The Evolution of the Heavenly Stems and Earthly Branches and the Transformation of Dun Jia: A Unified Mathematical Metaphysical Study of the Information Capacity in Bazi and Qimen Dunjia

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."