou-Chen breach, Yin-Hai breach, Mao-Wu breach, Si-Shen breach, Wei-Xu breach.
• Three Meetings (Sān Huì): Yin-Mao-Chen meet as Wood; Si-Wu-Wei meet as Fire; Shen-You-Xu meet as Metal; Hai-Zi-Chou meet as Water.

There are 6 pairs of relationships among the four Earthly Branches. Each pair may have multiple types of relationships listed above, and multiple relationships can coexist (e.g., simultaneously clashing and punishing). Furthermore, there are multi-party relationships among three or four Earthly Branches, such as Three Combinations and Three Punishments.

This relational network complexity far exceeds that of the Heavenly Stems. A preliminary estimate suggests that the relational information among the Branches alone could reach tens of bits.

3. Relationships between Heavenly Stems and Earthly Branches:

• Stem Seated on Branch: Each Heavenly Stem sits on the Earthly Branch below it, and the two have a generation or overcoming relationship.
• Stem Rooting: A Heavenly Stem finds its "root" within the Hidden Stems of the Earthly Branches (e.g., Jia Wood has a root in Yin).
• Capping and Intercepting: When a Stem overcomes a Branch, it is called Capping; when a Branch overcomes a Stem, it is called Intercepting.

Each of the four Pillars has its own Stem-Branch relationship, totaling 4 groups. Moreover, Stems and Branches across different pillars also have relationships.

4. The Ten Gods System (Shí Shén):

Taking the Day Master (Day Stem) as "Self" (Wǒ), the remaining seven characters determine the Ten Gods:

• Companion (Bǐjiān): Same Five Phase, same Yin/Yang as Self.
• Rob Wealth (Jiécái): Same Five Phase, different Yin/Yang from Self.
• Output God (Shíshén): Same Five Phase, same Yin/Yang as what Self generates.
• Hurting Officer (Shāngguān): Same Five Phase, different Yin/Yang as what Self generates.
• Indirect Wealth (Piāncái): Same Five Phase, same Yin/Yang as what Self overcomes.
• Direct Wealth (Zhèngcái): Same Five Phase, different Yin/Yang as what Self overcomes.
• Seven Killings (Qīshā): Same Five Phase, same Yin/Yang as what overcomes Self.
• Direct Officer (Zhèngguān): Same Five Phase, different Yin/Yang as what overcomes Self.
• Indirect Resource (Piānyìn): Same Five Phase, same Yin/Yang as what generates Self.
• Direct Resource (Zhèngyìn): Same Five Phase, different Yin/Yang as what generates Self.

The Ten Gods system transforms the Five Phase generation/overcoming relationships in Bazi into interpersonal relationships—this is the most core mechanism for information transformation in Bazi. The ten types of Gods each symbolize different social/humanistic symbols:

• Direct Officer: Reputation, status, constraint.
• Seven Killings: Authority, pressure, danger.
• Direct Resource: Mother, learning, protection.
• Indirect Resource: Esoteric learning, adoptive mother, isolation.
• Output God: Talent, enjoyment of food, ease.
• Hurting Officer: Outward display of talent, rebellion, verbal disputes.
• Direct Wealth: Legitimate income, wife, material goods.
• Indirect Wealth: Windfall, father, relations with the opposite sex.
• Companion: Siblings, competition, self-reliance.
• Rob Wealth: Seizing, cooperation, loss.

The Self is not assigned a God, so there are 7 God positions. Each position can take one of 10 Gods. Thus, there are 7 Ten God positions.

The information capacity of the Ten Gods system is approximately $7 \times \log_2(10) \approx 7 \times 3.32 \approx 23.24$ bits.

However, relationships exist among the Ten Gods—such as Officer and Killings mixed, Output/Hurting combined to generate Wealth, Killings generating Resource, etc.—these pattern relationships further increase the total information amount.

Section 3: Dynamic Information of Bazi—Great Cycles and Flowing Years

The information in Bazi is not only contained in the static Four Pillars at birth (static), but also in the subsequent Great Cycles (Dàyùn) and Flowing Years (Liúnián) (dynamic).

Great Cycles (Decades):

Great Cycles begin from the starting point of the Month Pillar, moving forward for Yang Males/Females and backward for Yin Males/Females, one step every ten years. Each step is a Stem-Branch combination, containing information on both the Heavenly Stem and Earthly Branch. A person typically experiences 8 to 10 Great Cycles (about 80-100 years), meaning a dynamic information trajectory spanning 80-100 years.

The information capacity of each Great Cycle step is about $\log_2(60) \approx 5.91$ bits. The total information capacity of 10 steps is about $59.1$ bits—but since adjacent Great Cycles have fixed intervals, the independent information capacity is smaller.

More importantly, the information in the Great Cycle lies in its interaction with the natal chart—the Harmony or Overcoming between the Great Cycle Stem and the natal Stems, the Clash, Harmony, or Punishment between the Great Cycle Branch and the natal Branches—these interactions generate a vast amount of "conditional information."

Flowing Years:

Each year is a Flowing Year Stem-Branch, cycling every 60 years. The triple interaction among the Flowing Year, Natal Chart, and Great Cycle generates extremely rich annual information.

Flowing Months, Flowing Days, Flowing Hours:

Further refinement shows that within a Flowing Year are Flowing Months (12 months), within Flowing Months are Flowing Days (about 30 days), and within Flowing Days are Flowing Hours (12 shí chén)—this recursive refinement can push temporal precision to the two-hour level.

Dynamic Information Estimation:

A lifetime of about 80 years, totaling $80 \times 12 \times 30 \times 12 \approx 345,600$ double-hours. The interaction of the Flowing Year, Flowing Month, Flowing Day, and Flowing Hour with the natal chart at each moment generates new information.

Of course, this dynamic information is not entirely independent—there is a great deal of redundancy and repetition. But the total amount of information is still substantial.

A preliminary estimate suggests that the "dynamic information capacity" of Bazi (including interactions from Great Cycles, Flowing Years, Flowing Months, Flowing Days) could reach several hundred bits.

Section 4: Estimation of Total Bazi Information Capacity

Summarizing the analysis above:

Taking the median value, the total information capacity of Bazi is about 250-300 bits.

Although this estimate is rough, it provides a reference order of magnitude. About 300 bits of information capacity implies that the Bazi system can distinguish about $2^{300} \approx 10^{90}$ different states—a number far exceeding the total number of atoms in the universe (about $10^{80}$).

Of course, in reality, the effective information capacity of Bazi is limited by its structural constraints—not all 300 bits are independent. After deducting redundancy, the effective information capacity might be around 100-200 bits.

Even so, this capacity is considerable—sufficient to provide a unique and detailed life description for every individual.

Chapter 6: Detailed Examination of the Mathematical Structure of Qimen Dunjia

Section 1: Enumeration of Basic Elements in Qimen Dunjia

A single chart in Qimen Dunjia contains the following basic elements:

1. Nine Palaces (Fixed Framework) 9 positions; information capacity $\log_2(9) \approx 3.17$ bits (used for location).

2. Earth Plate Curiosities/Instruments (Dìpán Qíyí) The nine symbols (Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui) distributed across 9 palaces, one per palace. This is a permutation of 9 symbols, but the arrangement is constrained by the "Bureau Number"—different Bureaus correspond to different Earth Plate arrangements, following specific rules (starting with Wu, arranged sequentially or reverse).

Yang Bureaus Nine + Yin Bureaus Nine = 18 Earth Plate arrangements. Earth Plate information capacity: $\log_2(18) \approx 4.17$ bits.

If we ignore this constraint, the theoretical number of arrangements of 9 distinct symbols across 9 palaces is $9! = 362,880$—an information capacity of $\log_2(362,880) \approx 18.47$ bits. The actual constraints compress this to $4.17$ bits.

3. Heaven Plate Curiosities/Instruments (Tiānpán Qíyí) Also the nine symbols, which rotate to different palaces following the rotation of the Value Symbol (Zhífú). Their arrangement is also a permutation of 9 symbols, constrained by the position of the Value Symbol.

For a given Bureau, the arrangement of the Heaven Plate is determined by the palace where the Value Symbol resides—the Star corresponding to that palace flies to the palace occupied by the Hour Stem, and the other Stars follow in sequence.

Each shí chén, the Value Symbol can reside in any of the 9 palaces (though the Central Palace may sometimes be treated specially or寄/temporarily assigned), resulting in about 8-9 basic rotational positions.

Heaven Plate independent information capacity: $\log_2(9) \approx 3.17$ bits (determined by the palace the Value Symbol enters).

However, when the Heaven Plate and Earth Plate are superimposed, the combination of Heaven Plate and Earth Plate Curiosities/Instruments in each of the 9 palaces is $9 \times 9 = 81$ possibilities—but due to permutation constraints, not all 81 combinations occur for every palace in every Bureau.

4. Nine Stars (Jiǔ Xīng) The 9 Stars each occupy a palace (original position) and redistribute across the 9 palaces after rotation. The arrangement of the Nine Stars is constrained by the position of the Star carried by the Value Symbol; it is a cyclic arrangement. Nine Stars independent information capacity: about $\log_2(9) \approx 3.17$ bits (determined by the palace the Value Symbol Star enters).

5. Eight Gates (Bā Mén) The 8 Gates are distributed in 8 palaces (no Gate in the center), redistributing after rotation. The arrangement of the Eight Gates is constrained by the position of the Value Embodiment (Zhíshǐ). Eight Gates independent information capacity: about $\log_2(8) = 3$ bits.

6. Eight Spirits (Bā Shén) The 8 Spirits are distributed in 8 palaces (no Spirit in the center or temporarily assigned), redistributing after rotation. The arrangement of the Eight Spirits is constrained by the position of the Value Symbol Spirit. Eight Spirits independent information capacity: about $\log_2(8) = 3$ bits.

7. Value Symbol and Value Embodiment:

The Value Symbol is the Star in command; the Value Embodiment is the Gate in command. Both are determined by the leader of the旬 (Ten-Day Period) corresponding to the Hour Stem. Value Symbol/Value Embodiment information capacity: about $\log_2(6) \approx 2.58$ bits (one of the Six Jia).

Section 2: Total Static Information Capacity of One Qimen Dunjia Chart

Summarizing the independent information elements, the independent information capacity of one Qimen Dunjia chart is:

However, there is a large amount of mutual constraint among the above elements—the Bureau Number determines the Earth Plate; the Shí Chén determines the Heaven Plate and the Value Symbol/Embodiment; the Value Symbol determines the position of the Nine Stars and Eight Spirits; the Value Embodiment determines the position of the Eight Gates. Thus, much of this information capacity is "redundant" (already determined by other elements).

Strictly speaking, the independent input information for one Qimen Dunjia chart is:

Bureau Number (18 types) $\times$ Shí Chén (12 types)

But we must also include the date information (which determines the Six Jia leader for the Value Symbol/Embodiment):

Day Stem/Branch (60 types)

Thus, the independent input information capacity is: $\log_2(18 \times 12 \times 60) = \log_2(12,960) \approx 13.66$ bits.

Wait—there is an error here. In reality, given a specific Year, Month, Day, and Hour, the Qimen chart initiation is completely deterministic. Thus, its independent input information is equivalent to a complete point in time.

However, Qimen Dunjia initiation does not use the Year information (only Solar Terms + Day Stem/Branch + Shí Chén), so its input information capacity is:

Solar Terms (24 types) $\times$ Upper/Middle/Lower Yuan (3 types) $\times$ Day Stem/Branch (60 types)

$= \log_2(24 \times 3 \times 60) = \log_2(4,320) \approx 11.98$ bits.

This input information capacity is about 12 bits—significantly less than Bazi's roughly 19 bits.

This means what$14 From the perspective of pure "input information," the information capacity of Qimen Dunjia appears smaller than that of Bazi. But the key to the problem lies here: Qimen Dunjia's mechanism for information amplification is far stronger than Bazi's.

Section 3: Information Amplification—The Core Advantage of Qimen Dunjia

What is "information amplification"$15 It means using a relatively small amount of input information, through the internal structure of the system, to generate an output information volume far greater than the input.

The mechanism of information amplification in Qimen Dunjia is manifested in:

1. Multi-Layer Plate Superposition

Given a point in time (about 12 bits of input), the Qimen system automatically generates a complete chart containing four layers of information:

• Earth Plate 9 Palaces $\times$ Curiosities/Instruments (9 symbols)
• Heaven Plate 9 Palaces $\times$ Stars + Curiosities/Instruments (9 + 9 = 18 symbols)
• Man Plate 8 Palaces $\times$ Gates (8 symbols)
• Spirit Plate 8 Palaces $\times$ Spirits (8 symbols)

Total: Earth Plate 9 + Heaven Plate 18 + Man Plate 8 + Spirit Plate 8 = 43 symbols simultaneously distributed across 9 palaces.

The arrangement state of these 43 symbols constitutes the "output information" of one chart. If constraints are ignored, the theoretical number of arrangements is immense.

2. Inter-Layer Interaction

The core judgment basis in Qimen Dunjia lies not in single-layer information but in inter-layer interactions—the combination of Heaven Plate Curiosities/Instruments and Earth Plate Curiosities/Instruments in the same palace (called "Qiyi Pattern" - Qíyí Géjú), which carries specific auspicious or inauspicious meanings.

For example:

• Heaven Plate Yi + Earth Plate Bing = "Qiyi Smooth Succession"
• Heaven Plate Geng + Earth Plate Yi = "Tai Bai Entering the Screen" (Tai Bai is Geng Metal)
• Heaven Plate Bing + Earth Plate Geng = "Screen Entering Tai Bai"
• Heaven Plate Geng + Earth Plate Geng = "Tai Bai Same Position" (Clash Pattern)

There are $9 \times 9 = 81$ possible combinations of the nine Heaven Plate Curiosities/Instruments and the nine Earth Plate Curiosities/Instruments—this constitutes the "Eighty-One Qimen Patterns." However, due to permutation constraints between the Heaven and Earth Plates, not all 81 patterns appear in every chart.

The actual number of Pattern combinations under constraint: Since the Heaven Plate and Earth Plate are each permutations of 9 different symbols, and the Heaven Plate arrangement is some rotation (or translation) of the Earth Plate arrangement, the total number of Heaven/Earth Plate combinations equals the number of Earth Plate arrangements $\times$ the number of Heaven Plate displacements = $18 \times 9 = 162$ (though this calculation is a simplification). More precisely, given an Earth Plate arrangement (18 types) and a Heaven Plate displacement (8-9 effective displacements), the total number of patterns is about $18 \times 9 = 162$ states.

Pattern information capacity $\approx \log_2(162) \approx 7.34$ bits (for 9 palaces).

3. Multi-Dimensional Cross-Judgment

For a specific palace, the interpreter must synthesize:

• The Heaven Plate Curiosities/Instruments of this palace (9 types).
• The Earth Plate Curiosities/Instruments of this palace (9 types).
• The Nine Stars of this palace (9 types).
• The Eight Gates of this palace (8 + 1 types, including the center having no gate).
• The Eight Spirits of this palace (8 + 1 types).
• The Trigram attribute of this palace (8 types, fixed).
• The Five Phase attribute of this palace (5 types, fixed).

If we treat each layer as an independent dimension (though they are not entirely independent), the information in one palace is:

$\log_2(9 \times 9 \times 9 \times 9 \times 9 \times 8 \times 5) \approx \log_2(2,624,400) \approx 21.32$ bits.

Total for 9 palaces $\approx 21.32 \times 9 \approx 191.9$ bits—but this is an overestimation because the distribution of symbols across palaces is strongly constrained.

A more reasonable estimation: Considering permutation constraints, the total independent arrangement information across the 9 palaces is about the sum of the arrangement information of each layer:

• Earth Plate Curiosities/Instruments Permutation: $\log_2(18) \approx 4.17$ (18 types corresponding to 18 Bureaus).
• Heaven Plate Curiosities/Instruments Displacement: $\log_2(9) \approx 3.17$ (Value Symbol can enter 9 palaces).
• Nine Stars Position: Follows the Heaven Plate, adds no independent information.
• Eight Gates Displacement: $\approx \log_2(8) = 3$.
• Eight Spirits Displacement: $\approx \log_2(8) = 3$.

Total independent permutation information $\approx 4.17 + 3.17 + 3 + 3 = 13.34$ bits.

But—this only covers "positional" information. More critical is the information derived from the interactions among layers.

Section 4: Information from Interactive Relationships

The primary basis for judgment in Qimen Dunjia relies on the following interactions:

1. Heaven-Earth Qiyi Patterns (81 Patterns)

Each palace has one Heaven/Earth Plate Qiyi pattern. The 9 palaces simultaneously present 9 patterns. The auspicious/inauspicious meanings of these patterns vary, and the patterns influence each other—e.g., one palace having an auspicious pattern, an adjacent palace having an inauspicious one, leads to mixed results.

The combination space of 9 Palace Patterns (theoretically): $81^9 \approx 1.5 \times 10^{17}$, information capacity about $\log_2(1.5 \times 10^{17}) \approx 57$ bits. But constrained by arrangement, the actual number of pattern combinations is far less.

The actual number of pattern states, given the constraints: The Heaven Plate arrangement is a rotation/translation of the Earth Plate arrangement, so the total number of Heaven/Earth combinations is constrained by the number of Earth Plate arrangements $\times$ number of Heaven Plate displacements $\approx 18 \times 9 = 162$ states (again, a simplification). More accurately, the total number of patterns is determined by the interaction of the two sets of permutations.

Pattern information capacity $\approx \log_2(162) \approx 7.34$ bits.

2. Gate and Star Combinations

Combination of Eight Gates and Nine Stars—which Gate falls into which Star's palace—has specific meaning. For instance, "Open Gate landing on Tianxin Star" differs greatly in meaning from "Open Gate landing on Tianpeng Star."

3. Gate and Qiyi Combinations

Similarly, the combination of the Eight Gates and the Heaven Plate Qiyi they fall upon—e.g., "Open Gate obtaining Yi Qi" (Auspicious), "Open Gate obtaining Geng" (Inauspicious)—this combination judgment is critical.

4. Star and Qiyi Combinations

Likewise, the combination of Nine Stars and Heaven Plate Qiyi has specific meanings.

5. Combinations with Spirits

The combination of the Eight Spirits with Gates, Stars, and Qiyi further enriches the information layers.

6. Five Phase Relationships Across Layers

The generation/overcoming between the Five Phases of the Heaven Plate and the Five Phases of the Earth Plate (the "Downward Adherence" relationship), the generation/overcoming between the Gate's Phase and the Palace's Phase, and the generation/overcoming between the Star's Phase and the Gate's Phase—this multi-layered network of Five Phase interactions is extremely complex.

Synthesizing the interactive relationship information from the above layers, the total relational information capacity of one Qimen Dunjia chart can be roughly estimated as follows:

Adding the positional information ($\approx 13$ bits), the total static information capacity of one Qimen Dunjia chart is about 90-133 bits.

Section 5: Dynamic Information of Qimen Dunjia

The dynamic information in Qimen Dunjia arises from:

1. Time Changes (Shí Chén)

A new chart is generated every two hours. 12 shí chén per day produce 12 different charts, each containing about 90-133 bits of information.

However, the charts in adjacent shí chén usually differ only slightly (only the Value Symbol moves one palace), so the independent information gained per change is small—about 10-20 bits.

Daily dynamic information increment: about $12 \times 15 = 180$ bits.

2. Date Changes

The Value Symbol and Value Embodiment change daily (depending on the sixty-year cycle's Ten-Day Period), meaning the basic chart for each day also changes.

3. Solar Term Changes

A new Yuan (5-day period) changes every 15 days, potentially causing a Bureau change (a shift in the underlying structure). This change is significant, adding tens of bits of independent information.

4. Annual Changes

Annual wandering stars (Zǐbái) and other factors also change annually.

Total dynamic information over a year: approximately $365 \times 12 \times 15 \approx 65,700$ bits (rough estimate, containing much redundancy). After deducting redundancy, the independent annual dynamic information might be several thousand bits.

Section 6: Estimation of Total Qimen Dunjia Information Capacity

Taking the median value, the total information capacity of Qimen Dunjia is about 250-400 bits.

Section 7: Preliminary Mathematical Comparison

From this preliminary comparison, we can see:

1. Input Information Capacity: Bazi is slightly greater than Qimen Dunjia.
2. Static Combination Information: Bazi is slightly greater than Qimen Dunjia.
3. Relational Information: The two are roughly equal.
4. Intrinsic System Information (Ten Gods vs. Multi-layer Patterns): Qimen Dunjia is significantly greater than Bazi.
5. Dynamic Information: Qimen Dunjia is slightly greater than Bazi (due to more frequent chart changes).
6. Total Information Capacity: The upper limit of Qimen Dunjia is clearly higher than that of Bazi.

This preliminary conclusion suggests: Mathematically, the information capacity of Qimen Dunjia is greater than or equal to that of Bazi.

However, this conclusion needs further verification from a metaphysical perspective.

Chapter 7: Deep Mathematical Analysis of Information Dimensions

Section 1: Concept of Information Dimensions

The preceding analysis focused mainly on the "quantity" of information (measured in "bits"). However, the "quality" of information is equally important, manifest in its "dimensions"—the number of independent aspects the information covers.

Using an analogy: A painting (two-dimensional) may have less information capacity than a book (one-dimensional text sequence), but the painting's information dimensions (color, shape, composition, depth, etc.) far exceed those of the book's text sequence. In certain judgment tasks, the painting's information might be more effective than the book's.

Information Dimensions of Bazi:

1. Temporal Dimension (Four levels of time: Year, Month, Day, Hour).
2. Five Phase Dimension (Five attributes: Metal, Wood, Water, Fire, Earth).
3. Yin/Yang Dimension (Yin/Yang Stems, Yin/Yang Branches).
4. Ten Gods Dimension (Ten types of interpersonal relationships).
5. Strength/Weakness Dimension (Strength/Weakness of the Day Master).
6. Pattern Dimension (Orthodox, Auxiliary, or Special Patterns).
7. Nayin Dimension (Thirty types of material images).
8. Divine Marker Dimension (Presence or absence of various markers).

Totaling about 8 primary information dimensions.

Information Dimensions of Qimen Dunjia:

1. Spatial Dimension (Nine Palaces, Eight Directions).
2. Temporal Dimension (Solar Terms, Shí Chén).
3. Heaven Plate Dimension (Nine Stars + Heaven Plate Qiyi).
4. Earth Plate Dimension (Earth Plate Qiyi + inherent palace attributes).
5. Man Plate Dimension (Eight Gates).
6. Spirit Plate Dimension (Eight Spirits).
7. Pattern Dimension (81 Heaven/Earth Plate Patterns).
8. Five Phase Dimension (Multi-layer Five Phase interactions).
9. Yin/Yang Dimension (Yin Bureau/Yang Bureau).
10. Strength/Vigor Dimension (Prosperity/Decline of each element).
11. Utility/Focus Star Dimension (The Utility Star for the person or the matter in question).
12. Directional Fortune Dimension (Overall auspiciousness of each direction).

Totaling about 12 primary information dimensions.

Dimension Comparison: Qimen Dunjia (12 dimensions) > Bazi (8 dimensions).

The fundamental reason for this difference is: Qimen Dunjia simultaneously encodes temporal and spatial information, whereas Bazi primarily encodes temporal information.

The Xici Zhuan states:

"The Yi as a writing is vast and fully equipped: there is the Way of Heaven, there is the Way of Man, there is the Way of Earth. Combining the Three Powers and multiplying them by two, thus we have Six." (Yì zhī wéi shū yě, guǎngdà xī bèi: yǒu tiāndào焉, yǒu réndào焉, yǒu dìdào焉. Jiān sān cái ér liǎng zhī, gù liù.)

The "Three Powers"—Heaven, Earth, Man—represent three fundamental dimensions. The hexagram, with its six lines, encodes the Three Powers by "multiplying by two" (i.e., assigning two lines each to Heaven, Man, and Earth).

The Heaven Plate, Man Plate (Eight Gates), and Earth Plate of Qimen Dunjia precisely correspond to this "Three Powers" structure. Adding the Spirit Plate, it can even be considered "Four Powers": Heaven, Earth, Man, Spirit—one dimension more than the Three Powers.

Bazi's Four Pillars, although having layers of time (Year for ancestors, Month for parents, Day for self, Hour for children), are fundamentally different scales of the "Time" dimension—not a spatial layering of Heaven, Earth, and Man.

This dimensional difference implies that Qimen Dunjia has a natural advantage in judgment tasks requiring spatial orientation information, which Bazi lacks directly. However, in pure "fate calculation" (inherent patterns determined by temporal flow), Bazi may be more specialized.

Section 2: Comparison of Information Density

Information Density = Information Quantity / Number of Symbols.

Bazi Symbols: 4 Heavenly Stems + 4 Earthly Branches = 8 characters. Bazi Information Quantity: $\approx 250-300$ bits. Bazi Information Density: $\approx 31-38$ bits/character.

Qimen Dunjia Symbols (in one chart): 9 Earth Plate Qiyi + 9 Heaven Plate Qiyi + 9 Stars + 8 Gates + 8 Spirits = 43 symbols. Qimen Dunjia Information Quantity: $\approx 250-400$ bits. Qimen Dunjia Information Density: $\approx 5.8-9.3$ bits/symbol.

The information density of Bazi (31-38 bits/character) is far higher than that of Qimen Dunjia (5.8-9.3 bits/symbol).

What does this mean$1

It means Bazi is a highly "compressed" information system—it conveys extremely rich information using the fewest possible symbols. The cost is that interpreting each symbol requires higher overall integrative judgment (as each character has multiple meanings).

Qimen Dunjia is a relatively "expanded" information system—it uses more symbols to present information in a more clearly layered manner—its advantage is that the layers of interpretation are more distinct.

In pre-Qin terms:

Bazi resembles the principle of "Simplicity" (Yì Jiǎn)—The Xici Zhuan states: "Qian is easy to know because of its simplicity (yì), Kun is capable because of its conciseness (jiǎn). Easy to know leads to affinity; easy to follow leads to achievement. Affinity allows longevity; achievement allows greatness. Longevity belongs to the virtue of the Sage; greatness belongs to the enterprise of the Sage."

Qimen Dunjia resembles the principle of "Vastness" (Guǎng Dà)—The Xici Zhuan states: "The Yi is vast and great. When speaking of the distant, it does not retreat; when speaking of the near, it is still and correct; when speaking between Heaven and Earth, it is complete."

Bazi uses the method of "Simplicity" to carry the most information—its merit is deep insight; Qimen Dunjia uses the method of "Vastness" to cover the comprehensive information of Heaven, Earth, and Man—its merit is broad utility.

Section 3: Comparison of Information Redundancy

Information redundancy is the proportion of repeated or derivable information within a system.

Bazi Redundancy:

• Year Stem determines Month Stem $\rightarrow$ Month Stem is redundant.
• Day Stem determines Hour Stem $\rightarrow$ Hour Stem is redundant.
• Five Phases of a Stem can be inferred from the Stem itself $\rightarrow$ Five Phase information is redundant.
• Five Phases of a Branch can be inferred from the Branch itself $\rightarrow$ Five Phase information is redundant.
• Ten Gods can be inferred from the Five Phases relative to the Day Master $\rightarrow$ Ten Gods information is redundant.
• Nayin can be derived from the Stems/Branches $\rightarrow$ Nayin is redundant.

High redundancy means: The effective (non-redundant) information capacity of Bazi is lower than its total information capacity. Deducting redundancy, the effective information capacity might be about 40-60% of the total, roughly 100-180 bits.

Qimen Dunjia Redundancy:

• Earth Plate Qiyi is completely determined by the Bureau Number $\rightarrow$ Earth Plate Qiyi is redundant (relative to the Bureau).
• Heaven Plate Qiyi is determined by the Value Symbol's displacement $\rightarrow$ Heaven Plate Qiyi is redundant.
• Nine Stars positions are determined by the displacement of the Value Symbol Star $\rightarrow$ Stars are redundant.
• Eight Gates positions are determined by the displacement of the Value Embodiment $\rightarrow$ Gates are redundant.
• Eight Spirits positions are determined by the displacement of the Value Symbol Spirit $\rightarrow$ Spirits are redundant.
• Heaven/Earth Patterns can be inferred from the Heaven and Earth Plates $\rightarrow$ Patterns are redundant.

Extremely high redundancy! In fact, the entire information of one Qimen chart can be completely determined by three independent parameters:

1. Bureau Number (1-18).
2. The Jia Leader (Jia Zi, Jia Xu... Jia Yin, 6 types).
3. The Earthly Branch of the Hour (Zi to Hai, 12 types).

Independent input information capacity = $\log_2(18 \times 6 \times 12) = \log_2(1,296) \approx 10.34$ bits.

This means the effective (non-redundant) information capacity of Qimen Dunjia is only about 10 bits!

This is a surprising result: although the Qimen chart appears information-rich (43 symbols), its effective input information capacity is even smaller than that of Bazi (about 12-15 bits after redundancy deduction).

However, this does not mean Qimen Dunjia's information capacity is "smaller" than Bazi's.

Why$2 Because "effective input information capacity" and "effective output information capacity" are different concepts.

A system can use very little input information and, through complex internal structure and operational rules, generate a much larger amount of output information. Qimen Dunjia is precisely such an "information amplifier"—10 bits of input, through the superposition and interaction of multiple layers, produce 90-130 bits of output information. Its "information amplification rate" is about 10 to 13 times.

Bazi's information amplification rate: 19 bits input $\rightarrow$ 250-300 bits output, about 13 to 16 times.

The two amplification rates are quite close—about 10 to 16 times. This may not be accidental but might reflect some inherent law of arcane systems.

Section 4: Comparison of Information Resolution

Information resolution refers to the system's ability to distinguish between different states, i.e., its level of detail.

Bazi Temporal Resolution: 2 hours (one shí chén). Inference: People born less than 2 hours apart might have the same Bazi chart.

Estimating the number of distinct Bazi time points: $60 \text{ years} \times 365 \text{ days} \times 12 \text{ hours} = 262,800$ different Bazi time points.

Deducting actual constraints (some Stem-Branch combinations don't appear), the effective number of Bazi types is about 518,400.

If we consider the number of people born globally (much smaller in ancient times), say 8.4 billion in 60 years, dividing by about 520,000 Bazi types gives an average of about 16,000 people sharing each type.

This means Bazi's resolution—in terms of temporal resolution—is quite coarse.

Qimen Dunjia Temporal Resolution: Also 2 hours (one shí chén).

But Qimen Dunjia adds Spatial Resolution—information about the auspiciousness of the Nine Palaces and Eight Directions. For the same Bazi chart (same time), Qimen Dunjia can give different judgments based on the spatial direction where the subject is located.

Spatial direction has 8 directions (or more precisely, 24 directions), so the actual resolution of Qimen Dunjia is:

Temporal Resolution $\times$ Spatial Resolution = $518,400 \times 8 = 4,147,200$

Or more precisely: $518,400 \times 24 \approx 12,441,600$

This resolution is 8 to 24 times greater than that of Bazi.

The fundamental reason for this difference is: Qimen Dunjia includes an additional "spatial dimension" compared to Bazi.

The Xici Zhuan states:

"It circumscribes the transformations of Heaven and Earth without going beyond them, and molds the myriad things without omitting them." (Fànwéi tiāndì zhī huà ér bù guò, qū chéng wànwù ér bùyí.)

"Circumscribing Heaven and Earth"—"Heaven and Earth" refer to time and space. Bazi "circumscribes Heaven" (time), while Qimen Dunjia "circumscribes Heaven and Earth" (time + space)—thus, the coverage of Qimen Dunjia's information is more complete.

Section 5: Re-examining the Model based on Trigram Structures

The preceding analysis focused heavily on numerical calculation. Let us now attempt to re-examine the comparison using the trigram model from pre-Qin Yi learning.

Bazi can be seen as a "Four-Line Trigram"—each Pillar corresponds to a "line." But the "lines" of Bazi are not simple Yin/Yang binaries; they take on sixty values from the Sixty Jiazi—so the information content of each "line" is far greater than that of a single line in a hexagram.

Analogy in terms of hexagrams:

Bazi's "Four Pillars" is equivalent to a four-line system, where each line has 60 possible values. Equivalent information capacity $= 4 \times \log_2(60) \approx 4 \times 5.91 \approx 23.6$ bits (This is the theoretical value, close to the $\approx 19$ bits calculated earlier—the difference comes from discounting inter-pillar constraints).

Qimen Dunjia's chart can be seen as a "Nine-Palace Grid"—nine positions, each carrying multiple layers of information.

Analogy in terms of hexagrams:

Qimen Dunjia is equivalent to a nine-line system (Nine Palaces), where each line has multi-dimensional values (Heaven Plate Qiyi, Earth Plate Qiyi, Stars, Gates, Spirits, etc.).

However, the structure of the Nine Palaces does not have a precedent in the Book of Changes—the Sixty-Four Hexagrams have six lines, never nine lines. Yet, the Xici Zhuan states:

"Varying with the interspersed Threes and Fives, interweaving their numbers. By penetrating their changes, one completes the patterns of Heaven and Earth. By exhausting their numbers, one determines the images of the world below." (Cānwǔ yǐ biàn, cuòzōng qí shù. Tōng qí biàn, suì chéng tiāndì zhī wén. Jí qí shù, suì dìng tiānxià zhī xiàng.)

"Interspersed Threes and Fives" (cānwǔ)—"Three" and "Five." $3 \times 5 = 15$, which is the sum of the rows/columns of the Luo Writing, and also the sum of the generating numbers of the River Chart. And $3 \times 3 = 9$, the number of the Nine Palaces.

The structure of the Nine Palaces can be understood as a "spatialization" of the Three-Line Trigram—where each line takes three values (not two), resulting in $3^3 = 27$ possible Three-Line Three-Element Trigrams. But the application of the Nine Palaces differs from hexagrams—the Nine Palaces are nine "parallel" positions, not three "superimposed" layers.

Perhaps it can be understood this way:

• Bazi = Four layers of "superposition" (Year $\rightarrow$ Month $\rightarrow$ Day $\rightarrow$ Hour, temporal superposition).
• Qimen Dunjia = Nine positions in "parallel" (Nine Palaces) $\times$ Four layers of "superposition" (Earth Plate $\rightarrow$ Heaven Plate $\rightarrow$ Man Plate $\rightarrow$ Spirit Plate).

Thus: Qimen Dunjia $\approx 9$ times the information volume of Bazi, depending on the degree of independence among the nine palaces.

Of course, this "nine times" is a rough estimate—the actual ratio depends on the correlation between the nine palaces. If the nine palaces were completely independent, it would be nine times; if they were highly correlated, it would be much less.

Based on the structure of Qimen Dunjia, the information in the nine palaces is not completely independent—once the Heaven Plate Qiyi of one palace is determined, the Heaven Plate Qiyi of the other palaces are also determined (due to circular permutation). Thus, the independence of the nine palaces is low—about equivalent to the information capacity of 1.5 to 2 independent palaces.

Therefore: Qimen Dunjia $\approx$ Bazi $\times 1.5$ to $2$ times.

This estimate roughly aligns with the results from the preceding numerical analysis.

Chapter 8: Comparing Information Capacity from the Perspective of "Images"

Section 1: The Information Science Essence of "Image" (Xiàng)

The preceding analysis focused mainly on the aspect of "Number" (shù). However, the essence of pre-Qin Yi learning lies not in "Number" but in "Image" (Xiàng).

The Xici Zhuan states:

"The Yi is Image; Image is likeness." (Yì zhě xiàng yě, xiàng yě zhě xiàng yě.)

"Therefore, the Sage perceived the intricacy (zé) of the world below, and modeled it on its forms, symbolizing its suitability for things; therefore, it is called Image. The Sage perceived the movement of the world below, and observed its connections and transformations, to enact its rites, and attached statements to judge its auspiciousness and inauspiciousness; therefore, it is called Line (Yáo)."

"Image" precedes "Number"—or rather, "Number" is the quantification of "Image." The information capacity of arcane arts lies in the capacity of "Image"—it cannot be exhausted by "bits."

Why is this so$3 Because the mapping of "Image" is open-ended—one trigram image can correspond to an infinite number of specific things. In contrast, "Number" mapping is closed—a number corresponds to a specific value.

For example:

The "Jia Wood" in Bazi includes, but is not limited to: tall trees, structural timber, upright objects, the head, the gallbladder, the East, Springtime, the color blue-green, the sour taste, benevolence, virtue... This list of mappings can extend infinitely.

The "Open Gate obtaining Yi Qi on the Tianxin Star" in Qimen Dunjia, its "Image" includes, but is not limited to: a good doctor, a skilled artisan, an affair initiating a new venture, construction work, favorable sounds from the Northwest... This too can extend infinitely.

If we measure information capacity by the volume of the "Image Semantic Space," then every character in Bazi and every symbol in Qimen Dunjia has an information capacity tending toward infinity. In such a case, calculating "bits" is no longer applicable—we need to introduce the concept of "Image Semantic Space."

Section 2: Comparison of Image Semantic Space

Definition: The "Image Semantic Space" of a symbol is the set of all concrete things it can map to.

Image Semantic Space of a Bazi Stem (e.g., Jia):

• Image of Five Phases (the entire set of Wood-related things).
• Image of Yin/Yang (the entire set of Yang-related things).
• Image of Ten Gods (ten types of interpersonal relationships when Jia is the Day Master).
• Image of Direction (the entire set related to the East).
• Image of Season (the entire set related to Spring).
• Image of Form (upright, tall, slender forms, etc.).
• Image of Five Tastes (sour taste related things).
• Image of Five Colors (blue-green related things).
• ...

The size of this semantic space depends on the precision of categorization. If categorized by Five Phases, all things are divided into five large classes, each containing about one-fifth of all things. If categorized by Heavenly Stems (ten), each contains one-tenth. If categorized by the Sixty Jiazi, each contains one-sixtieth.

Bazi Semantic Space Size $\approx$ Total Number of Things / Number of Categories.

• Categorized by Four Pillars: Total Things / 518,400.
• Categorized by adding Dimensions like Ten Gods: Total Things / $(518,400 \times 10^7)$ $\rightarrow$ extremely fine.

Qimen Dunjia Semantic Space Size $\approx$ Total Number of Things / Number of Categories.

• Categorized by one chart state: Total Things / 4,320 (number of independent charts per year).
• Categorized by adding Nine Palaces: Total Things / $(4,320 \times 9)$.
• Categorized by adding multi-layer interactions: Total Things / $(4,320 \times 9 \times 81 \times 8 \times 8)$ $\rightarrow$ extremely fine.

In terms of the fineness of semantic space, the multi-layer structure of Qimen Dunjia, upon superposition, can achieve classification precision comparable to, or even finer than, Bazi.

However, the "direction" of the semantic space of the two differs:

• Bazi Semantic Space leans toward Human Fate and Destiny—information regarding the trajectory of a person's life.
• Qimen Dunjia Semantic Space leans toward Time-Space Events—information about events that will occur in a specific time and specific space.

This difference in "direction" is fundamental—they cannot be simply compared by "size."

Section 3: Layers of Image and Recursion

In pre-Qin Yi learning, "Image" has layers of division.

The Xici Zhuan states:

"The Yi is Image; Image is likeness." (Yì zhě xiàng yě, xiàng yě zhě xiàng yě.)

"Therefore, the Yi has the Supreme Ultimate, which gives birth to the Two Modes; the Two Modes give birth to the Four Images; the Four Images give birth to the Eight Trigrams; the Eight Trigrams determine auspiciousness and inauspiciousness; auspiciousness and inauspiciousness give birth to great undertakings."

Supreme Ultimate $\rightarrow$ Two Modes $\rightarrow$ Four Images $\rightarrow$ Eight Trigrams $\rightarrow$ Auspicious/Inauspicious $\rightarrow$ Great Undertakings—this is the recursive unfolding of "Image." The "Image" at each layer is a refinement and elaboration of the layer above it.

Layers of Image in Bazi:

1. Supreme Ultimate Layer: Strength/Weakness of the Day Master (divided into two).
2. Two Modes Layer: Day Master strength/weakness + Pattern level (quadripartition).
3. Four Images Layer: Day Master + Pattern + Utility God Favorability + Great Cycle Trend.
4. Eight Trigrams Layer: Symbolic expansion of the images of the four Pillars.
5. Sixty-Four Hexagram Layer: Symbolic combination of the interactions among the Four Pillars.
6. 384 Lines Layer: Detailed analysis of every Hidden Stem and Ten God in the Eight Characters.

Bazi has about 6 layers of Image, with each layer recursively unfolding by a factor of about 2 to 8 times.

Layers of Image in Qimen Dunjia:

1. Supreme Ultimate Layer: Yin/Yang nature of the Bureau Number (division into two).
2. Two Modes Layer: Yin/Yang Bureau + Prosperity/Decline of the Utility Palace.
3. Four Images Layer: Comprehensive analysis of Utility Palace + Heaven Plate + Man Plate + Spirit Plate.
4. Eight Trigrams Layer: Symbolic expansion of the images of the Nine Palaces.
5. Eighty-One Pattern Layer: The 81 combinations of Heaven/Earth Plate Qiyi.
6. Multi-layer Interaction Layer: Comprehensive cross-analysis of Gates, Stars, Spirits, and Qiyi.
7. Positional Expansion Layer: Specific auspiciousness judgments for each direction.

Qimen Dunjia has about 7 layers of Image, with each layer unfolding by a factor of about 2 to 9 times.

The more layers and the larger the unfolding factor at each layer, the larger the "Image Semantic Space"—this implies that the Image Semantic Space of Qimen Dunjia is theoretically slightly larger than that of Bazi.

However, more layers also mean greater interpretive difficulty—more information, but limited by the cognitive capacity of the interpreter. This is the predicament described by Zhuangzi as "the skilled calculator cannot arrive."

Part Three: Metaphysical Unification

Chapter 9: Information Capacity from the Viewpoint of the Heavenly Way

Section 1: The "Great Information Capacity" of the Heavenly Way

Chapter 25 of the Laozi states:

"There exists a thing, undifferentiated, born before Heaven and Earth. Silent! Empty! It stands alone and does not change; it circulates eternally and is never exhausted. It can be the mother of all under Heaven. I do not know its name; I style it the Dao. If I am forced to name it, I call it the Great (Dà). Great implies passing; passing implies distance; distance implies returning." (Yǒu wù hùnchéng, xiān tiāndì shēng. Jì xī liáo xī, dúlì ér bù gǎi, zhōuxíng ér bù dài, kěyǐ wéi tiānxià mǔ. Wú bùzhī qí míng, zì zhī dào, qiǎng wéi zhī míng yuē dà. Dà yuē shì, shì yuē yuǎn, yuǎn yuē fǎn.)

The Dao is "Great" (Dà); the meaning of "Great" lies not just in magnitude, but in the infinity of its information. "Stands alone and does not change"—the information of the Dao is constant, incompressible; "circulates eternally and is never exhausted"—the information of the Dao is cyclical and endless.

All arcane arts—Bazi, Qimen Dunjia, Liuren, Taiyi...—are partial extractions and encodings of the Dao's information. They extract from different "angles," with different "precisions," and cover different "ranges," but none can exhaust the entire information of the Dao.

Chapter 1 of the Laozi states:

"The Dao that can be told is not the eternal Dao. The name that can be named is not the eternal Name. Nameless is the beginning of Heaven and Earth; named is the Mother of the myriad things." (Dào kě dào, fēi cháng dào. Míng kě míng, fēi cháng míng. Wú míng tiāndì zhī shǐ, yǒumíng wànwù zhī mǔ.)

The Dao that "can be told" is not the "eternal Dao." Any arcane art is a "tellable Dao"—a finite information encoding system—not the entirety of the "eternal Dao."

Therefore, from the perspective of the Heavenly Way, the comparison between Bazi and Qimen Dunjia is like comparing a dipper and a bushel in measuring the ocean—both are finite.

But even within the finite, there are differences in size. Our comparison aims to distinguish this difference in "finitude."

Section 2: Temporality and Spatiality of the Heavenly Way

The operation of the Heavenly Way includes both temporality and spatiality.

Temporality—the "Movement" (Xíng) of the Heavenly Way:

The Book of Changes, Qian Trigram, Image Commentary states:

"The movement of Heaven is vigorous; the superior man strives ceaselessly." (Tiānxíng jiàn, jūnzǐ yǐ zìqiángbùxī.)

"Heaven's movement" (Tiān xíng)—the ceaseless circulation of Heaven, which is the passage of time.

The "Canon of Yao" (Yao Dian) in the Book of Documents—"calculate the ephemerides of the sun, moon, and stars, and diligently present the proper times to the people"—the essence of the calendar is the encoding of the temporal information of the Heavenly Way.

Spatiality—the "Position" (Wèi) of the Heavenly Way:

The Book of Changes, Kun Trigram, Image Commentary states:

"The disposition of Earth is yielding; the superior man supports all things with deep virtue." (Dì shì kūn, jūnzǐ yǐ hòudé zài wù.)

"Disposition of Earth" (Dì shì)—the configuration of the Earth, which is the expanse of space.

Bazi mainly encodes the temporality of the Heavenly Way ("Heaven's Movement"), while Qimen Dunjia encodes both the temporality and spatiality of the Heavenly Way ("Heaven's Movement" + "Earth's Disposition").

From the perspective of the Heavenly Way, temporality and spatiality are equally important aspects—neither can be omitted.

The Xici Zhuan states:

"Images are formed in Heaven, and forms take shape on Earth; transformations are thereby revealed." (Zài tiān chéng xiàng, zài dì chéng xíng, biànhuà jiàn yǐ.)

"Heavenly Images" (Tiān Xiàng) primarily relate to time (movements of the sun, moon, stars); "Earthly Forms" (Dì Xíng) primarily relate to space (distribution of mountains and rivers). "Transformation" (the generation and flow of information) must be revealed through the interaction of Heavenly Images and Earthly Forms.

Pure temporal information (Bazi) is like "Images formed in Heaven" but not yet "Forms taken shape on Earth"—the transformation may not be fully revealed.

Information that includes both temporal and spatial information (Qimen Dunjia) is like "Images formed in Heaven" and "Forms taken shape on Earth"—the "transformation is revealed"—the manifestation of information is more complete.

This provides a metaphysical basis for judging the difference in information capacity between the two: Because Qimen Dunjia incorporates both temporal and spatial dimensions, its coverage of the information of the Heavenly Way is more comprehensive.

Section 3: Heaven-Man Resonance and Information Transmission

The concept of "Heaven-Man Resonance" (Tiānrén Gǎnyìng) in pre-Qin thought is essentially a theory of information transmission—the information of the Heavenly Way can be transmitted to human affairs.

The Shangshu, Hongfan records Jizi's words:

"If the King is solemn, timely rain will fall. If he governs, timely sunshine will fall. If he is clear, timely warmth will fall. If he plans, timely cold will fall. If he is sage, timely wind will fall. Signs of calamity: If he is wild, constant rain will fall. If he is presumptuous, constant sunshine will fall. If he is complacent, constant warmth will fall. If he is urgent, constant cold will fall. If he is obscure, constant wind will fall."

There is a correspondence between the King's actions (Solemn, Governing, Clear, Planning, Sage) and meteorological phenomena (Rain, Sunshine, Warmth, Cold, Wind)—this is the specific expression of "Heaven-Man Resonance."

From an information theory perspective, this "resonance" is the "transmission" of information—the information of human affairs is transmitted to celestial phenomena through some mechanism, and the information of celestial phenomena is transmitted to human affairs through some mechanism—the transmission mechanism is the resonance of "Yin, Yang, and the Five Phases."

The path of Heaven-Man Resonance in Bazi: Temporal information of the Heavenly Way (Stems/Branches at birth) $\rightarrow$ Innate Endowment of the individual (Bazi Pattern) $\rightarrow$ Trajectory of the individual's destiny.

This path is: Heaven $\rightarrow$ Man (Unidirectional, mediated by Time).

The path of Heaven-Man Resonance in Qimen Dunjia: Time-Space information of the Heavenly Way (Bureau number and chart layout at the time of initiation) $\rightarrow$ Current state of the event (chart analysis) $\rightarrow$ Future direction and optimal course of action.

This path is: Heaven $\leftrightarrow$ Earth $\leftrightarrow$ Man (Multi-directional, mediated by both Time and Space).

The information flow of a multi-directional path is greater than that of a unidirectional path—this also supports the conclusion that the information capacity of Qimen Dunjia is greater than that of Bazi.

However, it must be pointed out: the "unidirectional" path of Bazi is not truly unidirectional—the addition of Great Cycles and Flowing Years gives it a certain characteristic of dynamic feedback. It is just that this feedback mainly occurs in the temporal dimension and does not involve the spatial dimension.

Section 4: "Counting the Past is顺 smooth/forward, Knowing the Future is 逆 reverse"—The Directionality of Information

The Shuogua Zhuan states:

"Counting the past is smooth (shùn); knowing the future is reverse (nì); therefore, Yi uses reverse counting." (Shù wǎng zhě shùn, zhī lái zhě nì, shì gù yì nì shù yě.)

This speaks of the direction of information in the Book of Changes—"reverse counting" means inferring backward, deriving causes from results, and the present from the future.

The direction of information in Bazi is mainly "smooth"—starting from the time of birth, flowing forward through the Great Cycles and Flowing Years to predict future fortune. This is the application of "Counting the past is smooth"—using the established past (moment of birth) as a basis to project the future.

The direction of information in Qimen Dunjia is mainly "reverse"—starting from the present question, inferring the pattern of Heaven and Earth, and finding the optimal course of action. This is the application of "Knowing the future is reverse"—aiming at the unknown future, designing actions in reverse.

"Reverse" information capacity is generally greater than "smooth"—because "reverse" requires considering more possibilities (inferring multiple possible causes from one result), whereas "smooth" only follows a single path forward.

This again supports the conclusion that the information capacity of Qimen Dunjia is greater than that of Bazi from another angle.

Section 5: The Dao of the Three Powers and the Hierarchy of Information

The Xici Zhuan states:

"There is the Way of Heaven, there is the Way of Man, there is the Way of Earth. Combining the Three Powers and multiplying them by two, thus we have Six. The Six are nothing else but the Way of the Three Powers." (Yǒu tiāndào焉, yǒu réndào焉, yǒu dìdào焉. Jiān sān cái ér liǎng zhī, gù liù. Liù zhě fēi tā yě, sān cái zhī dào yě.)

The Three Powers—Heaven, Earth, Man—are the three fundamental layers of cosmic information.

The correspondence of the Three Powers in Bazi:

• Way of Heaven: Heavenly Stems of the Great Cycles/Flowing Years (external environmental changes).
• Way of Earth: Earthly Branches of the Four Pillars (material basis of innate endowment).
• Way of Man: The Day Master and the Ten Gods (the individual's subjectivity and social relations).

In Bazi, these Three Powers are "folded"—compressed within the Four Pillars of Stems and Branches, requiring the interpreter to unfold them.

The correspondence of the Three Powers in Qimen Dunjia:

• Way of Heaven: Heaven Plate (Nine Stars + Heaven Plate Qiyi).
• Way of Earth: Earth Plate (Nine Palaces + Earth Plate Qiyi).
• Way of Man: Man Plate (Eight Gates).

Adding the Spirit Plate, it can be considered "Four Powers": Heaven, Earth, Man, Spirit.

In Qimen Dunjia, these Three Powers (or Four) are "unfolded"—each has an independent plate, and the information layers are clearly distinguished.

The difference between "unfolded" and "folded" relates to the readability of information:

• "Unfolded" information is easy to analyze layer by layer, but the interactions between layers require separate inference.
• "Folded" information is hard to layer, but its internal integration is stronger.

In terms of information capacity, "unfolded" information is no less than "folded"—because the process of "folding" might result in information loss (like lossy compression). But if the "folding" process incurs no loss, its information capacity is equal to that of the "unfolded" state.

Does the "folding" of Bazi involve no loss$4

The answer is: Generally no loss—because the rules of Bazi interpretation can restore the information of the Three Powers from the folded state. However, certain subtle pieces of information might be obscured during folding—for instance, spatial information like "This person develops better in the East." Bazi might infer this indirectly through the strength of Wood, but it is not as explicit as the direct directional indication in Qimen Dunjia.

Thus, from the perspective of the Dao of the Three Powers: Qimen Dunjia has slightly superior information in the "Man" and "Earth" dimensions compared to Bazi, while in the "Heaven" dimension, they are comparable.

Chapter 10: Information Capacity as Seen Through the Methods of the Sages

Section 1: Fuxi Drawing Trigrams—The Original Information Encoding

The Xici Zhuan states:

"In the time when Bao Xi ruled the world, he looked up to observe the images in Heaven, and looked down to observe the laws on Earth. He observed the patterns of birds and beasts and the suitability of the Earth. He took examples from his own person for what was near, and took examples from things far away. From this, he first created the Eight Trigrams, to connect with the spiritual virtue of Heaven and Earth, and to categorize the feelings (qíng) of the myriad things."

Fuxi's method of drawing trigrams—"observing above and below"—was simultaneously collecting information from Heaven (observing above) and Earth (observing below). This method of information collection was inherently "encompassing Heaven and Earth."

Fuxi's encoding result—the Eight Trigrams—is a system of "Images," not purely "Number." This method of "Image" encoding allows a finite set of eight symbols to carry infinite information.

Bazi inherited Fuxi's Stems and Branches system (Stems + Branches), but it mainly utilizes the "Number" aspect (combination, inference).

Qimen Dunjia inherited Fuxi's Eight Trigrams system (Nine Palaces), and fully utilizes its "Image" aspect (the spatial image of each palace, the image of direction, the image of all things).

Therefore, from the perspective of the "quality" of information encoding (richness of symbolic meaning), Qimen Dunjia inherits the original encoding of Fuxi more completely.

Section 2: Wen Wang Developing the Yi—Expansion of Information

It is traditionally believed that when Wen Wang was imprisoned in Youli, he elaborated the Book of Changes, doubling the Eight Trigrams to form the Sixty-Four Hexagrams and attaching line statements and hexagram statements. This was the first major expansion of information.

Wen Wang's contribution was: expanding the original "Eight Images" system of Fuxi into a system of "Sixty-Four Images $\times$ Three Hundred Eighty-Four Lines," increasing the information capacity by about 48 times (on the order of $64/8 \times 384/24$).

Bazi's Four Pillar system can be seen as a "parallel development" of Wen Wang's Sixty-Four Hexagram system—both extracting information from combinations of Stems and Branches, but using different encoding methods (Hexagrams use Yin/Yang line superposition; Bazi uses Stems/Branches combination and sequence).

Qimen Dunjia's multi-layered chart system can also be seen as another "parallel development" of Wen Wang's system—the arrangement of the Nine Palaces is like the ordering of the Sixty-Four Hexagrams, and the multi-layer superposition is like the superposition of the Six Lines.

From the perspective of information expansion, both systems expand upon the original ancient encoding—but in different directions:

• Bazi's expansion direction: Refinement of the temporal dimension (Four layers: Year $\rightarrow$ Month $\rightarrow$ Day $\rightarrow$ Hour).
• Qimen Dunjia's expansion direction: Expansion of the spatial dimension (Nine Palaces) $\times$ Superposition of the layer dimension (Four layers: Earth, Heaven, Man, Spirit).

Information generated by "temporal refinement": $\approx 60^4 / \text{Constraints} \approx 518,400$ combinations.

Information generated by "spatial expansion $\times$ layer superposition": $\approx 18 \text{ (Bureau)} \times 12 \text{ (Hour)} \times 81 \text{ (Patterns)} \times 8 \text{ (Gates)} \times 8 \text{ (Spirits)} \approx 1,119,744$ combinations.

From this perspective, the combination space of Qimen Dunjia is about 2.16 times that of Bazi.

Of course, this factor of "nine times" mentioned earlier is only a rough estimate—the actual ratio depends on the correlation between the nine palaces. Since the information in the nine palaces is not completely independent—once the Heaven Plate Qiyi of one palace is determined, the Heaven Plate Qiyi of the other palaces are also determined (due to circular permutation)—the independent information capacity is lower, around 1.5 to 2 times that of Bazi.

This estimation roughly aligns with the results of the numerical analysis in the previous section.

Section 3: Duke Zhou's Establishment of Rites—Standardization of Information

Duke Zhou's establishment of rites and music integrated the principles of the Book of Changes into the social structure, transforming the information of the "Heavenly Way" into the norms of the "Way of Man."

The Zuo Zhuan, Duke Zhao Year 2 records the words of Han Xuanzi:

"Only now do I understand the virtue of Duke Zhou and why Zhou was able to reign."

Duke Zhou's system—rites, music, lineage laws, land division—are all standardized encodings of information. In this process of standardization, some of the original information is solidified (becoming institutions), and some is discarded (not incorporated into the institution).

The development of Bazi is similar—from ancient intuition to the systematization of Stems and Branches in the pre-Qin period, and then to the systemization of Ten Gods and Patterns later on. Each step of standardization makes the expression of information more precise, but it may also lead to the loss of some original information.

The development of Qimen Dunjia is similar—from ancient military intuition to the systematization of the Nine Palaces in the pre-Qin period, and then to the complete systemization of the four plates of Heaven, Earth, Man, and Spirit. Its degree of standardization is higher (more complex structure, clearer rules), so it likely retains more of the original information.

Section 4: Information Capacity in Pre-Qin Divination Cases

Let us compare the information capacity of "Four-Pillar style" inference versus "Multi-layer style" inference through actual pre-Qin divination cases.

Case: The Battle of Han Pass (Zuo Zhuan, Duke Xi Year 32-33)

Background: Before the battle between Qin and Jin at Han Pass, several divinations were performed within the Jin state.

"The tortoise said: 'Inauspicious.' The yarrow said: 'Auspicious.'"

Tortoise divination and yarrow divination gave opposite conclusions—a classic example of different information systems yielding different results.

Information from Tortoise Divination: One sign type (Inauspicious)—low information capacity, only one bit (Auspicious or Inauspicious).

Information from Yarrow Divination: Hexagram image, line text, hexagram change—high information capacity, potentially over ten bits.

Furthermore:

"They divined with yarrow, obtaining Gu (Corruption) transforming into Sui (Following). The diviner said: 'The firmness of Gu is Wind; its regret is Mountain. The year is Autumn, we are stripping its fruit and taking its wood. This is how we overcome. When the fruit is stripped and the wood is lost, what remains for failure$5'"

The interpretation process:

1. Obtaining Hexagram Gu—Xun (Wind) below, Gen (Mountain) above, Wind over Mountain.
2. Analyzing the inner trigram (Firmness) as Xun (Wind), the outer trigram (Regret) as Gen (Mountain).
3. Further symbolic inference based on the season (Autumn).
4. Symbolic action: Wind blowing down the fruit from the mountain—"stripping the fruit and taking the wood"—the image of overcoming the enemy.
5. But "fruit stripped and wood lost"—there will be losses after victory.

This process of inference involves multiple information layers:

• Hexagram Image Layer (Wind, Mountain).
• Seasonal Layer (Autumn).
• Material Image Layer (Fruit, Wood).
• Human Affairs Layer (Overcoming the enemy, suffering losses).

The interaction of these four layers produces a judgment of far richer meaning than the hexagram image itself.

This pattern of inference closely resembles the multi-layer interactive analysis of Qimen Dunjia—rather than the single-line temporal projection of Bazi.

Case: Divination of Nan Kuai before Rebellion (Zuo Zhuan, Duke Zhao Year 12)

"When Nan Kuai was about to rebel, someone in his village knew of it, sighed as he passed by, and said: 'So worried! What will come of it$6' Nan Kuai divined with yarrow repeatedly, obtaining Kun (Yielding) transforming into Bi (Clinging/Alliance). He said: 'Yellow skirt, fundamentally auspicious.' Thinking this meant great luck, he showed it to Zifu Huibo. Huibo said: 'I have studied this. If the matter involves loyalty and trust, it is possible; otherwise, it will surely fail. Yellow means the color of the center. Skirt means the adornment of the lower part. Fundamentally auspicious means the greatest good. If the center is not loyal, one cannot attain the color. If the lower part does not share, one cannot attain the adornment. If the undertaking is not good, it cannot be great. Furthermore, Kun transforming into Bi says, "To remain firm in peace brings auspiciousness," and its changing line says, "Straight, square, and great; without deviating from this, nothing is unfavorable." Thus, those who study this and do not deviate are auspicious. If one proceeds with disloyalty and untrustworthiness into great danger, can one escape$7''

Here, Zifu Huibo's interpretation is brilliant—not only interpreting the literal meaning of the hexagram and line texts but also judging based on moral dimensions like "center," "lower part," and "goodness." The depth of information extraction here is far beyond simple auspicious/inauspicious judgment.

Dimensions of information required for such a deep interpretation:

1. Hexagram Image Dimension (Kun, Bi).
2. Line Text Dimension ("Yellow skirt, fundamentally auspicious").
3. Color Image Dimension (Yellow—color of the center).
4. Adornment Image Dimension (Skirt—lower adornment).
5. Moral Dimension (Loyalty and Trust).
6. Situation Dimension (Rebellion—disloyalty and untrustworthiness).

The judgment based on the interaction of these six dimensions is closer to the multi-dimensional analysis of Qimen Dunjia than the simple analysis of Bazi.

This suggests that multi-dimensional interactive judgment was common practice in pre-Qin divination—not just simple fortune-telling as later simplified. Both Bazi and Qimen Dunjia emerged from this multi-dimensional tradition, but Qimen Dunjia retained this tradition more explicitly through its multi-layered structure.

Chapter 11: Information Capacity Verification in Historical Cases

Section 1: The Battle of Muye—Information Needs in Ancient Military Decision Making

The Shangshu, Oath of Muye records King Wu's oath before attacking Zhou:

"The King said: 'Alas! My allied feudal lords, magistrates, ministers of instruction (Sītú), ministers of war (Sīmă), ministers of works (Sīkōng), adjutants (Yǎlǚ), officers of the divisions (Shīshì), commanders of a thousand, commanders of a hundred, as well as the men of Yong, Shu, Qiang, Mao, Wei, Lu, Peng, and Pu peoples. Raise your spears, hold your shields, set up your spears; I shall now swear the oath.'"

King Wu's military decisions required the following information:

1. Temporal Information: When to depart, when to arrive, when to engage.
2. Spatial Information: Where to assemble, which route to march, where to deploy.
3. Celestial Information: Whether the celestial signs were auspicious.
4. Human Information: The morale of the troops, intelligence on the enemy.
5. Geographical Information: The terrain of Muye.

These five categories of information—Time, Space, Celestial, Human, Geography—correspond exactly to the five layers of Qimen Dunjia: Shí Chén (Time), Nine Palaces (Space), Nine Stars (Celestial), Eight Gates (Human Affairs), Earth Plate (Geography).

Bazi can provide only the first category (Time) and the third (Celestial, indirectly via the Five Phases of the Stems and Branches)—it is insufficient to meet all information needs for military strategy.

The Shiji, Basic Annals of the Five Emperors records the divination before King Wu's attack on Yin:

"When King Wu attacked Yin, he first embarked a boat to cross the River. His army had three hundred war chariots, three thousand tiger guards, and forty-five thousand armored soldiers. While crossing the river, a white fish jumped into the King's boat... After crossing, a fire descended from above onto the King's roof, turning into a black bird whose color was red... Although the Yin army was numerous, none had the will to fight."

This record shows that King Wu received various "Heavenly Omens" (Tiānzhào) while crossing the river (white fish jumping, fire turning into a bird), which were interpreted as auspicious signs—an instance of "Heaven-Man Resonance" information transmission.

Case Analysis:

If we assume the moment of departure could be calculated into a Bazi chart, and a Qimen Dunjia chart could also be initiated.

Bazi information provided:

• Five Phases strength/weakness at the time of departure.
• Strength/Weakness of the Day Master.
• Favorable direction inferred indirectly (Utility).
• Trend of Great Cycles and Flowing Years.

Qimen Dunjia information provided:

• The complete pattern of the chart at the time of departure.
• The auspiciousness of each direction.
• The optimal direction for advancement.
• The state of the enemy (auspiciousness of the Guest Palace).
• Favorable/unfavorable aspects of Heavenly timing (Heaven Plate information).
• Advantages/disadvantages of terrestrial location (Earth Plate information).
• Presence or absence of human accord (Man Plate information).
• Warnings about unexpected factors (Spirit Plate information).

Clearly, Qimen Dunjia provides far more information than Bazi—especially in scenarios requiring comprehensive information like military strategy.

Section 2: The Battle of Chengpu—Example of Pre-Qin Military Divination

The Zuo Zhuan, Duke Xi Year 28 records the divination before the Battle of Chengpu between Jin and Chu:

"Marquis Jin dreamed of grappling with the Duke of Chu; the Duke of Chu pinned him down and scraped his brain, causing him to be frightened. Zifan said: 'Auspicious. We have received Heaven; Chu has pinned down its guilt, and we shall treat it gently.' "

Here, Zifan interpreted the ominous dream image ("pining down and scraping the brain") as auspicious by transforming it through "Images"—"receiving Heaven" and "Chu admitting guilt"—this is the flexible application of the multi-faceted nature of "Image."

Furthermore:

"Marquis Jin sent Buyan to divine; he said: 'Auspicious. It resembles the sign of the Yellow Emperor fighting at Banquan.'"

The tortoise divination yielded the sign of "Yellow Emperor fighting at Banquan"—this sign directly referenced the ancient battle against Chiyou, predicting that Marquis Jin would win like the Yellow Emperor. This method of "using ancient events to symbolize present events" adds a historical dimension to the information.

Furthermore:

"Zifan said: 'We fight. If we win quickly, we will certainly gain the allegiance of the feudal lords. If we do not win quickly, we have mountains and rivers guarding us internally and externally, and there will be no harm.'"

Zifan's judgment synthesized multiple pieces of information:

1. Divination information (Auspicious omen).
2. Geographical information (Mountains and rivers guarding Jin—a territorial advantage).
3. Political information (Victory ensures allegiance of feudal lords).
4. Risk assessment (Even if defeated, there is no major harm).

This pattern of multi-dimensional comprehensive judgment closely matches the multi-layer analysis of Qimen Dunjia.

The Zuo Zhuan also records specific tactics during the Battle of Chengpu:

"Jin's army had seven hundred chariots, fully equipped with harnesses and traces. The Marquis of Jin ascended the heights of You Shen to observe the army, saying: 'The young and old have proper rites; they can be used.' He then ordered wood to be cut to augment the troops. On the Day Ji Si, the Jin army formed up north of Shen; Xu Chen and others commanded the auxiliary forces against Chen and Cai. Ziyu, commanding the six battalions of the central army, said: 'Today, Jin shall surely be ruined!' Zixi commanded the left wing, Shangfu commanded the right wing. Xu Chen covered his horses with tiger skins and charged first against Chen and Cai. Chen and Cai fled, and Chu's right wing collapsed. Hu Mao set up two banners and retreated, while Luan Zhi ordered carts to drag burning wood in a feigned retreat. The Chu army charged; Yuan Zhen and Xi Zhen, commanding the central army's clansmen, attacked horizontally. Hu Mao and Hu Yan attacked Zixi's central army from both flanks, and Chu's left wing collapsed. The Chu army was routed."

The information processing in this battle was extremely detailed:

• Temporal Information: "Ji Si" (Exact date—can be used to deduce Stems and Branches).
• Spatial Information: North of Shen, Heights of You Shen (Specific terrain).
• Military Strength Information: 700 chariots.
• Morale Information: Young and old have proper rites.
• Tactical Information: Horses covered in tiger skins, setting up banners to retreat, dragging burning wood in a feigned retreat, central army attacking horizontally, upper army attacking from flanks.

If analyzing this battle with Qimen Dunjia:

• Day Ji Si—Value Symbol/Embodiment can be determined.
• North of Shen—the direction corresponds to a specific Palace.
• Direction of Jin army movement, location of Chu army—can be pinpointed on the Nine Palaces.
• Auspiciousness of the Heaven Plate Stars—corresponds to the benefit of Heavenly timing.
• Opening/Closing of the Eight Gates—corresponds to the appropriateness of action.
• State of the Enemy (Guest Palace).
• Unexpected factors (Spirit Plate).

If only analyzing with Bazi: One could only determine the strength/weakness of the Day Stem Ji Earth on that day, and the general auspiciousness of the day—far insufficient to guide specific tactical deployment.

This case clearly demonstrates that in military (action-oriented) scenarios, the information capacity and dimensions of Qimen Dunjia far exceed those of Bazi.

Section 3: The Battle of Xiao—The Interaction of Time-Space Information

The Zuo Zhuan, Duke Xi Year 32-33 records the Battle of Xiao between Qin and Jin.

Jian Shu's Warning:

"Jian Shu wept and said: 'Mengzi! I saw the army depart but have not seen its return.' The Duke sent someone to ask him: 'What do you know$8 If you are of middle age, the trees on your grave will already be fully grown!' Jian Shu's son was with the army. Jian Shu wept as he sent him off, saying: 'The Jin people will certainly ambush the army at Xiao. Xiao has two ridges: the southern ridge is the tomb of Gao of the Xia Dynasty; the northern ridge is where King Wen first sheltered from wind and rain. They will surely die between them. I will collect your bones there.'"

Jian Shu's judgment was precisely localized to a spatial orientation—"certainly at Xiao," "surely die between them"—this is not merely derived from temporal projections (though the day of departure could be analyzed for fortune), but relies more on spatial analysis (the dangerous terrain of Xiao).

This spatial judgment is precisely the strength of Qimen Dunjia—analyzing the auspiciousness/inauspiciousness of the Nine Palaces and Eight Directions.

The facts proved Jian Shu entirely correct—the Jin army indeed ambushed at Xiao, and the Qin army was annihilated.

Analyzing this from an information capacity perspective:

Information used by Jian Shu:

1. Temporal Information: Unfavorable time of departure (possibly based on Stems/Branches).
2. Spatial Information: Terrain of Xiao as a deadly location.
3. Enemy Information: Jin would certainly set an ambush (based on political analysis).
4. Comprehensive Judgment: Both time and space conditions are unfavorable; defeat is certain.

This comprehensive judgment based on four layers of information cannot be fully provided by a single arcane art—but Qimen Dunjia can provide most of the required elements (Time + Space + Enemy Intelligence—reflected by the auspiciousness of the Guest Palace), whereas Bazi can only provide part of the temporal dimension.

Section 4: The Exile of Chong'er—Validation of Fate Information

The Zuo Zhuan records the story of Duke Chong'er of Jin, who was exiled for nineteen years before returning to rule.

"When he reached Cao, Duke Gong of Cao heard of his fused ribs and wished to see him naked. After bathing, he looked at him thinly. The wife of Xi Fuzi said: 'I observe Duke Chong'er's followers; all are capable of being prime ministers. If you treat them as ministers, the Duke will surely regain his state. When he regains his state, he will surely attain influence over the feudal lords. If he punishes those without propriety, Cao will be the first. Why not attach yourself to him earlier$9'"

The wife of Xi Fuzi's judgment was based on "Human Image" (analysis of personnel information)—analysis of human talent and destiny, which has corresponding elements in the Bazi system (Bazi can infer a person's talent, destiny) and in Qimen Dunjia (auspiciousness of the Man Plate Eight Gates).

Chong'er's Bazi chart (if calculable) must necessarily show an extremely noble fate—the image of an emperor. And his nineteen-year exile, his Great Cycles must show a turning point from initial misfortune to later fortune. This type of "life trajectory" information is precisely the specialty of Bazi—Qimen Dunjia is not skilled in such long-term fate divination.

Thus, this case demonstrates: In the domain of long-term destiny (the pattern of a person's entire life), the information specialization of Bazi surpasses that of Qimen Dunjia. Although Qimen Dunjia has a larger total information volume, it is less profound in this specific area.

This leads to an important correction—the comparison of information capacity cannot only look at the total volume but must also consider the effective information capacity for a specific question:

• On the question of "A person's destiny throughout life": Bazi's effective information capacity > Qimen Dunjia.
• On the question of "Decision making for a specific time-space event": Qimen Dunjia's effective information capacity > Bazi.
• On "Comprehensive questions concerning Heaven, Man, and Earth": The two arts each have their strengths, and the total volume is roughly comparable.

Chapter 12: Constructing the Unified Conclusion

Section 1: Re-examining the Core Question

The core question of this paper is: "Between Bazi and Qimen Dunjia, which one carries a greater volume of information$10"

After detailed analysis in the preceding chapters, we have reached the following preliminary conclusions:

1. From the Mathematical Combination Perspective: Qimen Dunjia's combination space is slightly larger than Bazi's (about 1.5-2 times).
2. From the Information Dimension Perspective: Qimen Dunjia's dimensions (12) exceed those of Bazi (8).
3. From the Information Density Perspective: Bazi's density (31-38 bits/character) exceeds that of Qimen Dunjia (5.8-9.3 bits/symbol).
4. From the Effective Information Perspective: This depends on the question asked—Bazi is superior for fate calculation, Dunjia is superior for decision-making.
5. From the Image Semantic Space Perspective: They are roughly comparable, with Qimen Dunjia slightly larger.
6. From the Heavenly Way Coverage Perspective: Qimen Dunjia is more comprehensive as it encompasses both time and space.

These conclusions seem to point toward the conclusion that "Qimen Dunjia's information capacity is greater than Bazi's"—but this needs a more rigorous distinction of the different meanings of "information capacity."

Section 2: The Triple Meaning of "Information Capacity" in Pre-Qin Context

In the pre-Qin context, "information capacity" can be understood in three layers:

First Layer: Number (Shù) (Quantity)

"Number" refers to quantifiable information—combination counts, bit counts, permutation counts.

From this perspective: Qimen Dunjia $\geq$ Bazi.

Reason: Qimen Dunjia's multi-layered plate structure (Heaven, Earth, Man, Spirit) and its simultaneous encoding of time and space produce a larger combinatorial space than Bazi's four-pillar structure.

Second Layer: Image (Xiàng) (Quality)

"Image" refers to perceptible information—material images, implied meanings, symbolism.

From this perspective: The two arts each have their own strengths, making direct comparison difficult.

Reason: Each character in Bazi carries extremely rich symbolic meaning (high information density), capable of mapping to countless things; Qimen Dunjia's individual symbols carry relatively simpler meanings, but the cross-mapping of multiple symbols generates an extremely rich set of combined meanings. Overall semantic space is roughly comparable.

Third Layer: Principle/Application (Lǐ) (Utility)

"Principle" refers to applicable information—the scope and depth of problems that can be answered.

From this perspective: Qimen Dunjia > Bazi.

Reason: Qimen Dunjia can answer a far wider range of questions than Bazi—not only destiny calculation but also military strategy, choosing directions, selecting auspicious times, judging events, finding lost objects, assessing illness, etc. (hence its title as the "Art of the Emperor"). Bazi is primarily used for questions centered on "Man"—destiny, marriage suitability, career choice, etc.

Synthesizing the analysis across the three layers:

From the perspective of the three layers: If measured by "breadth," Dunjia is greater; if measured by "density," Bazi is greater; if measured by "total quantity," Dunjia is slightly greater; if measured by "origin," both come from the Dao and are incomparable.

Section 3: Unifying the Conclusion with the Principles of the Book of Changes

The Xici Zhuan states:

"One Yin and one Yang is called the Dao. What follows it is Goodness; what completes it is Nature. The benevolent see it and call it benevolence; the wise see it and call it wisdom. The common people use it daily yet do not know it, thus the Way of the Superior Man is rare."

"One Yin and one Yang is the Dao"—the essence of the Dao lies in the interaction of Yin and Yang. Both Bazi and Qimen Dunjia are built upon Yin and Yang—this is their commonality.

"The benevolent see benevolence, the wise see wisdom"—the same Dao, seen from different angles by different people—results in different understandings. Bazi "sees the Dao" from the time perspective; Qimen Dunjia "sees the Dao" from the time-space perspective—the scope of what they see differs, thus their information capacities differ.

The Xici Zhuan further states:

"The Yi has four ways of the Sage: those who speak value its diction; those who act value its change; those who fashion implements value its image; those who divine value its judgment."

"Diction" (Cí), "Change" (Biàn), "Image" (Xiàng), "Judgment" (Zhàn)—These four aspects: Diction pertains to textual information (verbal layer); Change pertains to dynamic information (temporal layer); Image pertains to structural information (spatial layer); Judgment pertains to decision-making information (application layer).

Bazi excels in "Change" (dynamic temporal inference). Qimen Dunjia excels in "Image" (multi-layered structural presentation) and "Judgment" (multi-dimensional decision support).

Comparing the four aspects:

• Bazi coverage: Diction (partial, e.g., pattern names) + Change (Great Cycles/Years) = 2/4.
• Qimen Dunjia coverage: Diction (partial) + Change (chart rotation/bureau change) + Image (Nine Palaces multi-layer) + Judgment (comprehensive application) = 4/4.

From the perspective of coverage of the "Four Ways of the Sage," Qimen Dunjia is more comprehensive.

However, the Xici Zhuan immediately follows by saying:

"Therefore, when a Superior Man is about to act, about to proceed, he asks about it and receives the response as an echo. There is no distance near or far, hidden or deep, whereby he comes to know the future. If it is not the utmost subtlety of the world, who can participate in this$1"

"About to act," "about to proceed"—this specifically refers to moments requiring action and decision-making. At these times, one needs to "ask and receive the response as an echo"—requiring a comprehensive information system to support decision-making. "No distance near or far, hidden or deep"—regardless of spatial distance or hidden depth—information is provided. This description aligns more closely with the characteristics of Qimen Dunjia.

If the objective is not "action" or "acting," but purely understanding a person's inherent destiny pattern—then Bazi's specialized depth is more appropriate.

The Xici Zhuan also states:

"Interspersing Threes and Fives brings change, interweaving their numbers. By penetrating their changes, one completes the patterns of Heaven and Earth. By exhausting their numbers, one determines the images of the world below. If it is not the utmost change in the world, who can participate in this$2"

"Interspersing Threes and Fives brings change, interweaving their numbers"—the complex numerical operations involving variation and interweaving. This precisely describes the multi-layered interwoven operation of Qimen Dunjia—the "interweaving" (Cuò) of Heaven Plate and Earth Plate, the "interlacing" (Zōng) of the Man Plate and Spirit Plate, and the "interspersing" (Cānwǔ) of the Nine Palaces.

"Penetrating their changes, one completes the patterns of Heaven and Earth"—by mastering all changes, one completes the texture (the entirety of information) of Heaven and Earth.

This suggests: The size of the information capacity is directly related to the richness of the "Change"—and the more varied the "Change," the greater the information capacity.

Bazi's "Change": The sequential change of Heavenly Stems and Earthly Branches in the Great Cycles and Flowing Years—a one-dimensional change (linear progression along the time axis).

Qimen Dunjia's "Change": The rotation and superposition of multi-layered plates—a multi-dimensional change (cross-progression of Time and Space).

Multi-dimensional change is richer than one-dimensional change—thus, from the perspective of "Change," the information capacity of Qimen Dunjia is also greater than that of Bazi.

Chapter 13: Further Reflections and Deep Questions

Section 1: Why Did the Ancients Create Multiple Arcane Arts Instead of Just One$3

This question is very important. If one arcane art could "vastly and fully encompass" (guǎngdà xī bèi) the entire information of the Heavenly Way, why would multiple arts need to be created$4

The Xici Zhuan states:

"All under Heaven have the same destination but follow different paths; they have one goal but a hundred considerations." (Tiānxià tóng guī ér shū tú, yī zhì ér bǎi lǜ.)

"Same destination but different paths"—the goal is the same (to attain the Dao of Heaven), but the paths are different (various arcane arts). The reason multiple arts are needed is precisely because no single arcane art can completely carry the entire information of the Heavenly Way.

This is a mathematical necessity. As analyzed previously: The information of the Heavenly Way tends toward infinity ("the greatest without outside"), while the information capacity of any arcane system is finite—Bazi about 300 bits, Qimen Dunjia about 400 bits—both are only small fragments of the Heavenly Dao.

Therefore, multiple arcane arts are needed to "extract" the Dao's information from different angles—like needing multiple cameras to film a scene from different viewpoints to restore its entirety.

Bazi extracts information from the "Time" angle. Qimen Dunjia extracts information from the "Time-Space" angle. Liuren extracts information from the "Human Affairs" angle. Taiyi extracts information from the "National Destiny" angle.

Using multiple arts together can approximate (but never achieve) the full information of the Heavenly Dao.

Section 2: Does Greater Information Capacity Mean Greater Accuracy$5

This question is also crucial. Greater information volume does not equate to higher accuracy.

The Zuo Zhuan, Duke Xiang Year 9 records the divination of Lady Mu before her death:

"Lady Mu died in the Eastern Palace. When first they divined about her leaving, they obtained Gen transforming into Sui. The official said: 'This is called Gen transforming into Sui; it means she will leave. The Lord must leave quickly.' Lady Mu said: 'No. In the Zhou Yi, it says: "Sui, supremely prosperous, beneficial and firm, without blame." Yuan means the greatest in body. Heng means the confluence of good fortune. Li means the harmony of righteousness. Zhen means the foundation of affairs. Possessing the four virtues, following leads to no blame. Now I, a woman, am involved in chaos. I am in a low position and lack benevolence, so I cannot be Yuan. I do not settle the state, so I cannot be Heng. Acting rashly harms the body, so I cannot be Li. Abandoning my position for adultery, so I cannot be Zhen. I lack all four virtues, how can I follow without blame$6 I embrace wickedness, how can I be blameless$7 I shall surely die here and not get out.'"

This passage is extremely classic. The divination yielded Gen not changing (fixed Gen, or Gen transforming into Sui), and the official advised her to leave the Eastern Palace quickly. But Lady Mu, deeply understanding the doctrine, argued that she lacked the Four Virtues of Yuan, Heng, Li, and Zhen, so even if the hexagram implied "Following," she could not avoid blame—she ultimately died in the Eastern Palace as predicted.

This case reveals: The same hexagram image, interpreted by different readers, yields different conclusions—the accuracy of the interpretation depends on the reader's "Clarity" (mastery of doctrine), not the sheer "amount" of information.

By extension: Although Qimen Dunjia has a greater information volume than Bazi, if the interpreter cannot correctly process this large amount of information, they might be less accurate than an interpreter who has mastered the precision of Bazi.

The Laozi, Chapter 71, states:

"Knowing what one knows and knowing what one does not know—this is true knowledge. Not knowing what one knows and knowing what one does not know—this is delusion."

The information capacity of any arcane art, no matter how large, has its boundaries. Understanding these boundaries is the key to using the arts effectively.

Thus, the size of the information capacity is an objective mathematical problem, while the accuracy of the application is a subjective problem of skill. The two should not be conflated.

Section 3: Why is the Information Density of Bazi So High$8

This question deserves deep investigation. We previously established that Bazi's information density is about 2-3 times that of Qimen Dunjia—carrying extremely rich information with only 8 characters. Whence does this high density come$9

The answer lies in the "encoding efficiency" of the Stems and Branches.

A Heavenly Stem (Jia, Yi, Bing, Ding, etc.) carries information:

• Five Phases (one of 5 types).
• Yin/Yang (one of 2 types).
• Sequential Position (one of 10 positions).
• Direction (one of 5 directions).
• Season (one of 5 seasons).
• Five Tastes (one of 5 tastes).
• Five Colors (one of 5 colors).
• Five Tones (one of 5 tones).
• Five Viscera (one of 5 viscera).
• ...

Each Heavenly Stem carries information across at least 10 dimensions—each dimension being about 1-3 bits. Totaling about 10-20 bits/Stem.

An Earthly Branch carries even more information (due to Hidden Stems, Three Combinations, etc.), about 15-25 bits/Branch.

Thus, one Pillar (one Stem and one Branch) carries about 25-45 bits of information. Four Pillars total about 100-180 bits (after deducting redundancy).

The root of this high density lies in this: The Stems and Branches are a form of "multi-dimensional encoding"—a single symbol encodes information across multiple dimensions.

This is similar to a modern barcode—a single barcode encodes product category, origin, sequence number, etc. But the dimensions encoded by the Stems and Branches are far more numerous—and the dimensions have rigorous internal logical relationships (generation/overcoming of Five Phases, correspondence of Yin/Yang), making the encoding extremely compact.

This efficient encoding demonstrates the profound wisdom of the ancient Sages—to carry the richest information in the simplest form. The Xici Zhuan's statement "Simplicity leads to the understanding of the Dao under Heaven" (Yì jiǎn ér tiānxià zhī lǐ dé yǐ) refers precisely to this meaning.

Section 4: Why Does Qimen Dunjia Require Such a Complex Structure$10

Since the encoding efficiency of Stems and Branches is so high, why didn't Qimen Dunjia adopt a concise form like Bazi$11 Why construct the complex structure of four plates and nine palaces$12

The answer is: Because the problems Qimen Dunjia seeks to address are an order of magnitude more complex than those addressed by Bazi.

Bazi primarily addresses the core question: "What is the innate destiny pattern of this person$13"—a problem centered on "Man" and primarily utilizing the "Time" axis. The problem dimensions are about: 1 (Person) $\times$ 1 (Time Axis) = 1 Dimension.

Qimen Dunjia primarily addresses the question: "At this time and in this place, what action should be taken, and which direction should be pursued$14"—a problem centered on "Event" and utilizing "Time + Space" as dual axes. The problem dimensions are about: Multiple (types of events) $\times$ 2 (Time + Space) = Multi-dimensional.

The higher the dimensionality of the problem, the more complex the required information structure—this is inevitable.

Use the "Five Factors" from Sunzi's Art of War as an analogy:

"Hence the five fundamental factors must be compared when making plans: First, the Way (Dào); second, Heaven (Tiān); third, Earth (Dì); fourth, the Commander (Jiāng); and fifth, Method (Fă)."

The Five Factors—Dao, Heaven, Earth, Commander, Method—represent at least five dimensions. If one tries to address a five-dimensional problem with a simple structure like Bazi, the information is insufficient to support decision-making. Qimen Dunjia's four-plate structure—Heaven Plate (Heaven), Earth Plate (Earth), Man Plate (Commander/Man), Spirit Plate (Dao/Spirit)—coincidentally corresponds to most of the dimensions required by the Five Factors.

Thus, the complex structure of Qimen Dunjia is not over-engineering—it is necessitated by the complexity of the problem it aims to solve.

Section 5: Can the Two Arts Be Used Concurrently$15

Since Bazi and Qimen Dunjia each have their strengths, can they be used together$16

The answer is Yes, and this practice existed in antiquity.

In pre-Qin divination practice, there are examples of "using multiple methods concurrently":

The Zuo Zhuan, Duke Xi Year 4 records:

"Initially, Duke Xian of Jin wished to make Li Ji his principal wife, and divined about it: inauspicious. He used yarrow, which was auspicious. The Duke said: 'Follow the yarrow.' The diviner said: 'Yarrow is short, tortoise is long; it is better to follow the long one. Moreover, the line text says: "Focusing on deviation, pushing away the Lord's strength. One fragrance, one stench; after ten years, the smell remains." It must not be done.' The Duke did not listen."

This is an example of using both tortoise divination (Gui) and yarrow divination (Shi) simultaneously. The two methods gave contradictory conclusions—Tortoise inauspicious, Yarrow auspicious—the diviner advocated "following the long" (the tortoise), but Duke Xian chose the "yarrow"—and the result proved the tortoise divination correct.

This case shows that using multiple arcane arts concurrently can verify and supplement each other—when conclusions align, confidence increases; when they contradict, it signals the need for more cautious judgment.

Combining Bazi and Qimen Dunjia:

• Use Bazi to determine a person's innate pattern—to know the essence of their fate.
• Use Qimen Dunjia to determine the optimal time and space for action—to choose the advantageous path.

The combined information capacity = Bazi Info + Qimen Dunjia Info - Overlapping Information.

If the overlapping information is about 30% of each, then: Combined Capacity $\approx 300 + 400 - 0.3 \times \min(300, 400) \approx 300 + 400 - 90 = 610$ bits.

This is far greater than using either art alone—about twice that of Bazi and 1.5 times that of Qimen Dunjia.

Therefore, combining the two arts can yield the maximum information capacity and the most comprehensive decision support.

Section 6: The Limit of Information Capacity—The Boundary of Arcane Arts

Any arcane system has an upper limit to its information capacity. This limit is determined by its mathematical structure.

Bazi Information Limit: $\approx 518,400$ combinations $\times$ Ten Gods relations $\times$ Great Cycles/Years $\approx 10^8-10^{10}$ states.

Qimen Dunjia Information Limit: $\approx 18 \text{ Bureaus} \times 60 \text{ Hours} \times 81 \text{ Patterns} \times 8 \text{ Gates} \times 8 \text{ Spirits} \approx 10^6-10^8$ states.

However, the existence of an upper limit does not imply the inadequacy of the system—because the number of distinguishable states in the real world is also finite.

The number of distinct temporal segments in a human lifetime (using shí chén as the unit) is about 345,000—far less than the state limit of Bazi. The number of different decision options for a military engagement is a few hundred to a few thousand—far less than the state limit of Qimen Dunjia.

Therefore, in their respective fields of application, the information capacity of both arts is "sufficient"—the difference lies only in the amount of "redundancy" beyond what is sufficient.

Bazi Redundancy $\approx$ State Limit / Actual Need $\approx 10^8 / (3.5 \times 10^5) \approx 286$ times.

Qimen Dunjia Redundancy $\approx$ State Limit / Actual Need $\approx 10^7 / 10^3 \approx 10,000$ times.

Qimen Dunjia has much greater redundancy than Bazi—this implies that Qimen Dunjia has a larger margin for error in application; even if the interpretation has some error, it is less likely to lead to a catastrophic misjudgment.

Greater redundancy also means lower precision requirements—in some cases, Qimen Dunjia might not need to reach the level of detail required by Bazi to satisfy the need.

This confirms the conclusion from before: Bazi is deep but narrow, Dunjia is broad but wide—each is suited for its purpose.

Section 7: Returning to the "Dao"—The Ultimate Answer to the Information Capacity Question

Ultimately, we must return to the level of the "Dao" to examine the question of information capacity.

The Laozi, Chapter 1:

"The Dao that can be told is not the eternal Dao. The name that can be named is not the eternal Name. Nameless is the beginning of Heaven and Earth; named is the Mother of the myriad things."

All the information in arcane arts—whether the 300 bits of Bazi or the 400 bits of Qimen Dunjia—is "tellable Dao," "namable Name"—not the "eternal Dao," not the "eternal Name."

What is the information capacity of the "eternal Dao"$17 It is infinite.

No matter how much it is expanded, any arcane art can never reach the "eternal Dao"—this is a mathematical necessity (a finite number is always less than infinity) and the profound truth of metaphysics (the "Dao" cannot be completely encoded).

Therefore, when comparing the information capacities of Bazi and Qimen Dunjia, on the level of the "Dao," the comparison is between two finite quantities—and compared to infinity, both are effectively zero.

The Zhuangzi, Autumn Floods Chapter states:

"One cannot speak of the ocean to a frog in a well, for it is constrained by its habitat. One cannot speak of ice to a summer insect, for it is bound by its season. One cannot speak of the Dao to a narrow scholar, for he is constrained by his teachings."

The "teachings" (jiào) of the arcane arts, while individually marvelous, are all teachings of "narrow scholars"—partial knowledge, limited views. To "speak of the Dao"—to achieve a complete understanding of the Heavenly Way—is beyond the capacity of any single arcane art.

Thus, the final unified conclusion must add one final sentence:

On the level of the "Dao"—the information capacity of both Bazi and Qimen Dunjia is zero—compared to infinity, all finite quantities equal zero.

This is not nihilism—it is a reminder to those who study arcane arts to maintain humility. The arts are useful, but finite; they have areas where they reveal spiritual efficacy, but also areas where they fail to reach.

The final teaching of the Xici Zhuan:

"The Spirit has no fixed place, and Yi has no fixed form." (Shén wú fāng ér yì wú tǐ.)

The marvelous aspect is not confined to any one direction (it is not limited to the timeline of Bazi, nor to the Nine Palaces of Qimen Dunjia). The true essence of Yi resides not in any concrete structure (neither the form of the Four Pillars nor the body of the Four Plates).

To transcend the form and directly grasp the body of the Dao—this is the highest realm of arcane art cultivation.

Chapter 14: Summary of Mathematical Parameters

Section 1: Summary of Bazi Mathematical Parameters

Section 2: Summary of Qimen Dunjia Mathematical Parameters

Section 3: Comparative Summary Table

Chapter 15: Conclusion

Section 1: Concluding Remarks

This article has traced the origins from pre-Qin classics and ancient traditions, using mathematical structure as the warp and metaphysical doctrine as the weft, to systematically compare the information capacities of Bazi and Qimen Dunjia.

After fourteen chapters of detailed analysis, we arrive at the following unified conclusion:

Conclusion 1 (Mathematical Perspective): Qimen Dunjia's information capacity (total volume) is greater than Bazi's, by about an order of magnitude. This difference primarily stems from Qimen Dunjia's multi-layered plate structure (Heaven, Earth, Man, Spirit) and its encoding method that encompasses both time and space.

Conclusion 2 (Metaphysical Perspective): Bazi, through the method of "Simplicity" (Yì Jiǎn), embodies the essence of fate calculation with extremely high information density; Qimen Dunjia, through the method of "Vastness" (Guǎng Dà), covers the comprehensive information of Heaven, Earth, and Man with extreme breadth. The two arts grasp different aspects of the Yi and are mutually indispensable.

Conclusion 3 (Unified Perspective): The size of the information capacity depends on the standard of measurement—Dunjia is greater in "breadth" and "total volume"; Bazi is greater in "density"; in terms of "applicability," each has its strengths; on the level of the "Dao," both are finite.

Conclusion 4 (Practical Perspective): For questions of fate calculation, Bazi is superior—using simplicity to derive profound insight. For questions of action, Dunjia is superior—using vastness to master transformation. Using both arts concurrently yields the maximum information capacity and the most comprehensive judgment.

Section 2: Concluding with the Words of the Sages of Pre-Qin

The final words of the Book of Changes, Great Treatise (Part II) are aptly suited to conclude this paper:

"The Yi as a writing, traces the origin and grasps the end, serving as its substance. The six lines intermix, corresponding only to the time and the object. Its beginning is difficult to know, its culmination is easy to know—this is the beginning and the end. If one is to interweave the myriad things and record their virtues, distinguishing right from wrong, then the middle lines must be fully prepared."

"Tracing the origin and grasping the end" (yuánshǐ yàozhōng)—tracing the origin (the birth moment of Bazi), grasping the end (the action result of Qimen Dunjia).

"The six lines intermix, corresponding only to the time and the object" (liù yáo xiāngzá, wéi qí shí wù yě)—the intermixing of multiple layers of information (the multi-plate superposition of Qimen Dunjia) only reveals its meaning in specific time and circumstances.

"Its beginning is difficult to know, its culmination is easy to know" (qí chū nán zhī, qí shàng yì zhī)—the beginning of a matter is hard to predict (the profound nature of innate Bazi destiny is difficult to fathom), while the end result is relatively easy to know (the action outcome in Qimen Dunjia is easier to determine).

"Interweaving myriad things and distinguishing right from wrong, then the middle lines must be fully prepared"—to integrate various things and distinguish right from wrong, one must rely on complete information—and "complete" information requires both the "depth" of Bazi and the "breadth" of Dunjia.

Finally, concluding with a sentence from the Xici Zhuan:

"Thus the Yi is Image; Image is likeness." (Shì gù yì zhě, xiàng yě. Xiàng yě zhě, xiàng yě.)

All arcane arts are "Images" (Xiàng) of the Heavenly Dao—likenesses of the Heavenly Dao. The clarity of the likeness (information capacity) depends on the size and precision of the "mirror." Bazi is a small, precise mirror; Qimen Dunjia is a large, broad mirror. Using both mirrors together allows for a more complete reflection of the whole picture of the Heavenly Dao.

And the Heavenly Dao itself—that which the "Image" "likens"—always transcends the reflection of any mirror.

This is the final unified expression of the conclusion from mathematics and metaphysics.

References Cited:

The Book of Changes (Complete Text of Classic and Commentaries) The Book of Documents (Canon of Yao, Grand Plan, Oath of Mu Ye, Oath of Gan) The Zuo Zhuan (Duke Zhuang Year 22, Duke Xi Years 4, 15, 28, 32-33, Xuan Gong Year 3, Duke Xiang Year 9, Duke Zhao Years 2, 7, 12) Guoyu (Discourses of the States, Zhou Yu) Laozi (Chapters 1, 25, 42, 57, 71) Zhuangzi (Discussion on Making Things Equal, Heaven and Earth, Transcending Things, Autumn Floods) Guanzi (Inner Cultivation, Five Phases, Four Seasons, Minor Officials) Sunzi's Art of War (Laying Plans, Maneuvering for Position, Nine Grounds) Lüshi Chunqiu (Twelve Records, Records of Origin) Erya (Distinctions of Heaven) Heguanzi (Circular Flow Chapter) Shiji (Basic Annals of the Five Emperors, Basic Annals of Zhou) Shangshu Dazhuan (Relevant Chapters)

Drawn up by the Xuanji Editorial Department

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