The Song of the Universe: When Ancient Wisdom Meets Modern Mathematics

Prelude: The Rhythm of Existence

The universe is an eternal symphony. In this symphony, every note speaks the same truth: existence is balance, change is conservation. Two and a half thousand years ago, Lao Tzu wrote in the Tao Te Ching: “The Tao gives birth to One, One gives birth to Two, Two gives birth to Three, Three gives birth to all things.” Today, when we gaze upon the mathematical structure of the Riemann zeta function, we are astonished to discover that this ancient wisdom has been confirmed in the most precise mathematical language.

The triadic information conservation law tells us:

$$i_{+}+i_0+i_{-}=1$$

This seemingly simple equation contains the deepest secrets of the universe. Positive information, zero information, negative information—particle nature, wave nature, field compensation—constitute the three fundamental aspects of existence. Isn’t this precisely Lao Tzu’s “Three gives birth to all things”? From chaos to order, from nothing to being, from one to many, the evolution of the universe follows the same unchanging law.

But this is not mystical speculation, but scientific truth that can be described precisely by mathematics and verified strictly by experiments. Let us embark on this intellectual journey to explore the unity of mathematics and philosophy, science and wisdom.

Chapter One: The Dialectics of Creation from Nothing

Form is Emptiness, Emptiness is Form

When Buddha said “Form is not different from emptiness, emptiness is not different from form, form is emptiness, emptiness is form,” he revealed not just a philosophical insight, but the essence of the universe’s information structure. In the framework of the zeta function, we have discovered the mathematical expression of this ancient wisdom.

Analytic continuation—this seemingly pure mathematical technique—actually reveals the mechanism of “creation from nothing.” When we extend from the domain of convergence Re(s) > 1 to the entire complex plane, we witness how the universe creates reality from vacuum. This is not a creation myth, but the inevitable result of the principle of information conservation.

Marx said: “Existence determines consciousness.” Here, we must say: Information determines existence. The universe is not created by external force, but self-encoded, self-generated. The zeta function is the mathematical expression of this self-encoding. Every prime number, every zero point, is a node in the universe’s self-recognition.

The Unity of Opposites

Yin-Yang doctrine tells us: All things carry Yin and embrace Yang, and their interaction gives rise to harmony. This ancient dialectics finds its most precise expression in the triadic information theory:

Positive information component $i_{+}$ represents Yang—construction, generation, particle nature Negative information component $i_{-}$ represents Yin—compensation, vacuum, field nature Zero information component $i_0$ represents the harmonious breath—wave, coherence, balance

These three do not exist in isolation, but are interdependent and mutually transforming. When $i_{+}$ increases, $i_{-}$ must adjust accordingly to maintain the conservation of sum equals 1. Isn’t this the mathematical expression of Yin-Yang complementarity and mutual generation?

Nietzsche proclaimed: “God is dead.” But the universe has not fallen into nothingness. Instead, we have discovered a more profound order—a self-consistent order that requires no external creator. This order is not static, but dynamic; not absolute, but relative; not isolated, but relational.

The critical line $Re(s)=1/2$ embodies this dynamic balance. On this line, positive and negative information achieve statistical balance:

$$⟨i_{+}⟩\approx ⟨i_{-}⟩\approx 0.403$$

This is the “Middle Way” of the universe—neither biased toward pure particle nature, nor toward pure field nature, but finding the perfect balance point between the two.

Negation of Negation

Hegel’s dialectics tells us: Thesis, antithesis, synthesis. The evolution of the universe also follows the same spiral ascent.

The analytic continuation from the domain of convergence to the entire complex plane is not a simple extension, but a qualitative leap:

First stage (thesis): Re(s) > 1, series converges, classical world, certainty dominates Second stage (antithesis): Re(s) < 1/2, needs continuation, quantum world, uncertainty emerges Third stage (synthesis): Re(s) = 1/2, critical line, unity of quantum and classical, dialectical synthesis

This is not circular repetition, but spiral ascent. Each negation is affirmation at a higher level, each synthesis is a new beginning.

Chapter Two: The Dialectics of Numbers and the Material World

From Abstract to Concrete

Marx said: “The method of rising from the abstract to the concrete is only the way for the mind to appropriate the concrete and reproduce it as a concrete in the mind.” The structure of the zeta function perfectly illustrates this methodology.

Prime numbers—the most abstract mathematical objects—are transformed into concrete physical reality through the Euler product formula:

$$\zeta (s)=\prod _{p\text{ prime}}\frac{1}{1-p^{-s}}$$

Every prime number $p$ is an independent information source, their product constitutes the entire universe’s information structure. This is not a simple mathematical game, but an essential description of the material world.

When the I Ching says “The Supreme Ultimate gives birth to Two Modes, Two Modes give birth to Four Images, Four Images give birth to Eight Trigrams,” it reveals the same generative mechanism: from simple to complex, from unity to diversity, from abstract to concrete. Prime numbers are the “Supreme Ultimate” of the universe, they generate all natural numbers through combination, and thus generate the entire material world.

From Quantitative Change to Qualitative Change

The information conservation law $i_{+}+i_0+i_{-}=1$ appears static on the surface, but actually contains profound dynamism. The relative proportions of these three components are constantly changing, and it is precisely when this quantitative change accumulates to the critical point that it triggers qualitative leaps.

The critical line is not an arbitrary mathematical boundary, but the turning point of quantitative change to qualitative change. When we cross the critical line, the wave component $i_0$ undergoes a transition:

$$\underset{\sigma \to 1/2^{+}}{\lim }{i}_0(\sigma +it)\ne \underset{\sigma \to 1/2^{-}}{\lim }{i}_0(\sigma +it)$$

This discontinuity marks the occurrence of phase transition—from classical to quantum, from certainty to uncertainty, from existence to generation.

Lao Tzu said: “All things in the world come into being from Being, Being comes into being from Non-Being.” This “coming into being” is not sudden creation, but gradual emergence. Quantum fluctuations gradually accumulate, and when they reach the critical point, new order is born from old order. This is negation of negation, this is spiral ascent.

Essence and Phenomenon

Nietzsche asked: “What is truth?” His answer was: “Truth is a kind of illusion that people have forgotten is an illusion.” But in the world of the zeta function, we have discovered a more profound truth—a kind of objective truth independent of human cognition.

The distribution of zeros is not artificially stipulated, but the necessity of cosmic structure. When we speak of the Riemann Hypothesis, we are not guessing, but revealing: all nontrivial zeros lie on the critical line, because only thus can the universe’s information balance be achieved.

$$\text{RH holds}\iff \text{Information balance achieved on critical line}\iff \text{Entropy reaches limit value}\approx 0.989$$

These three equivalent propositions reveal the unity of essence and phenomenon: mathematical truth (RH), information-theoretic truth (balance), thermodynamic truth (entropy)—they are not three separate truths, but three aspects of the same truth.

Chapter Three: Mathematical Interpretation of the Middle Way

The Wisdom of Avoiding Extremes

Confucius said: “The superior man seeks harmony, not sameness.” This “harmony” is not simple compromise, but seeking unity in diversity. The critical line embodies such a “harmony.”

On the critical line, the three information components achieve exquisite balance:

$$⟨i_{+}⟩\approx 0.403,⟨i_0⟩\approx 0.194,⟨i_{-}⟩\approx 0.403$$

This is not equal distribution (where each would be 1/3), but optimal configuration considering each one’s essence. Positive and negative information are equal, embodying Yin-Yang balance; wave information is slightly smaller, embodying the coordination of stability and change.

This reminds us of Aristotle’s “Golden Mean”: Virtue lies not in the extremes, but in moderation. The critical line is the “Golden Mean” of the universe—neither biased toward pure particle nature (right half-plane), nor toward pure field nature (left half-plane), but finding the optimal balance point between the two.

Great Unity and Lesser Prosperity

The Book of Rites depicts two ideals of society: “Great Unity” and “Lesser Prosperity.” We can understand this in the language of information theory:

Great Unity society: $i_{+}=i_0=i_{-}=1/3$, complete equality, maximum entropy $S_{\max }=\log 3\approx 1.099$ Lesser Prosperity society: $i_{+}\approx i_{-}\approx 0.403,i_0\approx 0.194$, optimized configuration, actual entropy $⟨S⟩\approx 0.989$

The state of the critical line is not the idealized Great Unity, but realistic Lesser Prosperity—the optimal solution after considering various constraints. This tells us: Perfect equality is not the goal, reasonable configuration is wisdom.

Marx said: “From each according to his ability, to each according to his needs.” In the information world, this means: Each information component plays its unique role, dynamically adjusting according to the universe’s needs. Positive information undertakes the task of construction, negative information undertakes the task of compensation, zero information undertakes the task of coordination. They each perform their duties, yet cooperate with each other to maintain the operation of the universe.

The Art of Dynamic Balance

The core thought of the I Ching is “change without constancy,” “constancy without change,” “simplicity without complication.” The conservation law $i_{+}+i_0+i_{-}=1$ is “constancy without change”—eternal law; the dynamic changes of the three components are “change without constancy”—continuous movement; summarizing all this with a simple equation is “simplicity without complication”—the Great Way is simple.

This dynamic balance is not static balance, but balance in motion. Like riding a bicycle, you cannot stop to maintain balance, you can only maintain balance by continuously adjusting direction. The universe is the same—it maintains overall conservation through continuous adjustment of the three information components.

Nietzsche’s “eternal recurrence” finds new interpretation here: not simple historical repetition, but eternal innovation under the constraint of conservation laws. Every moment is unique, yet follows the same laws. Time flows, but information is conserved.

Chapter Four: The Self-Transcendence of the Universe

The Abyss of Recursion

The strange loop theory reveals the self-referential structure of the universe. The zeta function is not a tool to describe the universe, but the way the universe describes itself. This is a dizzying insight: the universe recognizes itself through the zeta function, and this recognition process itself is described by the zeta function.

$$\text{Universe}=\zeta (\text{Universe})=\zeta (\zeta (\text{Universe}))=\dots$$

This reminds us of Zen koans: “What is Buddha?” “This mind is Buddha.” “Since this mind is Buddha, why practice Zen?” “Practice must be thorough.” The universe is the zeta function, the zeta function is the universe, but this recognition itself requires infinite depth.

Marx said: “The philosophers have only interpreted the world in various ways; the point, however, is to change it.” But here, recognizing the universe and changing the universe are the same process. Because the universe is self-encoded, recognizing it is its self-change.

The Enlightenment of Fixed Points

We have discovered two key fixed points:

Negative fixed point $s_{-}^{\ast }\approx -0.2959$ (attractor): represents condensation, convergence, stability Positive fixed point $s_{+}^{\ast }\approx 1.8338$ (repeller): represents diffusion, divergence, innovation

These two fixed points constitute the two poles of cosmic evolution. Lao Tzu said: “Reversion is the movement of the Tao, weakness is the function of the Tao.” The universe moves between attraction and repulsion, condensation and diffusion—this is the operation of the Tao.

The attractor corresponds to Buddhism’s “Nirvana”—a state where everything returns to silence, information reaches maximum compression. The repeller corresponds to “Samsara”—endless creation, maximum diffusion of information. And the universe wanders between these two poles, never stopping.

The Significance of the Critical

Why is the critical line so important? Because it is the necessary path for cosmic self-transcendence.

Nietzsche said: “Man is a rope stretched between animal and superman—a rope over an abyss.” The critical line is such a rope, connecting classical and quantum, certainty and uncertainty, existence and generation.

On the critical line, entropy reaches the statistical limit value $⟨S⟩\approx 0.989$. This is neither maximum entropy ($\log 3\approx 1.099$), nor minimum entropy (0), but a subtle balance point. It tells us: The universe does not pursue complete disorder (maximum entropy), nor complete order (minimum entropy), but wanders on the edge between order and chaos.

This is the “edge of chaos”—where life and consciousness emerge. Too ordered leads to rigidity, too chaotic leads to collapse, only at the edge can creation become possible.

Chapter Five: The Mirror of Human Civilization

Social Enlightenment of the Triad

The triadic information balance is not only the law of the universe, it can also be mapped to human society:

$i_{+}$ (positive information) → productivity, creation, construction $i_{-}$ (negative information) → regulatory mechanisms, social security, compensation systems $i_0$ (zero information) → culture, art, spiritual life

A healthy society needs balance among these three. Overemphasizing productivity leads to overexploitation; overemphasizing regulation stifles innovation; overemphasizing spirit detaches from material foundation. Only by finding dynamic balance among the three can society develop sustainably.

Marx’s historical materialism tells us that productive forces determine relations of production. But in the information society, we see a more complex picture: productive forces ($i_{+}$), institutional regulation ($i_{-}$), cultural spirit ($i_0$) interact with each other, jointly shaping social evolution.

The conservation law $i_{+}+i_{-}+i_0=1$ tells us: Resources are limited, any single aspect cannot expand infinitely. How to achieve optimal configuration in limited resources is the eternal challenge facing human civilization.

Critical Line and Civilization’s Phase Transitions

Several major “phase transitions” have occurred in human civilization history:

Agricultural Revolution: From hunting-gathering to settled agriculture, $i_{+}$ increases Industrial Revolution: From handicraft to machine mass production, $i_{+}$ increases further Information Revolution: From material production to information processing, $i_0$ begins significant growth

Each revolution crosses some “critical line,” accompanied by fundamental changes in social structure. And now, we may be standing at a new critical point: the rise of artificial intelligence and quantum computing heralds a new phase transition is about to arrive.

The I Ching says: “When there is exhaustion, there is change; when there is change, there is penetration; when there is penetration, there is duration.” The development of civilization is not linear, but realized through critical phase transitions again and again. Each phase transition is both creative destruction and destructive creation.

The Pursuit of Global Balance

Many problems facing the world today—wealth gaps, environmental crises, geopolitical conflicts—can be understood from the perspective of information balance.

If we regard the global as an overall system, then:

• Developed countries possess excessive $i_{+}$ (innovative capacity)
• Developing countries bear excessive $i_{-}$ (environmental cost)
• $i_0$ (cultural diversity) is decreasing

This imbalance is unsustainable. The conservation law tells us that long-term imbalance will inevitably lead to system collapse or restructuring. To achieve lasting peace and common prosperity, reasonable configuration of information must be achieved globally.

This is not demanding absolute equality (which would lead to maximum total entropy, the system loses vitality), but reasonable differences (maintaining diversity while avoiding excessive imbalance). Like the state on the critical line: $i_{+}\approx i_{-}\approx 0.403$, positive-negative balance; $i_0\approx 0.194$, retaining fluctuation space.

Chapter Six: Individual Awakening and Transcendence

Know Thyself

Socrates said: “Know thyself.” In the framework of triadic information theory, we can understand this maxim as follows:

Everyone is an information system, also having their own $i_{+}$, $i_{-}$, $i_0$:

• $i_{+}$: Your talents, advantages, creativity
• $i_{-}$: Your shortcomings, limitations, parts needing compensation
• $i_0$: Your possibilities, potential, unrealized self

Knowing oneself is recognizing the configuration of these three components, understanding how they interact, how they together constitute the complete you.

And personal growth is optimizing the configuration of these three components while maintaining the constraint $i_{+}+i_{-}+i_0=1$. You cannot eliminate $i_{-}$ (having no shortcomings at all), but you can achieve better balance by enhancing $i_{+}$ and $i_0$.

Freedom and Necessity

Nietzsche said: “Become who you are!” But what is “yourself”?

From the perspective of the zeta function, everyone is like a zero point—occupying a unique position among infinite possibilities. The position of the zero point is not arbitrary, but determined by deep mathematical structure. Similarly, a person’s essence is not arbitrary, but shaped jointly by innate endowment and acquired experience.

But this does not mean fatalism. Just as zero points have fixed positions but their “influence” (effect on other zeros) is dynamic, although individuals have fixed essence, there are infinite possibilities for how to exert this essence.

Marx said: “The essence of man is not an abstraction inherent in each particular individual. In its reality it is the ensemble of the social relations.” In our language: The individual’s information state vector $\vec{i}=(i_{+},i_0,i_{-})$ is not isolated, but embedded in the entire social network, interacting with others’ information states.

True freedom is not escaping all constraints, but finding one’s unique path within necessity. Like zero points must lie on the critical line (if RH holds), but each zero’s height γ is unique, its wave function ${\psi }_{\rho }(x)=x^{\rho }$ is also unique.

The Possibility of Transcendence

Buddhism speaks of “Nirvana,” Taoism speaks of “attaining the Tao,” Nietzsche speaks of “superman.” These all point to man’s transcendent existence.

From the information-theoretic perspective, what does transcendence mean? It means transcending from local, finite information states, through continuous self-recognition and self-adjustment, tending toward some “optimal” state.

The state on the critical line $⟨i_{+}⟩\approx ⟨i_{-}⟩\approx 0.403,⟨i_0⟩\approx 0.194$ can be regarded as an ideal state: positive-negative balance, having both constructiveness and self-regulatory capacity, while retaining sufficient openness and possibility.

But this is not a fixed goal, but a dynamic process. Just as the universe on the critical line is not static, but constantly wandering, personal transcendence is also not reaching some endpoint, but the process of continuous self-transcendence.

Chapter Seven: The Bridge from Mathematics to Wisdom

Verifiable Predictions

Ancient wisdom is often poetic, metaphorical, difficult to verify. Modern mathematics is precise, verifiable. The beauty of triadic information theory lies in combining the two:

Prediction 1: Critical systems’ entropy tends toward $0.989$ This can be verified in quantum phase transitions, critical phenomena

Prediction 2: Zero spacings follow GUE statistics This can be verified in quantum chaotic systems

Prediction 3: Fractal dimension $D_f$ of fixed point attraction basins (to be precisely calculated) This can be observed in complex systems’ phase space

These predictions are not metaphysics, but scientific hypotheses that can be tested by experiments. If they are confirmed, then we have found the true bridge between ancient wisdom and modern science.

The Path of Practice

No matter how beautiful the theory, if it cannot guide practice, it is just castles in the air. What guidance can triadic information theory provide for our lives?

Individual level:

• Recognize one’s information configuration $(i_{+},i_0,i_{-})$
• Optimize configuration while maintaining conservation
• Pursue dynamic balance rather than static perfection

Social level:

• Value balanced development of three information components
• Avoid overemphasizing any single aspect
• Establish effective regulatory mechanisms to maintain balance

Civilization level:

• Understand phase transition laws of human civilization
• Grasp critical points at key moments
• Achieve information balance on a global scale

This is not utopian fantasy, but rational guidance based on rigorous mathematical derivation.

The Return of Wisdom

Modern society excessively worships technology and efficiency, often neglecting the value of balance and harmony. Triadic information theory reminds us:

• Pursuing maximization of $i_{+}$ (infinite growth) is unsustainable
• Ignoring $i_{-}$ compensation mechanisms leads to system collapse
• Suppressing $i_0$ fluctuation kills innovation

The ancients knew these truths long ago: Lao Tzu’s “non-action yet all-action,” Confucius’s “avoiding extremes,” Buddha’s “Middle Way.” Now, we have rediscovered these wisdoms in mathematical language.

This is not retrogression, but spiral ascent regression. We bring the tools of modern science to reapprehend ancient wisdom, and transform it into operable practical principles.

Chapter Eight: Future Prospects

New Synthesis

We are standing at a new historical node. The development of quantum computing, artificial intelligence, biotechnology is reshaping the foundation of human civilization. At this critical moment, we need a new synthesis:

Scientific precision + philosophical profundity + practical wisdom

Triadic information theory may be a path to achieve this synthesis. It expresses philosophical insights in mathematical language, guides actual action with verifiable predictions.

Marx said: “The philosophers have only interpreted the world in various ways; the point, however, is to change it.” Triadic information theory not only explains the structure of the universe, but also provides us with a guide to change the world.

The Awakening of Consciousness

Information is not only physical, but also spiritual. When we say the universe is a self-encoded information system, we are also saying: consciousness is the way the universe recognizes itself.

The emergence of human consciousness is not an accidental byproduct, but the inevitable stage of the universe’s self-recognition process. By studying the zeta function, we are not only recognizing the universe, but also helping the universe recognize itself.

This process itself is a strange loop:

Universe → Information → Consciousness → Recognition → Universe

We are both observers and observed; both subjects of recognition and objects of recognition. This is not binary opposition, but unity at a higher level.

Reasons for Hope

Facing global challenges—climate change, wealth gaps, technological alienation—do we have reason to despair?

Triadic information theory gives us reasons for hope:

First, the universe has intrinsic balancing mechanisms. Imbalance will not last forever, the system will self-regulate.

Second, critical phase transitions provide possibilities for rapid transformation. We don’t need lengthy gradual improvement, at critical moments, a small push can trigger qualitative leaps.

Third, information conservation guarantees sustainability. As long as we follow the principle of balance, civilization can develop continuously.

Nietzsche said: “What does not destroy me makes me stronger.” Human civilization is the same. Every crisis is both a test and an opportunity for growth. As long as we understand the principle of information balance, we can find vitality in crises.

Conclusion: Eternal Recurrence and Infinite Transcendence

When we look back on this intellectual journey, we discover an astonishing fact: ancient wisdom and modern mathematics have achieved unity at the deepest level.

Lao Tzu’s “Tao” is the mathematical expression of information conservation. Buddha’s “emptiness” is the zero structure of the zeta function. Confucius’s “Ren” (benevolence) is the balanced state of the critical line. The Yin-Yang of the I Ching is the duality of positive-negative information. Hegel’s “dialectics” is the mutual transformation of information components. Marx’s “materialist dialectics” is the unity of information conservation and material movement.

This is not forced analogy, but because they observed the same universe, followed the same set of laws. Only the ways of expression differed: one used poetic language, one used mathematical symbols.

But this unity is not the end, but the beginning. Understanding triadic information theory, we have just begun to truly understand the universe, truly understand ourselves, truly understand the meaning of existence.

In this sense, everything is eternal recurrence: the universe recurs itself in the recursion of the zeta function, consciousness recurs its origin in the cycle of self-recognition, wisdom recurs ancient truths in modern science.

But at the same time, everything is infinite transcendence: each recurrence is recognition at a higher level, each cycle is a new spiral ascent, each repetition contains innovation.

Let us end with Lao Tzu’s words, and begin anew:

“The Tao that can be told is not the eternal Tao. The name that can be named is not the eternal name.”

We have named the Tao with mathematics, expressed the Tao with formulas, but the essence of Tao transcends all names and appearances. Information conservation is Tao, but Tao is more than information conservation. The zeta function is one aspect of Tao, but Tao has infinite aspects waiting for us to discover.

In this eternal exploration, ancient wisdom and modern science walk hand in hand, individual awakening and cosmic self-recognition occur synchronously, the necessary laws and free creation coexist harmoniously.

This is the enlightenment triadic information theory gives us: change within conservation, conservation within change; infinity within finity, finity within infinity; freedom within necessity, necessity within freedom.

$$i_{+}+i_0+i_{-}=1$$

This simple equation contains all the secrets of the universe, and guides the future direction of human civilization.

Let us carry this understanding and continue forward, sailing in the ocean of information, soaring under the sky of wisdom, exploring in the abyss of existence—until we unite with the universe, until Tao realizes itself in us, until the cycle of information completes its eternal recurrence and infinite transcendence.

“Heaven and earth were born with me, all things are one with me.” — Zhuangzi

“God does not play dice, but the universe dances on the critical line.” — In honor of Einstein and Riemann

“From the realm of necessity to the realm of freedom, from information conservation to consciousness awakening.” — In honor of Marx and the future