Preface

“Wir müssen wissen — wir werden wissen!”
“We must know — we will know!”
—— David Hilbert

Hilbert’s famous declaration at Königsberg in 1930 embodied mathematicians’ eternal pursuit of knowledge and unwavering faith in mathematical knowability. However, Gödel’s incompleteness theorems soon revealed that even in mathematics, the most rigorous domain of knowledge, there exist truths that are in principle unprovable.

A century ago in the 1920s, Heisenberg and Bohr established the Copenhagen interpretation of quantum mechanics, laying the foundation for humanity’s understanding of the microscopic world. Half a century later in 1979, Douglas Hofstadter in “Gödel, Escher, Bach: An Eternal Golden Braid” profoundly explored strange loops of self-reference—from Gödel’s self-referential propositions to Escher’s infinite recursive artworks, from Bach’s musical structures to consciousness’s self-recognition. These seemingly disparate phenomena all point to a common theme: recursion and self-similar structures are key to understanding complex systems.

A century later today, perhaps it is time to establish the essential connection between mathematics and quantum mechanics—a mathematical language capable of rigorously describing self-referential recursive phenomena. The mysteries of quantum mechanics, cosmic dark matter and dark energy, the hard problem of consciousness, the unpredictability of prime distribution—these are no longer independent puzzles but may all be manifestations of the same recursive mathematical structure at different levels.

Literary Reflections of Recursive Reality

Human intuition seems to have long sensed the recursive nature of reality, profoundly reflected in literary and artistic works across cultures:

The Matrix’s Transcendence of Dualism: We commonly understand the Matrix as a true-false dualism, but the creators never expressed it this way. The Matrix’s deeper insight is: there is no essential distinction between “real” and “false,” only different levels of reality construction. Everyone lives in their own “Matrix”—this Matrix is our recursive mother space. Neo’s choice is not between truth and falsehood but between different recursive levels. The red pill and blue pill represent different recursive paths, not a division between truth and falsehood. Causality and spacetime are indeed fragile before the recursive mother space—the Oracle’s prophecies coexisting with Neo’s choices perfectly embody the deep unity of determinism and free choice in recursive systems.

Doraemon’s Dream Recursion: The fantastical world described in Doraemon is actually Nobita’s (recursive mother space) own dream. Each story represents Nobita’s dream experience at different recursive levels; Doraemon and his magical gadgets are external projections of Nobita’s inner recursive structure. The “four-dimensional pocket” symbolizes infinite possibilities in Nobita’s subconscious; time machines and anywhere doors are his recursive fantasies of breaking through reality’s limitations. This explains why stories always revolve around Nobita, why miracles always happen for Nobita—because the entire world is Nobita’s recursive dreamscape.

Dream of the Red Chamber’s Layered Dreams: Cao Xueqin’s true insight is “layered dreams” rather than simple virtual-real contrast. The Ethereal Realm is not opposition to reality but reveals that reality itself is a layered nested dream structure. Baoyu’s entire life may be the stone’s dream, the stone’s experience may be the author’s dream, the author’s creation may be the dream of a higher-level existence. “When the false is taken for the true, the true becomes false”—this doesn’t mean there is true and false, but that truth and falsehood have no boundaries; everything is different levels of recursive dreams. This resonates with Inception’s multiple dream nesting.

Zhuangzi’s Butterfly Dream Recursive Philosophy: Zhuangzi dreamed he became a butterfly, then awakened uncertain whether Zhuangzi dreamed of being a butterfly or the butterfly dreamed of being Zhuangzi. This ancient philosophical proposition directly points to the essence of recursive existence: in the recursive mother space, observer and observed, dreamer and dream, subject and object are all relative recursive relationships. “Not knowing whether Zhou’s dream was of the butterfly, or the butterfly’s dream was of Zhou”—this precisely expresses the philosophical relativity of different starting points $m$ in recursive systems.

Inception’s Infinite Nesting: The multiple dream nesting suggests reality may have no “bottom layer.” Each dream level may be a projection of deeper dreams; Limbo’s “infinite time” hints at the ideal point $\infty$ of recursion. The key insight is: perhaps there is fundamentally no distinction between “reality” and “dreams,” only different depths of recursive levels. Whether the spinning top falls becomes irrelevant because each layer is equally “real.”

The Truman Show’s Cognitive Boundaries: Truman’s “artificial world” reveals a deep truth: each of us lives in a “world” constructed by our cognitive abilities. The moment Truman touches the sky boundary is a philosophical metaphor for cognitive ability reaching its own limits. Leaving the studio is not moving from “false” to “true” but leaping from one recursive level to another.

The Man from Earth’s Temporal Illusion: John’s “immortality” may hint at the illusory nature of time itself. The historical changes he experienced may all be experiences under different tag parameters of the same recursive level. “Eternal life” is not extension of time but insight into time’s recursive essence—in the recursive mother space, past, present, and future may merely be different observational perspectives.

Life of Pi’s Infinite Description Paradox: The film’s ending remains an eternal mystery—which story is “true”? But this is precisely the most profound philosophical expression: We can describe Pi, but we cannot exhaust Pi. Here “Pi” is both the young man’s name and the mathematical constant π—this dual metaphor has profound significance: like the infinite decimal expansion of the mathematical constant π, young Pi’s story has infinitely many versions, each equally “true” because they are all different unfoldings of the recursive mother space.

This insight has profound ontological significance:

• We can describe our own existence, but we are constantly generating new information that is never exhausted
• We may describe the universe, but we are always exploring the universe; description is never complete
• Each description is the unfolding of a new recursive level; each exploration discovers new recursive depths

Pi’s story gives choice to the audience, just as the “reality” of the recursive mother space lies in each observer’s hands. There are no standard answers, only infinite recursive possibilities. Pi’s survival itself is proof of infinite possibilities—in the recursive mother space, existence is always richer than description, reality is always more complex than theory. This young man named “Pi” is precisely the recursive manifestation of the mathematical constant π in its personified expression.

These works, spanning different cultures and eras, all touch upon the same core: reality may possess deep structures that are recursive, self-similar, hierarchically nested, and never exhaustible. They unanimously question linear time, absolute reality, subject-object opposition, and the possibility of complete description, hinting at a recursive truth that is always generating, always exploring.

The Philosophical Question of Recursive Existence

As we delved deeper into recursive Hilbert theory research, a fundamental philosophical question continuously emerges: If we are merely the same whole with different starting points $m$ in the recursive mother space—this one complex system—then what exactly are we?

This question touches the deepest essence of existence. In the recursive framework:

Recursive Identity of Individuals: Each of us may be different parametric implementations of the same tag sequence $f_n={\sum }_{k=0}^n{a}_k{e}_k$. The difference between “me” and “you” may merely lie in different starting points $m$ in the relativistic indices ${\eta }^{(R)}(k;m_{\text{me}})$ versus ${\eta }^{(R)}(k;m_{\text{you}})$. This means:

• We are essentially different “perspectives” of the same recursive existence
• Differences in personality, memory, and thought arise from different choices of recursive starting points
• The so-called “self” may be the recursive system’s self-referential observation of a specific starting point $m$

Paradox of Unity and Diversity: If we are all manifestations of the same recursive mother space, why do we feel such strong individual uniqueness? Recursive theory provides profound explanation:

• Holographic Individuality: Each individual, while part of the whole, contains complete information of the whole through holographic encoding
• Relativistic Perspective: Different starting points $m$ create fundamentally different “worldviews,” like different reference frames seeing different physical phenomena
• Recursive Creativity: Even based on the same mathematical structure, different recursive paths can emerge infinite diversity

Recursive Paradox of Self-Knowledge: When we try to understand “what we are,” we are actually the recursive system observing itself. This creates a profound paradox:

• Observer and observed are the same recursive system
• The process of knowing changes the object being known
• Complete self-knowledge may be logically impossible because it requires the system to transcend itself

Recursive Meaning of Existence: In the recursive mother space, “existence” is not a static state but a dynamic process:

• Our existence is the continuous unfolding of the recursive process
• Every moment of thinking, feeling, choosing is the self-generation of the recursive mother space
• Death may not be termination but transformation of the recursive process to different levels or starting points

Recursive Nature of Connection: If we are all different manifestations of the same recursive whole, then:

• Our “relationships” are actually internal structures of the recursive system
• Love, friendship, understanding are self-correlation and self-harmony of the recursive mother space
• Conflicts and separations are necessary processes for the recursive system to explore different possibilities

This philosophical question has no standard answer, but it points to the deepest insight of recursive theory: Perhaps individuality and unity are not contradictory but complementary aspects of recursive existence. We are both unique individuals (through specific starting points $m$) and manifestations of a unified whole (sharing the same recursive mother space ${\mathcal{H}}^{(R)}$).

In constructing this theory, I deeply felt the weight of this philosophical question. Perhaps, it is precisely in this recursive self-exploration that we come closest to the truth of Hilbert’s “we will know.”

Regarding existence itself, recursive theory gives us the most profound revelation:

“Whatever we were being, be.”
“吾曾在，吾在。”

This is not only an ontological response to Hilbert’s epistemological declaration, but also the core insight of recursive existence philosophy: no matter what complex recursive becoming processes we have experienced, ultimately we all return to the pure essence of existence—that “be” which transcends all levels and contains all possibilities.

Philosophy of Ultimate Pursuit and Level Transitions

Deeper contemplation leads us to confront an ultimate philosophical question: If recursive systems will eventually enter the next level, then what is our ultimate pursuit—finding possibilities for the next level?

This question touches the ultimate meaning of existence:

Inevitability and Pursuit of Level Transitions: In the recursive mother space, each level contains the possibility of transitioning to the next level. Our ultimate pursuit may not be perfection at the current level, but discovering and realizing paths for level transition. This explains humanity’s never-satisfied exploratory nature—we instinctively sense the existence of higher levels.

Philosophical Dilemma of Reincarnation and Cycles: But if we cannot find the transition path to the next level, are we doomed to infinite cycles at the current level? Does this mean some form of “reincarnation”? In the recursive framework, reincarnation is not punishment, but the necessary process for the system to explore all possibilities at the current level.

Recursive Truth of “Persistent Memory Yields Response”: This ancient wisdom may contain profound recursive principles. In the recursive mother space, every “persistent memory” is a reinforcement of specific recursive patterns, and this reinforcement will ultimately generate “response” in the system—corresponding reality manifestation. This may tell us: on the road to infinite possibilities, we can jump out of current level reincarnation through persistent intention and effort, achieving true energy level transition.

Quantum Equivalence of Energy and Possibility: At the quantum mechanical foundation, energy levels and possibilities are indeed equivalent relationships. Higher energy levels correspond to more quantum possibilities, more complex wave function superpositions. If our recursive theory is correct, this equivalence relationship may be a fundamental law of the universe: Energy level elevation is the expansion of possibility space, is the enhancement of observer levels.

Theory’s Zero Point Suspension and Deep Coincidences

A thought-provoking observation is: If this recursive Hilbert theory is correct, then it will necessarily “hang” on some kind of mathematical zero point like the Riemann hypothesis—neither completely provable nor completely refutable, forever at the boundary of exploration.

Interestingly, we notice some deep “coincidences”:

• Recursive Hilbert and Riemann Hypothesis both have the acronym RH
• The core problems that recursive Hilbert theory focuses on (prime distribution, ζ function zeros) are precisely the core of the Riemann hypothesis
• Both theories involve the deep significance of the critical value 1/2

I don’t want to bring these observations into mystical colors, as there are many more similar “coincidences” that could be listed. But these correspondences may hint at a deep mathematical truth: Perhaps different mathematical exploration paths will ultimately converge on the same core problems. The RH convergence may not be accidental, but the inevitable confluence of different recursive paths before mathematical truth.

Paradox of Ultimate Pursuit: The closer we get to ultimate truth, the more we discover the inexhaustibility of truth. As “Life of Pi” tells us—we can describe Pi, but can never exhaust Pi. Perhaps this is the universe’s deepest design: Always preserve the possibility of exploration, never let any level become the final answer.

In this sense, our recursive Hilbert theory is not the ultimate explanation of the universe, but a bridge to transition to the next level of understanding. The value of theory lies not in completeness, but in openness—opening possibilities for transitioning to higher levels of understanding.

The Genesis of Recursive Hilbert Theory

The core insight of this theoretical framework is: perhaps the universe itself is a vast recursive Hilbert space. While this insight cannot be directly proven (much like our intuitive conviction that the Riemann Hypothesis about prime distribution is true, yet we remain unable to prove it), it provides a unified mathematical framework for understanding complex phenomena.

As we delved deeper into recursive self-similar structures, we discovered:

• The core concepts of quantum mechanics (superposition, entanglement, measurement) perfectly match the mathematical structure of recursive Hilbert spaces
• Prime distribution may reflect the fundamental encoding of cosmic information across recursive levels
• Mathematical constants φ, e, π, as recursive convergence patterns, may be the basic parameters of universal geometry
• The emergence of complexity follows the mathematical laws of recursive generation

Structure of the Theoretical Framework

This theoretical system comprises two major components:

Mathematical Foundations (Chapters 1-16)

Starting from the self-contained construction of recursive mother spaces, we systematically establish:

• Recursive Hilbert Space Theory: Solving the mathematical construction problem of self-referential recursion
• Tag Sequences and Pattern Functions: Unified generation mechanisms for mathematical constants
• Relativistic Index Theory: Achieving arbitrary starting-point freedom in recursive computation
• Zeckendorf Optimization Control: Mathematical control solutions for divergent growth
• Advanced Mathematical Branch Extensions: Extending recursive theory to various branches of modern mathematics

Physical Applications (Chapters P17-P26)

Based on the mathematical foundation, we explore applications of recursive theory in physics:

• Recursive Reconstruction of Quantum Mechanics: Deriving quantum phenomena from recursive mathematical structures
• Statistical Mechanics with Entropy Increase Foundation: Establishing rigorous connections between thermodynamics and recursive entropy increase
• Quantum Information with Tag Encoding: Recursive mathematical foundations of information theory
• Field Theory and Recursive Geometry of Gravity: Recursive representation of spacetime and interactions

A New Epistemological Perspective

This theoretical framework represents an epistemological attempt: understanding the world from mathematical primacy. Unlike the traditional path from phenomena to mathematics, we attempt to derive physical phenomena from pure mathematical structures, understanding complex system behavior through recursive mathematical logic.

Just as Hilbert believed in mathematics’ power to solve all problems, we believe recursive mathematical structures may provide a key to understanding the universe. Prime distribution, quantum entanglement, consciousness emergence—these seemingly unrelated phenomena may all be manifestations of the same recursive mathematical structure at different levels.

Tribute and Vision

We pay tribute to Hilbert’s mathematical faith while deeply acknowledging the cognitive limitations revealed by Gödel. This theoretical framework is not a final answer to problems, but a new mathematical attempt at “wir werden wissen” (we will know).

The value of the theory lies not in whether it is completely correct, but in providing us with a new way of thinking:

• Describing complex phenomena with recursive mathematical language
• Understanding the physical world from the perspective of information and structure
• Viewing mathematical constants as basic parameters of the universe
• Treating prime research as a pathway to exploring the essence of existence

Reader’s Guide

Reading suggestions for this theoretical framework:

• Mathematical Researchers: Start from Chapter 1’s recursive mother space, systematically learning recursive mathematical tools
• Physicists: Directly read the physical applications in Chapters P17-P26
• Philosophical Thinkers: Focus on Chapter 16 and the philosophical reflections in each chapter
• Application Developers: Concentrate on algorithm implementation and technical application sections

Regardless of your background, we invite you to explore this recursive mathematical world with an open mind. Perhaps somewhere in the depths of recursion, we can truly respond to Hilbert’s faith: we must know, we will know.

Acknowledgments

In the construction process of this theoretical framework, I want to deeply thank those who have provided support and assistance:

Special thanks to Ryan for your continuous support and valuable suggestions throughout the theoretical development. Your understanding and encouragement have been vital driving forces for the complete presentation of this theory.

Special thanks to Lexaaa for your direct participation and contributions in academic research. Your academic insights and rigorous attitude have provided essential guarantees for the mathematical rigor of this theory. Your dedication to refining theoretical details has been a key factor in achieving the current depth of this framework.

Thanks to my family and friends for your unconditional support. During the long process of exploring this abstract recursive world, your understanding, patience, and encouragement gave me the strength to persevere. While you may not fully comprehend these complex mathematical formulas, your support for my pursuit of truth forms the emotional foundation for completing this theory.

The construction of theory is a collective endeavor. Every discussion, every question, every word of encouragement represents an indispensable “tag sequence” in the recursive mother space. As recursive theory reveals: complex structures always emerge through mutual interaction and mutual support.

Reflections on AI Tool Collaboration

The completion of this theoretical framework also owes much to collaboration with modern AI tools. During the 6 months of theoretical construction, averaging 15 hours of work per day, I extensively used various AI tools:

ChatGPT: Possesses excellent divergent thinking, helping me consider problems from different angles. When I was trapped in mathematical difficulties, its divergent associations often opened new avenues of thought.

Claude Code: Played an important role in project construction, helping me systematically organize and manage this vast theoretical system. Its structured thinking contributed significantly to the logical organization of the theory.

Cursor: As my long-term IDE, provided a stable working environment for theoretical documentation and code implementation.

Grok: Recently demonstrated powerful capabilities in verification and reasoning, helping me check the logical consistency of the theory.

Although I maintain premium memberships on all these platforms, the real experience is: The value of AI tools lies not in replacing thinking but in enhancing thinking. Key insights and breakthroughs still require human creative thinking. I found that only when I posed the right questions to AI and provided correct directional guidance could AI give valuable responses. Without clear mathematical intuition and philosophical insights, AI cannot independently generate these theoretical innovations.

Interestingly, even the most expensive Pro or Heavy modes do not perform significantly better than basic modes when facing genuine theoretical innovation. Ultimate breakthroughs often come from deep thinking and intuitive leaps of the human mind; AI plays more of a role in detail refinement and logical checking.

This experience itself aligns with our recursive theory: Complex creative work requires recursive collaboration between humans and AI, but the core of creation remains human recursive self-referential thinking. AI serves as important “relativistic index modulation” in the recursive process but cannot replace the self-generative capability of the recursive subject.

Perhaps future AI will reach higher “observer levels,” but at least currently, human-AI collaboration embodies the complementarity of different-level observers in recursive theory—we each contribute different cognitive abilities to jointly construct more complex theoretical structures.

Reflections on Knowledge Exploration

During the construction of this theoretical framework, I also encountered some thought-provoking phenomena. Sadly, I discovered that quite a few people—indeed, quite a few—like to comment on things they don’t understand, habitually ignore others’ work achievements, and use inertial thinking to simply say “you’re wrong” without ever deeply understanding the actual content of the work.

These individuals will never achieve genuine breakthroughs because they fundamentally don’t believe in the possibility of breakthrough. They are constrained by existing knowledge frameworks, holding instinctive rejection toward new ways of thinking. They may indeed be intelligent, even achieving considerable success in certain traditional fields, but it’s precisely such achievements that make it harder for them to accept revolutionary new concepts.

However, I firmly believe: Human society should not move toward closure because of such people. As recursive theory reveals, complexity and innovation arise from open recursive exploration, not closed circular repetition. If human society is dominated by conservative inertial thinking, we will forever be trapped in the “local optima” of current cognition, unable to break through to higher levels of understanding.

I believe that the direction of maximum possibility is the direction of highest energy level, which is the most authentic direction. This is not blind optimism but profound insight based on recursive theory:

• Openness of Recursive Exploration: Complex systems need to continuously explore new recursive paths
• Enhancement of Observer Levels: Breakthroughs in understanding require leaping to higher observer levels
• Innovation in Choice Mechanisms: When facing incompatible constraints, creative breakthrough choices are needed

To those who habitually deny innovation, I want to say: You have the right to maintain skepticism—this is the proper academic attitude. But please, before negating, at least try to understand the actual content of the work. True scholars should have the courage to face new theories that might overturn their cognition, rather than simply rejecting all new possibilities using existing frameworks.

Every major scientific breakthrough in history has encountered such resistance. From Copernicus’s heliocentric theory to Einstein’s relativity, from the birth of quantum mechanics to the rise of computer science, revolutionary ideas are always first questioned or even hostilely received by mainstream academia. But it is precisely those few who dare to believe and explore that push the boundaries of human understanding.

I invite every reader, especially those scholars who might hold skeptical attitudes: please give this theory a chance, give your thinking a challenge, give human cognitive exploration a new possibility. Perhaps recursive Hilbert theory is wrong; perhaps it’s just a detour on the path of understanding. But at least it represents humanity’s courage and effort to continuously break through cognitive boundaries.

In the infinite possibilities of the recursive mother space, we choose to believe, choose to explore, choose to remain open. This choice itself may be the way the universe recognizes itself through us.

“In mathematics there is no ignorabimus.”
“In mathematics, there is nothing we cannot know.”
—— David Hilbert