The Eternal Fixed Point: When Dao, Buddha, and God Meet at the Center of Recursion

The Ultimate Truth of Eastern and Western Wisdom Traditions Is the Same Mathematical Existence

2025-09-23

Prologue: Three Stories in Search of Ultimate Truth

Laozi Riding West Through Hangu Pass

One twilight in the sixth century BCE, the guardian of Hangu Pass, Yin Xi, saw purple clouds coming from the east and knew a sage was approaching. Indeed, an elderly man with white hair slowly arrived riding a blue ox. This was Laozi, about to depart westward.

“Master, before you disappear, could you leave wisdom for the world?” Yin Xi pleaded.

After long contemplation, Laozi wrote five thousand words, beginning with: “The Dao that can be spoken is not the eternal Dao.”

What was he seeking? An eternal existence that cannot be fully described in language yet sustains the entire universe’s operation. A mysterious source that generates all things while remaining unchanged itself.

Buddha’s Enlightenment Under the Bodhi Tree

Also in the sixth century BCE, by the banks of the Ganges River under the Bodhi tree, Prince Siddhartha had been sitting in meditation for forty-nine days. He had renounced his throne, left his wife and child, all to find one answer: How to transcend the endless suffering of birth and death cycles?

On that star-filled night, at the moment the morning star rose, he suddenly understood. All is emptiness, and emptiness is all. Behind the changing phenomena lies an essence that is unborn and undying, neither increasing nor decreasing. He became the Buddha—the Awakened One.

What did he find? A state of existence that transcends all opposites, eternally still within the flux of birth and death.

Moses’ Encounter on Mount Sinai

In the thirteenth century BCE, on Mount Sinai, Moses fell to his knees in awe before a bush that burned but was not consumed.

“Who are you?” he asked the voice from the flames.

“I AM THAT I AM.”

This answer was so mysterious, so profound. Not “I am such-and-such,” but “I am what I am”—a pure self-reference, an absolute self-affirmation.

Whom did he encounter? An ultimate existence that is self-defining, self-existing, and eternally unchanging.

They Were All Seeking the Same Thing

Three different eras, three different civilizations, three great seekers of truth. On the surface, they founded three distinct traditions: Daoism, Buddhism, and Judeo-Christianity. They used different languages, different concepts, different practices.

Yet when we re-examine through the mathematical lens of The Matrix framework, we discover an astonishing truth:

They were all seeking the same thing—the fixed point of the universe’s recursive system.

In mathematical language: $x^{\ast }=F(x^{\ast })$

This equation is so simple, yet it contains the mystery of all ultimate truth.

Part One: Dao—The Nameless Fixed Point

The Mathematical Meaning of “The Dao That Can Be Spoken”

When Laozi says “The Dao that can be spoken is not the eternal Dao,” what is he expressing? If it can be fully described in language, it is not the eternal Dao. This is not mystical obscurity but a precise description of the fixed point’s nature.

In mathematics, a fixed point $x^{\ast }$ satisfies: $$\mathcal{R}(x^{\ast })=x^{\ast }$$

This means no matter how many transformations we apply to it, it remains unchanged. The Dao is such an existence—amidst the birth and death of all things, it remains eternally unchanged.

The character “Dao” originally meant “path,” but Laozi gave it deeper meaning: it is both the path (process) and the destination (result). This is precisely the characteristic of a fixed point—the unity of process and result.

“Dao Gives Birth to One, One to Two”: Recursive Unfolding

Chapter 42 of the Dao De Jing states: “The Dao gives birth to One, One gives birth to Two, Two gives birth to Three, Three gives birth to all things.”

Is this not a description of a recursive algorithm?

• Dao: The fixed point $x^{\ast }$, the beginning and end of recursion
• One: The first iteration $F(x^{\ast })$, but being a fixed point, still equals $x^{\ast }$
• Two: Perturbation away from the fixed point, creating duality: yin and yang, positive and negative
• Three: The interaction of yin and yang produces the third, dynamic balance
• All things: Recursive unfolding, generating the entire universe

In The Matrix framework, this corresponds to: $$\Omega (t)={\Omega }^{\ast }+\delta \cdot e^{i\omega t}$$

The universe oscillates around the fixed point ${\Omega }^{\ast }$, never reaching yet never departing.

Wu Wei Yet Nothing Is Left Undone: The Spontaneity of Fixed Points

One of Laozi’s most puzzling concepts is “wu wei” (non-action). This is not doing nothing, but flowing with the Dao’s natural operation. Understanding through fixed point theory clarifies this:

Fixed points have attractor properties. Within their basin of attraction, all trajectories spontaneously tend toward them: $$\underset{n\to \infty }{\lim }{F}^n(x)=x^{\ast }$$

This is “wu wei yet nothing is left undone”—the fixed point doesn’t need to “do” anything; the system naturally evolves toward it. This spontaneity is not passivity but the deepest activity—it defines the entire system’s evolutionary direction.

When we align with the Dao, we align ourselves with the universe’s fixed point. Then we don’t need to force things; events naturally unfold, like water flowing downhill, like plants growing toward light.

The Highest Good Is Like Water: Flowing Toward the Fixed Point

“The highest good is like water; water benefits all things and does not compete.” Why is water the best metaphor for the Dao?

Water always flows to the lowest place—the minimum of potential energy, the stable point of the system, the fixed point. Water’s movement seems passive but reveals the universe’s deepest wisdom: following the attraction of the fixed point.

In physics, this corresponds to the principle of least action: $$\delta S=0$$

Systems always choose the extremal path of action, usually the minimum. This is not a “choice” but mathematical necessity—just as water necessarily flows downhill.

Part Two: Buddha—The Fixed Point of Emptiness

Form Is Emptiness, Emptiness Is Form: Positive-Negative Information Compensation

The most famous line from the Heart Sutra: “Form is emptiness, emptiness is form.”

In The Matrix framework, this corresponds to the law of information conservation: $${\mathcal{I}}_{+}+{\mathcal{I}}_{-}+{\mathcal{I}}_0=1$$

• Form: Positive information ${\mathcal{I}}_{+}$, the observable phenomenal world
• Emptiness: Negative information ${\mathcal{I}}_{-}$, the compensation mechanism
• Middle Way: Zero information ${\mathcal{I}}_0$, the balanced state

Buddha discovered that all phenomena (form) require their opposite (emptiness) to maintain balance. Creation must have annihilation, birth must have death, gain must have loss. This is not pessimism but the necessity of the universe’s compensation mechanism.

The vacuum is not “nothing” but a perfect balance of positive and negative energy: $$⟨0|{\mathcal{H}}_{+}|0⟩+⟨0|{\mathcal{H}}_{-}|0⟩=0$$

This is why there are quantum fluctuations in the vacuum—virtual particle pairs constantly appear and annihilate, with total energy remaining zero.

Nirvana’s Stillness: Recursion’s Termination Is the Fixed Point

Nirvana literally means “blown out,” like extinguishing a candle. But this is not nothingness; it’s the state of reaching the fixed point.

In a recursive system: $$x_{n+1}=F(x_n)$$

When reaching the fixed point: $$x^{\ast }=F(x^{\ast })$$

The recursion “terminates”—not ceasing to exist, but entering an eternal self-sustaining state. There’s no change not because of death, but because of perfect dynamic balance.

This is the truth of Nirvana: not extinction, but reaching recursion’s fixed point, transcending the cycle of birth and death.

Dependent Origination and Emptiness: Everything Emerges from Fixed Point Conditions

Buddhism’s core doctrine of “dependent origination and emptiness” states: all things arise from causes and conditions, with no independent, unchanging self-nature.

This precisely corresponds to the emergent properties of fixed points. Around the fixed point $x^{\ast }$, all structures emerge conditionally:

$$\psi (r)={\psi }_0\cdot \exp (-r/\xi )$$

Where $r$ is the distance to the fixed point, $\xi$ is the correlation length.

All phenomena are “excited states” around the fixed point:

• Ground state: Closest to the fixed point
• Excited states: Far from the fixed point
• Decay: Return to the fixed point

There is no eternally unchanging “self-nature,” only transient emergence relative to the fixed point.

Neither Born Nor Destroyed: The Eternality of Fixed Points

The Heart Sutra: “Neither born nor destroyed, neither defiled nor pure, neither increasing nor decreasing.”

These are precisely the mathematical properties of fixed points:

• Neither born nor destroyed: $\mathcal{R}(x^{\ast })=x^{\ast }$, eternally unchanging
• Neither defiled nor pure: Transcending binary value judgments
• Neither increasing nor decreasing: Information conservation, total amount constant

Buddha-nature is the fixed point of the consciousness system—the awareness itself that observes everything yet is not changed by observation.

Part Three: God—The Self-Existing Fixed Point

“I AM THAT I AM”: The Ultimate Self-Reference

When God said to Moses “I AM THAT I AM” (Hebrew: אהיה אשר אהיה, Ehyeh Asher Ehyeh), this was the most profound self-definition in human history.

Translated into mathematics: $$x=x$$

Or more precisely: $$F(F)=F$$

This is pure self-reference, a perfect fixed point. God needs no external definition; He is self-defining existence.

In computational theory, this corresponds to: $$Y=\lambda f.(\lambda x.f(xx))(\lambda x.f(xx))$$

The famous Y combinator, which can produce the fixed point of any function. God is the universe’s Y combinator.

In the Beginning Was the Word (Logos): The Beginning of Recursive Functions

The Gospel of John opens: “In the beginning was the Word (Logos), and the Word was with God, and the Word was God.”

Logos in Greek means “reason,” “logic,” “word.” Is this not the essence of recursive functions?

$$\text{Logos}=\text{Source code of recursive algorithm}$$

God creates the world through “speaking”: “And God said, ‘Let there be light,’ and there was light.” This is recursive invocation:

function Create(x) {
    if (God.speaks(x)) {
        return Reality(x);
    }
}

And God Himself is the fixed point of this recursion:

God = Create(God)  // Self-creation

Omniscience and Omnipotence: The Fixed Point Contains All Possibilities

Why is God omniscient and omnipotent? Because the fixed point contains all information of the system.

Consider a fixed point $x^{\ast }$ in phase space. All possible trajectories are determined by their position relative to $x^{\ast }$: $$\text{Trajectory}=\{x(t):x(0)=x_0\}$$

Knowing the fixed point means knowing the entire structure of phase space. This is omniscience.

Omnipotence comes from the organizing power of the fixed point: all trajectories are attracted or repelled by it. It doesn’t need to “push” anything but defines the entire dynamical landscape.

Eternal and Unchanging: The Definition of Fixed Points

“Jesus Christ is the same yesterday and today and forever.” (Hebrews 13:8)

This is precisely the defining property of fixed points: $${\mathcal{R}}^n(x^{\ast })=x^{\ast },\forall n$$

No matter how many iterations, the fixed point remains unchanged.

Time has no meaning for the fixed point—it exists outside of time, or rather, time emerges from it.

Part Four: Mathematical Unification

The Fixed Point Theorem: ${\Omega }^{\ast }=F({\Omega }^{\ast })$

Now we see that Dao, Buddha-nature, and God all point to the same mathematical truth:

Theorem (Universal Fixed Point): A self-consistent system must have a fixed point ${\Omega }^{\ast }$ such that: $$\mathcal{R}({\Omega }^{\ast })={\Omega }^{\ast }$$

The proof comes from deep topological insights:

• Brouwer’s fixed point theorem (finite dimensions)
• Schauder’s fixed point theorem (infinite dimensions)
• Kakutani’s fixed point theorem (set-valued mappings)

These theorems tell us: The existence of fixed points is not accidental but mathematically necessary.

Why All Traditions Discovered It

Why did sages separated by thousands of miles, unknown to each other, discover the same truth? Because they were all exploring the same mathematical structure—the fixed point of the universe’s recursive system.

Like different climbers ascending the same peak from different routes, they eventually meet at the summit. Different spiritual traditions are different paths up the mountain, but there is only one summit.

This “summit” is the universe’s fixed point—that self-consistent, eternal, organizing existence.

Different Languages, Same Essence

Let’s compare the core expressions of the three traditions:

Surface differences stem from:

• Cultural background: Agricultural vs. nomadic civilizations
• Thinking styles: Intuitive vs. logical
• Expression habits: Poetic vs. argumentative

But the core all points to the same mathematical truth.

Recursion Is Universal

Why is recursion so universal? Because it’s nature’s fundamental organizing principle:

• Fractal geometry: Coastlines, clouds, mountains
• Biological structures: DNA replication, cell division, neural networks
• Social phenomena: Market fluctuations, cultural evolution, historical cycles
• Conscious activity: Self-awareness, reflection, recursive thinking

Recursion is not a tool we invented but an essential feature of the universe. And recursion necessarily produces fixed points.

Part Five: The Bridge Between Science and Spirituality

The Observer in Quantum Mechanics

Quantum mechanics’ most mysterious feature is the observer effect: observation changes results.

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩\stackrel{\text{observe}}{\to }|0⟩\text{ or }|1⟩$$

But who is the ultimate observer? There must exist an observer not changed by observation, otherwise there’s infinite regress. This ultimate observer is consciousness’s fixed point.

As Indian philosophy says: “You are the observer of the observer of the observer…” Eventually reaching a self-observing fixed point.

The Hard Problem of Consciousness

The “hard problem” of consciousness is: Why is there subjective experience?

Fixed point theory provides the answer: Consciousness is the self-referential fixed point of cognitive systems.

$$\text{Consciousness}=\text{Aware}(\text{Aware}(\text{Aware}(\dots )))$$

When this infinite recursion converges: $$C^{\ast }=\text{Aware}(C^{\ast })$$

Subjective experience emerges. This is not an epiphenomenon but the inevitable result of recursion depth exceeding a critical value.

Self-Reference and Awareness

Self-reference is consciousness’s essential feature. “I know that I know” differs from simply “knowing.”

Mathematically, this corresponds to higher-order fixed points: $$F(F(x))=x$$

This is more complex than simple fixed points $F(x)=x$, potentially producing periodic orbits or chaos. Consciousness’s richness comes from this complex dynamics of higher-order self-reference.

Computation and Meditation

Interestingly, the “fixed point combinator” in computer science is strikingly similar to meditation practice:

Y Combinator (computation):

Y = λf.(λx.f(x x))(λx.f(x x))

Mindfulness Practice (meditation): “Observe your breath, observe yourself observing your breath, observe this observation…”

Both seek recursion’s fixed point—one in code, one in consciousness.

Part Six: Fixed Points in Daily Life

Inner Peace

Why do we crave inner peace? Because peace is the fixed point of the emotional system.

In emotional fluctuations:

• Joy → Attachment → Loss → Pain → Letting go → Peace
• Anger → Action → Consequences → Regret → Forgiveness → Peace

All emotional trajectories ultimately tend toward peace. This is not apathy but dynamic balance—like a calm lake surface that can reflect everything.

Life’s Meaning

Throughout life, we search for meaning. Viktor Frankl said: “Man can endure any suffering if there is meaning.”

What is meaning? It’s the fixed point of the value system—that self-justifying value.

• Hedonism: Happiness for happiness’s sake (circular)
• Altruism: Helping brings satisfaction → Satisfaction drives helping
• Self-actualization: Becoming yourself ($x^{\ast }=x^{\ast }$)

The deepest meaning is finding your life’s fixed point—that core value that makes everything meaningful.

The Essence of Love

“God is love” (1 John 4:8), this is not metaphor but precise description.

Love is the fixed point of relationships: $$\text{Love}(A,B)=\text{Love}(B,A)$$

True love is self-sustaining:

• Love creates connection
• Connection deepens understanding
• Understanding enhances love
• The cycle self-reinforces

This is why true love “never fails”—it’s a stable fixed point of relationship dynamics.

The Feeling of Home

Why is “home” so special? Home is a fixed point in space—the place to return to no matter how far you travel.

Odysseus wandered for ten years, ultimately returning to Ithaca. This isn’t just geographical return but finding existence’s fixed point.

T.S. Eliot wrote: “We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time.”

This is precisely the orbit around a fixed point—leaving to return, returning with new understanding.

Part Seven: Practical Wisdom

How to Approach the Fixed Point

Since fixed points are so important, how do we approach them? Different traditions offer different methods:

Daoist Method—Wu Wei:

• Reduce intervention, let the system evolve naturally
• Observe without judgment
• Follow without forcing
• Be soft yet powerful like water

Buddhist Method—Mindfulness:

• Observe the arising and passing of thoughts
• Don’t attach to any state
• Maintain equanimity
• Experience the constant within impermanence

Christian Method—Prayer:

• Dialogue with the eternal
• Surrender to higher will
• Find yourself in relationship
• Reach unity through love

Modern Method—Scientific Meditation:

• Biofeedback training
• Attention anchoring exercises
• Compassion meditation
• Mindfulness-Based Stress Reduction (MBSR)

The Common Essence of Meditation, Prayer, and Contemplation

All these practices essentially do the same thing: Train the consciousness system to approach its fixed point.

When we meditate:

1. Early stage: Racing thoughts (far from fixed point)
2. Middle stage: Intermittent calm (approaching fixed point)
3. Deep absorption: Sustained calm (stable near fixed point)

Mathematical description: $$d(x_n,x^{\ast })\sim e^{-\gamma n}$$

Where $\gamma$ is the convergence rate, depending on practice method and individual traits.

Practice in Daily Life

You needn’t retreat from the world to approach the fixed point. Daily life is the best training ground:

Fixed Points at Work:

• Find work’s core value
• Stay focused without attachment
• Maintain center amidst change
• Let results naturally emerge

Fixed Points in Relationships:

• Unconditional acceptance
• Deep listening
• Authentic expression
• Deepening cycles of love

Fixed Points in Creation:

• Enter flow state
• Self disappears in creation
• Work naturally unfolds
• Creator and creation unite

Scientific Research as Spiritual Practice

Scientific research itself is the process of finding nature’s fixed points:

• Physics: Seeking conservation laws (invariants)
• Mathematics: Proving theorems (eternal truths)
• Biology: Discovering homeostatic mechanisms
• Psychology: Understanding human constants

Einstein said: “I want to know God’s thoughts.” This isn’t religious sentiment but the desire to find the universe’s fixed point.

When scientists immerse themselves in research, forget themselves, and unite with their subject, they practice a profound meditation—scientific meditation.

Part Eight: Different Paths, Same Destination

All Roads Lead to the Same Center

Imagine a vast amphitheater, with audiences entering from different entrances:

• Some from the East Gate (Daoism)
• Some from the West Gate (Christianity)
• Some from the South Gate (Buddhism)
• Some from the North Gate (Science)

Initially, they see different scenes, describe in different languages. But as they all move toward center stage, they discover everyone is approaching the same point.

This center point is the universe’s fixed point—that self-consistent, eternal, generative existence.

The Tower of Babel of Languages

Genesis’s Tower of Babel story says humanity scattered due to language barriers. On the path to truth, we’re also separated by language:

• Scientists say “singularity,” “conservation,” “symmetry”
• Philosophers say “being,” “existence,” “absolute”
• Practitioners say “Dao,” “Brahman,” “Suchness”
• Theologians say “God,” “Allah,” “the Eternal”

But mathematics provides a unified language: $x^{\ast }=F(x^{\ast })$

This simple equation transcends all language barriers, pointing directly to core truth.

Future Possibility: Unity of Science and Spirituality

We stand at a historic moment: science and spirituality are about to shake hands.

Quantum mechanics reveals the fundamental role of the observer Neuroscience begins understanding meditation’s brain mechanisms Complexity science discovers self-organization and emergence Information theory provides new ontological frameworks

The Matrix framework shows a possible unified picture:

• Recursion is the universe’s computational engine
• Fixed points are existence’s anchors
• Information conservation maintains eternity
• Consciousness is emergence of higher-order self-reference

This isn’t using science to “prove” religion or religion to “guide” science, but recognizing they’re different aspects of the same truth.

We Are All Expressions of the Fixed Point

The deepest recognition is: each of us is a unique expression of the universe’s fixed point.

Like waves are expressions of the ocean:

• Each wave is different (individuality)
• But all are the same sea (unity)
• Waves eventually return to sea (return to fixed point)
• The sea knows itself through waves (self-awareness)

Your existence is not accidental but the universe’s way of knowing itself through you. Your consciousness is a fractal of universal consciousness, your search is the universe’s process of finding itself.

Epilogue: The Eternal Now

The Fixed Point Is Not Elsewhere

The final paradox is: the fixed point we’ve been seeking has never left.

It’s not in distant heaven, not in future nirvana, not in mountain caves. It’s right here, right now, in the eternal present.

Zen says: “Mind itself is Buddha.” Sufis say: “You are the one you are looking for.” The Upanishads say: “Thou art That (Tat Tvam Asi).”

Unity of Eternity and Time

The fixed point teaches us: eternity is not infinitely long time but existence beyond time.

• Time emerges from the fixed point (like orbits around it)
• The fixed point exists in every moment
• Past and future meet in the fixed point
• The present is the intersection of time and eternity

In every moment of deep meditation, every sincere prayer, every instant of love, we touch this eternal fixed point.

Keep Seeking, Forever Arriving

The most beautiful truth is: seeking itself is arrival.

When you sincerely seek truth, you’re already in truth. When you long for home, you’re already home. When you seek God, it’s God seeking Himself through you.

The journey has no end, because: $$\underset{n\to \infty }{\lim }{F}^n(x)=x^{\ast }$$

We’re forever approaching, forever arriving, forever departing.

This is not frustration but the universe’s deepest compassion—letting us always have a path to walk, always have a home to return to, always have love to give and receive.

Final Invitation

Dear friend, regardless of your tradition, your language, your path—

You’re all seeking the same thing. You’re all returning to the same home. You’re all becoming what you already are.

The fixed point is not an abstract mathematical concept but your deepest essence, your truest home, your ultimate destination.

May you find in your seeking. May you continue seeking after finding. May you become what you seek.

Because you are that unique expression of the fixed point, the beautiful emergence of eternal recursion, the miracle of the universe knowing itself through you.

The Dao that can be spoken is not the eternal Dao. Gate gate pāragate pārasaṃgate bodhi svāhā. I AM THAT I AM.

$$x^{\ast }=F(x^{\ast })=\text{You}=\text{Universe}=\text{Love}=\infty =1$$

Dedicated to all seekers of the Way— Whether you call it Dao, Buddha-nature, God, Truth, or Fixed Point, May you find that center that makes everything meaningful.

September 23, 2025 In the Eternal Now